Firm Objectives and Decision Making Under Uncertainty

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Foundations of Managerial Economics

Learning Objectives

A�er reading this chapter, you should be able to:

Explain that the economic way of thinking considers both monetary and nonmonetary variables and is consistent with the "triple bo�om line" of contemporary business firms. Discuss the economic principles of rela�ve scarcity, the market mechanism, and opportunity costs and revenues. Discuss how simple models can be used to explain and predict the ac�ons of individuals and business firms who are pursuing their objec�ve func�ons. Dis�nguish between the explicit and the implicit costs associated with a decision. Dis�nguish between accoun�ng costs and profits and economic cost and profit concepts. Explain how costs and revenues that occur in the future must be discounted back to their net present value. Explain how costs and revenues that are uncertain, due to risk, may be quan�fied in expected net present value terms.

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Businesses take on greater risk when the expected return on investment is also greater.

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1.1 What Is Managerial Economics About?

Managerial economics concerns the applica�on of economic concepts and the economic way of thinking to the decision-making processes of managers. Managers make decisions in both the private and the public sectors at the household (individuals), organiza�on (firms), and society (governments) levels. Managerial economics facilitates be�er decision making: that is, it helps managers make decisions that best serve the objec�ves of the individual, the organiza�on, and/or society. In this book our main interest is in decision making by managers of profit-seeking business firms, but you will see that the same principles also apply to decisions made by not-for-profit organiza�ons, households, and governments.

The Economic Way of Thinking

When we refer to "the economics" of something, such as a new product proposal or a new infrastructure project, we are usually referring to the costs, revenues, and profit characteris�cs of that something. Costs are the ou�low of funds associated with a decision or ac�on; revenues are the inflow of funds associated with a decision or ac�on; and profits are the excess of revenues over costs—if nega�ve, this difference is known as a loss, of course. These monetary issues are the tradi�onal concern of managerial economics, but increasingly we must also take into account the nonmonetary costs and benefits of our decisions. For example, will the decision contribute posi�vely or nega�vely to individual sa�sfac�on, business reputa�on, societal welfare, and/or environmental protec�on? Nonmonetary cost impacts of a decision are also known as psychic costs. That is, the individual, the firm, or society at large suffer a psychological (or emo�onal) cost because the decision and its consequences cause what psychologists call nega�ve affect, or a loss of psychic sa�sfac�on, which economists have tradi�onally called disu�lity. Conversely, the nonmonetary benefits of a decision or its consequences are known as psychic revenues when they generate posi�ve affect or psychic sa�sfac�on (which we call u�lity).

Almost all decisions have both monetary and nonmonetary consequences. To gain be�er economic outcomes for individuals, firms, and society, we must take into account the monetary and nonmonetary outcomes. O�en there is a simple monetary equivalent for these nonmonetary costs and benefits. For example, individuals expect to be paid more for working in a dirty job. Likewise, we pay an insurance premium to avoid not only the financial cost of replacing our car but also to avoid the psychic dissa�sfac�on that would be associated with losing the services of our car. In business, we accept higher business risk only if the expected return on investment is also higher. In general, people understand these monetary trade-offs: People are prepared to accept less revenue if it comes with psychic benefits, or conversely they will want more revenue if it comes with psychic costs.

The economic way of thinking can best be summarized as thinking at the margin; that is, considering marginal costs, marginal revenues, and marginal profits rather than focusing on average or total costs, revenues, or profits. The marginal unit of something is the last unit that is either consumed or produced. For example, the marginal cost of produc�on is the change in total costs that is incurred when an addi�onal unit of output is produced, as compared to the average cost of produc�on, which is the total cost of produc�on divided by the total number of units produced. By thinking at the margin, economists ask whether an addi�onal unit of output, or consump�on of that output, would improve or reduce individual sa�sfac�on, business profits, and/or societal welfare. The presump�on is that if an addi�onal unit will improve the situa�on, then the decision should be made to do it because we assume that individuals want to maximize their sa�sfac�on, firms want to maximize profits, and socie�es want to maximize social welfare.

In this book, we will encounter a variety of economic concepts that allow us to apply the economic way of thinking in our decision making. As these economic concepts are introduced, many of them will be familiar to you because you already have plenty of experience as an "economic actor" in the economy and in society. You make economically based decisions daily and, consciously or subconsciously, apply economic concepts in your personal and professional decision making. So, managerial economics should be an extension of your intui�ve way of making decisions in your personal life into the interac�ons and decisions you make for your business organiza�on.

The conceptual heritage of managerial economics can be found in the tenets of microeconomics, which is the study of the behavior of individual people and organiza�ons in a market economy. Macroeconomics, conversely, is the study of the aggregate behavior of consumers, investors, and governments. Macroeconomics generates many of the variables that enter the decision making of individuals and business firms such as unemployment rates, interest rates, infla�on rates, and foreign exchange rates. Governments are fundamentally concerned with managing the macroeconomy, of course. In this book, our main concern is the decision making of business firms who take these macroeconomic variables as givens, as these variables are outside firms' control as individual economic en��es. Hence, managerial economics examines the decision-making processes of individuals and other economic en��es at the microeconomic level that in aggregate determine the levels of important variables in the macroeconomy.

The Use of Models in Managerial Economics

In managerial economics, we frequently use simplified representa�ons of reality—known as models—to analyze decision-making problems. A model depicts and defines the rela�onships among the major variables in a decision problem while abstrac�ng away from (i.e., ignoring) minor influences on the outcome. Professionals in other fields also use models: Architects and planners use scale models of buildings; engineers use scale models of automobiles and aircra�; and fashion designers use human models as representa�on of you and me!

In economics we use symbolic models with words and other symbols that have a specialized meaning. You already know that words like costs and profits have special, more precise meanings in managerial economics. Jargon words are verbal models of things or phenomena. They allow us to communicate

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Revenues > Costs = Profit (called "surplus" in non-profit firms)

more efficiently because they offer a concise and precise means of conveying the informa�on we wish to convey. For example, "u�lity" is much quicker than saying, "the psychic sa�sfac�on that a consumer expects to derive from the consump�on of a product or service." Communica�on between economists, between managers, and between you and me through the medium of the printed word is substan�ally enhanced by the use of jargon. Diagrams model a situa�on by the use of lines, shading, and other features, while abstrac�ng from the finer details. Similarly, a mathema�cal equa�on simplis�cally expresses one (dependent) variable as a func�on of one or more (independent) variables and does not account for the variability that is likely to occur due to the minor and perhaps random variables that also impact the dependent variable.

In this book, we will use a lot of diagrams and a few equa�ons as a quick and effec�ve means of demonstra�ng the main variables in par�cular decision problems and the assumed rela�onship between and among these variables. Our models incorporate assumed rela�onships among the variables and assumed mo�va�ons of the economic actors. For example, we assume that firms want to maximize their profits and that consumers want to maximize their u�lity. If these assumed rela�onships are inaccurate, or we have overlooked important variables, these inaccuracies will be revealed by informa�on search and empirical tes�ng—by gathering data to test the model against reality. If our models are found to be excessively inaccurate or misleading, these models may consequently be modified to become more realis�c for future applica�ons.

There are three main purposes of models. First, models are useful for teaching purposes. They are a useful device for teaching individuals about the opera�on of complex systems because they allow the complexity of reality to be reduced by abstrac�ng from (i.e., ignoring) variables that have a minor effect on the outcome of a decision. A simple model allows us to deal with the central issues of a decision problem without the added complexity of rela�vely minor influences. As noted above, models can always be made more complex (and more accurate depic�ons of reality) by adding the minor variables into the model. Second, models are used for explanatory purposes. They allow us to discover the causal rela�onships between and among variables. Using prior informa�on and logic we can hypothesize, for example, that sales will increase by 40% if we set up a Facebook page for our business and reward individuals with discount coupons if they recruit their friends to "like" our Facebook page. We would then conduct an empirical test of that hypothesis to confirm (or reject) the hypothesis. If supported by the data, we can then claim to explain the increase in sales as being the result of having a Facebook page and issuing discount coupons to those who recommend our page to their friends.

Third, models are used for predic�ve purposes. We might predict that fuel economy is a nega�ve func�on of both vehicle weight and engine capacity and develop a simple formula that reasonably predicts the actual fuel consump�on of any par�cular vehicle. This kind of predic�ve model is typically based on prior empirical studies, and would be periodically adjusted as engine efficiency con�nues to improve, for example. Another class of predic�ve model might predict behavior using a model that is completely unrealis�c but nonetheless is able to predict outcomes reasonably accurately. An example is the u�lity- maximizing model of consumer behavior (see Chapter 3) that assumes individuals choose among different items to buy a�er carefully calcula�ng the u�lity they will derive from consuming specific goods and services. No consumer actually calculates expected u�lity, but virtually all consumers act as if they do. Through a complex intui�ve reasoning process they decide which products to buy, and typically, these choices are accurately predicted by a simple model that assumes individuals will buy those goods that are expected to give them the most u�lity (psychic sa�sfac�on) per dollar. So business decision makers are able to focus on what aspects of the product or service will give the customer greater sa�sfac�on and how to reduce costs so that their price can be more compe��ve.

Integrated Managerial Economics

Managerial economics is usually taught in the context of a business or MBA degree program alongside other business courses such as accoun�ng, finance, human resource management, marke�ng, and business strategy. Students may think that these are separate silos of informa�on but in reality economics is the glue that binds them. Any business decision—whether a marke�ng, financial, accoun�ng, human resource, or a strategic decision—must take the economics of that decision into account. It makes no sense to make decisions in these other business areas without regard to the impact of that decision on cost, revenues, and profits. It is also poten�ally detrimental to one's objec�ves to make decisions without regard to the nonmonetary psychic costs and revenues and the monetary trade-offs that are involved.

Accordingly, in this book we will integrate examples rela�ng to marke�ng, finance, accoun�ng, human resource management, marke�ng, and strategy into our discussion of managerial economics. We shall also u�lize some rela�vely simple quan�ta�ve methods (also studied by business students), since we will need to use data to make es�mates of demand and costs in later chapters.

The Profit Concept

The profit concept is central to the pursuit of business and is thus central to the study of managerial economics. As discussed previously, profit is defined as the excess of revenues over costs. For not-for-profit and public-sector organiza�ons, an excess of revenues over costs is called a "surplus." Conversely, if costs exceed revenues, there is a loss, which is known as a "deficit." Regardless of the terms used, no firm or organiza�on can sustain losses or deficits forever. The decision-making problems facing managers of for-profit firms and not-for-profit organiza�ons are essen�ally similar, involving revenue enhancement if possible and cost control wherever possible. The a�ainment of profit/surplus and the avoidance of loss/deficit are generally seen as measures of managerial effec�veness, and the market for managers generally rewards (with higher salaries) those who are be�er at making decisions that raise profit or surplus.

It follows that decision making in the areas of revenue enhancement and cost reduc�on are major themes in managerial economics. Concerning the revenue side, we will study consumer decision making and how we can increase demand for the goods and services provided by firms and other organiza�ons. On the cost side, we consider the produc�on process and the costs associated with producing a product or service for sale. Se�ng price levels in different market situa�ons is an important decision related to revenue enhancement, and our study of the pricing decision will extend to four chapters.

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Costs > Revenues = Loss (called "deficit" in non- profit firms)

But revenues can also be augmented by improving product design and by be�er marke�ng of the firm's products, so we also examine these sources of revenue enhancements, which of course also cause the firm or organiza�on to incur costs. Finally, we bring it all together in the final chapter, which considers the economics of compe��ve strategy for the business firm. But first we will consider some fundamental concepts of economics that pervade the economic way of thinking and underpin the opera�on of the na�onal economy and the global economic system.

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With the current push towards social and environmental responsibility, more companies are working to achieve the triple bo�om line, which is concerned with people, planet, and profit.

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1.2 Fundamental Concepts of Economics

Underlying the economic way of thinking are several concepts that form the founda�on on which economic thinking rests. First, we note that economists explain the behavior of economic en��es by assuming that they are ac�ng to pursue a par�cular objec�ve, rather than ac�ng in a random or irregular manner. Second, the principle of rela�ve scarcity—meaning that money, �me, and all other resources are in limited supply—underlies all economic analysis. Third, and because resources are in limited supply, the value of those resources in an alterna�ve use (i.e., their opportunity cost) must always be considered. We now consider these in more detail.

Economic Entities Have Objective Functions

Economists assume that consumers, organiza�ons, and society make decisions purposefully to achieve their specific objec�ves, which are target outcomes that the decision maker wants to a�ain. We o�en talk of the consumer's objec�ve func�on, the organiza�on's objec�ve func�on, or society's objec�ve func�on. We use the word func�on in the mathema�cal sense, that is, the level or magnitude of the outcome is a func�on of the level of the determining variables that the decision maker decides to use.

What is the objec�ve of consumers? Economists view consumers as hedonists who act to best serve their own well-being or psychic sa�sfac�on, which is called "u�lity." Thus, we assume that consumers decide which goods and services to purchase (the inputs to their objec�ve func�on) such that they maximize their expected u�lity. They decide to buy par�cular goods and services depending on how much sa�sfac�on they expect to gain by consuming each one of those goods and services. As a consumer, you make decisions like this all the �me, whether consciously or subconsciously. Think about it: Would you rather spend $500 for a larger TV or for a new rug for your living room? Your answer will depend on how much you think you would enjoy either the new TV or the new rug. If you already have a large TV, or a perfectly good rug, your enjoyment (i.e., your psychic sa�sfac�on) from having a new one would be rela�vely small because your perceived need for the item would be rela�vely low. Conversely, if your old TV has a bad picture, or your rug has worn thin, those issues would factor into your purchasing decision. So we proceed on the assump�on that consumers make decisions (among alterna�ve combina�ons of goods and services) to maximize their u�lity.

What is the objec�ve of an organiza�on? Tradi�onally economists have assumed that the owners or shareholders

of the business firm will want to maximize the firm's monetary profit. Shareholders1

(h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec1.2#Ch1footNote1) want the firm to make a profit to pay out dividends and capital gains so that the shareholders can buy the things they want. But, more recently economists have recognized the triple bo�om line concept that says environmentally conscious and socially responsible firms

will want to achieve a balance between profits, avoidance of damage to the environment, and achieving social benefits. However, note that firms will pursue the triple bo�om line only if their owners and managers want them to, and thus it comes back to the objec�ve func�on of individuals. Economists incorporate the triple bo�om line into their models of business decision making by assuming that many individuals will buy shares in companies that achieve the triple bo�om line outcomes they want and will sell shares in companies that do not. Thus, these individuals drive up the stock prices of firms

that are environmentally and socially conscious while driving down the stock price of firms that are not.2

(h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec1.2#Ch1footNote2)

Many not-for-profit organiza�ons exist to do good things for society and/or the natural environment. Examples include organiza�ons that want to help disadvantaged individuals, families or groups; organiza�ons that want to save endangered species; and organiza�ons that want to stop the degrada�on of the natural environment. Again, these organiza�ons operate to achieve what their owners and other stakeholders want them to achieve, so again, it comes back to individual hedonism. Individuals who gain u�lity from helping others, from saving endangered species, and/or from reducing environment degrada�on will operate and support such organiza�ons, and the managers of these organiza�ons will need to make decisions that deliver what the owners

and stakeholders want.3 (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec1.2#Ch1footNote3)

At the societal level, governments exist to serve the wants and needs of the people, and accordingly argue about and make decisions that are intended to improve the psychic sa�sfac�on of individuals, the profitability of firms, the welfare of society, and the health of the natural environment. Poli�cal differences underlie the arguments about which groups should be the major beneficiaries of policy changes, of course, and there are always winners and losers. For example, passing a new law to restrict pollu�on will increase the costs and thereby reduce firms' profits (and reduce the u�lity of shareholders who care only about profits) while increasing the u�lity of individuals who are more environmentally conscious. Laws that apply domes�cally but not in foreign countries will also affect the balance of compe��on between domes�c firms and interna�onal firms, and thus impose further costs on profit-seeking shareholders while conferring psychic benefits on other stakeholders.

Relative Scarcity

If everything that people wanted was plen�ful, there would be no economic problem. The economic problem is that resources are rela�vely scarce and must be allocated to best serve the wants and needs of individuals, which are effec�vely unlimited. Put another way, individuals, organiza�ons, and socie�es must make alloca�on decisions because their wants and needs are effec�vely infinite, whereas the resources to produce and serve those wants and needs are available only in finite quan��es. For individuals, except perhaps for the super-rich, needs and wants are typically greater than the income available to

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Economists are concerned with the explicit and implicit costs. The explicit cost is the s�cker price of an item, whereas the implicit cost is the opportunity cost or trade-off associated with buying that item.

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purchase goods and services to sa�sfy those needs and wants. Thus, individuals must make decisions to allocate their limited incomes among a wide variety of goods and services such that their u�lity is maximized.

But even the super-rich have to make alloca�on decisions. For example, should a billionaire buy a new private jet (for $2.5 million) or fund a charity for homeless people, starving children, threatened species, or the like? It is useful to dis�nguish between needs and wants. Needs are goods and services that individuals find necessary to survive and to conduct their daily lives. Wants are goods and services that are nice to have but are not necessary to have; individuals gain u�lity from the consump�on of wants but could do without them. Economists like to illustrate the concept of scarcity, and the difference between wants and needs, with the diamond–water paradox. Diamonds are rela�vely scarce, while water is rela�vely abundant. Accordingly, water is rela�vely cheap to buy and diamonds are rela�vely expensive. The price of diamonds (per carat) is much higher than the price of water (per gallon) because of rela�ve scarcity, not because of necessity.

For business firms, managers have to make alloca�on decisions, such as whether to use $1 million to buy more machines (for a more capital-intensive manufacturing process) or pay more wages (for a more labor-intensive process) in determining the least cost of producing a given output level (so as to maximize the firm's profits). Managers also have to allocate the limited produc�on capacity of their produc�on processes to different products or services to meet the increasing market demand for one product and the decreasing demand for another. For example, should Toyota produce more cars and fewer trucks, or more of one model car and less of another model? Should a hairdresser allocate less floor space to hairdressing chairs and more floor space to places for filing and pain�ng fingernails? Managers of not-for-profit organiza�ons also must decide whether to allocate their scarce funds among compe�ng social and environmental needs. For example, managers of the Save the Children interna�onal charity must decide whether to save the children in Eritrea, Somalia, or the jungles of the Amazon. Their limited resources mean they cannot possibly save all the children in all the places where their health and welfare is threatened, so they must make alloca�on decisions.

At the societal level, governments must similarly make alloca�on decisions. For example, in which region should they spend scarce public funds to reduce poverty, or upgrade the transporta�on infrastructure, or relocate a government office so as to provide local employment? Should they increase public spending on educa�on or on military preparedness? On the revenue side, should they raise profit taxes or raise individual income taxes? The la�er is also an alloca�on decision because it involves a trade-off of votes (at the next elec�on) from those who earn most of their income from wages or salaries compared to those who earn most of their income from dividends and capital gains. Poli�cians know that voters who suffer from a government decision like this are likely to vote against the party in power at the next elec�on while those who gain are likely to support that party in a subsequent elec�on. Thus, they effec�vely trade-off the interests of one group against those of the other and suffer the poli�cal consequences of their decisions.

Opportunity Costs, Economic Costs, and Accounting Costs

Scarcity of �me and of resources gives rise to what are called opportunity costs. The opportunity cost of something is what you have to give up in order to have that thing. As you know, an opportunity is something that you could do if you had the �me and the resources to do it. Because there are only 24 hours in the day and an infinite variety of things you could do with that �me, you have to give up one ac�vity (use of the �me) to spend �me doing an alterna�ve ac�vity. Similarly, to u�lize part of your money buying one thing you have to forego the opportunity of spending that money on another thing. For example, if I want to spend $20,000 on a long holiday, then I will have to forego buying a new car.

But note that opportunity costs are not simply monetary costs. Economists are concerned not only with the explicit cost of an item (its s�cker price) but also want to factor in the implicit cost (other expenses also incurred) associated with buying that item. For example, suppose you took �me off from work to drive to a store to buy something on sale for $50, and your tank was empty so you had to buy $10 worth of fuel—just enough to get you there and back. Your explicit costs would be $60. But in addi�on, you would have an opportunity cost associated with using your owned resources in this way. Owned resources are things that you already own, such as your �me, your car, and your other possessions, so they do not directly cost you money to use. Suppose it took an hour to drive to the store and back. First, your �me has an opportunity cost when you consider you could have earned $20 had you worked that hour. Second, suppose that the addi�onal wear-and-tear on your car would cause its value to decline by $5. Thus, your implicit costs are $25. The economic cost of the item to you is $85—the sum of the explicit cost ($60) plus the implicit costs ($25) due to spending your �me and other owned resources to buy that item. In summary, the economic cost is equal to the opportunity cost of all resources involved in the decision, including both explicit and implicit costs.

Similarly, a business firm has to consider both explicit and implicit costs. A firm will pay explicit costs for labor and materials, which should be equal the opportunity costs of those items because, for example, a worker or supplier of materials would refuse to sell labor or materials for a lower price than he could get from another firm. We must also consider the implicit costs of the owned resources of the firm, such as management �me, real estate, machinery and equipment, and other items that the firm already owns. If these physical resources were rented or leased, their explicit cost would equal their opportunity cost, but if they are owned resources, they might alterna�vely be used to produce a more profitable product or be sold for a higher value than they contribute in their current produc�on process. For example, a furniture maker's equipment and skilled labor might be more profitable making custom- designed kitchen units, or the firm's land might be sold to a real estate developer for more than it is worth underneath a factory that makes furniture or kitchen units. Thus, firms must also consider the economic costs of the resources they use if they are to maximize profits for their shareholders.

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Economic cost = explicit cost + implicit cost

Accoun�ng cost = explicit cost only

Untaxed carbon emissions produced as a byproduct of manufacturing is an example of market failure because neither the seller nor the buyer repays the economic value.

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It is important to understand that accoun�ng costs may differ from economic costs. Accoun�ng costs are the costs that firms must use to report their costs and consequent profits to the public. Accountants are constrained by the Generally Accepted Accoun�ng Principles (GAAP) laid down by the Financial Accoun�ng Standards Board (FASB) and the Securi�es Exchange Commission (SEC). Accountants must follow specific rules so that customers, suppliers, and people who buy and sell shares on stock exchanges can be assured that the costs, revenues, and profits announced to the public have been calculated using accepted accoun�ng principles and procedures. Thus, for financial accoun�ng purposes, the cost of an hour's labor is the explicit cost of the salary paid to the worker plus superannua�on payments and labor taxes, if any, and ignores any implicit costs (which might have made the economic cost to the firm higher). Similarly, the accoun�ng cost of a machine will be recorded as the deprecia�on charge (e.g., 20% of the machine's historical cost rather than the actual cash ou�low for the purchase of that machine in the accoun�ng period) plus the costs of repairs and maintenance (which are explicit costs captured

in a separate cost category)4 (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec1.2#Ch1footNote4) The opportunity cost of the equipment is what it could earn in a different produc�on process, or its resale value, or its scrap value, whichever is the highest. For financial repor�ng purposes, (financial) accountants ignore these opportunity costs, but for decision-making purposes (managerial) accountants will incorporate the opportunity costs of owned resources into their calcula�ons.

Profits are equal to revenues minus costs, but economic costs include both explicit and implicit costs while accoun�ng profits are constrained by the GAAP to include only explicit costs. It follows that if economic costs are likely to exceed accoun�ngs costs, then accoun�ng profit may overstate economic profit to the extent that implicit costs are not accounted for. Henceforth in this book when we use the term profit we will mean economic profit, which is a concept that both managerial economists and managerial accountants accept as the appropriate measure of profit for decision- making purposes.

The Market Mechanism

Economists place great faith in the market mechanism to ensure that buyers can trade money for goods and services, and sellers can trade goods and services for money. The term market mechanism refers to the process by which when demand exceeds supply, prices will rise, and conversely when supply exceeds demand, prices will fall. The market mechanism works because, when demand exceeds supply at any price level, some poten�al buyers will miss out and will offer slightly higher prices to buy the product or service, which will soon induce suppliers to set price at a higher level. Conversely, when supply exceeds demand at any given price level, some suppliers will be unable to sell their goods or services and will offer to sell at a slightly reduced price, which will soon bring the prices of other suppliers to a lower level (or they would not be able to sell their goods or services). If prices rise when demand exceeds supply, and fall when supply exceeds demand, it follows that when supply and demand are equal the price will be the market equilibrium price. There will be neither seller pressure to reduce prices nor buyer pressure to raise prices, such that the price will remain constant un�l either the demand or the supply situa�on changes. The equilibrium price is also known as the market-clearing price because at that price there are no goods and services le� in the market unable to be sold, and no buyer le� unable to buy.

Stock prices on the stock exchange are the best example of a compe��ve market at work: Stock prices for any par�cular company rise and fall on a daily basis reflec�ng the balance of stock supplied for sale and bids made to purchase that stock. Poten�al buyers (or sellers) of that stock bid to buy (or sell) stock at the price at which they are willing to buy (or sell) and the stock exchange matches people willing to buy at a par�cular price with sellers willing to sell at that price. Because the matching process starts from the highest bid to buy and proceeds to lower bids, trading con�nues un�l there is no seller le� who is willing to sell at the highest bid to buy, and thus there is neither excess demand nor excess supply remaining at the equilibrium price.

The stock exchange is an example of a purely compe��ve market, or a price-takers market, in which sellers have to accept the market price even though they may wish for a higher price, and buyers have to pay the market prices even though they might wish for a lower price. The market mechanism ensures that the combined forces of supply and demand determine the equilibrium market price level and this price must be accepted by both buyers and sellers (but they do not have to buy or sell at that price unless they want to). In other market forms, the sellers are pricemakers; that is, they set the price and the buyers must take it or leave it. In Chapter 7, we examine market structure, which is defined in terms of the rela�ve number of buyers and sellers and the degree of product differen�a�on. We will see that monopolies (single sellers, such as the electricity company) and oligopolies (rela�vely few sellers, such as the passenger airplane makers), in both cases facing many buyers, have the market power to raise their prices and make extraordinary profits. Monopolis�c compe�tors (many sellers facing many buyers, such as restaurants in a big city) are also price makers but can only make excess profits in the short term un�l others copy their point of differen�a�on. Pure compe��on features many sellers each selling an undifferen�ated product (like people selling stock in a par�cular company) to many buyers. Because each seller is selling an iden�cal item, price compe��on prevents one seller from gaining a higher price than others (at any point of �me in that market; subsequently excess demand might push prices up or excess supply might press prices down).

In some markets there will be market failure, meaning that the market price does not repay the seller for the full economic value of the item being produced and sold. For example, as a byproduct of manufacturing, firms produce carbon, which (in the absence of a tax on carbon produc�on or a carbon-trading system) is not paid for by either the seller or the buyer. Similarly, individuals who drive cars, or use electricity for household appliances, cause carbon to be produced but do not pay a tax on this carbon. Because carbon is effec�vely free to produce and consume, society consequently gets too much carbon (and this accelerates global warming). To rec�fy the failure of the market system to put an appropriate price on carbon, governments need to implement a tax on the produc�on and consump�on of items that produce carbon. Other instances of market failure include the

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market for the protec�on of the environment, endangered species, and human beings who are disadvantaged in some way. Governments and/or socially conscious and environmentally concerned individuals are needed to set up and operate charitable and not-for-profit organiza�ons to rec�fy such market failures.

All of the decisions made by individuals, organiza�ons, and socie�es are made with an expecta�on of achieving a par�cular outcome that will best serve the objec�ve func�on of the individual, the organiza�on, or society, respec�vely. Arguably, organiza�onal and societal objec�ves will reflect the objec�ves of the relevant stakeholders. But good decisions are not always followed by good outcomes. For many decisions, we cannot be certain that the desired outcome will follow the decision we make because most decision making takes place in an environment of risk and uncertainty.

1. It is useful to dis�nguish "shareholders" from "stakeholders." A stakeholder is anyone who has an interest in the opera�ons of the firm, and thereby incurs or receives either monetary or nonmonetary costs or benefits as a result of the firm's opera�ons. Thus, shareholders are stakeholders, but so too are non-shareholders who incur monetary or psychic costs, or who gain monetary or nonmonetary benefits, as a result of the firm's opera�ons. Thus, stakeholders include suppliers and buyers and governments (e.g., tax collectors) and anyone else in society whose well-being is reduced or improved because of the firm's opera�ons. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec1.2#return1) ]

2. This is a simplis�c treatment of a complex issue. The monetary trade-off between profits and nonmonetary concerns for par�cular business firms might be quite high, such that paying a�en�on to the la�er might reduce profits substan�ally and thus reduce the stock price of the firm. Proponents of the triple bo�om line argue that socially and environmentally conscious customers will choose to buy (and even pay more for) products and services from firms that pursue the triple bo�om line and thus those firms will earn higher profits. Others argue that due to "market failure" the adverse social and environmental effects will not be priced appropriately such that firms will not be adequately compensated (by customers) for paying a�en�on to the social and environmental "externali�es" of their produc�on processes, and so will make lower profits unless governments or not-for-profit organiza�ons step in to fix the market failure. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec1.2#return1) ]

3. The shareholders and supporters of organiza�ons can exert pressure on the managers of those organiza�ons to pay more or less a�en�on to social and environmental outcomes, and typically do this via the Board of Directors (who monitor management decisions on behalf of all the shareholders) or by transferring their investment to organiza�ons that more closely reflect their preferences. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec1.2#return3) ]

4. Accountants make a deprecia�on charge against revenue earned in the current period to reflect the por�on of the asset's life represented by the present period. For example, if the asset's usable life is five years, the deprecia�on charge would be 20% of the asset's ini�al purchase price per year for five years. Note that the explicit cost of the asset is equal to the purchase price in the first year and is zero in the next four years, while the deprecia�on charge spreads that purchase cost over the expected life of the asset. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec1.2#return4) ]

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1.3 Certainty Versus Uncertainty

Certainty means that we know what the outcome of a decision will be before we make it. For example, if I drop a ball, I know the outcome is that it will hit the floor (because I know about gravity). Indeed, I could calculate exactly how many milliseconds it would take for it to hit the floor if I also know the weight of the ball and the resistance to gravity that would be due to wind resistance as the ball fell.

But most economic decisions are not this simple—instead they are complicated by the fact that we do not know the outcome in advance. Instead, a range of unknown factors— known but immeasurable factors—and/or poten�al interven�ons to the decision outcome (such as the ac�ons of rival firms) cannot be accurately predicted. Managerial decision making takes place in an environment of risk and uncertainty, requiring a manager to apply economic concepts in the context of risk and uncertainty.

An early economic theorist, Frank H. Knight (Knight, 1921), made the technical dis�nc�on between risk and uncertainty. Risk is involved when the poten�al outcomes and the probability of each outcome are known in advance of a decision. In the simplest case of dropping a ball, I know the outcome will be that it will hit the floor, and the probability of that happening is 100%. Or, when tossing a coin I know the outcomes are either heads or tails and that there is a 50% probability for heads and 50% for tails (on average, if the coin is tossed many �mes). Uncertainty, on the other hand, is involved when the poten�al outcomes are not en�rely predictable and/or the probabili�es of these are not es�mable in advance. Suppose I need to decide whether to drive to work or to take the train. The train runs on a rela�vely reliable schedule, and I es�mate it will take me between 40 and 42 minutes to get to work (door to door). Alterna�vely, driving my car in city traffic could take as li�le as 20 minutes or as long as 60 minutes, depending on the (basically unpredictable) degree of traffic conges�on due to accidents or street repairs. I need to get to work by 9:00 a.m. for an important mee�ng. It is already 8:05 a.m., and the train leaves at 8:10 a.m. The poten�al outcomes of my decision (train or car) are the possible �mes of arrival at work. The train costs $5 while the economic cost of driving my car would be only $2. What should my decision be? As we shall see, managerial economics provides the tools to make this decision.

Business firms face much more complex decisions and must make these in an environment that usually includes both risk and uncertainty. To solve these managerial decision problems, we o�en "model" the problem and let the economic principles suggest the best decision. For example, an insurance company wants to answer the ques�on: "What premium will allow the company to make a sa�sfactory rate of return on our business?" The insurance industry sets premiums for par�cular risks insured on the basis of es�mated risk because they take the same risk many �mes. For example, consider an 18-year-old man seeking accident insurance for his turbocharged sports car. The insurance companies know that the poten�al outcomes are "no accidents" or "accidents" and from their prior experience insuring 18-year-old males driving sports cars they know the probability of an accident happening within a year is (let's say) 60%. The insurance company models its decision problem by assuming that all 18-year-old males driving a turbocharged sports car are equally liable to crash. Past data also shows that the average cost of repairing these cars is $10,000. This model delivers the answer required: There is a 60% probability of a $10,000 cost of repair (and associated overhead costs including a margin for profit), so the insurer will want to charge an insurance premium of $6,000 to cover the young man's sports car for one year.

The Expected Value of Uncertain Events

In the previous example above we have calculated the expected value of the possible outcome (that a young male will crash his sports car). The expected value of an outcome is the value of the outcome mul�plied by the probability that the outcome will occur. Note that not all young males will crash their sports cars—the probability is that 6 out of 10 actually will crash in any year. Moreover, some will barely scratch their cars while others will absolutely trash their cars, such that the repair cost might vary from nearly nothing to the cost of a new car replacement. The insurance company cannot predict which young males will actually crash, or the extent of the damage in each case (due to limited prior data on individuals coupled with the seemingly random occurrence and severity of car crashes). Thus, the company shares the cost of repairs over all those who take out an insurance policy and sets the premium just high enough to earn a sa�sfactory profit. In any given year, fewer or more than 60% of these drivers might crash and their repair bill might average more or less than $10,000, but over many cases the best price decision is provided by using the expected value model of the decision problem.

As an example of a decision-making problem under condi�ons of uncertainty, let's consider the pricing decision of a storekeeper who is contempla�ng the purchase of 200 mangos at the fruit market for $1 each and wants to sell them at a profit before they become over-ripe and thus unsalable. She expects them to last for only three days before they are too ripe to sell, so she has to decide on a price that is sufficiently a�rac�ve to her customers so they will buy the mangos within three days. The storekeeper faces several unknown variables. How many customers will actually come into the store in the next three days? Will they be cashed-up or buying only the essen�als while wai�ng for their next paycheck? At what prices are other stores selling their mangos, if they indeed have any to sell? What is the rela�ve quality of these other available mangos? What other foodstuffs will be put on sale in the next few days to tempt customers to spend their money elsewhere? Thus, there are many unknown variables that cannot easily be given a probability of happening. To model this pricing decision we adopt a simplified model that combines all these factors into a single probability—namely, what is the probability that all 200 mangos will be sold at several alterna�ve prices? Table 1.1 shows the data we need to solve this pricing problem. Note that because all other costs of the storekeeper are constant, regardless of this pricing decision, we consider only the incremental costs of the decision.

Table 1.1: Expected value of profit contribu�on at alterna�ve price levels Alterna�ve possible price levels

Contribu�on to profit per mango (cost was $1 ea.)

Probability of selling all 200 mangos

Expected value of contribu�on to profit per mango

Expected value of total contribu�on to profit

$1.50 $0.50 100% $0.50 $100

$1.75 $0.75 80% $0.60 $120

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Some outcomes (such as four aces when playing cards or double sixes when rolling dice) have very low probabili�es, and are thus very risky to gamble on. Individuals who are less risk-averse may be willing to make such gambles, while other more riskaverse individuals may not.

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$2.00 $1.00 70% $0.70 $140

$2.25 $1.25 50% $0.625 $125

$2.50 $1.50 30% $0.45 $90

From the table it is clear that the price that is expected to maximize profit contribu�on (at $140) is $2.00 per mango. The only difficult part of this pricing decision is es�ma�ng the probabili�es of selling all the mangos at each price. This es�mate has to be provided by the storekeeper, who should be able to make an informed guess about these probabili�es from her prior experience with mangos and other food items and the prior behavior of her regular customers and rival stores.

These examples demonstrate that the expected value model is a simplified version of a complex situa�on. In general, models abstract from reality by ignoring the finer details that are not essen�al to the decision at hand. They concentrate on the major variables and rela�onships without obscuring the picture with the less important details that vary across individuals and/or are unknowable.

Risk Analysis

Where a decision might be followed by one of several outcomes, the decision maker faces the risk that the expected outcome might not be the one that actually happens. For example, in the above illustra�on, when price was set at $2.00 each, there was only a 70% probability that all the mangos would be sold, so there is a 30% probability that all the mangos would not be sold. Thus, although the storekeeper's expected value is $140, the actual outcome might be as high as $200 (if all 100 mangos are sold) or as low as, say, $50 (if only 25 mangos are sold). The actual outcome depends on how many mangos actually are sold.

If the storekeeper had set the price at $1.50 per mango the probability of selling all was 100%, the mango pricing decision would be risk free, with a certain payoff of $100. By choosing a higher price (to make more profit) the storekeeper took the risk that not all mangos would be sold.

Apparently our storekeeper is rela�vely risk tolerant—she was willing to take the gamble of pricing at $2 per mango, which could pay off a maximum of $200 (with probability 70%) or some lesser sum, rather than taking the certain bet of se�ng price at $1.50 with payoff $100 (with probability of 100%). In Chapter 2 we will consider the individual's degree of risk aversion and see that some less-risk-averse individuals will prefer the gamble (to make more or less money at the higher price) while other more-risk-averse individuals would prefer the risk-free alterna�ve.

Information Search Costs

No�ce that risk and uncertainty is fundamentally due to the absence of knowledge. We do not know which of the possible outcomes will happen, or what the exact probabili�es are that the alterna�ves will happen, because we do not know enough about the "system" that causes the outcome to occur a�er a decision has been made. We learn about a system by collec�ng and making sense of informa�on to be�er understand the mechanisms within the system that form the linkage between the decision that is made and the outcome that is observed. Collec�ng and interpre�ng informa�on typically costs money, and this expense is called informa�on search cost.

Decision makers should incur informa�on search costs if they expect the increased revenue (from making a be�er decision) to exceed the search costs that allow that be�er decision to be made. As a simple example, a quick phone call to see if the store has a desired item in stock would save the consumer from was�ng �me and fuel on a fruitless drive across the city to buy that item if it is already out of stock. Similarly, the storekeeper with the mangos might conduct a simple market survey of customers to ascertain what price would be high enough to maximize profit while not so high as to leave her with a pile of over-ripe mangos.

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1.4 Multiyear Scenarios

Future cash flows must be discounted by a factor (known as the discount rate) that is dependent on the available interest rate if they are to be compared with current cash flows. The opportunity discount rate is equal to the rate of interest that cash held today could earn if invested elsewhere at the same risk. In many managerial decision problems the revenues and costs will be received and incurred over a �me interval of more than one year, so it is necessary to discount future-period cash flows to make them comparable (and thus addi�ve) to present-period cash flows. By reducing future cash flows to their present value equivalent, we are able to compare like with like, rather than trying to add or compare cash flows from different �me periods, which is like comparing apples and oranges.

Net Present Value Analysis

A dollar received in the present period is worth more than a dollar received in a future period because an amount less than a dollar received today could be put into a bank and grow to equal one dollar by earning interest before the end of the future period. For example, if I had 91 cents today, and I could earn 10% interest per annum on my 91 cents, it would grow to about $1 in a year (i.e., 91 + 9.1 = 100.1). Thus, if I can earn 10% on any funds that I hold today, a dollar to be received in a year's �me should be valued at no more than 91 cents in present value terms. Conversely, if I expect to receive $1.10 in a year's �me, it would be worth only $1 in present value terms if I could earn 10% on funds put in a bank today. Cash flows from years 2, 3, 4, 5, and further into the future must be discounted progressively more heavily since even smaller sums held presently would grow to a dollar if allowed to earn interest for more years (i.e., the interest compounds) from the present period out to year 2, 3, 4, or 5 and beyond. Note that discoun�ng back to find the present value of a future period's dollar is the converse process of compounding a present period's dollar up to find its value in a future period.

Let us look into the rela�onship between present value and future value in more detail, using a li�le symbolic nota�on. The future value at a point one year hence (FV1) of a present value (PV) is FV1 = PV(1+r) where r represents the rate of interest. Thus, the FV1 of $1 is equal to $1(1.1) = $1.10 in one year

(when r = 10%). If that future value were to be re-invested for another year, it would earn 10% on the principal and the interest already earned. That is: FV2

= PV(1+r)(1+r) which is equal to PV(1+r)2. If we were to re-invest the money for a third year we would find FV3 = PV(1+r)(1+r)(1+r) which is equal to

PV(1+r)3. More generally, the future value of a dollar to be invested for n years (where n might be 1, 2, 3, 4, 5, etc.) at r percent interest is FVn = PV(1+r) n.

Conversely, the present value of a dollar to be received n years into the future is PV = FVn/(1+r) n. Note that the la�er element in this expression, 1/(1+r)n, is

the discount factor, which is the frac�on by which the future value must be mul�plied to find the present value of the future sum. As you can see, the discount factor depends on the specific discount rate (r) and the specific period (n) in which the future funds are received or disbursed. For example, when

n = 1 (i.e., the next period) and r =10%, the discount factor is equal to 1/(1.1)1 = 0.9091, consistent with the 91-cent example used above. Note further that

the discount factor is 1/(1.1)2 = 0.8232 when n = 2, and 1/(1.1)3 = 0.7566 when n = 3, and so on. Thus, amounts to be received further into the future are mul�plied by progressively smaller discount factors.

Let us illustrate present value analysis with a business example. Suppose a firm has asked for tenders to build a new factory and receives two quotes. Supplier A would charge a total of $3.8 million, payable $1.8 million immediately, $1 million in one year, and $1 million at the end of two years when the project will be finished. Supplier B would charge $4 million, payable $500,000 immediately, $1.5 million in one year and $2 million at the end of two years when the project would be finished. Suppose the firm has sufficient cash to pay for either deal, but has an opportunity cost of 12%, which is the rate of interest it could earn on its cash balances if loaned out at equal risk. Which supplier is offering the be�er deal? To answer this we must calculate the present value (PV) of each offer, as in Table 1.2.

Table 1.2: Present value calcula�on for cost of new factory

Year Tender A cash flows

Discount factors* (12%) PVA Tender B cash flows Discount factors* (12%) PVB

0 $1,800,000 1.000 $1,800,000 $500,000 1.000 $500,000

1 $1,000,000 0.8929 $892,857 $1,500,000 0.8929 $1,339,286

2 $1,000,000 0.7972 $797,719 $2,000,000 0.7972 $1,594,388

Totals $3,800,000 $3,490,051 $4,000,000 $3,433,673

*Discount factors are rounded to four decimal places.

Using the equa�on PV = FVn/(1+r) n for each payment, we first calculate the discount factors, which are equal to 1/(1+r)n, where r is the opportunity

discount rate (in this case 12%) and n = 0, 1, and 2 in turn. These are shown in the third column for Supplier A and repeated in the sixth column for Supplier B. Mul�plying the future value (FV) by the discount factor we find the present value (PV) for each payment, and summing these present values we find that Supplier B actually offers the be�er deal, being $56,378 cheaper in present value terms, despite having a larger total cost of the project in undiscounted terms.

More generally, the present value calcula�on will involve both revenues and costs, and we will want to net (or subtract) the costs from the revenues in each

period to find the net present value (NPV) of each decision alterna�ve. Thus the formula for net present value becomes NPV = FVn/(1+r) n – Cn /(1+r)

n ,

which simplifies to NPV = NCFn/(1+ r) n where NCFn signifies net cash flow in year n. Usually mul�year streams of revenue will require an ini�al investment

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The expected value of a decision is used when more than one possible outcome exists. Expected value is calculated by mul�plying the value of the outcome by the probability of its occurrence.

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cost at the beginning (year 0) with revenues occurring in subsequent years, in which case NCF will be nega�ve in year 1 and posi�ve subsequently.5

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We illustrate the calcula�on of net present value in Table 1.3. Suppose that a new housing development is planned that would cost $15 million in the first year to set up the necessary infrastructure (roads, drainage, electricity supply) and the developer would then build and sell houses over the following five years un�l the development is complete. As you can see in Table 1.3, there is cash ou�low of $10 million in year 1 with zero cash inflows—thus net cash flow (NCF) is –$10 million in year 1. In year 2 the developer spends $3 million building houses and sells those that are completed for a total of $5 million, so the NCF = $2 million in year 2. Similarly, we can verify the NCF values for years 2, 3, 4, and 5. The developer's discount rate is 15% per annum because this is the rate of interest the developer could earn by lending the funds to another developer who wants to build a similar new housing development in

another city. To calculate the discount factor for year 1 we set DF = 1/(1+0.15)1 to find DF = 1/1.15 = 0.8696. For year 2 we find DF = 1/(1.15)2 = 0.7561, and so on for years 3, 4 and 5 to arrive at the values shown in the DF column. Mul�plying the NCF by the DF for each year we find the NPV of each year's NCF in the final column, and summing these at the bo�om of the table we find that the NPV for the housing development project is $4.355 million.

Table 1.3: Net present value analysis of the housing development project Year Cash ou�low ($m) Cash inflow ($m) NCF ($m) DF @ 15% NPV ($m)

0 10 0 −10 1.0000 –10.000

1 3 5 2 0.8696 1.739

2 5 8 3 0.7561 2.268

3 8 13 5 0.6575 3.288

4 12 20 8 0.5718 4.574

5 5 10 5 0.4972 2.486

Total 43 56 13 4.355

Note that we have totaled the cash flow columns in Table 1.3 to show the difference between nominal net cash flows ($13 million) over the five years and discounted net present value of those cash flows ($4.355 million). I hope you can see that it would be managerial folly to make decisions based on nominal dollars rather than net- present-value dollars—cash inflows that are received several years away are worth a lot less (e.g., 49.72 cents in the dollar in year 5 when the discount rate is 15%). If the net cash flows (or the profit rate) had not been as high as those shown in Table 1.3 in the la�er few years, this project could have lost money in NPV terms even if the nominal net cash flows remained posi�ve. For example, if the cash ou�lows remain the same in years 3–5 but the cash inflows fell to $12 million, $15 million, and $8 million (due to a global financial crisis, for example), the nominal net cash flows would be only $5 million over the five-year period and the NPV would fall to −$155,580 represen�ng a loss on the en�re project compared with the next best opportunity (i.e., inves�ng the funds elsewhere at 15% per annum).

A second important thing to no�ce from the above examples is that as the opportunity discount rate (ODR) increases, the discount factors become increasingly smaller. From the above examples you will see that the discount factor for cash received in one year is 0.9090 when the ODR is 10%; 0.8929 when the ODR is 12%; and 0.8696 when the ODR is 15%. We know that the ODR is based on the rate of interest that could be earned if the funds were invested at equal risk, so these differences in ODR are due to differences in the risk associated with investment projects—more risky projects should be discounted using higher ODRs. We will return to the issue of differing degrees of investment risk in Chapter 2.

Measuring Profit in Different Scenarios

Since managers might face either certainty or uncertainty, and profit may occur either only in the present period or in both the present and future periods, managers might be opera�ng in one of four different scenarios as shown in Table 1.4. First, if there are only present period cash flows and these are certain, the managers' decision rule is simply to maximize profit. Second, if they face only present period cash flows but these are subject to uncertainty, they need to maximize expected value (EV) of profits. Third, if they face both present and future period cash flows and these are certain, they need to calculate the net present value (NPV) of profits. And, fourth, finally if they face future period cash flows and these are subject to uncertainty, they need to calculate the expected net present value (ENPV) of profits. In managerial decision making the final scenario is the most common situa�on.

Table 1.4: The decision criterion for profit maximiza�on under different decision scenarios Present period Future periods

Certainty Maximize Profit Maximize NPV of Profit

Uncertainty Maximize EV of Profit Maximize ENPV of Profit

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Using models for predic�ve purposes enables business decision makers to focus on what aspects of a product or service will provide the customer greater sa�sfac�on and how to reduce costs to maintain compe��ve prices.

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To calculate the ENPV of uncertain future revenues and costs we would first calculate the EV for each future period and then use the appropriate discount factor to calculate the present value of each expected value. Subtrac�ng costs from revenues in each year would then give us the ENPV of future profits. This is best demonstrated using a decision tree format.

Decision Trees

A decision tree is useful to depict decisions that involve mul�year net profits with poten�al variability of the outcomes in each year. A decision tree shows the poten�al outcomes of a decision like branches on a tree (that is, lying sideways!), as shown in Table 1.5. This example relates to a young entrepreneur who is considering establishing a microbusiness to print and sell souvenir T-shirts in the two years prior to the Olympic Games. Depending on the success of his designs and the quality of rival designs, he expects that demand will be high, medium, or low in each year, with probabili�es 0.2, 0.3, and 0.5 in the first year and probabili�es 0.4, 0.4, and 0.2 in the second year, respec�vely. He will need to invest $2,000 in the necessary equipment right now, and his opportunity cost of the funds involved is 10%.

Table 1.5: Decision tree analysis to calculate ENPV of profits (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

Year 0 Year 1 Year 1 Year 1 Year 2 Year 2 Year 2

Cost ($)

Demand probability

Profit ($)

PV (DF = 0.909)

Demand probability

Profit ($)

PV (DF = 0.826)

NPV of Profit ($)

Joint probability

ENPV of branches ($)

High (P = 0.4) $12,500 $10,325 $17,415 0.08 $1,393

High (P = 0.2) $10,000 $9,090 Medium (P = 0.4)

$5,000 $4,130 $11,220 0.08 $898

Low (P = 0.2) $1,000 $826 $7,916 0.04 $317

High (P = 0.4) $12,500 $10,325 $11,961 0.12 $1,435

− $2,000

Medium (P = 0.3)

$4,000 $3,636 Medium (P = 0.4)

$5,000 $4,130 $5,766 0.12 $692

Low (P = 0.2) $1,000 $826 $2,462 0.06 $148

High (P = 0.4) $12,500 $10,325 $7,416 0.20 $1,483

Low (P = 0.5) −$1,000 −$909 Medium (P = 0.4)

$5,000 $4,130 $1,221 0.20 $244

Low (P = 0.2) $1,000 $826 −$2,083 0.10 −$208

ENPV = $6,402

You can see that the table looks something like a tree lying on its side—the trunk of the tree in year 0 (column 1) splits into three branches in year 1 and each of these branches splits into three more branches in year 2 (column 5), making nine possible outcomes at the end of year 2 (column 8). These terminal branches each have a joint probability of occurring, equal to the joint probability of demand being high, medium, or low in the first year in

combina�on with being either high, medium, or low in the second year.6

(h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec1.4#Ch1footNote6) Weigh�ng the NPV (in column 10, which is the sum of columns 1, 4, and 7) by the joint probabili�es gives us the ENPV for each terminal branch, and summing these

ver�cally in column 10 gives us the ENPV of the entrepreneur's project.7

(h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec1.4#Ch1footNote7) Because the ENPV is posi�ve and of significant magnitude, we conclude that the young entrepreneur should certainly invest in this project, unless he could u�lize his �me and money in an even more lucra�ve project (i.e., one with a higher ENPV).

We should make it perfectly clear that the expected value analysis presumes that the individual makes many similar decisions, such that over all these decisions the total outcome would approximate the sum of the expected values for all the decisions. So, for a construc�on manager who con�nually tenders quotes for building new buildings, she can win some and lose some and expect to be be�er off at the end of the year by using the expected value approach. But for a one-shot deal, such as the T-shirt project, the actual outcome might be as high as $17,415 or as low as −$2,083 in NPV terms, and there are no other similar ventures the entrepreneur could use to "average out" the profit outcomes. But, if this entrepreneur did con�nue to undertake similar projects he would, in effect, be conduc�ng many trials of this gamble and should expect to earn the sum of the ENPV of those many projects. The ENPV analysis we have conducted here essen�ally assumes the decision maker is risk neutral with respect to any one project, which may not be true, of

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course. In Chapter 2, we will consider different a�tudes to risk and risk-adjusted decision making, and you will be equipped to advise the entrepreneur whether or not he should undertake this project.

5. Note that because we are talking about economic profits, these costs in each period must be economic costs (i.e., both explicit and implicit costs) rather than simply the actual (explicit) costs. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec1.4#return5) ]

6. The joint probabili�es in column 9 are found by mul�plying the probability in column 2 by the probability in column 5. The joint probability of an event is the probability of two events occurring together, and is found by mul�plying the probability of one event by the probability of the second event. For example, the probability of demand being high in the first year (0.2, or 20%) is mul�plied by probability of demand also being high in the second year (0.4, or 40%) to find the joint probability of demand being high in both years to be 0.2 x 0.4 = 0.08. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec1.4#return6) ]

7. The ENPV is effec�vely a weighted average of the NPVs, where the weights a�ached to each branch of the decision tree is the joint probability of being on that branch. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec1.4#return6) ]

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Summary

In this chapter we have introduced and considered the nature of, and the fundamental building blocks of, managerial economics. Managerial economics is based on microeconomics, the study of individual economic en��es, such as consumers and business firms. The economic way of thinking is to consider changes at the margin and to incorporate monetary trade-offs for nonmonetary costs and benefits. The study of economics makes liberal use of models, or simplified depic�on of reality, to explain, predict, and teach people about complex systems. We have argued that managerial economics should be integrated into the accoun�ng, financial, human resource, marke�ng, and strategic decisions that managers make, since in all cases managers are trying to increase the profitability of the firm, or are considering monetary trade-offs to achieve the nonmonetary objec�ves of the firm (when pursuing the triple bo�om line outcome of a suitable balance between economic, social, and environmental net benefits).

We noted that economic actors each pursue their objec�ve func�on and must make alloca�on decisions because of rela�ve scarcity. Consumers have limited means (income and/or assets to sell) but have effec�vely unlimited appe�tes for u�lity. Firms have limited funds to allocate amongst different resource combina�ons but must choose the combina�on that maximizes their profits. If firms observe the triple bo�om line objec�ve func�on, they will seek a balance between profits and benefits to society and to the natural environment. The extent of this balance, which typically involves a trade-off between profit and the other nonmonetary benefits, will be driven by shareholders who push the firms to pay more (or less) a�en�on to the social and environmental outcomes. Other organiza�ons, such as chari�es and not-for-profit organiza�ons, typically seek to maximize the social and/or environmental benefits while earning enough revenue to avoid a deficit.

Economic costs are defined to include both explicit and implicit costs, both of which should be valued at their opportunity costs. These o�en differ from accoun�ng costs, which typically neglect the implicit costs in order to adhere to the "generally accepted accoun�ng principles" required by the Securi�es and Exchange Commission (in the United States, or equivalent body in other na�ons). It follows that the economic profits of firms may differ from the accoun�ng profits, since costs may be measured differently.

Understanding the market mechanism is fundamental to an understanding of managerial economics. While different market forms will be discussed in Chapters 7 and 8, we introduced the no�on of compe��ve markets, like the stock exchange, where the forces of demand and supply combine to determine the equilibrium market price, where buyers and sellers have to be price takers. In other market forms, with fewer sellers and/or differen�ated products, sellers can be price makers. The price chosen in these markets is important because the price charged for goods and services (the explicit cost) should be equal to its opportunity cost, since this underlies the calcula�on of economic profit.

Next we defined certainty, risk, and uncertainty. Business managers typically must make their decisions in a context of risk and uncertainty. Risk can be narrowly defined as the situa�on where alterna�ve outcomes are known, with known probabili�es, whereas in uncertainty, all outcomes may not be foreseen and probabili�es cannot be reliably es�mated. We commonly treat risk and uncertainty as a composite concept, o�en simply referring to it as "risk." Informa�on search cost can be incurred to obtain more informa�on, which will usually allow more reliable es�mates of the outcome magnitudes and of the probabili�es.

When there is more than one possible outcome to a decision, we need to calculate the expected value (EV) of the decision, which is the value of the outcome mul�plied by the probability of its occurring. Where the financial outcomes of a decision are spread over more than one year, we need to calculate the present value (PV) of the decision, which is the sum of the products of the cash flows in any year mul�plied by the appropriate discount factor. The appropriate discount factor is the opportunity cost of the funds involved, at equal risk. When there are both mul�ple possible outcomes and these occur over mul�ple years we need to calculate the expected net present value (ENPV) of the decision. We calculate the ENPV by first summing the present values of the net cash flows for each terminal branch on the decision tree, then mul�plying this by the joint probability of its occurring, and finally summing these ENPVs of the terminal branches to find the overall ENPV of the decision. The ENPV approach will be appropriate for most real-world business decision problems because revenues and costs are received to be incurred into the future, and the values of these cash flows are not known with certainty.

Ques�ons for Review and Discussion

Click on each ques�on to reveal the answer.

1. Why do some business firms pursue a triple bo�om line outcome while others focus only on profit maximiza�on? (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

Firms pursue triple-bo�om line (TBL) outcomes to the extent that their managers and shareholders want them to. If the human and ins�tu�onal owners of shares in the firm do not pressure the managers to seek beneficial social and environmental outcomes, or if the managers are not mo�vated independently to do so, TBL outcomes are less likely to happen.

2. In what ways can customers influence a firm to pay more a�en�on to the preserva�on of the natural environment? (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

Customers can communicate directly to firms that they prefer to purchase from firms that pursue TBL outcomes; withdraw their purchasing from firms that do not achieve sa�sfactory TBL outcomes; seek to publicize using TV, print, and social media that par�cular firms are, or are not, pursuing TBL outcomes; sell any shares they hold in firms that do not pursue TBL outcomes.

3. What do we mean when we say that consumer needs and wants are unlimited? Do we mean they are greedy and would not give part of their income to chari�es? (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

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We mean that rela�ve to the limited (finite) wealth of individuals, wants and needs are unlimited (infinite). However, note that the individual consumer might want to donate money to a charity or an environmental cause (gaining psychic u�lity from that). So, while consumers are hedonis�c (u�lity seeking), they are not necessarily opposed to helping others or the natural environment.

4. Why are the explicit costs of an item that you could purchase usually equal to the opportunity cost of that item? Can you envision a situa�on where the explicit cost of an owned resource would be less than its opportunity cost? (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

When you buy an item, you are effec�vely preven�ng the seller from selling that item to another customer – thus the seller should expect to gain from you a price at least equal to what the seller could gain by selling it to the next customer. The seller's opportunity cost of the item is what it is worth in the next-best alterna�ve usage. The explicit cost of an owned resource could exceed its opportunity cost if you paid above market value for it, or if its market value fell a�er you bought it; or if the item is temporarily on sale at less than its fair market value in order for the seller to quickly reduce excess stock of that item.

5. When will economic profits be less than accoun�ng profits and why? (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

Economic profit will be less than accoun�ng profit when the accoun�ng costs do not reflect one or more implicit or future economic costs, or where the accoun�ng revenues do not reflect one or more implicit or future economic revenues, or some combina�on thereof. This would occur because accountants are bound by their "Generally Accepted Accoun�ng Principles" to include only explicit present period costs or alloca�ons of prior period explicit costs (such as deprecia�on allowances).

6. What would be the result, in a price-maker market, if a decision was made to price an item (its explicit cost) at a level higher than its opportunity cost? (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

The seller's opportunity cost is the cost of replacing the item to be sold, e.g., by making or buying another one. If the seller can replace the item in inventory at less than the price it can obtain from a buyer, the seller would make a pure (economic) profit on the sale, since economic revenue would exceed economic costs. The buyer's opportunity cost is the price at which he or she can buy the same item elsewhere (inclusive of all search, transac�on and delivery costs). If the buyer can buy the same product elsewhere for less, he or she will not buy from this seller.

7. Dis�nguish among certainty, risk, and uncertainty, and explain how informa�on search could reduce uncertainty or even change uncertainty to certainty. (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

Certainty means the outcome of a decision (or ac�on) is known in advance (i.e., it can be fearlessly predicted). Risk means the outcome is not known in advance but the alterna�ve possible outcomes are known, and their probabili�es of occurring are known. Uncertainty means the outcome is not known in advance and that the alterna�ve outcomes are not known in advance and the probabili�es of these outcomes occurring are also not known in advance (and must be es�mated). Note we generally lump together "risk and uncertainty" to include any situa�on in which an ac�on might lead to one of several possible outcomes.

8. What is the profit-maximizing decision criterion when there is uncertainty and the costs and revenue outcomes are spread over several years? Why? (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

In uncertainty we calculate the expected value (EV) of the outcome, which is the sum of the products of the predicted value of each outcome and its probability of occurring. When costs and revenues occur over several years, we must discount the future sums back to present value terms and add the products of the cash flows (posi�ve or nega�ve) and the discount factors to find the present value (PV) of those cash flows. When there is both uncertainty and future period cash flows, we first find the PV of the cash flow and then mul�ply that by its probability to find the expected present value (EPV) of that cash flow, and summing the nega�ve EPVs (rela�ng to costs) and the posi�ve EPVs (rela�ng to revenues) we find the expected net present value (ENPV) associated with the decision or ac�on.

9. Using the concept of a decision tree, explain what we mean by the "path-dependency" of outcomes. (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

Path-dependency means that achieving an outcome is dependent upon following a par�cular path to that outcome. To arrive at a par�cular branch of a decision tree, one must have travelled along a path comprising the trunk and lower (earlier) branches of that decision tree.

10. Why is the joint probability of two uncertain events always smaller than the individual probabili�es of those events occurring separately? (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

The joint probability of two uncertain events must be smaller than the probabili�es of two separate events. This is because the probabili�es are frac�ons of one (or percentages less than 100%) such that when they are mul�plied together to find the probability of them occurring jointly, the arithme�c product must be less than either of the two separate probabili�es.

Decision Problems

1. A global so� drink company has announced it will establish a founda�on to provide scholarships to students at a local university. It proposes two alterna�ve �melines, due to a current cash-flow problem: Either it will provide an immediate $10m fund, or it will provide $11.5m over two years, payable $2.5m immediately, $4m next year, and $5m in two years. The university president announces to the faculty that he will accept the $11.5m alterna�ve. As a managerial economics student you are concerned that he has not made the best decision.

a. Assuming the opportunity interest rate is 14%, what is the present value of the $11.5m alterna�ve? b. Would your decision change if the opportunity interest rate was 16% or 12% instead of 14%? c. Explain to the president, in a memo of 200 words or less, which alterna�ve should be accepted.

2. The Pulitzer Publishing Company is considering offering a contract to an author who has wri�en a book on the European Debt Crisis. This project would involve reviewing, edi�ng, designing artwork, and layout of the book at an es�mated cost of $160,000, payable at the end of year 1 before a single book is

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printed or sold. The publisher expects to sell 40,000 copies by the end of year in year 2, 35,000 copies in year 3, 25,000 copies in year 4, and 5,000 copies in year 5. It plans to sell the rights to an Internet bookseller at the end of year 5 for $20,000. Its produc�on, distribu�on and royalty costs are expected to be constant at $3 per book and it will receive $7 per book (wholesale price to booksellers). These prices are expected to stay the same over the five-year period. The funds could alterna�vely be invested in a Greek Government bond issue of similar risk that would pay 18% per annum.

a. Calculate the NPV of the book project, assuming that all cash flows are spent or received at the end of the year nominated. b. How would the NPV vary if the opportunity discount rate was much higher, say 25%? c. Advise the publisher whether it should proceed to offer the author a contract on the proposed terms.

3. The owner of a restaurant approaches you and offers you the following deal. You would give up your current $55,000 per annum job to manage his restaurant while he returns home to his country for two years to complete his compulsory military service. He shows you all his records and you see that the restaurant had total sales of $200,000 last year with costs of only $120,000. You expect the revenue and cost situa�ons to remain stable for the next two years. The restaurant owner wants you to pay him $50,000 up front but would allow you to keep all the profits you earn over the next two years (before handing the restaurant back to him upon his return). You know that you could alterna�vely loan your $50,000 to another restaurant business for a 20% annual rate of return. But this opportunity does appeal to you, since you would gain experience opera�ng a restaurant without incurring the full cost of buying one.

a. What is the net present value of this opportunity based on the figures provided? b. What other implicit costs or revenues might you consider before making your decision? c. What would be your response to the restaurant owner?

4. The University of Dingbat is considering whether to buy an "off-the-shelf" student management system or have its IT people build a similar system in-house. There are several such systems on the market and the one that is most like what the university wants is priced at $200,000. The IT manager says she can replicate that system using current employees for as li�le as $50,000, but admits there might be cost and delivery-�me varia�ons due to unexpected problems that might arise. She provides a probability distribu�on of cost levels and delivery �mes as follows.

Possible cost for in-house delivery of new system ($) Month of delivery Probability

50,000 12 0.25

100,000 24 0.50

150,000 36 0.25

This kind of student management system is expected to save the university about $125,000 per year in labor costs, as well as deliver increased student sa�sfac�on with beneficial impact on future revenues. The off-the-shelf system could be opera�onal within 12 months, whereas the in-house alterna�ve might take anywhere from 12 to 36 months to become opera�onal, as indicated in the table. The university considers that its opportunity discount rate is 8%.

a. Calculate and compare the expected value of the "off-the-shelf" system with the expected value of the "in-house" alterna�ve system (in undiscounted dollars).

b. Calculate and compare the expected present value of the two alterna�ves— ignoring the future revenue impact of a be�er student management system.

c. By how much would future revenues need to increase (due to this system) to make the two alterna�ves equal in EPV terms? d. Advise the university president which op�on should be adopted, with your reasoning.

5. Home Goods Stores operates a chain of retail supermarkets across the state. The chief execu�ve officer (CEO) wants to open a new store in your area, but seeks guidance as to whether it should be a regular-sized store or a super-sized store. The ini�al costs, expected demand scenarios, associated profit data, and the probabili�es are shown in the table below. The CEO is considering only a two-year horizon, since he expects to be re�red by then. He feels that the opportunity discount rate should be 10%. Note that demand condi�ons could vary from year to year—demand could be high in the first year and low in the second year, due to condi�ons outside the firm's control.

Store size Ini�al cost ($m) Demand situa�on Year 1 profit ($m) Probability Year 2 profit ($m) Probability

Low 30 0.2 50 0.2

Regular 85 Medium 50 0.5 80 0.4

High 80 0.3 100 0.4

Low 20 0.4 60 0.3

Super 100 Medium 80 0.4 90 0.5

High 100 0.2 120 0.2

a. Which store promises the larger expected net present value?

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b. What is your advice to the CEO of Home Goods Stores?

Key Terms

Click on each key term to see the defini�on.

accoun�ng costs (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The debits or subtrac�ons recorded in an accoun�ng book, including taxes, interests on loans, deprecia�on, and other business-related expenses.

accoun�ng profit (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The profit that remains a�er the explicit costs of produc�on and selling have been subtracted from the revenues earned from the firm’s produc�on and sale of goods or services.

average cost of produc�on (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The total cost of produc�on divided by the total number of units produced.

costs (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Expenses incurred by the firm when procuring resources, maintaining debt, paying for property, and so on.

decision tree (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Helps a manager make a business decision by showing the outcomes of a given choice. It is a diagram with branches, like a tree lying sideways, which can include sta�s�cal probability values for each of the possible branches occurring.

diamond–water paradox (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Illustrates the concept of scarcity and the difference between wants and needs. Diamonds are rela�vely scarce, while water is rela�vely abundant. The price of diamonds (per carat) is much higher than the price of water (per gallon) because of rela�ve scarcity, not because of necessity.

discount factor (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The frac�on by which the future value must be mul�plied to find the present value of the future sum.

disu�lity (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The nega�ve u�lity, or loss of psychic sa�sfac�on, derived from an unpleasant outcome.

economic cost (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The sum of the opportunity costs of all resources involved in a decision, including both explicit and implicit costs associated with that decision.

economic problem (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Occurs when all needs and wants cannot be sa�sfied simultaneously because resources are rela�vely scarce and must be allocated to best serve the wants and needs of individuals, which are effec�vely unlimited.

economic profit (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Is different from the accoun�ng profit because it is not calculated simply by looking at explicit revenues and costs recorded in the accoun�ng books, but also takes into account the implicit opportunity costs associated with conduc�ng the business.

expected net present value (ENPV) (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Before a business decision maker can make a significant investment, he or she has to calculate the ENPV of this investment decision by conver�ng uncertain future revenues and profits into expected net present value to determine if the resul�ng figure is posi�ve or nega�ve, and thereby if the decision is beneficial or not.

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expected value (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The predicted worth of an investment or other ac�on, computed by mul�plying the possible outcome by the probability of that outcome. Where there are mul�ple possible outcomes, the expected value is sum of the outcomes each mul�plied by its probability of occurring.

explicit cost (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Actual “out-of-pocket” payments made by a person or organiza�on to another party for products or resources provided.

implicit cost (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

An expense that reflects the lost opportunity of u�lizing par�cular resources for some other purpose. For example, money that is le� in a savings account can make a small amount of interest income, which would be lost if that money is invested into buying a new piece of machinery.

informa�on search cost (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

An expense suffered by a consumer or manager in spending energy, money, or �me searching for informa�on about products or resources.

macroeconomics (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The study of broad economic problems, such as monetary infla�on, na�onal unemployment, and economic growth, which informs governments to determine the best economic policies for their countries.

marginal cost of produc�on (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The cost of producing one more unit of a par�cular product. This cost calcula�on helps businesses to make profit-maximizing decisions by calcula�ng the change in the total cost of produc�on due to genera�ng one more item.

market equilibrium price (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The price at which the supply of goods or services to a market equals the demand for these same goods or services. You can visualize a store where a farmer ships nine boxes of tomatoes on a given day, and each day consumers come in and buy exactly nine boxes of tomatoes in total, without anybody wan�ng more tomatoes to buy, and without the farmer wishing to sell more tomatoes.

market failure (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Occurs when a market system is unable to properly account for the external costs of produc�on (e.g., pollu�on) such that the market price of the firm’s output is too low and thus too much is demanded rela�ve to the amount that would be socially op�mal (if firms were forced to internalize their external costs of produc�on).

market mechanism (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Refers to a system in which when demand exceeds supply, prices will rise, and conversely, if supply exceeds demand, prices will fall.

market-clearing price (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The market price that leaves no buyer wan�ng to pay any more and no seller wan�ng to pay any less, otherwise known as the market equilibrium price.

microeconomics (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The economics of individual consumers and producers within the larger economy.

model (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A simplified representa�on of reality used to analyze decision-making problems.

net present value (NPV) (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Is calculated by first subtrac�ng the costs from the revenues in each period to find the net cash flow, then mul�plying this value by a discount factor reflec�ng the appropriate opportunity discount rate and the number of periods into the future that the net cash flow occurs. If net cash flows occur in

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mul�ple future periods (to cons�tute a stream of cash flows), the present values of the net cash flows (i.e., their NPVs) are added together to find the NPV of the stream of net cash flows. If the NPV value is nega�ve, the manager is facing an unprofitable investment decision.

opportunity costs (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The implicit costs associated with choosing to do something, and as a result foregoing the opportunity to do something else that would be the next-best usage of one’s �me or resources.

opportunity discount rate (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The rate of interest that cash held today could earn if invested elsewhere at the same risk.

price-makers (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Firms that can set their price independently because they have the economic power due to the differen�a�on of their product rela�ve to other suppliers’ products in the same product category.

price-takers (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Firms that sell undifferen�ated products and are forced to trade their goods or services at the equilibrium price set by the market because they hold an insignificant share of the market and cannot directly influence the market price.

probabili�es (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The percentage chance that a par�cular outcome or event will happen. Probabili�es may be es�mated by calcula�ng how frequently the event has happened before rela�ve to the �mes it could have occurred.

profits (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Are calculated by subtrac�ng the total costs of a business (including salaries, furniture, buildings, loan payments, etc.) from the total revenues (sales of goods and services) made by a given company.

revenues (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The amount of money a company makes from the sale of its goods or services, minus the refunded value of any returns in a given period.

risk (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The variability of outcomes that may follow an ac�on or decision. Risk means the results may not be what one an�cipates when one makes a decision or starts a new project.

triple bo�om line (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

An economic impact calcula�on that takes into account not only financial costs and benefits of running a business, but also the environmental and social impacts a company makes.

uncertainty (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The doubts associated with predic�ng the outcome of a decision, in terms of unpredictable costs, benefits, and effects.

u�lity (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The psychological sa�sfac�on people expect to receive from doing something. People are mo�vated to increase their u�lity and their ac�ons and decisions are driven by this need.

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2

© Phillip and Karen Smith/Ge�y Images

Decision Making Under Risk and Uncertainty

Learning Objectives

A�er reading this chapter, you should be able to:

Explain that risk is measured by the varia�on of poten�al returns around the mean or expected value of the poten�al returns. Describe what is meant by the expressions risk taking, risk seeking, risk aversion, risk preference, and risk neutrality. Discuss why the degree of risk aversion (or conversely risk tolerance) can lead different people to different decisions. Explain how risk-return "indifference curves" can be used to demonstrate why different people make different choices in the same decision scenarios. Explain several decision criteria that allow individuals to adjust for risk in their decision making. Describe how the shape of the distribu�on of possible outcomes will change the probabili�es associated with the most-likely scenario and the worst-case scenario.

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Standard Devia�on

Introduction

In Chapter 1, we saw that managers wan�ng to make decisions that best serve their objec�ve func�ons will need to first define the metric for their objec�ve func�on. We argued that the firm’s objec�ve will be to maximize profits, and that managers must make decisions under condi�ons of either certainty or uncertainty, and might foresee returns that accrue in the current period or in future periods. In the real world, where risk and uncertainty is the norm, and where expenses and revenues are incurred and received both in the present period and into the future, the appropriate decision criterion is the expected net present value (ENPV). We also argued in Chapter 1 that decisions will usually have both monetary and nonmonetary outcomes that are of interest, or concern, to the decision maker. Accordingly, decision makers will make trade-offs against profit to compensate for the nonmonetary costs or benefits that are associated with the decision. If the psychic pain (disu�lity) of nonmonetary costs is greater than the psychic gain (u�lity) of the nonmonetary benefits, we say there is net disu�lity associated with the decision and the decision maker will require addi�onal profit to compensate for the net disu�lity associated with the decision. Conversely, if the nonmonetary benefits exceed the nonmonetary costs, there is net u�lity associated with the decision, and the decision maker will be willing to give up some profit to compensate for the nonmonetary aspects of the decision.

Risk causes disu�lity for most business decision makers and so they will want to be compensated for bearing risk. In this chapter we integrate risk analysis into the decision-making process and consider several decision criteria that adjust the monetary outcomes of a decision for the risk that is associated with those returns. In Chapter 1, we noted that the ENPV criterion is only really appropriate if the manager con�nually makes the same type of decision in the same environment, such that the manager could reasonably expect that over many trials the aggregate outcome would be approximately equal to the sum of the individual ENPVs. Approximately half the �me the actual outcome will be higher than the ENPV and the other half of the �me the actual outcome will be below the ENPV. Although facing risk in each specific decision, the repe��veness of the decision allows the chances of below-average outcomes to be offset by above-average outcomes, and over many similar decisions the total profit outcome would approximate the sum of the ENPVs of all the decisions.

But in many business situa�ons the manager faces a variety of different decisions from day to day and most types of decision are not repeated o�en enough to make the ENPV criterion an appropriate decision criterion since it does not adjust for differing degrees of risk associated with individual decision problems. In this chapter, we recognize that the decision maker will want to incorporate risk analysis into the decision-making process for those decisions that are not repeated frequently and will want to adjust each decision to take account of the degree of risk involved in each par�cular decision.

How Is Risk Measured?

Risk can be expressed as a measure of the chance that the value of the ENPV of a decision will not be the actual outcome. To calculate a measure of variability of the poten�al outcomes rela�ve to the ENPV we need to understand the sta�s�cal concept known as the standard devia�on, which is a measure of the devia�ons of the possible outcomes from the central tendency (or mean) of those possible outcomes. First note that the ENPV is a measure of central tendency of the poten�al outcomes—indeed the ENPV is the weighted mean (where the weights are the probabili�es of occurring) of the possible outcomes. The mean of any series of numbers has an associated standard devia�on, which indicates the extent to which the mean value is representa�ve of all the data points that enter the calcula�on of that mean. The standard devia�on is higher if the possible outcomes are more widely dispersed around the mean value, or is lower if the actual outcomes lie rela�vely close to the mean value. To calculate the standard devia�on, the devia�ons of each data point from the mean are squared and then summed to find the variance of the distribu�on, and the standard devia�on is then simply the square root of the variance. In effect the standard devia�on indicates the average absolute devia�on of the outcomes from the mean outcome. Thus, the standard devia�on provides a suitable measure of the risk that the ENPV will not be a�ained.

Any good calculator can instantly deliver the standard devia�on of a series of simple numbers. Similarly, it is easy in an Excel spreadsheet to type (for example) = stdev(c2-c24) into a vacant cell to indicate the range of data points (cells c2 to c24 in this example) over which the computer can calculate the standard devia�on. Note that it is more complex to calculate the standard devia�on of a probability distribu�on. We need to (1) find the ENPV of that distribu�on; (2) subtract each outcome from the ENPV to find the devia�ons from the mean; (3) weight each devia�on by its probability of occurring; (4) sum these weighted devia�ons to find the variance; and (5) take the square root of the variance to find the standard devia�on. This is demonstrated for a simple case in Table 2.1, in which we suppose that three outcomes are possible (column 1) with probabili�es as shown (in column 5), which you can verify gives an expected value of 10 as shown (in column 2). For simplicity here, we assume the cash flows all take place in the present period such that ENPV = EV.

Table 2.1: Calcula�on of the standard devia�on of a probability distribu�on

Possible outcome ($)

Expected value ($)

Devia�on of possible outcome from the mean (EV) outcome

Squared devia�on from the mean outcome

Probability of each possible outcome

Weighted devia�on from EV

−10  10  30

10 10 10

−20  0 20

400   0 400

0.25 0.5  0.25

100   0 100

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To find whether the wins offset the losses, variance or standard devia�on can be used to measure the risk associated with uncertain outcomes.

© Photodisc/Thinkstock

Entrepreneurs, skydivers, and motorcyclists all voluntarily take risks with the expecta�on that the u�lity of the reward will outweigh the disu�lity of the risk.

© iStockphoto/Thinkstock

Variance = 200

Std. Devia�on = 14.14

Half of the variance around the ENPV is quite desirable, and of course I am referring to the outcomes that are greater than the ENPV. This part of the risk is known as the upside risk and represents be�er outcomes than can generally be expected in mul�ple trials of this decision. Some risk analysts have suggested that no one is worried about these posi�ve devia�ons from the mean because we would "laugh all the way to the bank" if one of these were to occur. On the other hand, the downside risk represents the outcomes that are worse than the ENPV, and we definitely worry about these. Accordingly, some analysts have suggested that we calculate only the semi-variance of the outcomes by including only those outcomes with nega�ve devia�on from the ENPV to measure only the downside risk. Although intui�vely appealing, the semi-variance approach is not commonly used because it ignores the upside risk; a�er all, if "some�mes you win and some�mes you lose," you need to know the extent of the wins to see whether they would offset the losses. Thus, we tend to use the standard devia�on as our measure of the risk associated with uncertain outcomes. So, now we have a measure of risk, but before we adjust for risk, we need to consider the decision maker’s a�tude toward risk.

Attitudes Toward Risk

People have different a�tudes toward risk. Some people seem to enjoy doing risky things, while others are extremely unhappy to be exposed to risk, and, of course, there are those who do not seem to care. Indeed, individuals will have one of three a�tudes toward risk: risk preference, risk aversion, or risk neutrality. Risk preference means that the individual prefers more risk to less risk, with other things (such as reward or profit) being equal. A risk preferer would therefore choose the riskier of two equally profitable investments. This is only ra�onal behavior if the individual’s objec�ve func�on is to maximize risk rather than to maximize profit, or if the person is so rich that he or she places very li�le value on the money he or she might lose while placing more value on the thrill he or she will get by taking the risk; perhaps high-roller gamblers might fit this profile. Risk preferers might get lucky and have a series of wins (despite the odds) but sooner or later will be bankrupted if they con�nue to make large ENPV decisions this way.

Risk aversion means that the individual prefers less risk to more risk, other things being equal, and will therefore choose the less risky of two equally profitable investments. Risk-averse individuals may choose the more risky alterna�ve if they expect to be adequately compensated for the addi�onal risk undertaken. They weigh up the monetary trade-off of the extra income against the psychic dissa�sfac�on of addi�onal risk bearing and make their decision between less-risky-but-less-rewarding investments and more-risky-but-more-rewarding investments. Finally, some individuals are risk neutral, not caring about risk, in which case they do not need to adjust their decisions for risk. Risk neutrality can occur due to a genuine lack of concern for risk and subsequent losses (which makes it almost as dangerous to your wealth as risk preference), or due to the repe��on of the same or similar decision many �mes, so that on any one occasion the decision makers can act as if they are risk neutral. Thus, as we have noted, the ENPV measure of profit is appropriate if the same (or sufficiently similar) decision is to be repeated many �mes. In this situa�on the decision makers can act as if they are risk neutral for any one of those decisions.

We need to clarify a few more terms that are commonly used in discussions of risk. People o�en talk about entrepreneurs, for example, being risk seekers. Risk seekers seek to do risky things (like entrepreneurship, skydiving, and motor racing), because they expect that risk and return are posi�vely correlated: The higher the risk the higher the return. Whether the return is simply monetary, or is both monetary and nonmonetary (i.e., includes psychic sa�sfac�on), most risk seekers only take the risk if they expect the payoff to be greater to compensate them for risk bearing. Risk seekers are therefore not

risk preferers but are actually risk-averse.1

(h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/ch02introduc�on#footernote1) Another term that probably needs clarifying is risk taker. We are all risk takers, like it or not. Every day we are subject to the risks of global warming, asteroids, tsunamis, earthquakes, global financial crises, traffic accidents, and physical violence, to name just a few sources of the risks we con�nually take. What is important is not that we take risks but what our a�tude is toward taking risks. As you now know, our a�tude either will be risk preference, risk aversion, or risk neutrality, depending on our prior knowledge of the situa�on and our cogni�ve processing of the psychic costs and benefits associated with taking specific risks. We should also dis�nguish between voluntary risk taking and involuntary risk taking. The everyday risks listed earlier in this paragraph are imposed on

us by nature or by our fellow man and are borne involuntarily. Risk seekers, however, take risk voluntarily in the expecta�on that the u�lity of the reward will outweigh the disu�lity of the risk. Thus entrepreneurs, skydivers, racing drivers, and business decision makers voluntarily undertake risky projects and make risky decisions even though they are averse to risk.

Degrees of Risk Aversion

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Risk aversion can range from almost zero degrees of risk aversion (i.e., being almost risk neutral) to being extremely risk-averse. For someone who is slightly risk-averse, bearing risk causes rela�vely li�le psychic dissa�sfac�on. We say they are highly tolerant of risk. People like this will require only a rela�vely small amount of monetary compensa�on for bearing addi�onal risk. For others, bearing risk causes much more psychic dissa�sfac�on. We say they are highly intolerant of risk—they will try to avoid voluntary risk taking as much as possible. For these people, it will require much greater monetary compensa�on to induce them to accept addi�onal risk. Since different decision makers exhibit different degrees of risk aversion (or conversely, risk tolerance), the extent to which they will want to adjust their decision for risk will differ. Accordingly, we must take into account the decision maker’s degree of risk aversion as well as the extent of risk involved in any par�cular decision.

Risk Percep�on

Similarly, each person might perceive risk differently. Individuals perceive the risk in a decision situa�on more or less accurately depending on their prior

knowledge and their cogni�ve biases. Greater prior knowledge of the situa�on, or greater informa�on search ac�vity,2

(h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/ch02introduc�on#footernote2) may provide the decision maker with useful informa�on that others do not have, such that she might (correctly) say the situa�on is not very risky while others might say it is highly risky because of their ignorance of the situa�on. The old saying that "fools rush in where angels fear to tread" reflects the percep�on of li�le or no risk by those who have less knowledge about the situa�on compared to those who have more knowledge. Next, a cogni�ve bias such as overconfidence may cause one person to overlook risks that a less confident person might perceive because the la�er looks more carefully into the situa�on or spends more �me and money on informa�on search ac�vity to reveal the hidden dangers. Another cogni�ve bias is the tendency of decision makers to use heuris�cs, or simplis�c decision rules. While economizing on �me and search costs, heuris�cs could actually increase the decision maker’s exposure to risk, since they consider only some of the informa�on that is poten�ally available. For example, entrepreneurs have been shown to be more overconfident and to use heuris�cs more than employed managers of firms (Busenitz &

Barney, 1997).3 (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/ch02introduc�on#footernote3) When others see entrepreneurs taking extraordinary risks they o�en presume that these entrepreneurs must be highly tolerant of risks, when in fact many entrepreneurs are highly risk-averse; they do indeed take greater risks, but this may be because they have be�er informa�on, have stronger desire for income, or they did not perceive some of the risks in the first place.

1. Probably most gamblers are risk-averse because they know that the odds are stacked in favor of the house or person offering the gamble. If they con�nue to take the same gamble, they know that the ENPV of repeated gambles (e.g., rolling the dice) is ul�mately nega�ve, but they hope to get lucky and experience one of the upside-risk outcomes. When be�ng on horses or sports teams, gamblers may believe they possess be�er informa�on than the person accep�ng the bet. Decision making is akin to gambling, of course, since the decision maker must choose one decision out of two or more possible decisions and then wait for the roll of the dice to see which of the possible outcomes actually happens. Thus, we o�en use the word "gamble" (as a noun) to refer to the decision-making problem facing a business decision maker. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/ch02introduc�on#return1) ]

2. As we saw in Chapter 1, informa�on search ac�vity is the purposeful search for informa�on to inform the decision making process. It includes gathering data and analyzing that data to reduce the decision maker’s ignorance of the factors and issues that might otherwise cause the outcome of the decision to vary. In Chapters 4 and 6 we look into informa�on search ac�vity rela�ve to consumer demand and to cost levels, respec�vely. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/ch02introduc�on#return2) ]

3. Entrepreneurs tend to anchor their es�mates on past outcomes and to not revise their es�mates on the basis of new informa�on. Second, they tend to base their decision making on the most recently acquired or most easily recalled informa�on, this being known as the availability heuris�c, but of course such data may not be representa�ve of the range of outcomes that should be expected. Third, the representa�ve heuris�c is where people base decisions on a rela�vely small number of observa�ons (rather than a representa�ve sample), which introduces risk because the limited sample might not be representa�ve of the range of probable outcomes. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/ch02introduc�on#return2) ]

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2.1 Adjusting for Risk Using the Certainty Equivalent

The certainty equivalent of a decision is the amount of money, available with certainty, that a person would consider equivalent to the expected value of a risky decision. In this sec�on, we will introduce risk–return trade-off curves and show how these differ according to the decision maker’s degree of risk aversion. This will allow us to demonstrate that different individuals typically have different certainty equivalents.

Risk–Return Trade-off Curves

As noted, risk causes disu�lity to be incurred by the risk-averse decision maker. We have argued that people with different degrees of risk aversion will require different amounts of compensa�on to induce them to bear an addi�onal quantum of risk. Using simple graphical analysis we can depict the risk– return trade-off curves of a par�cular risk-averse individual (whom we shall call Mr. X) shown in Figure 2.1.

Figure 2.1: Risk–return trade-off curve for risk-averse decision maker, Mr. X.

Suppose that Mr. X must decide where (at which loca�on) he will build a new restaurant. The points A, B, and C shown in Figure 2.1 relate to three different risk–return combina�ons that represent different restaurant loca�ons. We depict these three decision alterna�ves with risk measured by standard devia�on (SD) and return measured by ENPV. Their risk and return outcomes differ because of differences in popula�on density, passing traffic, proximity to public transport, and so on. As you can see, decision A has ENPV = 100 and SD = 50; decision B has ENPV = 100 and SD = 30; and decision C has ENPV = 60 and SD = 30. It should be immediately clear that Mr. X, and indeed any risk-averse profit-maximizing decision maker, will prefer B to A, because A is equally profitable but has more risk (higher SD) than B. Similarly, all risk averters will prefer B to C, because these two op�ons have the same amount of risk but B is more profitable (higher ENPV) than C.

We now know that decision B is the best choice for Mr. X, but which would he consider to be the second-best loca�on? In fact, I have prejudged the answer by drawing the risk–return (RR) curves such that A and C lie on the same RR curve (shown as RR2) so the answer is that they are both equal in the case

depicted (i.e., reflec�ng Mr. X’s feelings about risk and return). Each RR curve depicts those combina�ons of risk and return that give the same level of u�lity. These curves are more commonly known as indifference curves, which are lines drawn to pass through combina�ons of variables among which the

decision maker is indifferent, that is, receives the same amount of u�lity.4 (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.1#footernote4) Thus, Mr. X will be indifferent between A and C, or indeed any other combina�on of risk and return that lies on RR2. Now, since point B is preferred to both point A and C, it

follows that every combina�on of risk and return on RR3 is preferred to any combina�on on RR2. Similarly, any risk–return combina�on on RR1 is considered

inferior to any combina�on on any higher indifference curve. Thus, we can say that any point on a higher indifference curve will be preferred to any point on a lower indifference curve and that the direc�on of preference is shown by the arrow; more return is preferred when risk is the same, or conversely, less risk is preferred when return is the same, and the decision maker prefers combina�ons that have both more return and less risk. Note that we do not need to know the actual value to the u�lity represented by the RR1, RR2, and RR3 curves, we just need to know the order of preference—thus indifference curve

analysis is concerned with ordinal (i.e., simply in order) preferences rather than cardinal (i.e., measurable) preference differences.

The RR indifference curves demonstrate the decision maker’s trade-off between risk and return. This trade-off is also known as the marginal rate of subs�tu�on (MRS) between risk and return, which is equal to the amount of risk the decision maker will accept for an addi�onal measure of return. This trade-off is indicated by the slope of the RR curve, which is equal to the "rise over the run." In Figure 2.1, we saw that the decision maker considers points A and C to be equivalent. Now if Mr. X was asked to change from C to A, we can see that he wants 40 more units of return (the rise from 60 to 100) to compensate for the 20 extra units of risk (the run of 30 to 50). Thus, the slope of the RR2 indifference curve between points C and A is 40/20 = 2 and this

value is rather typical of this individual’s MRS at other risk–return combina�ons in the vicinity of decisions A, B, and C.5

(h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.1#footernote5)

Figure 2.2: Differing degrees of risk aversion for two different decision makers

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In Figure 2.2, we show the RR curves of two other individuals (Mr. Y and Ms. Z) who have quite different degrees of risk aversion, and thus quite different marginal rates of subs�tu�on. These people are considering restaurant loca�ons A and C, because loca�on B has already been taken by Mr. X. Note that Mr. Y prefers decision A because, for him, it lies on a higher RR indifference curve. Conversely, Ms. Z prefers decision C because, for her, it is on a higher indifference curve.

Looking carefully at Mr. Y’s indifference curves we no�ce that his risk–return trade-off (i.e., his MRS) is rela�vely low in the vicinity of point C; to move from

30 units to 50 units of risk (along RRY1) he would require only about $5 more (from $60 to about $65) to compensate for the 20 addi�onal units of risk.

Thus, his MRS for return and risk is 5/20 = 0.25. Because decision A offers $40 more return for those 20 extra units of risk, it is u�lity maximizing for him to

take decision A rather than decision C. Conversely, the MRS for Ms. Z, moving along RRZ2, is about 4 (i.e., a $40 change in return from $60 to $100 is

necessary to compensate for a 10 change in risk from 30 to about 40, along RRZ2), so the addi�onal risk associated with decision A (20 units) is not

compensated for by the addi�onal 40 units of return offered by A, and thus Ms. Z prefers decision C.

What we have demonstrated is that risk-averse decision makers will make different decisions according to their degree of risk aversion. Note that both Mr. Y and Ms. Z would have preferred restaurant loca�on B if it were s�ll available, since its risk–return outcomes would fall on a higher indifference curve for both of them. Once that decision op�on was gone, they had different preferences for the remaining two op�ons. We saw that Mr. X moved first and chose his preferred alterna�ve, which was loca�on B. Subsequently, Mr. Y and Ms. Z chose differently, because their risk–return trade-offs were different. Mr. Y, being less risk-averse, chose loca�on A, while Ms. Z, being more risk-averse, chose loca�on C.

The Certainty Equivalent as a Decision Criterion

The analysis above allows us to consider the certainty equivalent (CE) of a risky decision. The CE of a risky decision (or gamble) is the amount of return, available with certainty (i.e., zero risk), that the decision maker will consider is equivalent to the risk–return combina�on of the risky decision. Looking at Figures 2.1 and 2.2, you will see that the ver�cal axis in each figure represents a series of levels of return (ENPV) that have zero risk a�ached to them. Now no�ce that each of the indifference curves terminates at the ver�cal axis, at a point of zero risk, thus revealing the CEs for all of the risk–return combina�ons on each indifference curve.

In Figure 2.1 for example, for Mr. X the CE of decision B seems to be about 80, while the CEs of both decision A and C seem to be about 40 (i.e., where the indifference curves hit the ver�cal axis). Thus, for Mr. X the CE of decision B is much greater than the CE for either A or C, so he prefers op�on B over the other two op�ons. In Figure 2.2 we see that the CE for Mr. Y seems to be about 85 for decision A and 55 for decision C. Finally, for Ms. Z, the CE is about 22 for decision C and much lower for decision A. In each case the individual prefers the decision alterna�ve with the highest certainty equivalent.

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Put in simple terms, the certainty equivalent factor, which is the ra�o of the perceived value of the risk-free alterna�ve to the risky alterna�ve, expresses how many cents in the dollar a decision maker would consider to be equivalent to the risky decision.

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The Certainty Equivalent Factor

The Certainty Equivalent Factor (CEF) is the ra�o of the perceived monetary value of the risk-free alterna�ve (i.e., the CE) to the risky alterna�ve (i.e., the ENPV). In the case of Mr. X, the CEF for decision B is 80/100 = 0.8. The CEF effec�vely tells us what propor�on of the risky ENPV would be considered equivalent to the risky ENPV, if it were risk-free. Put another way, the CEF tells us how many cents in the dollar, available with certainty, the decision maker will consider to be equivalent to the risky decision. Thus, Mr. X values decision B at 80% of the dollar value of the ENPV. So, the CE criterion will tell us not only which is the preferred alterna�ve but will also tell us how many cents in the dollar would be just sufficient to trade for the risky decision, which tells us just how risk-averse the decision maker is. In this case Mr. X is willing to take a 30% reduc�on in monetary value to

compensate for the risk involved in decision B.6 (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.1#footernote6)

No�ce that the CE values are different for each individual—we cannot compare the psychic value of either risk or return across people. That is why the RR curves are labeled differently for the three people depicted: Each person makes his or her own, personal, internal psychic evalua�on of the disu�lity of risk and the u�lity of income and makes his own decision accordingly.

Of course, it is unrealis�c to think we would plot out risk–return indifference curves for all decision makers to see which decision they will choose. The graphical model of the decision-making process that we have u�lized here is primarily intended to facilitate your learning about risk–return trade-offs in decision making. But note that the

model has brought us to the point of a rather simple decision rule for decision making under risk and uncertainty; namely, risk-averse managers should choose the decision alterna�ve that has the highest certainty equivalent. A li�le introspec�on on the part of decision makers will lead them to an intui�ve preference for one decision alterna�ve over the others, which will reflect their personal risk–return trade-off.

4. Indifference curve analysis is used widely in economic analysis. We will encounter them again in Chapter 3 when we examine the decision-making process of consumers as they allocate their limited incomes among compe�ng goods and services. Indifference in an economics context means not preferring any one combina�on over the others, in contrast to vernacular usage where it might mean you do not care about any of the combina�ons. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.1#return4) ]

5. Although the slopes of Mr. X’s indifference curves are generally about 2, you can see that they are convex from below (or equivalently, concave from above) and thus the MRS value ranges from below 2 (where the curves are slightly fla�er) to above 2 (where they are slightly steeper). The convexity of these risk–return indifference curves reflects the individual’s increasing MRS for risk and return as more risk is undertaken. That is, the individual will require an increasing quantum of return to compensate for constant increments of risk, as the total amount of risk borne is increased. Increasing MRS of risk and return reflects the individual’s diminishing marginal u�lity of income and increasing marginal disu�lity of risk, which seems characteris�c of most human beings. We shall revisit these concepts in Chapter 3, in the context of consumer behavior. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.1#return5) ]

6. No�ce that for Mr. X, the CEF for loca�on A is 40/100 = 0.4; while for loca�on C the CEF = 40/60 = 0.67, and thus appears to rank loca�on C ahead of loca�on A, which is not what the indifference curves and the CE criterion previously told us (and we assume that they are the true reflec�on of Mr. X’s preferences). Instead the CEF provides an insight into the rela�ve degree of risk aversion exhibited by the individual for the op�on selected. For Mr. Y, the CEF of loca�on A is about 85/100 = 0.85 and for Ms. Z the CEF of loca�on C is about 22/60 = 0.37. Thus, as we noted earlier, Mr. Y is less risk-averse, followed by Mr. X, and then followed by Ms. Z, who is the least risk-tolerant of the three. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.1#return6) ]

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Simple and transparent, the maximin criterion is appropriate for risk-averse people making risky decisions and is an effec�ve way to avoid incurring losses.

© Neil Leslie/Ge�y Images

2.2 More Transparent Decision Rules for Managers

If decision makers are self-employed and are the sole owner of their own business firms, they can make their business decisions this way, but if they are employed managers of firms that are owned by other shareholders they will have to be more accountable to those shareholders for the decisions they make, and, accordingly, will have to adopt a more transparent decision rule than, "I made that decision because it made me feel be�er." Thus, we need to consider some decision rules that can be argued somewhat more objec�vely by managers to shareholders.

The Maximin Decision Rule

When the shareholders of a firm are risk-averse, as we expect they are, they will want managers to adopt decision-making rules or policies that take the risk

associated with different decisions into account.7 (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.2#footernote7) One such decision rule is maximin—that is, choose the alterna�ve that has the highest (maximal) worst (minimum) outcome. In the examples discussed earlier, we were concerned with the standard devia�on of the poten�al outcomes associated with restaurant loca�ons A, B, and C. The maximin rule is concerned with only one of those poten�al outcomes for each of A, B, and C—the worst one. It is based on the principle of affordable loss—can the firm afford to suffer the worst outcome associated with a risky decision? Shareholders will not want the manager to make decisions that could possibly bankrupt the firm (and cause shareholders to lose their investment), so they may put pressure on managers to make rela�vely conserva�ve decisions.

As an example of the maximin decision rule, consider the choice between an investment Project A and an investment Project B. For Project A the ini�al investment is $1 million in year 1 with possible final outcomes in year 2 ranging from a loss of $200,000 to a profit of $5 million. Project B has an ini�al investment cost of $2 million with possible outcomes in year 2 ranging from a loss of $500,000 to a profit of $10 million. For the maximin decision rule, we simply compare the two minimum outcomes and therefore choose Project A because its worst outcome is a loss of $200,000 compared to Project B’s worst outcome, which is a loss of $500,000. If the worst outcome were to occur, the firm would be be�er off taking a hit of $200,000 rather than $500,000.

As you can see, the maximin criterion is appropriate for risk-averse people making risky decisions that are not repeated enough �mes to allow the law of averages to work out in the firm’s favor. This decision-making rule is designed to be a simple and very transparent way to avoid incurring losses that cannot be tolerated by the firm and its shareholders. But, by refusing to consider the other poten�al outcomes it may be a very poor decision criterion. What if the probability of Project B’s worst outcome occurring was only 10% and the probability of Project A’s worst outcome was 40%? In that case, by taking decision A the managers have chosen to risk a worst outcome with four �mes the chance of occurring than the worst outcome of Op�on B. Or, what if the other possible outcomes for Project A were posi�ve but rela�vely small while the other outcomes for Project B were posi�ve and rela�vely large?

The maximin criterion does not consider these other outcomes at all.8

(h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.2#footernote8) So let’s look at some decision rules that do.

Coefficient of Variation Decision Rule

The coefficient of varia�on (CV) is a sta�s�c of a probability distribu�on and is calculated as the ra�o of the standard devia�on to the mean. Going back to the example of restaurant loca�on A, B, and C in the earlier decision-making problem, we can calculate the CV of A as 50/100 = 0.5; for B it is 30/100 = 0.3; and for C it is 30/60 = 0.5. In effect the CV criterion provides a measure of the risk per dollar of return, and the decision rule is to choose the op�on that has the smallest CV. So according to this rule, Mr. X would consider loca�ons C and A as equal but inferior to loca�on B, thereby agreeing with his certainty- equivalent-based decision. For Mr. Y and Ms. Z, the CV rule would say the two remaining op�ons are equal, but we saw that Mr. Y preferred op�on A (higher CE for him) while Ms. Z preferred op�on C (higher CE for her). Thus, the CV criterion does not take into account the differing degrees of risk aversion that individuals may have and is, therefore, an inferior decision rule for individuals making decisions when taking into account only their own risk– return preferences. But for managers making decisions on behalf of shareholders, the CV criterion may be more suitable because some shareholders (like Mr. Y) will be less risk-averse while others (like Ms. Z) will be more risk-averse. On average, shareholders might be happy enough with the CV decision rule, and they can always sell their share in this firm and buy shares in a more (or less) conserva�ve alterna�ve business if they want to.

In Figure 2.3 we show the CV decision rule as it applies to the restaurant loca�on decision problem. The CVs associated with decision A and C are equal to 0.5 in both cases, and the CV associated with decision B is 0.3. No�ce that the slope of the CV lines emana�ng from the origin (these lines are known as rays) are each equal to the reciprocal of the CV value, since the slope is equal to the rise (ENPV) over the run (SD), while CV is equal to the run (SD) divided by the rise (ENPV). Also note that in effect the CV rays are like indifference curves since every combina�on on a par�cular CV ray is equally preferred. These CV rays have constant MRS between risk and return, but as we have seen, individuals do not. Their risk–return indifference curves are concave from above, exhibi�ng increasing MRS as more and more risk is taken on. As we saw in the case of our three restaurateurs, individual preferences might agree with the CV criteria (as Mr. X did) or not. While both Mr. Y and Ms. Z agreed that loca�on B was the best loca�on, Mr. Y ranked loca�on A superior to C while Ms. Z ranked loca�on C superior to A. Thus, the CV criterion is not generally suitable for individual decision making.

Figure 2.3: The coefficient of varia�on decision criterion

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As a more complex applica�on of the CV criterion, let us now reconsider the investment Project A and Project B decision introduced above. In Tables 2.2a and 2.2b we show the probability distribu�on of outcomes associated with these projects and calculate the ENPV, SD, and CV for each project (behind the scenes I have used an Excel spreadsheet to calculate these numbers). You will see that I have assumed a discount rate of 10% and that the ini�al cost is paid at the end of year 1 while the possible outcomes (cash inflows and ou�lows) are realized at the end of year 2. Whereas earlier we selected Project A using the maximin decision criterion, by applying the CV criterion we find that Project B is preferred. Although it is riskier (SD = 1.4374 compared to 1.0414), its ENPV is much higher ($3.5124 million compared to $0.7603 million) such that the CV ra�o is only 0.4092 for Project B compared with 1.3697 for Project A.

Table 2.2a: Calcula�ng the coefficient of varia�on for Project A

(1) Ini�al Cost year 1 (millions)

(2) Present value of ini�al cost

(3) Year 2 outcomes (millions)

(4) Present value of year 2 outcomes

(5) Net present value (millions)

(6) Probability of year 2 outcomes

(7) ENPV of outcomes (millions)

DF = 0.9091 DF = 0.8264

 10 8.2644 6.4463 0.4 2.5785

−2 −1.8182   5 4.1322 2.3140 0.5 1.1570

−0.5 −0.4132 −2.2314 0.1 −0.2231

ENPV = 3.5124

SD = 1.4374

CV = 0.4092

Table 2.2b: Calcula�ng the coefficient of varia�on for Project B

(1) Ini�al cost year 1 (millions)

(2) Present value of ini�al cost

(3) Year 2 outcomes (millions)

(4) Present value of year 2 outcomes

(5) Net present value (millions)

(6) Probability of year 2 outcomes

(7) ENPV of outcomes (millions)

DF = 0.9091 DF = 0.8264

5 4.1322 3.2231 0.3 0.9669

−1 −0.9091 2 1.6529 0.7438 0.3 0.2231

−0.2 −0.1653 −1.0744 0.4 −0.4298

ENPV = 0.7603

SD = 1.0414

CV = 1.3697

Thus, the CV decision criterion is an extension of the ENPV profit-maximizing rule and is appropriate when (1) outcomes are uncertain; (2) cash flows occur beyond the current �me period; (3) similar decisions are not made repeatedly; and (4) managers are risk-averse (hopefully reflec�ng their shareholder’s preferences). Note that in this example the CV criterion agrees with the ENPV criterion but disagrees with the maximin criterion, which neglected most of the informa�on available and made the decision based simply on the best of the worst outcomes. The CV criterion is thus a more sophis�cated decision criterion that, while not generally suitable for individual decision making, does have value for decisions made by managers of firms where the decision made

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Government bonds are regarded as the ul�mate risk-free security because repayment is absolutely certain.

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must be jus�fied to risk-averse shareholders on some objec�ve basis. Moreover, it can be argued that in the context of managerial decision making within the firm, the linear indifference rays implied by the CV criterion might be a sufficient approxima�on of shareholders’ preferences in aggregate since these shareholders are free to build a por�olio of shares (in different companies) that best serves their overall risk–return preferences. The CV decision rule is transparent and easily communicable to shareholders. If they want to hold shares in a more- or less-risk-taking firm they are usually free to sell their shares in this firm and buy shares in another firm that be�er fits their risk preferences. And, in any case, they can buy shares in a variety of firms such that the overall risk exposure of their investment por�olio be�er suits their risk and return preferences.

The ENPV Criteria Using Risk Premiums

Another commonly used method to adjust uncertain cash flows for risk is to adjust the discount factor to reflect the degree of risk. We do this by adding a risk premium to the discount factor such that projects with higher risk are discounted at higher opportunity discount rates. A risk premium is an addi�onal amount that is propor�onate to the addi�onal risk perceived. Recall that we defined the opportunity discount factor as the rate of interest that could be earned on an alterna�ve opportunity of equal risk. When the decision alterna�ves are clearly not equally risky it follows that their opportunity discount rate should not be the same for each alterna�ve.

At this point it is appropriate to examine the two main component parts of the opportunity discount rate (ODR). The first main part is the risk-free rate of return that one could earn on a loan that was absolutely certain to be repaid, for example, the purchase of government bonds. Although the government may change from �me to �me, the newly elected poli�cians would respect the previous government’s obliga�on to repay lenders who had bought government bonds, so government bonds are regarded as the

ul�mate risk-free security.9

(h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.2#footernote9) The risk-free rate is made up of two subparts, the real rate of interest and the premium for expected infla�on. The real rate of interest is the rate that would cause the supply and demand for loanable funds to be equal in a market for funds without risk and without infla�on. But when lenders expect infla�on to occur, they expect the purchasing power of the funds returned (a�er the loan is se�led) to be lower than the amount loaned. For example, if prices are expected to rise by 5% over a year, the goods and services that could be purchased with $100 at the start of the year will probably cost about $105 at the end of the year. Thus,

the infla�on premium charged needs to be about 5% to compensate the lender for the loss of purchasing power due to infla�on, in this case where the

expected rate of infla�on is 5% per annum.10 (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.2#footernote10)

The second main part of the ODR is the risk premium, which is the addi�onal return the lender will require to cover the risk that the borrower might default on the loan and not pay the money back. To es�mate the appropriate risk premium, the lender must ask, "What is the probability that the borrower will not repay the loan?" To answer this ques�on, the lender (like the insurance manager in Chapter 1) must consider what propor�on of people, in roughly the same risky situa�ons, have previously defaulted on their loans. Suppose the answer is 20%. That means that one out of five borrowers did not pay the lender back the loaned funds and the interest that should have been earned on those funds. Because the lender cannot tell in advance which one in every five borrowers will be unable to repay the loan, the lender must set a risk premium on all loans that is high enough to allow the funds received back (from borrowers who do in fact repay their loans) to compensate the lender for the funds lost due to borrowers who cannot repay the loan. In this case, since only four of the five are expected to repay, all will be charged a 25% risk premium to ensure that the four borrowers repaying the loan allow the lender to recoup 4 x 25% = 100% of the loan advanced to the borrower who ul�mately defaults. The formula for the risk premium is thus the ra�o of the probability of default (PD) to its complement, the probability of repayment (PR). That is, PD/PR. In Table 2.3, we show the risk premiums for a range of default

probabili�es, and you can see that the risk premium increases exponen�ally as the probability of default increases.

Table 2.3: Default risk and calcula�on of the applicable risk premium Probability of Default PD Probability of Repayment PR Risk Premium PD/PR

 5% 95%  5.26%

10% 90% 11.11%

20% 80% 25.00%

30% 70% 42.85%

40% 60% 66.67%

50% 50% 100.00%

Now let’s revisit the Project A versus Project B decision that we considered above. Using the CV criterion we adjusted for risk by finding the risk-per-dollar- of-return and we selected Project B despite it being more risky (higher SD). But note that the expected cash flows of both projects were discounted by the same 10% discount factor. Now that we know Project B is more risky, we should discount it by a higher rate. Suppose that 10% was indeed the correct ODR

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for Project A, being the real rate of interest (say 2%) plus an expected infla�on (say 3%), plus a risk premium of 5%. Also suppose that for Project B the appropriate risk premium is about 15%, causing the ODR to be 20%. In Table 2.4 we recalculate the ENPV for Project B.

Table 2.4: Recalcula�ng the ENPV for Project B, with ODR = 20%

(1) Ini�al Cost Year 1 (millions)

(2) Present value of ini�al cost

(3) Year 2 outcomes (millions)

(4) Present value of Year 2 outcomes

(5) Net present value (millions)

(6) Probability of outcomes

(7) ENPV of outcomes (millions)

DF = 0.8333 DF = 0.6944

 10 0.6944 5.2778 0.4 2.1111

−2.000 −1.6667   5 3.4722 1.8056 0.5 0.9028

−0.5 −0.3472 −2.0139 0.1 −0.2014

ENPV = 2.8125

SD = 1.2078

CV = 0.4295

Now compare the ENPVs of the two Projects A and B. Of course the ENPV is s�ll $760,300 for Project A, but it is now about $2.8 million for Project B (down from $3.5 million) due to being more heavily discounted. So, Project B is s�ll the preferred alterna�ve using the risk-adjusted ENPV decision criteria. Also note that while the CV for Project B has increased due to the higher ODR, the CV criterion also s�ll favors Project B over Project A.

A Simplification for More Complex Situations

In prac�ce, we are typically confronted by more complex situa�ons than the above examples. Fortunately, we can simplify these examples by assuming only three outcomes (high, medium, and low) in each year and by assigning what seem to be reasonable "guess�mates" of the different monetary outcomes and of the probabili�es of these different outcomes occurring. If these es�mates are inaccurate, scru�ny by others who have different informa�on will lead us to revise them to more accurately reflect the consensus of opinion about what the values should more likely be.

If we suppose that the possible outcomes are symmetric around the medium outcome each year, and also suppose the probability distribu�ons are symmetric around the medium outcome, then a useful simplifica�on becomes possible. To find the ENPV of the decision alterna�ve we need only add the medium NCF outcomes in each year and subtract the ini�al cost outlay. This is because the high outcomes would be exactly offset by the low outcomes when they are symmetric around the medium outcome. This simplifica�on is hardly necessary for the rela�vely simple two- and three-year �me horizons that we have considered, but think about a decision with a five-year horizon—with a NCF stream stretching over five years with high, medium, and low

outcomes in each year. This would involve 35 = 243 terminal branches on the decision tree! Although one could build a very large spreadsheet or write a computer program to do all the hard work, it is generally not necessary to do so. This simplifica�on will give a sufficiently robust indica�on of the ENPV as long as there is not substan�al asymmetry of the high, medium, and low outcomes. For the most part, asymmetry one way (e.g., toward the high outcome) in one year will be offset by asymmetry the other way (toward the low outcome) in another year. Next, the future outcomes and their probabili�es are es�mates anyway, and these es�mates are increasingly like guesswork in the "out years" (i.e., beyond the present period), so it is false accuracy to place too much credence on the precise value we find for the ENPV.

Thus, it is generally a sufficient approxima�on to consider only the medium NCF outcome for each of the out years when the �me horizon is three to five years or longer. For decision alterna�ves that have longer �me horizons the "sum of the medium outcomes" approach is likely to be a sufficient approxima�on for the ENPV of the decision alterna�ve.

7. Shareholders may buy stock in a variety of firms and thus build up a por�olio of risky stocks in which the downward varia�ons in one stock’s price are likely to be offset by the upward varia�ons in another stock’s price. Notwithstanding this, risk-averse individuals will want each firm in which they hold stock to make decisions that adjust for risk. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.2#return7) ]

8. Other decision rules of the maximin type are maximax, which says you should select the alterna�ve that has the highest of the maximal outcomes, and minimax, which says you should choose the alterna�ve with the smallest of the largest outcomes. These rules similarly focus on only the largest, or the smallest, outcomes and do not consider other possible outcomes or their probabili�es. These rules seem even less applicable in real-world business situa�ons although maximax may be useful for risk-preferring decision makers and minimax might be useful for people trying to reduce their taxable income in the current year, for example. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.2#return8) ]

9. This may not apply to governments in all countries, of course. If a revolu�on or military takeover were to overthrow a government, the incoming government may decide not to honor the borrowings of the prior government. Also, in countries with excessive government debt (such as in the European debt crisis that erupted in 2011) those governments at risk of defaul�ng on their debts need to add a risk premium to induce investors to be willing to take on any further debt issues. But generally, for poli�cally stable na�ons without excessive government debt, it is safe to regard government bonds as a risk-free investment. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.2#return9) ]

10. Strictly, the infla�on premium is calculated as k/(1–k), where k is the expected rate of infla�on. Thus, for example, if expected infla�on is 5%, the infla�on premium is 0.05/0.95 = 5.26%. For expected infla�on of 10%, the infla�on premium is 0.1/0.9 = 11.11%. It can be seen that as the rate of expected infla�on increases, the infla�on premium increases faster. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.2#return9) ]

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2.3 Most-Likely Scenario and the Best- and Worst-Case Scenarios

What we have been calling the medium outcome is alterna�vely called the most-likely scenario. You will note that it had the highest probability in each year, so it is indeed more likely to occur than the high (best-case) or the low (worst-case) scenario. Now we can view the medium outcome as being representa�ve of the middle part of the probability distribu�on, and similarly view the high outcome as being representa�ve of the upside-risk side of the probability distribu�on and the low outcome as being representa�ve of the downside-risk side of the probability distribu�on. Thus, the point es�mates of high, medium, and low should be viewed as representa�ve of the three regions of the probability distribu�on.

The Normal Distribution

We can illustrate these scenarios in the context of the special case of the normal distribu�on.11

(h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.3#footernote11) The first property of a normal distribu�on is that it is symmetric around its mean value. It is o�en called a bell curve because it looks somewhat like an old-fashioned bell, as displayed in Figure 2.4. As you would guess, the mean value of the normal distribu�on is also the median (or the 50th percen�le) value. That is, 50% of the observa�ons lie above, and 50% lie below, the mean value. The standard devia�on of a normal distribu�on is such that almost all (about 99.7%) of the outcomes lie within plus or minus three standard devia�ons from the mean. Thus, the second property of a normal distribu�on is that the bell curve is just tall enough to cause 99.7% of all outcomes to lie within plus or minus three SDs from the mean. Moreover, the shape of the bell curve is such that 95% of all outcomes will lie within plus or minus two SDs from the mean, and 68% will lie within plus or minus one SD from the mean.

Now no�ce that the mean value (i.e., the ENPV) of the probability distribu�on effec�vely represents the middle part of the distribu�on, or 68% of all outcomes. It is most likely (with 68% probability) that the actual outcome will fall within the range of plus or minus one SD from the mean outcome. With repeated trials of the same decision we would expect the actual outcome to some�mes be more than the mean, and some�mes less, such that the average outcome over many trials would be the ENPV.

Note that what we call the best-case scenario is not the absolute best outcome out at the extreme right-hand side of the probability distribu�on. Instead it is representa�ve of the range of outcomes that are more than one SD above the mean. In Figure 2.4, we have depicted the best-case scenario as the outcome that roughly bisects the area under the curve to the right of the outcome that is more than one SD above the mean. Similarly, the worst-case scenario is not the worst possible outcome at the extreme le� of the probability distribu�on, but represents all the outcomes that have values more than one SD below the mean outcome, so we posi�on it at approximately the point that bisects the area under the curve to the le� of the outcome that is one SD below the mean outcome.

Figure 2.4: The proper�es of a normal distribu�on

Since the most-likely scenario (MLS) represents 68% of the outcomes (when the outcomes are normally distributed) the best-case scenario (BCS) must represent half of the remainder (i.e., 16%) and the worst-case scenario (WCS) must represent the other half of the remainder (16%). Be aware that these specific probabili�es for the high, medium, and low outcomes may or may not be appropriate for par�cular business decision problems. If you have informa�on that indicates that the outcomes seem likely to be approximately normally distributed around the mean, then these probabili�es will be appropriate. On the other hand, you might have informa�on that indicates that the probability distribu�on is definitely not normally distributed and so you should use the probabili�es that seem to be more appropriate.

Skewness of the Probability Distribution

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Skewness refers to the degree of asymmetry of the probability distribu�on. A distribu�on that is perfectly symmetric is said to be nonskewed.

© Biwa Studio/Ge�y Images

Skewness refers to the degree of asymmetry of the probability distribu�on. A distribu�on that is perfectly symmetric is said to be nonskewed. But if the bell shape is distorted with more outcomes lying to the le� of the mean outcome, with a longer tail stretching out to the right-hand side, the distribu�on is said to be posi�vely skewed. In this case the median outcome (the 50th percen�le outcome) will lie to the le� of (below) the mean outcome. The modal outcome, which is the single outcome with the highest probability of occurring, will lie to the le� of both the mean and the median outcome, as shown in Figure 2.5a. A nega�vely skewed distribu�on will have the bulge on the right-hand side of the distribu�on with a long tail to the le�. The median outcome will lie to the right of the mean outcome, and the modal outcome will lie to the right of both the mean and the median outcomes, as shown in Figure 2.5b.

Figure 2.5a: Posi�vely skewed probability distribu�on

Figure 2.5b: Nega�vely skewed probability distribu�on

The implica�on of posi�ve skewness for managerial decision making is that while the majority of the possible outcomes will be below the mean, there will be a significant number of upside-risk outcomes that lie more than three standard devia�ons above the mean (ENPV). Repeated trials of such decisions would not generate an average outcome equal to the mean outcome, but would tend to average the median outcome (i.e., below the weighted mean outcome). The probabili�es associated with the worst-case, most-likely scenario, and best-case scenario would be different to the normal distribu�on, of course. Something like 10%, 60%, and 30% might be more appropriate for the worst-case, most-likely-case, and the best-case scenarios, respec�vely.

Conversely, when the distribu�on is nega�vely skewed, the majority of the possible outcomes will be above the mean, but there will be significant number of downside-risk outcomes that lie more than three standard devia�ons below the mean. Repeated trials of decisions with nega�vely skewed distribu�ons would tend to result in an average outcome that is above the weighted mean (ENPV) outcome. The probabili�es associated with the worst-case, most-likely scenario, and best-case scenario would be different to the normal distribu�on, of course. Something like 30%, 60%, and 10% might be more appropriate for the worst-case, most-likely-case, and the best-case scenarios, respec�vely.

Kurtosis of the Probability Distribution

Kurtosis refers to another aspect of the shape of a probability distribu�on, specifically its height. A distribu�on that is taller than a normal distribu�on is said to be leptokur�c and would have more than 68% of the outcomes falling within one SD each side of the mean. Conversely, a distribu�on that is platykur�c (like a plate, i.e., fla�er) would have less than 68% of the outcomes within one SD each side of the mean. Kurtosis of probability distribu�ons is demonstrated in Figure 2.6.

Figure 2.6: Kurtosis of probability distribu�ons

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You can see from the shapes of the two probability distribu�ons in Figure 2.6 that for a leptokur�c distribu�on, the propor�on of outcomes lying within one SD from the mean will be substan�ally above 68%, perhaps 80% in the example shown. This means that the probabili�es of the best-case and worst-case scenarios are rela�vely small, about 10% each in the situa�on depicted. Conversely, for the platykur�c distribu�on, the propor�on of the outcomes lying within one SD of the mean would be substan�ally below 68%, perhaps only 40–50% in the example shown. Thus, the probabili�es of the best- and worst- case scenarios might be rela�vely large, about 30% each in the platykur�c distribu�on shown in Figure 2.6.

What is the point of all this for the managerial decision maker? First, the decision maker needs to consider whether the probability distribu�on of outcomes is likely to be approximately normal or not. Remember that many decision situa�ons are likely to deliver approximately normal distribu�ons of outcomes since the independent ac�ons of many people (e.g., buyers) typically result in a normal distribu�on. Second, if the decision maker has reason to believe that the distribu�on will be skewed to one side or the other, or taller or fla�er than a normal distribu�on due to factors that he or she suspects are characteris�c of the popula�on or sample, the probabili�es need to be adjusted in the direc�on that reflects the decision maker’s best es�mate of the actual shape of the probability distribu�on. In the absence of any informa�on to indicate that the normal distribu�on is not appropriate, it is usually a good first approxima�on to assume normality of the distribu�on. It is useful to remember that unless you have data on the distribu�on of prior outcomes of similar decisions, assigning probabili�es to possible outcomes is an art, not a science—the decision maker needs to think about the situa�on and go with his or her best guesses. As a decision maker, it is useful to check your own best guesses against the opinions of others who have knowledge of the decision scenario. Their scru�ny may reveal informa�on that was not known to you and allow a more accurate probability distribu�on to underlie your decision. And finally, if this type of decision scenario is repeated again and again, data will build up and the probability distribu�on can be corrected subsequently.

11. It is called the normal distribu�on because this par�cular distribu�on of outcomes occurs in thousands of situa�ons, across many different fields of the natural and the social sciences. It does not imply that distribu�ons that are not normally distributed around their mean are abnormal in any pejora�ve sense. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.3#return11) ]

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When data is not available on the normal distribu�on of prior outcomes of similar decisions, the decision maker must rely on best guesses.

© Paul Taylor/Ge�y Images

Summary

In this chapter, we introduced adjustment for risk into our decision-making toolkit. Since business managers typically must make decisions under condi�ons of risk and uncertainty, we needed to incorporate a measure of the dispersion of the possible outcomes as well as the expected net present value (ENPV) of the probability distribu�on into the decision rule. We noted that risk is best measured by the standard devia�on of the probability distribu�on of the possible outcomes that might follow a decision. But we also needed to consider how decision makers feel about risk, since they might be risk-averse, risk preferring, or unconcerned by risk (risk neutral). We concluded that decision makers need to be risk-takers but are most likely risk-averse, preferring less risk when all other things (such as ENPV) are equal. For repe��ve decisions of the same type and in the same context, decision makers can act as if they are risk neutral. Similarly, if shareholders of firms hold stock in a large number of different firms, they would have a por�olio of stocks that would not be suscep�ble to wide swings in total value (except in a global financial crisis) and could act as if they were risk neutral with respect to any par�cular firm’s decisions. Notwithstanding that, these shareholders will want the managers of each of the firms (in which they hold stock) to adjust their decisions for risk.

Because risk causes the risk-averse owners of businesses to experience disu�lity, a trade- off is made against the monetary value of the decision (i.e., the ENPV) to compensate for bearing risk. For individuals making decisions on their own behalf (including sole owners of businesses) the certainty equivalent (CE) criterion is applicable. We introduced risk– return indifference curves to demonstrate that the individual will take addi�onal risk for addi�onal return, or conversely would give up part of the poten�al return in order to avoid risk. The certainty equivalent (CE) is the amount of money available with certainty that the decision maker will feel is equivalent to the weighted average of the risky gamble (i.e., the ENPV). Without plo�ng out all risk–return indifference curves, the individual decision maker simply needs a moment of introspec�on to conclude which one of the possible alterna�ve decisions makes him or her feel most comfortable, taking into account both the expected u�lity from money and the expected disu�lity from risk.

This process of introspec�on and personal choice is not usually sufficient for managers of other peoples’ money. So managers of firms in which there are other shareholders must u�lize decision rules that can be jus�fied on some objec�ve basis. The coefficient of varia�on (CV) decision criterion is the ra�o of the standard devia�on to the weighted

mean (i.e., ENPV) of the probability distribu�on and the CV ra�o effec�vely measures the risk-per-dollar-of-return. This is a transparent decision rule that shareholders can easily understand, such that they can invest in a different firm if the firm in which they have previously invested is taking (in their view) too much risk-per-dollar of poten�al return. Another transparent adjustment to the ENPV criterion to explicitly take risk into account is to use an appropriate risk premium as part of the opportunity discount rate (ODR). Choosing a risk premium that is based on the expected prior probability of failure of the investment or other decision causes the ENPV to be smaller, the higher the risk premium is. Riskier gambles are discounted more heavily, reducing their ENPV to lower levels. So, when comparing decision alterna�ves with different risk profiles, the decision maker should choose the one with the highest ENPV remaining a�er the net cash flow streams have been discounted by an ODR containing the appropriate risk premium.

Many decision problems can be reduced to rela�vely simple terms, such as three poten�al outcomes (high, medium, and low) in each of one or two years. Decision tree analysis can be u�lized for these rela�vely simple problems, where the number of terminal branches is equal to the number of outcomes (in each year) to the power of the number of years. But decisions that have outcomes con�nuing into the third and subsequent years have too many terminal branches to be realis�cally solved using a decision tree. Instead, we can simplify these more complex decisions in one way or another and s�ll arrive at a risk-adjusted-ENPV valua�on of the decision alterna�ve that should be sufficiently reliable (given that we are guess�ma�ng about future outcomes in any case). One way is to treat the decision tree as being approximately symmetrical, with the high outcomes being balanced out by the low outcomes. If this is a sufficiently realis�c assump�on, we can simply add up the net present value (NPV) of the medium case outcomes in each year to approximate the ENPV of the en�re probability distribu�on. We then argued that the medium outcome is actually representa�ve of the most-likely scenario—that is, it is the midpoint of a rela�vely narrow range of outcomes (plus or minus one SD from the mean) and that the actual outcome will fall within that range about two thirds (68%) of the �me; falling above that range (best-case scenario) about one sixth of the �me; and falling below that range about one sixth of the �me. Thus, we can safely make our decisions based on the ENPV as being the best predictor of the actual outcome, as long as the probability distribu�on is approximately a normal distribu�on.

If the distribu�on is skewed to one side or the other, the ENPV is no longer the most probable (or modal) outcome. If the distribu�on is posi�vely skewed (with a long tail to the right) the most likely scenario will have probability less than two thirds and the best-case scenario with have probability more than one sixth. Conversely, if the skew is nega�ve with a long tail to the le�, the most likely scenario will have probability less than two thirds and the worst-case scenario with have probability more than one sixth. Further, if the probability distribu�on exhibits leptokurtosis, being taller than a normal distribu�on, the most-likely scenario will have probability significantly higher than two thirds and oppositely, if it exhibits platykurtosis, the most-likely scenario will have probability significantly lower than two thirds. It is up to the decision maker to ensure that the use of the normal distribu�on is appropriate before proceeding on that simple but rela�vely reliable assump�on.

So, in the first two chapters of this book we have paved the way for decision making in the real world where decisions are made in the context of risk and uncertainty and where there are cost and revenue implica�ons that stretch beyond the present period. We have established that we need to use both expected value analysis to account for risk and uncertainty, and net present value analysis to account for the future value of revenues and costs. Combining these we have expected net present value (ENPV) as our measure of the firm’s objec�ve func�on. Then we recognized that people tend to be risk-averse

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and incur disu�lity from risk and uncertainty and thus, will wish to make risk-adjusted decisions that reflect the monetary trade-off they are prepared to make to avoid risk, or, conversely, the monetary gains they expect to receive if they have to bear risk. Having spent this �me learning the ground rules, we are now ready to proceed to learn more about the costs and revenue sides of the decisions that managers must make. In the following chapters, we will focus on the demand side of markets to gain a strong understanding of the revenue implica�ons of decisions.

Ques�ons for Review and Discussion

Click on each ques�on to reveal the answer.

1. Explain why the standard devia�on of a probability distribu�on is an appropriate measure of the risk involved in a real-world decision problem. (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

A measure of risk needs to measure the extent by which the actual outcome is likely to diverge from the expected value (EV) of a probability distribu�on. The standard devia�on (SD) is a measure of the average absolute devia�on (i.e., both above and below the EV of a probability distribu�on). The SD measures how far to one side or the other of the EV the actual outcome would fall, on average over many trials, and is thus a suitable measure of the risk because the SD is higher when the risk of an outcome not being the EV is higher. Conversely, the SD is lower when the risk of the actual outcome not being the EV is lower.

2. Define risk preference, risk aversion, risk seeking, and risk taking. (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

Risk preference means that other things being equal (such as rewards) the person will prefer the op�on with the highest risk. Oppositely, risk aversion means that other things being equal, the person will prefer the op�on with the lowest risk. Risk seeking means the person is mo�vated to seek out risky situa�ons either because the person is risk preferring, or if risk averse, because the person knows that higher-risk situa�ons generally offer higher rewards. Risk taking means that the person takes risks, either voluntarily or involuntarily, as we all do.

3. Explain why a risk averter facing two choices might prefer to take the higher risk alterna�ve. (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

When facing two choices, one with higher risk than the other, the risk averter will choose the higher-risk alterna�ve if the u�lity from the reward associated with the higher-risk alterna�ve is greater than the disu�lity associated with the risk of the higher-risk alterna�ve.

4. How might a risk-averse decision maker adjust the expected net present value of a decision to take into account its risk? (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

Several methods were suggested. (i) The person might introspec�vely consider his or her risk-return trade-offs in risk-reward space and choose the alterna�ve that lies on the highest indifference curve. (ii) The certainty equivalent (CE) is the sum of money available with certainty that the person will consider equal to the EV of the risky decision—the person would choose the alterna�ve with the highest CE. (iii) The CE factor (CEF) is the ra�o of the CE to the ENPV and effec�vely measures how many cents in the dollar, available with certainty, that the person would accept instead of taking the gamble. (iv) The maximin rule avoids the worst outcome by choosing the alterna�ve with the largest of the smallest outcomes. (v) The coefficient of varia�on (CV) rule is to choose the alterna�ve with the highest ra�o of SD/ENPV and effec�vely measures risk per dollar of return. (vi) Finally, if the ENPV is calculated using opportunity discount factors reflec�ve of the appropriate risk premium, the risk-averse decision maker will choose the alterna�ve with the highest ENPV.

5. Some risk-adjus�ng decision rules work well for the individual but are not sufficient for decisions made on behalf of others, such as shareholders of the firm. Please explain. (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

The subjec�ve criteria that require the person to reflect on his or her personal risk aversion and profit preferences (such as risk-reward curves and certainty equivalents) cannot easily be explained or jus�fied to other shareholders or stakeholders of the firm. Criteria that are more objec�ve, such as the CV rule and the ENPV using risk premiums, are more readily explained and defended to shareholders and stakeholders.

6. What is the certainty equivalent of a decision alterna�ve? Use this rule to explain your choice between a 70% chance of winning $10 and a 50% chance of winning $20. (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

The certainty equivalent (CE) of a decision alterna�ve is the sum of money available with certainty that the person will consider to be equal to the EV of the risky decision. The EV of $10 with probability 70% is $7.00 and similarly the 50% chance of $20 is $10. The la�er gamble has higher EV, but is more risky (i.e., 50% chance of not happening vs. 30%). If you plot these in risk-return space you would see that your indifference curves through these points would need to be very flat for the $20 alterna�ve to be preferred (i.e., to lie on a higher indifference curve than the $10 gamble). If so, this would mean you would have a very low marginal rate of subs�tu�on (MRS)—i.e., the amount of risk you would accept for an addi�onal unit of return is very low, meaning you are highly risk averse. Conversely, if your MRS is higher, the indifference curves will be steeper and the CE of the $20 gamble will be lower than the CE of the $10 gamble.

7. What are the main two components of the opportunity discount rate, and how do you find the value of these components? (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

The opportunity discount rate (ODR) is composed of the risk-free rate and the risk premium. The risk-free rate is the sum of the "real" rate of interest and the expected rate of infla�on. The real rate is the "price" of money (expressed as a percentage rate of interest) determined in the market for loanable funds by the intersec�on of the demand for loanable funds (i.e., loans demanded by individuals and organiza�ons) and the supply of loanable funds (i.e., deposits put in banks by individuals and organiza�ons). The expected rate of infla�on is the market consensus regarding the expected rate of decline (expressed as a percentage) of the purchasing power of the dollar over a specified period. The risk-free rate is found by checking the rate of return on government bonds, since these are generally considered to be risk free. The risk premium is calculated as the ra�o of the probability of default over the probability of repayment, and this must be es�mated based on the experience of similar loans or investments in the recent past.

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8. Why would a symmetric decision tree analysis of a complex decision problem give the same result as a simple summa�on of the medium net cash flows associated with that decision problem? (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

A symmetric decision tree means that the upside variance is the same in absolute terms as the downside variance, such that posi�ve devia�ons from the expected value are exactly offset by nega�ve devia�ons from the expected value. If the decision tree is symmetric, the mean, median and model values will coincide, and thus the expected value will be the median outcome.

9. Define the most-likely, best-case, and worst-case scenarios in terms of a probability distribu�on of the outcomes of a decision alterna�ve. (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

The most-likely outcome is the modal outcome, the one with the highest probability of occurring. In decision analysis, we use the expected value (equal to the modal outcome if the probability distribu�on is symmetric) to be representa�ve of the range of outcomes that extends from one standard devia�on (SD) either side of the expected value (EV) outcome. The best-case scenario is a number chosen to represent the outcomes that are more than one SD higher than the EV. The worst-case scenario is a number chosen to represent the outcomes that are more than one SD below the EV.

10. Explain how the probabili�es associated with a probability distribu�on would need to be adjusted if the probability distribu�on was (i) platykur�c, and (ii) nega�vely skewed. (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

If the distribu�on is significantly platykur�c (i.e., fla�er than a normal distribu�on), the percentage of observa�ons that lie within plus or minus one SD from the EV will exceed 68% and the probability of the best and worst case scenarios will be less than 16% each. Conversely, if the distribu�on is significantly leptokur�c (i.e., taller than a normal distribu�on), the percentage of observa�ons that lie within plus or minus one SD from the EV will be less than 68% and the probability of the best and worst case scenarios will exceed 16% each.

Decision Problems

1. While comple�ng your studies you want to try out as an entrepreneur and are considering two different entrepreneurial opportuni�es that you have iden�fied. On the one hand, you could invest all your money in a project to print T-shirts and sell them to visitors to the county fair in your home town. You expect that no one else would be doing that, yet you are fairly sure there will be a market for these T-shirts for people a�ending the fair and wan�ng a commemora�ve keepsake or a gi� for others. Your es�mate of ENPV for this project is $3,000 with standard devia�on of $1,000. On the other hand, you could invest all your money in a project to sell umbrellas outside the football stadium at your old high school for the annual East versus West game, which would be in high demand if it is raining, but would sell very poorly if it is not raining. You es�mate that the ENPV of this is alterna�ve is $4,000 with standard devia�on of $2,000.

a. Apply the coefficient-of-varia�on decision criterion to these alterna�ves to find which is preferred using that method of risk adjustment. b. Apply the maximin criterion, supposing that the worst outcome for the T-shirt alterna�ve is that you would make only $500 profit while for the

umbrella op�on the worst outcome (no rain) would cause you to lose $1,000. c. Indulge in some introspec�on and es�mate your personal certainty equivalent for these two projects.

2. The Sounds True Music company is considering the introduc�on of a new memory storage device to compete in the market for mini devices that allow recorded music to be replayed via Bluetooth through smart TVs and stereo systems. Sounds True Music has been taking losses due to heavy compe��on from rivals and is concerned about making the best choice among two alterna�ves being considered. Op�on A is to make a minor faceli� to an exis�ng product, while op�on B is to introduce a totally new product. The managers have determined that the net cash flow outcomes for the current year will depend on the state of the economy, as shown in the following table. They es�mate that the probabili�es of these macroeconomic outcomes are 30% for a downturn, 50% for constant, and 20% for an upturn.

State of the economy Op�on A – minor faceli� (NCF) Op�on B – new product (NCF)

Downturn $10 $−20

Constant $30 $20

Upturn $80 $150

a. Calculate the expected value, standard devia�on, and coefficient of varia�on for each decision alterna�ve (use a spreadsheet and the embedded formula for standard devia�on).

b. Apply the expected value, coefficient of varia�on, and maximin decision criteria to this decision problem. c. Which alterna�ve is likely to have the greater certainty equivalent, and why?

3. The Express Delivery Company operates a courier and parcel delivery service between the major ci�es in southern California. Business has been booming and it needs to add another very large truck to the fleet. Management is considering whether to lease or to buy the addi�onal truck. Careful analysis of costs and the poten�al demand situa�on has led to the following es�mates of net cash flow for each alterna�ve. The applicable opportunity discount rate is 15%.

NCF Year 1 (millions) Probability Year 1 NCF Year 2 (millions) Probability Year 2

LEASE OPTION

−5 0.25  5 0.30

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 5 0.40 10 0.50

15 0.35 15 0.20

BUY OPTION

−10 0.20 10 0.30

 0 0.50 15 0.50

10 0.30 20 0.20

a. Using decision tree analysis, find the expected net present value (ENPV) of each alterna�ve. b. Calculate a measure of risk for each alterna�ve. (See Table 2.1 for the basic method, which you will need to modify to accommodate the two-year

scenario here.) c. Apply several decision criteria and make your recommenda�on to management. d. State any qualifica�ons or reserva�ons you would want to add to your recommenda�on.

4. Your firm is considering the introduc�on of a new product, and you are required to set the price. You are considering three price strategies: high ($6), medium ($4), and low ($2.50). Your market research team has provided the following es�mates of sales at each price level over the next two years. The ini�al investment will be $22,000 and your costs per unit of output will be $1, regardless of volume. Your finance manager says that you could otherwise invest these funds at comparable risk in a forthcoming bond issue at 12.5% per annum.

HIGH PRICE MEDIUM PRICE LOW PRICE

Sales volume Probability Sales volume Probability Sales volume Probability

First year First year First year

3,500 0.1 5,000 0.2 10,000 0.4

2,500 0.3 4,000 0.5  7,500 0.3

1,500 0.6 3,000 0.3  5,000 0.3

Second year Second year Second year

5,000 0.2 8,000 0.3 12,000 0.3

4,000 0.3 6,500 0.4  9,000 0.5

3,000 0.5 5,000 0.3  5,000 0.2

a. Using decision tree analysis, find which alterna�ve promises the highest ENPV over the two-year period. b. Should the investment funds be used to buy the bonds instead? Why? c. Rank the alterna�ves in order of their risk, and explain the basis for your ranking. d. Rank the alterna�ves in order of their risk-adjusted expected present value.

5. The manager of the Highfields Real Estate Development Company is faced by a complex decision problem that has outcomes stretching over five years with mul�ple outcomes possible each year. There are two alterna�ve plans under considera�on for a large parcel of land that the company owns. Plan A is to seek city approval to subdivide the land into small housing lots, install all the necessary u�li�es and other infrastructure, and offer the lots for sales to individual homeowners wan�ng to build their own homes. Plan B is to set up the required infrastructure and then sell the land to a major home-building company that would develop the land into a new execu�ve suburb with higher quality homes of similar appearance on larger lots. In both cases the revenues would come to Highfields only as the lots are sold (plan A) or as completed houses are sold by the major home-building company (plan B). Highfields would need to borrow at a risk premium that depends on which project is undertaken. Suppose the current risk-free five-year bond rate is 3%, and that plan A carries a probability of a low outcome causing bankruptcy of 25%, whereas plan B has a probability of a low outcome (and bankruptcy) of only 20%. Highfields considers the probability distribu�ons to be symmetric in both cases and has es�mated the most-likely outcomes for each plan as shown in the following table.

Plan A – net cash flows in the most-likely scenario ($m) Plan B – net cash flows in the most-likely scenario ($m)

Year 1 −20 −10

Year 2 10 5

Year 3 40 30

Year 4 60 80

Year 5 30 100

a. Calculate the ENPV of each plan to find the preferred alterna�ve by that criterion.

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b. What addi�onal informa�on would be required to allow applica�on of the maximin criterion? c. What addi�onal informa�on would be required to allow applica�on of the coefficient of varia�on criterion? d. Given that there is no more informa�on available, make your recommenda�on and support it with reasoning, and list any other informa�on that you

suggest Highfields should seek before taking ac�on.

Key Terms

Click on each key term to see the defini�on.

best-case scenario (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A scenario that is representa�ve of the range of outcomes that are more than one standard devia�on above the mean of all the possible scenarios.

cardinal (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A system of measurement that allows the differences between things, such as the height of two fences, to be compared using a linear scales (such as inches), versus ordinal measures that simply rate one situa�on as being greater (higher) or less (lower) than another without specifying how much higher or lower one fence is.

certainty equivalent (CE) (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The amount of money available with certainty (that is, without risk) that the decision maker feels is equivalent to the risky gamble (the ENPV with an associated standard devia�on).

Certainty Equivalent Factor (CEF) (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The ra�o of the perceived monetary value of the risk-free alterna�ve (the CE) to the risky alterna�ve (the ENPV).

coefficient of varia�on (CV) (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A measure of the risk-per-dollar-of-return, derived by dividing the standard devia�on by the expected net present value.

direc�on of preference (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The direc�on that a person would want to go to increase total u�lity or to achieve higher return and lower risk.

downside risk (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The risk that the outcome that occurs will be less than the expected net present value.

highly intolerant of risk (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Those who are in this category try to avoid voluntary risk taking as much as possible. Risk-intolerant people need much greater monetary compensa�on to induce them to accept addi�onal risk.

highly tolerant of risk (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Are those who are slightly risk-averse. Bearing risk causes them rela�vely li�le psychic dissa�sfac�on, so they will take on addi�onal risk for rela�vely small increases in expected value.

indifference curves (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Lines that represent combina�ons of goods and services that give a consumer an iden�cal quan�ty of u�lity or represent a set of choices among which the buyer is indifferent.

infla�on premium (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

That part of the opportunity discount rate that is intended to compensate the lender for the expected fall in the purchasing power of the monetary unit (the dollar) during the period of the loan. It is calculated by the equa�on: k/(1 − k) where k is the expected rate of infla�on.

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involuntary risk taking (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Everyday risks that are imposed on us by nature, or other external circumstances, and which cannot easily be avoided.

kurtosis (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The height of a probability distribu�on; the degree to which it is peaked or flat.

leptokur�c (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A feature of a probability or frequency distribu�on that is evident as a rela�vely tall and narrow peak surrounding the mean because the data values have rela�vely li�le variance and are mostly similar to the average value.

marginal rate of subs�tu�on (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Reflected by the slope of the indifference curve and represents the rate of subs�tu�on between two compe�ng items that leaves a person at the same level of u�lity.

maximin (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A decision rule, where a consumer chooses the alterna�ve that has the largest (maximal), of the smallest (minimum) outcomes.

median outcome (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The 50th percen�le outcome, calculated by ranking the outcomes in order of magnitude and selec�ng the outcome that falls at 50% of the total number of outcomes.

modal outcome (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The single outcome with the highest probability of occurring, or the outcome that repeats the most amount of �mes in a given set of data.

most-likely scenario (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A scenario representa�ve of the middle range of the probability distribu�on, usually defined as the range of outcomes that are plus or minus one standard devia�on from the mean outcome.

nega�vely skewed (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A distribu�on pa�ern that has a bulge on the right-hand side of the distribu�on with a long tail stretching to the le�.

normal distribu�on (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A symmetrical distribu�on with a peak in the center of the range of possible outcomes, and with shape such that 68% of the observa�ons lie within plus or minus (+/–) distribu�ons that are not one standard devia�on from the mean; 95% lie within +/– two standard devia�ons; and 99.7% lie within +/– three standard devia�ons.

opportunity discount factor (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The rate of interest that could be earned on an alterna�ve opportunity of equal risk.

ordinal (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

An evalua�on system that relies on rela�ve loca�on (such as higher or lower, larger or smaller), as opposed to cardinal measures that would place a value on the differences (such as how much higher or how much larger).

platykur�c (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A distribu�on that has rela�vely wide dispersion around the mean value and thus has a wider and fla�er distribu�on surrounding the mean.

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posi�vely skewed (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

When results are posi�vely skewed the mean is greater than the median, which is greater than the mode, and therefore the right tail of the distribu�on is stretched. In this type of distribu�on, the expected outcome is likely to be below the mean, but there is a higher chance of a few extremely posi�ve results.

real rate of interest (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A measure of how much it really costs to borrow money when the infla�on rate is subtracted from the actual nominal interest rate of a given loan.

risk aversion (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A condi�on in which an investor or other decision maker exhibits a preference for less risk, other things (such as expected returns) being equal.

risk neutral (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A state where investors are unconcerned by the amount of risk associated with a given investment and don’t allow risk to enter their investment decisions.

risk preference (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A state where the individual prefers more risk to less risk, with other things (such as reward or profit) being equal. A risk preferer would therefore choose the riskier of two equally profitable investments.

risk premium (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

This is the amount by which the opportunity discount rate (ODR) is increased to reflect the higher risk of a risky decision, compared to one that is devoid of risk.

risk seeker (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

An investor that deliberately seeks out investments that have higher risk, generally because they also have higher expected returns.

risk taker (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

While risk takers do not go out of their way to deliberately find risky ventures like risk seekers do, they are ready to accept extra risk if the an�cipated returns are sufficiently higher than for a less risky investment.

risk-free rate (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Government bonds issued by a federal Treasury, according to standard economic theory, are examples of risk-free investments because they offer a very small rate of return in exchange of a lack of default risk.

semi-variance (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A calcula�on of the variance of a distribu�on including only those data points that fall below the mean.

skewness (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Indicated by asymmetry in the distribu�on curve around the mean and can be either nega�ve (skewed to the le�) or posi�ve (skewed to the right).

slope (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A measure of the incline or decline of a surface, or a line such as the risk–return (RR) curve, which is equal to the "rise over the run," or the length versus height of the line over a specific distance.

standard devia�on (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The extent of dispersion of the observa�ons in a data set from the data’s mean value. Standard devia�on increases when the data observa�ons are spread more widely around the mean value.

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upside risk (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The measure of the probability that value of the actual outcome will be higher than the expected present value of the prior distribu�on of possible outcomes.

variance (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A measure that evaluates the spread of data observa�ons in comparison to the loca�on of the mean for the given data set.

voluntary risk taking (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

An ac�vity engaged in by investors and individuals with the expecta�on that the u�lity of the reward will outweigh the disu�lity of the risk.

weighted mean (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A measure of central tendency that weights each observa�on by its probability of occurring, such as the ENPV of a probability distribu�on of poten�al outcomes.

worst-case scenario (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A situa�on that represents a range of outcomes that lie more than one standard devia�on below the mean or ENPV outcome.