Case study
13
Risk and Capital Budgeting
LEARNING OBJECTIVES
|
LO 13-1 |
The concept of risk is based on uncertainty about future outcomes. It requires the computation of quantitative measures as well as qualitative considerations. |
|
LO 13-2 |
Most investors are risk-averse, which means they dislike uncertainty. |
|
LO 13-3 |
Because investors dislike uncertainty, they will require higher rates of return from risky projects. |
|
LO 13-4 |
Simulation models and decision trees can be used to help assess the risk of an investment. |
|
LO 13-5 |
Not only must the risk of an individual project be considered, but also how the project affects the total risk of the firm. |
Nobody understands the meaning of risk better than Apache Corp., a firm that drills for natural gas and oil on properties in the United States, Canada, Australia, Egypt, and the North Sea.
Over the last 20 years, the price of oil has vacillated from less than $11 per barrel to over $145 per barrel. Natural gas has been even more volatile. What appears to be a great opportunity for drilling and discovery when energy prices are at their peak can turn out to be a disaster when those prices drop by 25 to 50 percent or more.1 An even greater threat to Apache Corp. is the proverbial “dry hole,” in which millions of dollars are spent only to discover that there is no oil to be found.
Energy producers such as Apache Corp. are much more vulnerable to changing circumstances in the market than fully integrated oil companies such as ExxonMobil, which not only drills for oil and gas, but refines it and sells it at the retail level through its service stations. For ExxonMobile, lower profits at the discovery level are often offset by higher profits at the retail level and vice versa.
But the upside for oil producers such as Apache Corp. is enormous when they discover oil and gas in a previously untested area. The risk and rewards of this business exceed those in almost any other.
In this chapter, we examine definitions of risk, its measurement and its incorporation into the capital budgeting process, and the basic tenets of portfolio theory.
Definition of Risk in Capital Budgeting
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Risk may be defined in terms of the variability of possible outcomes from a given investment. If funds are invested in a 30-day U.S. government obligation, the outcome is certain and there is no variability—hence no risk. If we invest the same funds in a deep-sea oil drilling venture above the Arctic Circle, the variability of possible outcomes is great and we say the project is extremely risky.
Risk is measured not only in terms of losses but also in terms of uncertainty.2 We say gold mining carries a high degree of risk not just because you may lose your money but also because there is a wide range of possible outcomes. Observe in Figure 13-1 examples of three investments with different risk characteristics.
In each case, the distributions are centered on the same expected value of $20,000, but the variability (risk) increases as we move from Investment A to Investment C. Because you may gain or lose the most in Investment C, it is clearly the riskiest of the three.
Figure 13-1 Variability and risk
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The Concept of Risk-Averse
A basic assumption in financial theory is that most investors and managers are risk-averse—that is, for a given situation they would prefer relative certainty to uncertainty. In Figure 13-1, they would prefer Investment A over Investments B and C, although all three investments have the same expected value of $20,000. You are probably risk-averse too. Assume you have saved $1,000 toward your last year in college and are challenged to flip a coin, double or nothing. Heads, you end up with $2,000; tails, you are broke. Given that you are not enrolled at the University of Nevada at Las Vegas or that you are not an inveterate gambler, you will probably stay with your certain $1,000.
This is not to say investors or businesspeople are unwilling to take risks—but rather that they will require a higher expected value or return for risky investments. In Figure 13-2, we compare a low-risk proposal with an expected value of $20,000 to a high-risk proposal with an expected value of $30,000. The higher expected return may compensate the investor for absorbing greater risk.
Figure 13-2 Risk-return trade-off
Actual Measurement of Risk
A number of basic statistical devices may be used to measure the extent of risk inherent in any given situation. Assume we are examining an investment with the possible outcomes and probability of outcomes shown in Table 13-1.
Table 13-1 Probability distribution of outcomes
|
Outcome |
Probability of Outcome |
Assumptions |
|
$300 |
0.2 |
Pessimistic |
|
600 |
0.6 |
Moderately successful |
|
900 |
0.2 |
Optimistic |
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The probabilities in Table 13-1 may be based on past experience, industry ratios and trends, interviews with company executives, and sophisticated simulation techniques. The probability values may be easy to determine for the introduction of a mechanical stamping process in which the manufacturer has 10 years of past data, but difficult to assess for a new product in a foreign market. In any event, we force ourselves into a valuable analytical process.
Based on the data in Table 13-1, we compute two important statistical measures—the expected value and the standard deviation. The expected value () is a weighted average of the outcomes (D) times their probabilities (P).
The expected value () is $600. We then compute the standard deviation—the measure of dispersion or variability around the expected value:
The following steps should be taken:
The standard deviation of $190 gives us a rough average measure of how far each of the three outcomes falls away from the expected value. Generally, the larger the standard deviation (or spread of outcomes), the greater is the risk, as indicated in Figure 13-3. You will note that in Figure 13-3 we compare the standard deviation of three investments with the same expected value of $600. If the expected values of the investments were different (such as $600 versus $6,000), a direct comparison of the standard deviations for each distribution would not be helpful in measuring risk. In Figure 13-4 we show such an occurrence.
The investment in panel A of Figure 13-4 appears to have a high standard deviation, but not when related to the expected value of the distribution. A standard deviation of $600 on an investment with an expected value of $6,000 may indicate less risk than a standard deviation of $190 on an investment with an expected value of only $600 (panel B).
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Figure 13-3 Probability distribution with differing degrees of risk
Figure 13-4
We can eliminate the size difficulty by developing a third measure, the coefficient of variation (V). This term calls for nothing more difficult than dividing the standard deviation of an investment by the expected value. Generally, the larger the coefficient of variation, the greater is the risk. The formula for the coefficient of variation is numbered 13-3.
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For the investments in panels A and B of Figure 13-4, we show:
We have correctly identified investment B as carrying the greater risk.
Another risk measure, beta (β), is widely used with portfolios of common stock. Beta measures the volatility of returns on an individual stock relative to the stock market index of returns, such as the Standard & Poor’s 500 Stock Index.3 A common stock with a beta of 1.0 is said to be of equal risk with the market. Stocks with betas greater than 1.0 are riskier than the market, while stocks with betas of less than 1.0 are less risky than the market. Table 13-2 presents a sample of betas for some well-known companies from 2008 to 2013. We note that betas are not stable over time.
Table 13-2 Average betas for a five-year period (ending February 2015)
|
Company Name |
Beta |
|
Walmart Stores Inc. |
0.36 |
|
Coca-Cola Co. |
0.47 |
|
Philip Morris International |
0.57 |
|
Exxon Mobil Corp. |
0.79 |
|
Nike Inc. Cl B |
0.89 |
|
Apple Inc. |
0.97 |
|
Intel Corp. |
1.09 |
|
The Walt Disney Co. |
1.15 |
|
Starbucks Corp. |
1.24 |
|
Fedex Corp. |
1.34 |
|
Apache Corp. |
1.45 |
|
Alcoa |
1.59 |
|
Ford Motor Co. |
1.80 |
|
Bank of America Corp. |
2.56 |
Risk and the Capital Budgeting Process
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How can risk analysis be used effectively in the capital budgeting process? In Chapter 12 we made no distinction between risky and nonrisky events.4 We showed the amount of the investment and the annual returns—making no comment about the riskiness or likelihood of achieving these returns. We know that investors and managers care about both risk and expected returns. A $1,400 investment that produces “certain” returns of $600 a year for three years is not the same as a $1,400 investment that produces returns with an expected value of $600 for three years, but with a high coefficient of variation. Investors, being risk-averse by nature, will apply a stiffer test to the second investment. How can this new criterion be applied to the capital budgeting process?
Risk-Adjusted Discount Rate
A favored approach to adjust for risk is to use different discount rates for proposals with different risk levels. Thus we use risk-adjusted discount rates. A project that carries a normal amount of risk and does not change the overall risk composure of the firm should be discounted at the cost of capital. Investments carrying greater than normal risk will be discounted at a higher rate, and so on. In Figure 13-5, we show a possible risk–discount rate trade-off scheme. We assume that risk is measured by the coefficient of variation (V).
Figure 13-5 Relationship of risk to discount rate
The risk of the typical project undertaken by the firm is represented by a coefficient of variation of 0.30 (normal risk) on the bottom of Figure 13-5. An investment with this normal risk would be discounted at the firm’s normal cost of capital of 10 percent. As the firm selects riskier projects, for example, with a V of 0.90, a risk premium of 5 percent is added to compensate for an increase in V of 0.60 (from 0.30 to 0.90). If the company selects a project with a coefficient of variation of 1.20, it will add another 5 percent risk premium for this additional V of 0.30. This is an example of being increasingly risk-averse at higher levels of risk and potential return. Management requires higher expected returns (by using higher discount rates) when the firm is considering projects with higher risks.
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Increasing Risk over Time
Our ability to forecast accurately diminishes as we forecast farther out in time. As the time horizon becomes longer, more uncertainty enters the forecast. The decline in oil prices sharply curtailed the search for petroleum and left many drillers in serious financial condition in the 1980s after years of expanding drilling activity. Conversely, the users of petroleum products were hurt in 1990 when the conflict in the Middle East caused oil prices to skyrocket. Airlines and auto manufacturers had to reevaluate decisions made many years ago that were based on more stable energy prices. September 11, 2001, dealt another blow to the already fragile economy. The collapse of the housing market caused a terrible shock to the economy in 2007–2009. The inability of Congress to agree on tax reform and spending cuts lingered throughout Obama’s presidency and caused a great deal of uncertainty for all businesses. Then oil prices again declined from $107 in June 2014 to $53 in July of 2015. These unexpected events create a higher standard deviation in cash flows and increase the risk associated with long-lived projects. Figure 13-6 depicts the relationship between risk and time.
Figure 13-6 Risk over time
Even though a forecast of cash flows shows a constant expected value, Figure 13-6 indicates that the range of outcomes and probabilities increases as we move from year 2 to year 10. The standard deviations increase for each forecast of cash flow. If cash flows were forecast as easily for each period, all distributions would look like the first one for year 2.
Qualitative Measures
Rather than relate the discount rate—or required return—to the coefficient of variation or possibly the beta, management may wish to set up risk classes based on qualitative considerations. Examples are presented in Table 13-3. Once again we are raising the discount rate to reflect the perceived risk.
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Finance in ACTION Managerial Energy: A High-Risk Industry
One industry that affects world economies is energy. This industry includes coal, natural gas, oil, wind, solar, and even the Canadian tar sands. The prices of all these types of energy are important to every person and company that uses electricity, buses, cars, airplanes, trains, plastics, fertilizer, and more. The search for energy sources can be high risk and can result in either large deposits that generate high returns or no returns if nothing is found. Additionally, as companies search for oil in deeper and deeper waters, the technology used is more sophisticated and the chance of a disaster becomes higher. Just ask British Petroleum about the more than $20 billion dollars in fines and claims that it will cost them to settle the Gulf of Mexico Deepwater Horizon rig explosion that occurred in April of 2010. Did they include the potential cost of disasters in their worldwide drilling program?
Besides the risk of a disaster, every producer of energy is affected by the cost of alternative energy sources. For example, the International Energy Agency now predicts that the U.S. will be energy independent by 2020. An article in The Journal of the International Energy Agency made this comment:
Five years ago no one would have been talking about the prospect of U.S. energy independence. But this year, domestic crude oil production should rise by 10%, and within five years the United States is likely to break the record output high reached more than two decades ago, to flirt with the position of top world producer. 5
The amazing part of the story is that new technology, such as horizontal drilling and hydraulic fracturing, has found ways to get more oil and gas out of old deposits and to recover oil and gas from places that were previously unrecoverable. Estimates are that in the last five years, the U.S. has found enough natural gas for 50 years and is still finding more.
Natural gas is priced per one thousand cubic feet of natural gas or by its equivalent, 1 million BTUs (British Thermal Units). The price of natural gas has gone from $12 per thousand cubic feet in 2008, to $6 in 2011, to $2 in 2012, and $2.79 in 2015. Now if anyone could have predicted this price movement, capital budgeting decisions would have been greatly influenced. Go back and read the quote. Five years ago, no one knew that the U.S. would find so much natural gas in the Marcellus Shale of Pennsylvania and West Virginia and other shale formations in Texas, Colorado, North Dakota, and other western states.
Now this should all seem like good news, but environmentalists are worried that these low natural gas prices will set back wind and solar power projects as natural gas is now more cost competitive. This affects the capital budgeting projects for electric utilities and changes the balance between coal-, oil-, and gas-generating plants. It also affects the budding solar industry and companies like General Electric and Siemens that make electric generating wind turbines.
The ability of the United States to become energy independent also has a geopolitical impact as it reduces U.S. dependence on Middle East oil, which at times can be unpredictable and used as a political tool. The newfound sources of oil and gas will reduce some of the uncertainty of supply constraints. At its peak, the U.S. imported $300 billion dollars of oil per year, and so more domestic oil production will help the U.S. reduce its trade imbalance. This will affect the economy and the value of the dollar and other unpredictable benefits and costs.
We have presented an ever-changing picture of energy and its costs, which needs to be forecast by many companies around the world. The point is that even though we can try to put probabilities on potential outcomes, the accuracy of our forecasts is probably not too high. Given the economic interactions of energy on multiple industries, all the consequences are difficult to predict.
5Michael Cohen, IEA Energy, The Journal of the International Energy Agency, © OECD/IEA 2012, Issue 3, p. 18, January 2013.
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Table 13-3 Risk categories and associated discount rates
|
|
Discount Rate |
|
Low or no risk (repair to old machinery) |
6% |
|
Moderate risk (new equipment) |
8 |
|
Normal risk (addition to normal product line) |
10 |
|
Risky (new product in related market) |
12 |
|
High risk (completely new market) |
16 |
|
Highest risk (new product in foreign market) |
20 |
Example—Risk-Adjusted Discount Rate In Chapter 12, we compared two $10,000 investment alternatives and indicated that each had a positive net present value (at a 10 percent cost of capital). The analysis is reproduced in Table 13-4.
Table 13-4 Capital budgeting analysis
Though both proposals are acceptable, if they were mutually exclusive, only Investment B would be undertaken. But what if we add a risk dimension to the problem? Assume Investment A calls for an addition to the normal product line and is assigned a discount rate of 10 percent. Further assume that Investment B represents a new product in a foreign market and must carry a 20 percent discount rate to adjust for the large risk component. As indicated in Table 13-5, our answers are reversed and Investment A is now the only acceptable alternative.
Other methods besides the risk-adjusted discount rate approach are also used to evaluate risk in the capital budgeting process. The spectrum runs from a seat-of-the-pants “executive preference” approach to sophisticated computer-based statistical analysis.
All methods, however, include a common approach—that is, they must recognize the riskiness of a given investment proposal and make an appropriate adjustment for risk.
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Table 13-5 Capital budgeting decision adjusted for risk
Simulation Models
Computers make it possible to simulate various economic and financial outcomes, using a large number of variables. Thus simulation is one way of dealing with the uncertainty involved in forecasting the outcomes of capital budgeting projects or other types of decisions. A Monte Carlo simulation model uses random variables for inputs. By programming the computer to randomly select inputs from probability distributions, the outcomes generated by a simulation are distributed about a mean, and instead of generating one return or net present value, a range of outcomes with standard deviations is provided. A simulation model relies on repetition of the same random process as many as several hundred times. Since the inputs are representative of what one might encounter in the real world, many possible combinations of returns are generated.
One of the benefits of simulation is its ability to test various possible combinations of events. This sensitivity testing allows the planner to ask “what if” questions: What will happen to the returns on this project if oil prices go up? Go down? What effect will a 2 percent increase in interest rates have on the net present value of this project? The analyst can use the simulation process to test possible changes in economic policy, sales levels, inflation, or any other variable included in the modeling process. Some simulation models are driven by sales forecasts with assumptions to derive income statements and balance sheets. Others generate probability acceptance curves for capital budgeting decisions by informing the analyst about the probabilities of having a positive net present value.
For example, each distribution in Figure 13-7 will have a value picked randomly and used for one simulation. The simulation will be run many times, each time selecting a new random variable to generate the final probability distribution for the net present value (at the bottom of Figure 13-7). For that probability distribution, the expected values are on the horizontal axis and the probability of occurrence is on the vertical axis. The outcomes also suggest something about the riskiness of the project, which is indicated by the overall dispersion.
Decision Trees
Decision trees help lay out the sequence of decisions that can be made and present a tabular or graphical comparison resembling the branches of a tree, which highlights the differences between investment choices. In Table 13-6, we examine a retailer considering two choices: (a) opening additional physical stores in a new geographic region but using a format that has already proven successful elsewhere, or (b) developing a new online-only retail venture. The cost of both projects is the same, $60 million (column 4), but the net present value (NPV) and risk are different. If the firm expands its physical store count (Project A), it has a high likelihood of a modest positive rate of return. This market has some uncertainty, but long-run success seems to be likely. If the firm launches a new online store, it faces stiff competition from many established firms. It stands to lose more money if expected sales are lower than it would under option A, but it will make more if sales are high. Even though Project B has a higher expected NPV than Project A (last column in Table 13-6), its extra risk does not make for an easy choice. More analysis would have to be done before management made the final decision between these two projects. Nevertheless, the decision tree provides an important analytical process.
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Figure 13-7 Simulation flow chart
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Table 13-6 Decision trees
The Portfolio Effect
Up to this point, we have been primarily concerned with the risk inherent in an individual investment proposal. While this approach is useful, we also need to consider the impact of a given investment on the overall risk of the firm—the portfolio effect. For example, we might undertake an investment in the building products industry that appears to carry a high degree of risk—but if our primary business is the manufacture of electronic components for industrial use, we may diminish the overall risk exposure of the firm. Why? Because electronic component sales expand when the economy does well and falter in a recession. The building products industry reacts in the opposite fashion—performing poorly in boom periods and generally reacting well in recessionary periods. By investing in the building products industry, an electronic components manufacturer could smooth the cyclical fluctuations inherent in its business and reduce overall risk exposure, as indicated in Figure 13-8.
The risk reduction phenomenon is demonstrated by a less dispersed probability distribution in panel C. We say the standard deviation for the entire company (the portfolio of investments) has been reduced.
Portfolio Risk
Whether or not a given investment will change the overall risk of the firm depends on its relationships to other investments. If one airline purchases another, there is very little risk reduction. Highly correlated investments—that is, projects that move in the same direction in good times as well as bad—do little or nothing to diversify away risk. Projects moving in opposite directions (building products and electronic components) are referred to as being negatively correlated and provide a high degree of risk reduction.
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Figure 13-8 Portfolio considerations in evaluating risk
Finally, projects that are totally uncorrelated provide some overall reduction in portfolio risk—though not as much as negatively correlated investments. For example, if a beer manufacturer purchases a textile firm, the projects are neither positively nor negatively correlated; but the purchase will reduce the overall risk of the firm simply through the “law of large numbers.” If you have enough unrelated projects going on at one time, good and bad events will probably even out.
The extent of correlation among projects is represented by a new term called the coefficient of correlation—a measure that may take on values anywhere from −1 to +1.6 Examples are presented in Table 13-7.
In the real world, few investment combinations take on values as extreme as −1 or +1, or for that matter exactly 0. The more likely case is a point somewhere between, such as −0.2 negative correlation or +0.3 positive correlation, as indicated along the continuum in Figure 13-9.
The fact that risk can be reduced by combining risky assets with low or negatively correlated assets can be seen in the example of Conglomerate Inc. in Table 13-8. Conglomerate has fairly average returns and standard deviations of returns. The company is considering the purchase of one of two separate but large companies with sales and assets equal to its own. Management is struggling with the decision since both companies have a 14 percent rate of return, which is 2 percentage points higher than that of Conglomerate, and they have the same standard deviation of returns as that of Conglomerate, at 2.82 percent. This information is presented in the first three columns of Table 13-8.
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Table 13-7 Measures of correlation
Figure 13-9 Levels of risk reduction as measured by the coefficient of correlation
Table 13-8 Rates of return for Conglomerate Inc. and two merger candidates
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Real Options Add a New Dimension to Capital Budgeting Finance in ACTION Managerial
According to traditional net present value analysis, the expected yearly inflows are discounted back to the present at the cost of capital, and their present value is compared to the cost of the investment. If the present value of the inflows is larger than the investment, the investment is considered acceptable; if not, it is rejected.
But some would argue this process is incomplete because it fails to consider the flexibility to revise decisions after a project has begun. For example, assume an oil company decides to drill for oil in 10 adjacent sites over the next five years. Under traditional capital budgeting analysis, the present value of expected cash flows would be discounted back for five years and compared to the cost of the venture. But traditional capital budgeting does not consider the intermittent decisions that can be made during the life of the project.
Let’s initially assume that the oil drilling project has a negative net present value. But in further analyzing the project, we include the option that if after hitting two successful oil wells, we encounter three dry holes, we will abandon the project and cut the size of our investment. This could lead to a positive net present value, especially if the last five drilling attempts were going to be particularly expensive.
Alternatively, let’s assume that if the first two sites turn out to be much more productive than initially anticipated, we expand the project to 15 sites by including other nearby potential oil wells. We might also have two or three other options. By including these options in the initial planning, a negative net value project may take on a positive return.
The options discussed here are termed real options because they involve assets as opposed to financial options, which relate to stocks and bonds.
Real options might include the flexibility of terminating a project, taking a more desirable route once initial results are in, greatly expanding the project if there is unexpected success, and so on. Such elements are particularly likely to be present in natural resource discovery, technology-related investments, and new product introductions. But the list does not stop there. Almost every capital budgeting project contains a potential element of flexibility once it’s put into place. There is a real option to change the course of action, and this real option has a monetary value just as a financial option does.
The value of the real option is the difference in the net present value of the project with the flexibility included in the analysis versus the traditional static net present value analysis. An analogy can be drawn to playing poker where you can return part of your hand and draw again in five-card draw, versus five-card stud in which you must stay with the cards that are initially dealt to you.
Including real options in a capital budgeting analysis sounds good, but the process is not widely used. A recent study showed that only 14.3 percent of the Fortune 1000 companies use real options in any form in their analysis.7 The primary reasons for this low utilization are lack of sophistication and distrust that the options will actually be properly used in the future. This low utilization rate is likely to change in the future as sophistication increases.
7Stanley Block, “Are ‘Real Options’ Actually Used in the Real World?” The Engineering Economist 52, no. 3 (2007), pp. 255–267.
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Since management desires to reduce risk (σ) and to increase returns at the same time, it decides to analyze the results of each combination.8 These are shown in the last two columns in Table 13-8. A combination with Positive Correlation Inc. increases the mean return for Conglomerate Inc. to 13 percent (average of 12 percent and 14 percent) but maintains the same standard deviation of returns (no risk reduction) because the coefficient of correlation is +1.0 and no diversification benefits are achieved. The +1.0 value is shown on the bottom line in black. A combination with Negative Correlation Inc. also increases the mean return to 13 percent, but it reduces the standard deviation of returns to 0.63 percent, a significant reduction in risk. This occurs because of the offsetting relationship of returns between the two companies, as evidenced by the coefficient of correlation of −.9 (bottom line of Table 13-8 in black). When one company has high returns, the other has low returns, and vice versa. Thus a merger with Negative Correlation Inc. appears to be the best decision.
Evaluation of Combinations
The firm should evaluate all possible combinations of projects, determining which will provide the best trade-off between risk and return. In Figure 13-10, we see a number of alternatives that might be available to a given firm. Each point represents a combination of different possible investments. For example, point F might represent a semiconductor manufacturer combining three different types of semiconductors, plus two types of computers, and two products in unrelated fields. In choosing between the various points or combinations, management should have two primary objectives:
1. Achieve the highest possible return at a given risk level.
2. Provide the lowest possible risk at a given return level.
Figure 13-10 Risk-return trade-offs
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All the best opportunities will fall along the leftmost sector of the diagram (line C–F–G). Each point on the line satisfies the two objectives of the firm. Any point to the right is less desirable.
After we have developed our best risk-return line, known in the financial literature as the “efficient frontier,” we must determine where on the line our firm should be. There is no universally correct answer. To the extent we are willing to take large risks for superior returns, we will opt for some point on the upper portion of the line—such as G. However, a more conservative selection might be C.
The Share Price Effect
The firm must be sensitive to the wishes and demands of shareholders. To the extent that unnecessary or undesirable risks are taken, a higher discount rate and lower valuation may be assigned to the stock in the market. Higher profits, resulting from risky ventures, could have a result that is the opposite from that intended. In raising the firm’s risk, we could be lowering the overall valuation of the firm.
The aversion of investors to nonpredictability (and the associated risk) is confirmed by observing the relative valuation given to cyclical stocks versus highly predictable growth stocks in the market. Metals, autos, and housing stocks generally trade at an earnings multiplier well below that for industries with level, predictable performances, such as drugs, soft drinks, and even alcohol or cigarettes. Each company must carefully analyze its own situation to determine the appropriate trade-off between risk and return. The changing desires and objectives of investors tend to make the task somewhat more difficult.
SUMMARY
Risk may be defined as the potential variability of the outcomes from an investment. The less predictable the outcomes, the greater is the risk. Both management and investors tend to be risk-averse—that is, all things being equal, they would prefer to take less risk, rather than greater risk.
The most commonly employed method to adjust for risk in the capital budgeting process is to alter the discount rate based on the perceived risk level. High-risk projects will carry a risk premium, producing a discount rate well in excess of the cost of capital.
In assessing the risk components in a given project, management may rely on simulation techniques to generate probabilities of possible outcomes and decision trees to help isolate the key variables to be evaluated.
Management must consider not only the risk inherent in a given project, but also the impact of a new project on the overall risk of the firm (the portfolio effect). Negatively correlated projects have the most favorable effect on smoothing business cycle fluctuations. The firm may wish to consider all combinations and variations of possible projects and to select only those that provide a total risk-return trade-off consistent with its goals.
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REVIEW OF FORMULAS
1.
|
D |
is outcome |
|
P |
is probability of outcome |
2.
|
D |
is outcome |
|
|
is expected value |
|
P |
is probability of outcome |
3.
|
σ |
is standard deviation |
|
|
is expected value |
LIST OF TERMS
risk 418
risk-averse 420
expected value 421
standard deviation 421
coefficient of variation 422
beta 423
risk-adjusted discount rates 424
simulation 428
decision trees 428
portfolio effect 430
coefficient of correlation 431
efficient frontier 435
DISCUSSION QUESTIONS
1. If corporate managers are risk-averse, does this mean they will not take risks? Explain. (LO13-2)
2. Discuss the concept of risk and how it might be measured. (LO13-1)
3. When is the coefficient of variation a better measure of risk than the standard deviation? (LO13-1)
4. Explain how the concept of risk can be incorporated into the capital budgeting process. (LO13-3)
5. If risk is to be analyzed in a qualitative way, place the following investment decisions in order from the lowest risk to the highest risk: (LO13-1)
a. New equipment.
b. New market.
c. Repair of old machinery.
d. New product in a foreign market.
e. New product in a related market.
f. Addition to a new product line.
6. Assume a company, correlated with the economy, is evaluating six projects, of which two are positively correlated with the economy, two are negatively correlated, and two are not correlated with it at all. Which two projects would it select to minimize the company’s overall risk? (LO13-5)
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7. Assume a firm has several hundred possible investments and that it wants to analyze the risk-return trade-off for portfolios of 20 projects. How should it proceed with the evaluation? (LO13-5)
8. Explain the effect of the risk-return trade-off on the market value of common stock. (LO13-3)
9. What is the purpose of using simulation analysis? (LO13-4)
PRACTICE PROBLEMS AND SOLUTIONS
Coefficient of variation
(LO13-1)
1. Hanson Auto Supplies is examining the following probability distribution. What is the coefficient of variation?
|
|
|
|
Cash Flow |
Probability |
|
$20 |
0.30 |
|
40 |
0.40 |
|
60 |
0.30 |
|
|
Risk-adjusted discount rate decision
(LO13-3)
2. Hanson Auto Supply in the prior problem uses the risk-adjusted discount rate and relates the discount rate to the coefficient of variation as follows:
|
|
|
|
Coefficient of Variation |
Discount Rate |
|
0–0.30 |
8% |
|
0.31–0.60 |
10 |
|
0.61–0.90 |
12 |
|
0.91–1.20 |
16 |
|
|
It will invest $120 and receive the expected value of cash flows you computed in problem 1 of $40. Assume those cash flows of $40 will be earned for the next 5 years and are discounted back at the appropriate discount rate based on the coefficient of variation computed in problem 1. What is the net present value?
Solutions
1. The formula for the coefficient of variation (V) is:
First compute the expected value ().
Page 438
Then compute the standard deviation (σ).
Next, divide the standard deviation (σ) by the expected value () to determine the coefficient of variation (V).
2. First you must determine the appropriate discount rate based on the coefficient of variation. The coefficient of variation computed in the solution to practice problem 1 is 0.3875. Using the table for this problem, the discount rate should be 10 percent since 0.3875 falls between 0.31 and 0.60.
Now take the present value of an annuity of $40 for 5 years and compare it to the investment of $120. The interest rate is the risk-adjusted discount rate of 10 percent identified above. The $40 is considered an annuity because it is an equal amount each year. This is not always the case, but eases the computation for this example.
|
|
|
|
FINANCIAL CALCULATOR |
|
|
PV of Annuity |
|
|
Enter |
Function |
|
5 |
N |
|
10 |
I/Y |
|
0 |
FV |
|
−40 |
PMT |
|
Function |
Solution |
|
CPT |
|
|
PV |
151.63 |
|
|
|
PV of cash flows (inflows) |
$151.63 |
|
Investment |
−120.00 |
|
Net present value |
$ 31.63 |
The investment has a positive net present value of $31.63.
PROBLEMS
Selected problems are available with Connect. Please see the preface for more information.
Basic Problems
Risk-averse
(LO13-2)
1. Assume you are risk-averse and have the following three choices. Which project will you select? Compute the coefficient of variation for each.
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|
||
|
|
Expected Value |
Standard Deviation |
|
A |
$2,200 |
$1,400 |
|
B |
2,730 |
1,960 |
|
C |
2,250 |
1,490 |
|
|
Expected value and standard deviation
(LO13-1)
Page 439
2. Myers Business Systems is evaluating the introduction of a new product. The possible levels of unit sales and the probabilities of their occurrence are given next:
|
|
||
|
Possible Market Reaction |
Sales in Units |
Probabilities |
|
Low response |
20 |
0.10 |
|
Moderate response |
40 |
0.30 |
|
High response |
55 |
0.40 |
|
Very high response |
70 |
0.20 |
|
|
a. What is the expected value of unit sales for the new product?
b. What is the standard deviation of unit sales?
Expected value and standard deviation
(LO13-1)
3. Sampson Corp. is evaluating the introduction of a new product. The possible levels of unit sales and the probabilities of their occurrence are shown next:
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|
||
|
Possible Market Reaction |
Sales in Units |
Probabilities |
|
Low response |
30 |
0.10 |
|
Moderate response |
50 |
0.20 |
|
High response |
75 |
0.40 |
|
Very high response |
90 |
0.30 |
|
|
a. What is the expected value of unit sales for the new product?
b. What is the standard deviation of unit sales?
Coefficient of variation
(LO13-1)
4. Shack Homebuilders Limited is evaluating a new promotional campaign that could increase home sales. Possible outcomes and probabilities of the outcomes are shown next. Compute the coefficient of variation.
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|
||
|
Possible Outcomes |
Additional Sales in Units |
Probabilities |
|
Ineffective campaign |
40 |
0.30 |
|
Normal response |
100 |
0.30 |
|
Extremely effective |
120 |
0.40 |
|
|
Coefficient of variation
(LO13-1)
5. Al Bundy is evaluating a new advertising program that could increase shoe sales. Possible outcomes and probabilities of the outcomes are shown next. Compute the coefficient of variation.
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|
||
|
Possible Outcomes |
Additional Sales in Units |
Probabilities |
|
Ineffective campaign |
40 |
0.20 |
|
Normal response |
60 |
0.50 |
|
Extremely effective |
140 |
0.30 |
|
|
Coefficient of variation
(LO13-1)
6. Possible outcomes for three investment alternatives and their probabilities of occurrence are given next.Page 440
Rank the three alternatives in terms of risk from lowest to highest (compute the coefficient of variation).
Coefficient of variation
(LO13-1)
7. Five investment alternatives have the following returns and standard deviations of returns:
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|
||
|
Alternatives |
Returns: Expected Value |
Standard Deviation |
|
A |
$ 5,000 |
$1,200 |
|
B |
4,000 |
600 |
|
C |
4,000 |
800 |
|
D |
8,000 |
3,200 |
|
E |
10,000 |
900 |
|
|
Using the coefficient of variation, rank the five alternatives from the lowest risk to the highest risk.
Coefficient of variation
(LO13-1)
8. Five investment alternatives have the following returns and standard deviations of returns:
|
|
||
|
Alternatives |
Returns: Expected Value |
Standard Deviation |
|
A |
$ 1,980 |
$ 970 |
|
B |
820 |
1,190 |
|
C |
12,700 |
3,100 |
|
D |
1,140 |
630 |
|
E |
62,700 |
14,100 |
|
|
Using the coefficient of variation, rank the five alternatives from lowest risk to highest risk.
Coefficient of variation and time
(LO13-1)
9. Digital Technology wishes to determine its coefficient of variation as a company over time. The firm projects the following data (in millions of dollars):
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|
||
|
Year |
Profits: Expected Value |
Standard Deviation |
|
1 |
$180 |
$ 62 |
|
3 |
240 |
104 |
|
6 |
300 |
166 |
|
9 |
400 |
292 |
|
|
a. Compute the coefficient of variation (V) for each time period.
b. Does the risk (V) appear to be increasing over a period of time? If so, why might this be the case?
Risk-averse
(LO13-2)
Page 441
10. Tim Trepid is highly risk-averse while Mike Macho actually enjoys taking a risk.
a. Which one of the four investments should Tim choose? Compute coefficients of variation to help you in your choice.
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|
||
|
Investments |
Returns: Expected Value |
Standard Deviation |
|
Buy stocks |
$ 9,140 |
$ 6,140 |
|
Buy bonds |
7,680 |
2,560 |
|
Buy commodity futures |
19,100 |
26,700 |
|
Buy options |
17,700 |
18,200 |
|
|
b. Which one of the four investments should Mike choose?
Risk-averse
(LO13-2)
11. Mountain Ski Corp. was set up to take large risks and is willing to take the greatest risk possible. Lakeway Train Co. is more typical of the average corporation and is risk-averse.
a. Which of the following four projects should Mountain Ski Corp. choose? Compute the coefficients of variation to help you make your decision.
b. Which one of the four projects should Lakeway Train Co. choose based on the same criteria of using the coefficient of variation?
|
|
||
|
Year |
Returns Expected Value |
Standard Deviation |
|
A |
$527,000 |
$834,000 |
|
B |
682,000 |
306,000 |
|
C |
74,000 |
135,000 |
|
D |
140,000 |
89,000 |
|
|
Coefficient of variation and investment decision
(LO13-1)
12. Kyle’s Shoe Stores Inc. is considering opening an additional suburban outlet. An aftertax expected cash flow of $130 per week is anticipated from two stores that are being evaluated. Both stores have positive net present values.
Which store site would you select based on the distribution of these cash flows? Use the coefficient of variation as your measure of risk.
Risk-adjusted discount rate
(LO13-3)
13. Waste Industries is evaluating a $70,000 project with the following cash flows:
|
|
|
|
Years |
Cash Flows |
|
1 |
$11,000 |
|
2 |
16,000 |
|
3 |
21,000 |
|
4 |
24,000 |
|
5 |
30,000 |
|
|
Page 442
The coefficient of variation for the project is 0.847.
Based on the following table of risk-adjusted discount rates, should the project be undertaken? Select the appropriate discount rate and then compute the net present value.
|
|
|
|
Coefficient of Variation |
Discount Rate |
|
0–0.25 |
6% |
|
0.26–0.50 |
8 |
|
0.51–0.75 |
10 |
|
0.76–1.00 |
14 |
|
1.01–1.25 |
20 |
|
|
Intermediate Problems
Risk-adjusted discount rate
(LO13-3)
14. Dixie Dynamite Company is evaluating two methods of blowing up old buildings for commercial purposes over the next five years. Method one (implosion) is relatively low in risk for this business and will carry a 12 percent discount rate. Method two (explosion) is less expensive to perform but more dangerous and will call for a higher discount rate of 16 percent. Either method will require an initial capital outlay of $75,000. The inflows from projected business over the next five years are shown next. Which method should be selected using net present value analysis?
|
|
||
|
Year |
Method 1 |
Method 2 |
|
1 |
$18,000 |
$20,000 |
|
2 |
24,000 |
25,000 |
|
3 |
34,000 |
35,000 |
|
4 |
26,000 |
28,000 |
|
5 |
14,000 |
15,000 |
|
|
Discount rate and timing
(LO13-1)
15. Fill in the following table from Appendix B. Does a high discount rate have a greater or lesser effect on long-term inflows compared to recent ones?
|
|
||
|
|
Discount Rate |
|
|
Years |
5% |
20% |
|
1 |
__________ |
__________ |
|
10 |
__________ |
__________ |
|
20 |
__________ |
__________ |
|
|
Expected value with net present value
(LO13-1)
16. Debby’s Dance Studios is considering the purchase of new sound equipment that will enhance the popularity of its aerobics dancing. The equipment will cost $27,900. Debby is not sure how many members the new equipment will attract, but she estimates that her increased annual cash flows for each of the next five years will have the following probability distribution. Debby’s cost of capital is 15 percent.
|
|
|
|
Cash Flow |
Probability |
|
$4,570 |
0.1 |
|
5,550 |
0.3 |
|
7,400 |
0.4 |
|
9,930 |
0.2 |
|
|
Page 443
a. What is the expected value of the cash flow? The value you compute will apply to each of the five years.
b. What is the expected net present value?
c. Should Debby buy the new equipment?
Advanced Problems
Deferred cash flows and risk-adjusted discount rate
(LO13-1)
17. Highland Mining and Minerals Co. is considering the purchase of two gold mines. Only one investment will be made. The Australian gold mine will cost $1,649,000 and will produce $353,000 per year in years 5 through 15 and $503,000 per year in years 16 through 25. The U.S. gold mine will cost $2,054,000 and will produce $282,000 per year for the next 25 years. The cost of capital is 13 percent.
a. Which investment should be made? (Note: In looking up present value factors for this problem, you need to work with the concept of a deferred annuity for the Australian mine. The returns in years 5 through 15 actually represent 11 years; the returns in years 16 through 25 represent 10 years.)
b. If the Australian mine justifies an extra 2 percent premium over the normal cost of capital because of its riskiness and relative uncertainty of cash flows, does the investment decision change?
Coefficient of variation and investment decision
(LO13-1)
18. Mr. Sam Golff desires to invest a portion of his assets in rental property. He has narrowed his choices down to two apartment complexes, Palmer Heights and Crenshaw Village. After conferring with the present owners, Mr. Golff has developed the following estimates of the cash flows for these properties:
a. Find the expected cash flow from each apartment complex.
b. What is the coefficient of variation for each apartment complex?
c. Which apartment complex has more risk?
Decision tree analysis
(LO13-4)
19. Allison’s Dresswear Manufacturers is preparing a strategy for the fall season. One alternative is to expand its traditional ensemble of wool sweaters. A second option would be to enter the cashmere sweater market with a new line of high-quality designer label products. The marketing department has determined that the wool and cashmere sweater lines offer the following probability of outcomes and related cash flows:Page 444
The initial cost to expand the wool sweater line is $142,000. To enter the cashmere sweater line, the initial cost in designs, inventory, and equipment is $102,000.
a. Diagram a complete decision tree of possible outcomes similar to Table 13-6. Note that you are dealing with thousands of dollars rather than millions. Take the analysis all the way through the process of computing expected NPV (last column for each investment).
b. Given the analysis in part a, would you automatically make the investment indicated?
Probability analysis with a normal curve distribution
(LO13-4)
20. When returns from a project can be assumed to be normally distributed, such as those shown in Figure 13-6 (represented by a symmetrical, bell-shaped curve), the areas under the curve can be determined from statistical tables based on standard deviations. For example, 68.26 percent of the distribution will fall within one standard deviation of the expected value ( ± 1σ). Similarly, 95.44 percent will fall within two standard deviations ( ± 1σ), and so on. An abbreviated table of areas under the normal curve is shown next.
|
|
||
|
Number of σ’s from Expected Value |
+ or − |
+ and − |
|
0.5 |
0.1915 |
0.3830 |
|
1.0 |
0.3413 |
0.6826 |
|
1.5 |
0.4332 |
0.8664 |
|
1.65 |
0.4505 |
0.9010 |
|
2.0 |
0.4772 |
0.9544 |
|
|
Assume Project A has an expected value of $24,000 and a standard deviation (s) of $4,800.
a. What is the probability that the outcome will be between $16,800 and $31,200?
b. What is the probability that the outcome will be between $14,400 and $33,600?
c. What is the probability that the outcome will be at least $14,400?
d. What is the probability that the outcome will be less than $31,900?
e. What is the probability that the outcome will be less than $19,200 or greater than $26,400?
Increasing risk over time
(LO13-1)
Page 445
21. The Oklahoma Pipeline Company projects the following pattern of inflows from an investment. The inflows are spread over time to reflect delayed benefits. Each year is independent of the others.
The expected value for all three years is $70.
a. Compute the standard deviation for each of the three years.
b. Diagram the expected values and standard deviations for each of the three years in a manner similar to Figure 13-6.
c. Assuming 6 percent and 12 percent discount rates, complete the following table for present value factors:
d. Is the increasing risk over time, as diagrammed in part b, consistent with the larger differences in PVIFs over time as computed in part c?
e. Assume the initial investment is $135. What is the net present value of the investment at a 12 percent discount rate? Should the investment be accepted?
Portfolio effect of a merger
(LO13-5)
22. Treynor Pie Co. is a food company specializing in high-calorie snack foods. It is seeking to diversify its food business and lower its risks. It is examining three companies—a gourmet restaurant chain, a baby food company, and a nutritional products firm. Each of these companies can be bought at the same multiple of earnings. The following table represents information about all the companies:
Page 446
a. Using the last two columns, compute the coefficient of variation for each of the four companies. Which company is the least risky? Which company is the most risky?
b. Discuss which of the acquisition candidates is most likely to reduce Treynor Pie Company’s risk. Explain why.
Portfolio effect of a merger
(LO13-5)
23. Hooper Chemical Company, a major chemical firm that uses such raw materials as carbon and petroleum as part of its production process, is examining a plastics firm to add to its operations. Before the acquisition, the normal expected outcomes for the firm were as follows:
|
|
||
|
|
Outcomes ($ millions) |
Probability |
|
Recession |
$20 |
0.30 |
|
Normal economy |
40 |
0.40 |
|
Strong economy |
60 |
0.30 |
|
|
After the acquisition, the expected outcomes for the firm would be:
|
|
||
|
|
Outcomes ($ millions) |
Probability |
|
Recession |
$ 10 |
0.3 |
|
Normal economy |
40 |
0.4 |
|
Strong economy |
80 |
0.3 |
|
|
a. Compute the expected value, standard deviation, and coefficient of variation before the acquisition.
b. After the acquisition, these values are as follows:
|
|
|
|
Expected value |
43.0 ($ millions) |
|
Standard deviation |
27.2 ($ millions) |
|
Coefficient of variation |
0.633 |
|
|
Comment on whether this acquisition appears desirable to you.
c. Do you think the firm’s stock price is likely to go up as a result of this acquisition?
d. If the firm were interested in reducing its risk exposure, which of the following three industries would you advise it to consider for an acquisition? Briefly comment on your answer.
(1) Chemical company
(2) Oil company
(3) Computer company
Efficient frontier
(LO13-5)
24. Ms. Sharp is looking at a number of different types of investments for her portfolio. She identifies eight possible investments.
Page 447
|
|
||
|
|
Return |
Risk |
|
(a) |
11% |
2% |
|
(b) |
11 |
2.5 |
|
(c) |
13 |
3.0 |
|
(d) |
13 |
4.2 |
|
(e) |
14% |
5.0% |
|
(f) |
16 |
5.0 |
|
(g) |
15 |
5.8 |
|
(h) |
18 |
7.0 |
|
|
a. Graph the data in a manner similar to Figure 13-10. Use the following axes for your data:
b. Draw a curved line representing the efficient frontier.
c. What two objectives do points on the efficient frontier satisfy?
d. Is there one point on the efficient frontier that is best for all investors?
Certainty equivalent approach
(LO13-1)
25. Sheila Goodman recently received her MBA from the Harvard Business School. She has joined the family business, Goodman Software Products Inc., as vice president of finance.
She believes in adjusting projects for risk. Her father is somewhat skeptical but agrees to go along with her. Her approach is somewhat different than the risk-adjusted discount rate approach, but achieves the same objective.
She suggests that the inflows for each year of a project be adjusted downward for lack of certainty and then be discounted back at a risk-free rate. The theory is that the adjustment penalty makes the inflows the equivalent of risk-less inflows, and therefore a risk-free rate is justified.
A table showing the possible coefficient of variation for an inflow and the associated adjustment factor is shown next:
|
|
|
|
Coefficient of Variation |
Adjustment Factor |
|
0–0.25 |
0.90 |
|
0.26–0.50 |
0.80 |
|
0.51–0.75 |
0.70 |
|
0.76–1.00 |
0.60 |
|
1.01–1.25 |
0.50 |
|
|
Page 448
Assume a $184,000 project provides the following inflows with the associated coefficients of variation for each year:
|
|
||
|
Year |
Inflow |
Coefficient of Variation |
|
1 |
$32,200 |
0.12 |
|
2 |
59,500 |
0.28 |
|
3 |
79,900 |
0.45 |
|
4 |
59,200 |
0.79 |
|
5 |
65,600 |
1.15 |
|
|
a. Fill in the following table:
b. If the risk-free rate is 5 percent, should this $184,000 project be accepted? Compute the net present value of the adjusted inflows.
COMPREHENSIVE PROBLEM
Gibson Appliance Co.
(Portfolio effect of a merger)
(LO13-5)
Gibson Appliance Co. is a very stable billion-dollar company with a sales growth of about 7 percent per year in good or bad economic conditions. Because of this stability (a coefficient of correlation with the economy of 10.4, and a standard deviation of sales of about 5 percent from the mean), Mr. Hoover, the vice president of finance, thinks the company could absorb a small risky company that could add quite a bit of return without increasing the company’s risk much. He is trying to decide which of the two companies he will buy, using the following figures. Gibson’s cost of capital is 12 percent.
a. What is the expected cash flow from both companies?
b. Which company has the lower coefficient of variation?
c. Compute the net present value of each company.
d. Which company would you pick, based on the net present values?
e. Would you change your mind if you added the risk dimensions to the problem? Explain.
Page 449
f. What if Genetic Technology Co. had a coefficient of correlation with the economy of 20.2 and Silicon Microchip Co. had one of 10.5? Which of these companies would give you the best portfolio effects for risk reduction?
g. What might be the effect of the acquisitions on the market value of Gibson Appliance Co.’s stock?
COMPREHENSIVE PROBLEM
Kennedy Trucking Company
(Investment decision based on probability analysis)
(LO13-1)
Five years ago, Kennedy Trucking Company was considering the purchase of 60 new diesel trucks that were 15 percent more fuel-efficient than the ones the firm was using. Mr. Hoffman, the president, had found that the company uses an average of 10 million gallons of diesel fuel per year at a price of $1.25 per gallon. If he can cut fuel consumption by 15 percent, he will save $1,875,000 per year (1,500,000 gallons times $1.25).
Mr. Hoffman assumed that the price of diesel fuel is an external market force that he cannot control and that any increased costs of fuel will be passed on to the shipper through higher rates endorsed by the Interstate Commerce Commission. If this is true, then fuel efficiency would save more money as the price of diesel fuel rises (at $1.35 per gallon, he would save $2,025,000 in total if he buys the new trucks). Mr. Hoffman has come up with two possible forecasts as shown next—each of which he feels has about a 50 percent chance of coming true. Under assumption number 1, diesel prices will stay relatively low; under assumption number 2, diesel prices will rise considerably. Sixty new trucks will cost Kennedy Trucking $5 million. Under a special provision from the Interstate Commerce Commission, the allowable depreciation will be 25 percent in year 1, 38 percent in year 2, and 37 percent in year 3. The firm has a tax rate of 40 percent and a cost of capital of 10 percent.
a. First compute the yearly expected costs of diesel fuel for both assumption 1 (relatively low prices) and assumption 2 (high prices) from the following forecasts.
Forecast for assumption 1 (low fuel prices):
Forecast for assumption 2 (high fuel prices):
Page 450
b. What will be the dollar savings in diesel expenses each year for assumption 1 and for assumption 2?
c. Find the increased cash flow after taxes for both forecasts.
d. Compute the net present value of the truck purchases for each fuel forecast assumption and the combined net present value (that is, weigh the NPVs by 0.5).
e. If you were Mr. Hoffman, would you go ahead with this capital investment?
f. How sensitive to fuel prices is this capital investment?
WEB EXERCISE
1. Alcoa was listed in Table 13-2 as a company that has a relativity high beta (a measure of stock price volatility). Alcoa produces aluminum and aluminum products. Go to its website at www.alcoa.com, and follow these steps.
Under “About,” select “Annual Report.” Download the latest annual report. Scroll all the way down to “Selected Financial Data” (page 45).
2. One of the characteristics of high beta stocks is that they often have volatile earnings performances. Let’s check out Alcoa. Compute the year-to-year percentage change in “Diluted-Income from Continuing Operations” for each of the five years. Do the earnings appear to be volatile?
3. Companies with high betas and inconsistent performance are encouraged to keep their debt ratios low (under 50 percent). Compute the ratio of long-term debt to total assets for each of the five years for Alcoa. What does the pattern look like to you?
Note: Occasionally a topic we have listed may have been deleted, updated, or moved into a different location on a website. If you click on the site map or site index, you will be introduced to a table of contents that should aid you in finding the topic you are looking for.
1This has happened many times worldwide.
2We use the term uncertainty in its normal sense, rather than in the more formalized sense in which it is sometimes used in decision theory to indicate that insufficient evidence is available to estimate a probability distribution.
3Other market measures may also be used.
4Our assumption was that the risk factor could be considered constant for various investments.
6Coefficient of correlation is not to be confused with coefficient of variation, a term used earlier in this chapter.
8In Chapter 20, you will evaluate a merger situation in which there is no increase in earnings, only a reduction in the standard deviation. Because the lower risk may mean a higher price-earnings ratio, this could be beneficial.