Funding and Debt Management

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Chapter 9 Pro Forma Statement and Financial Forecasting

Entrepreneurial Finance: Fundamentals of Financial Planning and Management for Small

Business, First Edition. M. J. Alhabeeb.

© 2015 John Wiley & Sons, Inc. Published 2015 by John Wiley & Sons, Inc.

AS IT WAS MENTIONED in the last chapter, a financial plan, as a major element in profit

planning, would require a series of projections and preparation of the pro forma statement.

The financial projections are not only required by lending institutions, potential investors,

and the Small Business Administration (SBA), but they also represent the key to successful

management, and they stand as a test of the project feasibility. Perhaps the most important

projections of all is the sales projection which forms the basis for other projections, and

their accuracy and reliability. Based on the predicted sales levels and the expected expenses,

the firm's profitability would be assessed. On the same basis, other requirements can be

assessed such as the size and type of financing, capital, services, obligations, personnel, and

size and status of cash flow. The crucial question on how much earnings a firm would

expect can be answered based on simple math using the projected values that would start

with the expected sales and the expected market price of the product. Figure 9.1 confirms

that a projected sale would be the start. A projected revenue would be obtained by

multiplying the projected sales by the expected market price of the product. All costs of

producing the goods and services would be subtracted from the projected revenue to get the

gross profit or as it is sometimes called the gross earnings. The gross profit would be subject

to three major deductions.

1. The operating costs which often include administrative cost, marketing, advertising, and

selling costs, in addition to depreciation expenses

2. Interests paid on all financing loans

3. Taxes due.

We should note that after the first deduction that would turn gross profit to operating profit,

the latter is often called earnings before interest and taxes (EBIT). After the interests have

been paid we get the net profit before taxes or earnings before taxes (EBT), and the last

deduction would be the taxes. Net profit or earnings after interests and taxes (EAIT) is what

we get at the end, which would be the basis for the firm real profitability.

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Figure 9.1 Steps to Estimate Net Profit

9.1 Basic Pro Forma Statements

A pro forma statement is a financial statement, the entries of which are projected values

based on a specific economic forecasting method, and a certain set of assumptions. These

assumptions are usually related to the general conditions, as well as to the specific state of

the project. The pro forma analysis is one of the most common and practical ways to predict

the feasibility and worth of a firm in the near future. Although it is useful for both new and

established firms, it is particularly valuable for a new venture, especially when it can be

presented as a yardstick to show lending banks and potential investors the future

performance of a project to prove how worthwhile it is for funding. An established firm

would also need the pro forma analysis to catch up with changes and better prepare for

them. For example, if the market conditions are good enough for a firm to expect an

increase in sales by a certain percentage, it would be imperative to predict a series of

consequence changes in production capacity, work shifts, inventories, transportation,

storage, selling expenses, as well as the additional fund needed to undertake all of these

changes.

For the startup venture, most if not all needed variables for the pro forma statements have

to be estimated. Well-educated estimations can be made based on either the entrepreneur's

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personal knowledge and expertise or based on specific industry standards known to the

public. The vast majority of entrepreneurs would not start in any venture unless they know

about it, are familiar with it, or even had a past experience in it. This, by itself, would qualify

them or their associates to deliver reasonable estimations to most of the needed variables.

Nowadays, and with the computer revolution, there is a wide access to all kinds of

information through the published reports and budgets and all kinds of documents on

almost anything. It is not terribly difficult, for example, to come up with a reasonable

average for the first year sales for a specific product in a certain market or time. It is also

not difficult to come up with the cost of anything based on the published prices of almost

any

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product you can think of, new or used. From these estimations that are based on the reality

somewhere else, a pro forma statement can be completed. Once the firm goes through the

first year of production and sales in the real market, it would accumulate all the actual data

that would be used to project the next year's data. Over time and with the availability of

good records, and for many years, prediction for the future becomes not only easier but also

more accurate and reliable, given that an appropriate method of forecasting is used and the

results are checked for reliability.

Table 9.1 One Year Pro Forma Income Statement

Changes:

Sales revenue, cost of goods sold, and gross profit: 32% ↑

Operating expenses: 2%↑

Utilities: 25%↑

Insurance: 20%↑

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Tables 9.1 and 9.2 show general examples of an income statement and a balance sheet for 1

year. Regardless of the source and method of getting the estimated values, the pro forma

statement would project the next set of values based on certain percentages concluded out

of the estimation and based on certain set of assumptions.

Pro Forma Income Statement

The pro forma income statement assumes, based on the firm's records of the last 3 years,

that the sale revenues for the next year will be up by 32%. This would increase the current

year's revenue of $450,000 to $594,000.

Depreciation: 15%↑

Operating profit: 438%↑

Interests: 22%↑

Taxes: 30%↑

Net Profit: 989%↑

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Table 9.2 One Year Pro Forma Balance Sheet

Changes:

Cash: 110%↑

Account receivable: 33%↑

Salaries: 5%↑

Inventory: 25%↑

Total current liabilities: 20%↑

Total current assets: 38%↑

Loans: 66%↑

Equipment: 43%↑

All liabilities: 46%↑

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The second line is the cost of production or basically the cost of all the goods produced,

which is assumed here for simplicity, to increase by the same percentage as the sales, and so

would gross profit, as a result. The gross profit is obtained by subtracting the cost of

production from the revenue. Operating expenses would include a variety of both fixed and

variable expenses. Some do not change for next year and some do. Salaries, rent, and

equipment stay the same. Utilities increase by 25% from $8000 to $10,000, insurance

increases by 20% from $5400 to $6480, and depreciation increases by 15% from $11,250 to

$12,937. Total operating cost would increase slightly from $230,525 to $235,292 or by only

2%. Subtracting the total operating expenses from the gross profit would yield the operating

profit or, as it is also called, EBIT. This operating profit would increase dramatically in one

year from $16,975 to $91,408 that is, an increase of 438%. The operating profit would be

subject to the subtraction of the interest paid for the firm's loans and the taxes due. Both

interest and taxes increase by 22% and 30%, respectively. Taking both interest

All assets: 40%↑

Owner's equity: 21%

Account payable: 35%↑

Notes payable: 7%↑

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and taxes away from the operating income would give us the net profit or what is called

earnings after interest and taxes (EAIT). This net profit would have a huge increase from

$7275 to $79,238, that is by 989%.

Pro Forma Balance Sheet

The example shown in Table 9.2 was constructed under the assumption that when sales are

predicted to increase next year, it would mean that the current asset would need to increase

as a consequence. Cash is assumed to increase by 110%, from $6600 to $13,860; account

receivable would increase by 33%, from $35,343 to $47,124; and the value of inventory

would increase from $24,750 to $30,937 or by 25%. As a result, the total current asset value

would increase by 38%, or from $66,693 to $91,921. As for the fixed assets, the example gives

only the equipment, which increased by 43%, from $49,500 to $70,950. This would complete

the list of all assets making the total value $162,871, an increase from $116,193 or by 40%.

All the increase in assets and in sales would not happen without the firm's ability to pay for

the cost incurred and finance the expanded operations. The firm would do that, at least

partially, by buying on credit and obtaining more loans. As a result, account payable would

increase from $18,562 to $24,749 or by 33%; notes payable would increase from $14,850 to

$16,000 or by 7%; and loans would increase from $49,500 to $82,500 or by 66%. Also, the

firm has to pay its employees with a slight increase in the total salaries from $4125 to $4330

or only by a 5% increase. All the increases in the firm's liabilities would push the total

liabilities to $127,579 from $87,037. That is an increase of 46%. Finally, subtracting all the

liabilities from all the firm's assets would give us the firm's net worth or its owner's equity,

which also would increase from $29,156 to $35,292 or by 21%.

9.2 Pro Forma and the Sales Ratio

A pro forma balance sheet can be constructed by depending only on the ratio of sales

revenue in the income statement. The ratio of the sales revenue in the projected years (S )

to the sales revenue in the current year (S ) is called the sales ratio (S%).

Multiplying the sales ratio (S%) by any entry of the current year on the balance sheet would

give us the pro forma value in the next year for that specific entry.

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t+1

t

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Looking at Table 9.3, we can see that all the entries on the next year column are obtained by

multiplying S% by the corresponding entries on the current year column. From the income

statement (Table 9.1), the sales ratio would be

Now we can verify all the projected entries in the next year (t+1) column on Table 9.3. For

example, here are some selected entries.

1. Next year's inventory of $32,670 is obtained by

Table 9.3 Pro Forma Balance Sheet as Generated by Using % Sales

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2. Next year's account payable is $24,501. It is obtained by

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3. Next year's owner's equity is $38,486. It is obtained by

and so on for all entries.

9.3 Change in Sales (ΔS) and the Needed Fund

To determine whether the owner's equity on the balance sheet is enough to cover the

increase in assets and whether the firm needs a new financing or not, we can use the

following formula:

where

NF: the needed fund

ΔS: change in sales revenue between the current and next year. ΔS = S − S (from the income statement) A: current year total assets obtained from the balance sheet

L: current year total liabilities obtained from the balance sheet

NP: net profit from the income statement

OPO: owner's payout (the portion of profit which the owner takes as earnings. Let us

assume here that the owner would take 58% of profit).

S : sales in the next year

We can plug in all these values in the formula above as we obtain them from the income

statement and the balance sheet.

Financial Forecasting

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t+1

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A forecast is an informed estimation of a future value. Financial forecasting is the process of

predicting the future state and value of financial and economic variables and

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their changes using current knowledge and present and past data. In a simpler sense,

forecasting is all about projecting the past and present into the future. Although forecasting

has been described as an adventure into the practically unknown, it is ironically justified by

the pressing need to reduce life uncertainties and minimize risks. We can logically say that

all decisions taken today are somehow connected to past experiences, and are most likely to

be connected to potential future development. This premise forms the compelling reason

why most decision makers in public and private enterprises, as well as on an individual

level, need to predict the future events.

Financial forecasting has proven to be critical for the efficient management of both the new

venture and the established firm. Most business managers know that forecasting can be a

difficult and daunting process but they also recognize the need to it as one of the most

important tasks to facilitate decision-making. Managerial choice to predict sales, for

example, would have a significant impact on planning for material and equipment

capacities for production, employment size, inventories and storage requirements, and

maintenance services. It would impact the financial manager's plans for the firm's cash

flow, investment and capital budgeting, and projected profits and losses. In marketing, the

sales forecast would be essential to plan for a sensible distribution program, promotional

strategy, and marketing budget. Forecast of future sales would also be utilized in personnel

plans for hiring, promotion, staff structure, and reward and retirement programs, and so

on. The major point to emphasize here is that any business decision would not be sound if it

is isolated from the future development of the contributing and related factors.

9.4 Role of Financial Forecasting

Financial forecasting acquires its importance for being instrumental in helping the firm

with

• Clarifying its mission and presenting solid evidence to potential investors and lending

institutions on the merits of the proposed project, its feasibility and expected

profitability. This would be critical in securing the needed capital and obtain the

suitable resources. It would also help in determining the proper value of the venture.

• Determining how much cash is needed for the present and future operations, and how

would the cash flow changes according to the other predicted changes.

• Devising standard and aspired benchmarks for the firm's performance in order to be

well prepared for the expectations, and be able to take the right and timely decisions.

• Drawing a road map for the firm's alternative business strategies as to facilitate taking

the best choices and realizing the highest possible efficiency to achieve the firm's

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objectives.

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9.5 Basic Steps of Forecasting

Although forecasting can practically be subjective and intuitive, and can rely a great deal on

personal experience and philosophy, it can also be a systematic and scientific process that

would apply the standard scientific research methods and procedures. This is especially true

in the case of quantitative type of forecasting that would depend on data analysis and

mathematical and statistical models. Generally speaking, we can summarize the basic steps

of forecasting process by the following, which are also shown in Figure 9.2.

• Deciding on the major rationale and objectives of forecasting. That is to answer the

questions: why are we predicting, and what do we want to achieve?

• Determining the span of time between the current time and the time in which the

forecast is made. This interval is called the forecast horizon. It is known that the longer

the horizon, the less the forecast accuracy. So, choosing a certain horizon is a balance

act between achieving the forecast objective and keeping it as accurate and reliable as it

can be. There are several factors that would play some roles into the determination of

the horizon such as

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Figure 9.2 Basic Steps of Forecasting

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– Nature of the product and its life span

– State of demand on the product

– Availability of the needed data

– Existence of the firm's strategic plan with a predetermined length

• Choosing the variables which are to be forecasted and those which are related directly

to achieving the stated objectives. This step requires data collection which can be in any

number of ways depending on what the forecaster sees fit.

• Selecting the appropriate forecasting model. As Figure 9.2 shows, deciding on the model

would come as a result of deciding the previous three elements: the objectives, the

horizon, and the variables. In the following section we will detail the many types of

forecasting models. The forecaster would decide which model is more suitable based on

many considerations.

• Applying the selected theoretical model and putting it into practice. In this step,

calculations are made and estimates are obtained.

• Testing and evaluating the model to make sure that the forecast estimates are plausible

and as close as possible to the actual data. The test also should assure if the method used

is valid and appropriate. The results of one method or one model can also be evaluated

in comparison to alternative methods in the light of getting the most reliable.

• Validating or modifying the model. Testing and evaluating the model should lead to

– either validating the model if it turns out to be a reliable model which can deliver

the expected accuracy, or

– modifying the model if it fails to provide what it was set to do.

Either way, the lessons would be directed to the subsequent application of the model.

Sometimes the forecasters would like to try a new model if they cannot validate or modify

the previous one. This is illustrated in Figure 9.2 by the dotted arrow which goes toward the

application phase. This new model would require a new set of data and that is shown again

by the dotted arrow from data to application.

9.6 Types of Forecasting Models

It has been said that forecasting is an art and science. Its major objective is to help make the

best possible judgment about the future circumstances and conditions as they are predicted

by a variety of methods. The most sensible approach is to combine the hard statistical facts

with thoughtful, unbiased, and fair subjective judgment that utilizes solid experience and

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practical wisdom. There are many types of forecasting models. Each can be more

appropriate than the others, depending on a host of determining factors such as

• what is being predicted and whether the purpose is to examine the trend continuation

or certain turning points;

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• what time considerations are being considered. Time factor may include

– projection time whether it is for a short or long term,

– time frame available to conduct the forecasting,

– lead time during which the firm can make decisions using the estimations of the

forecasted model;

• how much cost of the process is considered;

• what level of accuracy level is needed for a specific forecast;

• what data are available for use in the forecasting process;

• how complex is the forecasting model.

As for the scale of the process and its variables, forecasting can be macro forecasting where

the subjects would be at the aggregate level such as the gross domestic product (GDP),

national employment, national inflation rate, interest rate fluctuation, national money

supply, and so on. On the other hand, micro forecasting involves variables at the levels of

industry, firm, or specific market such as predicting the demand on a certain product or line

of a product. Forecasting can also be conducted at subset micro or individual levels of

consumer, market, and product.

As for the forecasting types according to the analytical methods, we can categorize them into

the following:

Qualitative Models

Qualitative models use non-numerical examination and interpretation of observations for

the purpose of discovering the underlying patterns of relationships and inferring their

meanings and significance. They are also called judgmental models for relying on value

judgment as the major tool in the analysis, as opposed to relying on the objective technical

facts in the quantitative models. The best representative of the qualitative forecasting

models is those which depend on expert opinion, polls and market research, and consumer

surveys.

Generally speaking, judgmental models depend on people's intuitiveness and experience in

determining what product sells, what consumers want or do not want, and what conditions

and circumstances the market will experience. Based on that, they would decide the size of

production that will be needed and the sales or profits that will be expected.

Opinions and Polls

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Firms often seek the opinions and views of others who are in touch with the market and

knowledgeable about the product and its customers. It can take the form of polling several

groups, such as the executives, the sales staff, the customers, and the market experts. Firms

can also conduct a Delphi method which uses a panel of corporate executives and experts

but questioning them separately which makes it different from the jury of executives

mentioned above which meet together and form a collective view. If the firm has an

international connection through its product, it can also form a council of distinguished

foreign dignitaries and business

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people to get their global perspectives on events and issues related to its market share and

international consumer behavior.

Surveys and Market Research

The idea of utilizing surveys in forecasting stems from the fact that most major decisions of

individuals and firms are usually preconceived and mostly predetermined. The required

expenditures are planned and allocated before the actual spending. This constitutes a good

rationale for surveying people on their plans and intentions which are most likely to reveal

valuable information for forecasting the future. This is why it is justified for US firms to

spend billions of dollars on surveys that enlist millions of people to ask them on a variety of

questions which ultimately have strong economic relevance. Among the famous surveys are

Surveys of Consumer's Expenditures (SCE) which are conducted by the Bureau of the

Census; Surveys of Business Executives (SBE), which are conducted by a consortium of

institutions such as the US Department of Commerce, the Securities and Exchange

Commission, the National Industrial Conference Board, and McGraw-Hill Publishing

Company. The third famous set of surveys is the Surveys of Inventory Changes and Sales

Expectations (SICSE), which are conducted by the US Department of Commerce, The Institute

for Supply Management, and Dun & Bradstreet Company. Yet, another important survey

related to consumers in the national context is the Survey of Consumer Confidence (SCC),

which produces three monthly indexes: the Consumer Confidence Index, the Present

Situation Index, and the Expectations Index.

When it comes down to the short-term projections, the greatest value of surveying people

would remain, as they are assumed, to provide insights into their intended actions on

financial and economic matters. This technique is perhaps the only available way to predict

consumer's responses on a new product. Over time, surveys have proven their large

capacity to reveal the nature and direction of changes in consumer tastes and preferences.

As for market research, it is a general area that may include questionnaires, observations,

“clinical tests,” field interviews, and focus groups. The primary purpose is to recognize the

consumer with whom a firm is dealing. The identification and recognition of consumers

could be demographically, economically, socially, and psychologically. Knowing the

consumer means knowing what to produce and how to market the product. Market research

is often used to introduce a new product, a new improvement on an existing product, or

even to introduce a new business or new market. Data obtained through the various

techniques are often extrapolated qualitatively and quantitatively to form certain

predictions.

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Quantitative Models

Quantitative models which involve utilizing historical data and relying on numerical

representation of the observations for the purpose of describing and explaining the trends

and changes that the observations reflect. There are two kinds of the quantitative models.

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Structural Models

These models focus on the dependent–independent relationships between variables for the

purpose of quantifying the impact of the independent variables on the variations in the

dependent variable. Econometric models of forecasting are the best representative for the

structural type.

Non-Structural Models

Models that focus on observing the patterns of change in the variables over time. The best

representative of this type is the time series models as well as the barometric models. We

shall start here with the time series model followed by a brief description of the barometric

model. Time series is the most common analysis in forecasting. It constitutes more than two-

thirds of the forecasting techniques used by businesses in the United States, according to a

recent survey by the Institute of Business Forecasting (IBF).

9.7 The Analysis of Time Series

Time series models represent the most common analytical method in the field of

forecasting. The central premise of the analysis is the use of historical data and the

application of the ordinary least squares statistical techniques to obtain predictions of

future values. It basically extrapolates data from the past and present into the future period

for the purpose of identifying general patterns in the development of a single variable

throughout time. This method allows to make forward projections out of data that have

long-term trends through the utilization of a simple bivariate regression model where time

trend (t) serves as the explanatory variable for the changes in the dependent variable.

Time Series and Data Variations

The long–term development of any variable can be plain and smooth if it follows a

straightforward secular trend but that is not the case very often. The development of most

variables historically contains several sorts of variation that would affect the real value of

the variable if it is predicted out of the general unadjusted progress. This is why an essential

part of this analysis is to recognize and count for these possible fluctuations within the

general flow of data. Calculating the impact of these variations would allow the adjustment

183

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of the forecasts made based on the estimates that the regression equation yields. Most of the

variations in the time series data are due to the following common patterns:

Secular Trends

Secular trend is a long–term consistent development in a variable value that is often

characterized by a general steady increasing or decreasing pattern and most likely

represented by a solid smooth line going upward or downward. Population growth and per

capita income are typical examples for the increasing secular trend while goods that go out

of date over time such as typewriters, cable phones, and personal train transportation are

typical decreasing secular trends.

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Seasonal Variations

Seasonal variations are recurring rhythmic fluctuations which are usually to reflect certain

seasons, weather conditions, or specific periodic occasions such as school time or holidays

like Christmas or Thanksgiving. The sales of many products can reflect typical seasonality

such as air conditioners, fans, swim suits, shorts, and tourism services in summer and snow

blowers, snow tires, heavy coats in winter, garden products in spring, turkey at

Thanksgiving, and gifts in Christmas times.

Cyclical Fluctuations

Cyclical fluctuations are long–term patterns of expansion and contraction in the economic

activity in general that reflects the recurring conditions of the economic business cycle that

characterizes the free market economy. The construction and housing sector, for example,

exhibits a typical long–term cycle in its activity that could last 10 years or more while other

sectors may experience shorter cycles.

Random Changes

Random changes are the reflection of the irregular unpredictable fluctuations that are due

to events such as war, political instability, natural disasters, strikes, and the like. This type of

variation is very difficult to count for in any modeling due to the random nature of its

occurring and the degree of its impact.

Mathematically, the actual value of a variable in the time series data (Y ) can be expressed as

a function of all of the above variations.

where

Sec is the secular effect for t period

Sea is the seasonal effect for t period

Cyc is the cyclical effect for t period

Ran is the random effect for t period

Geometrically, Figure 9.3 shows an example of each of the patterns.

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185

t

t

t

t

t

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9.8 Fitting the Model

Time series data are different in terms of their appropriateness to fit into a certain function

for the purpose of the regression estimation. Fortunately, most of the economic data can

easily fit into the linear function.

But some may fit better into the quadratic function

where the coefficient of t reflects the type of growth. Positive coefficient (γ > 0) refers to an increasing rate of growth and negative coefficient (γ < 0) refers to a decreasing rate of growth. Figure 9.4 shows both quadratic functions against the linear trend.

2

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Figure 9.3 Time-Series and Data Variations

Some data may need to be fitted into an exponential function of the form

which has to be transformed to linear by the logarithmic function in order to be estimated

by the ordinary least squares method:

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Figure 9.4 Quadratic Functions and Linear Trends

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and with a slight rearrangement, we can better identify the parameters

where Log α is a constant term and Log β is the time coefficient. The antilog function would return the parameters to their original state after estimation.

The following example shows how supportive information about a product can be used in

the fitting of a certain function.

Example

This example is using the linear function. It is about forecasting the size of membership in

an athletic gym of a club. Suppose that the firm owning the club has conducted a survey

and collected a list of related information including the following.

– At any time period (t) number of members of the club who are resuming their

membership from a previous period is a certain percentage (x) of the total number in

the previous period [M ]. Therefore, we can say that the number of members of the

club at any time period t [M ] would contain first those who were carried over from the

previous period (xM ).

– Also, at any period (t), there would be a number of new members just joining the club

for the first time. Suppose this number is a certain percentage (y) of the estimated total

market demand on this particular service (D). Therefore, this number can be expressed

as a y percentage of D after excluding those who already joined before.

– Total number of members at period (t) can then be obtained as a summation of the two

previous parts.

combining the similar terms, we get

Now, let us go back to the firm's survey. Suppose that the survey revealed that the total

demand on this service in the market is estimated by 1200 members and that those who

carry their membership over to the next period is averaging 76% of those who already

have membership last period, and that those who join the club for the first time is

usually an average of 18% of the market demand.

186

187

t–1

t

t–1

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Plugging these real figures in the equation above will give us the numeric form of the

function

or

This is the linear function format that is used to forecast for the next period using the

previous period. So if, for example, we look at the record of membership for the last year

and find it 450 members, we can immediately extrapolate an estimation of the following

year membership and probably for a few more years into the future.

But this kind of function is usually obtained by running a simple bivariate regression

analysis on two columns of data. The first is the membership size for a number of previous

periods, say years, and the trend of time from 1 to n. The 216 above would be the y-

intercept of the equation or the constant term, and the 0.58 would be β value or the slope of the regression line. If last year's membership was 450, we can project the membership

status for the next 7 years to see that it is a secular trend that is increasing but in a

decreasing rate. Table 9.4 shows the membership forecasts for the next 7 years.

It is essential for the manager to know that the growth of membership is decreasing and

that 7 years from now there will be no single person joining the club when the number of

members in the current year would equal the number in the previous year. It is a wakeup

call to change strategies and try to turn the tide around into an increasing rate growth.

The function to estimate the data at hand could also be non-linear had the conditions and

information be different. Let us assume the data would fit an exponential function such as

Table 9.4 Membership Forecast for 7 Years

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where the coefficients that would be estimated are β and N. The value of M or the membership size at any period (t) would grow proportionally with time if N is larger than 1

(N > 1). However, it would decline proportionally with time if N value is less than:

For example, if running the regression yields that N is 1.23, it would mean that M or annual

membership would increase by (N − 1) or by 23% annually. But if the estimate of N is 0.89, it would mean that membership would decline annually by 11%.

As we have seen before, an exponential function should be transformed to a linear function

by applying the logarithms in order to be estimated by the ordinary least squares method.

To estimate M , we need to take the antilog of both coefficients.

and the figures of antilog will be the values of the parameters.

9.9 Adjusting for Seasonality

Since time series data is inherently characterized by the sorts of variations mentioned

above, it is essential to adjust the estimation of the regression equation for those variations

so that we can get forecasts that reflect the realities of data. We will focus on the seasonal

variation as it is the most common variation in the time series data, and we will calculate its

impact using more than one method. It is worth noting here that isolating these variations

and adjusting the value of data according to their impacts is what is called the

“decomposition process.”

Let us first consider some quarterly data on a product sales for 20 quarters and let us use

the following fitted regression equation to calculate quarterly forecasts out of the actual

data.

188

189

t

t

t

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where S = 479.1 is the constant coefficient that stands for an average for the initial sales

and β = 3.01 is the coefficient for t. Table 9.5 shows the actual and forecasted sales for the first quarter in 2001 to the fourth quarter in 2010.

0

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The following are three methods to capture the impact of seasonality:

The Simple Average of Errors Method

According to this method, we calculate the simple average of error for each quarter and

adjust the next prediction by its value. So, let us rearrange Table 9.5 into Table 9.6 by the

following steps:

Group the similar quarters together first.

Create another column for the errors. The error is the difference between the forecasted

and the actual values.

Calculate the simple average for each group of similar quarters.

Table 9.5 Sales Forecasts for 20 Quarters

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If we want to predict the four quarters of 2011, their t values would be obtained by

continuing the sequence in Table 9.5. So, they would be 21, 22, 23, 24. Plugging these t values

into the regression equation gives us the forecast for 2011 quarters as in Table 9.7.

But these forecasts are not adjusted for any seasonal variation as it is reflected by the simple

average errors. This is why we should perform the adjustment. Looking at the average of

errors, we can see that the forecast for the first quarter was, in general,

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less than the actual by an average of 8.85. So we can add back this difference to the forecast

of the first quarter in 2011, as in Table 9.8.

For the second quarter of 2011, the forecasted value was more than the actual by an average

of 7.82. So this difference should be discounted.

Table 9.6 Simple Average Errors

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For the third quarter, the average error is 4.49. So the third quarter of 2011 would be

adjusted by discounting 4.49.

and the fourth quarter has a −4.28 average error which means it is underestimated by 4.28. So, adjusting the fourth quarter of 2011 requires adding this amount.

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Table 9.7 Regression Analysis Results

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192

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Figures 9.5, 9.6 and 9.7 show the difference between the actual and the interpolated data, as

well as the path of the residual values.

The Actual to Forecast (A/F) Ratio Method

We are arranging our data just like in the previous table where similar quarters of all years

are grouped together. This time we calculate the A/F ratio by dividing the actual data (A) by

the forecasted (F). Then we obtain a 5-year average of the ratio for the similar quarters

individually. It is the A/F ratio for a specific quarter that would serve as a multiplier of the

predicted values to adjust them for seasonality.

where F is the seasonally adjusted forecast, F is the unadjusted forecast, and (A/F) is the

quarterly average ratio as it is calculated in Table 9.9.

Table 9.8 Forecast Adjustment for the Four Quarters of 2011

192

193

sa q

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Figure 9.5 Actual Data and Interpolation I

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Figure 9.6 Actual Data and Interpolation II

The Dummy Variables Method

We can adjust for seasonality by using the regression equation obtained from the run that

includes dummy variables for the quarters.

where Q is the estimated coefficient for each quarter (Q , Q , Q ) while the fourth quarter is

represented by the intercept only. So, the estimation of the forecasted value of sales for the

first quarter of 2011 is going to use the intercept t = 21, and the Q coefficient (5.39). The

forecasted value for the second quarter would use the intercept t = 22, and the Q coefficient

(–11.33), and the value for the third quarter would use

193

194

i 1 2 3

1

2

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the intercept t = 23, and the coefficient for Q = −8.06. The fourth quarter would not have any Q coefficient since it is considered the base period.

Figure 9.7 Residual Values

Table 9.9 Five-year Average Quarterly Ratio of (A/F)

194

197 3

4

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Table 9.10 Forecasted Sales Using Regression with Dummy Variables

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Table 9.10 shows the calculations of the forecast sales for the four quarters of 2011 as they

are adjusted for the seasonal variation according to the regression with dummies method.

Table 9.11 shows the results of the Multiple Regression run.

The results of the adjusted forecasts in the three methods would have been much closer if it

was not for the practical rounding.

9.10 The Smoothed Forecasts

Smoothing techniques are other ways to produce forecasts values based on past

observations. It is more suited for time series data that have slow and infrequent changes in

the underlying pattern, and data which exhibits noticeable degree of randomness or

irregularity. The objectives of the smoothed forecast techniques are to even out such

irregularities as much as possible and reduce or eliminate the distortions arising from the

random variations. The forecasted value of a variable according to these techniques is

Table 9.11 Multiple Regression Results

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basically an average of the previous data. The most common of these techniques are the

moving average, and the exponential smoothing.

Simple Moving Average Method

This method calculates the predicted value of a variable as a simple average of a number of

observations (p) that would overlap throughout the prediction list by adding the latest

observation while dropping off the earliest. For example, if we are at the end of July and we

want to predict the sales for August, we can use the data we have for the previous month to

get an average to represent a predicted value for August. A 3-month average would be

adding the sales of July, June, and May and dividing by three. If at the end of August and

after recording the actual sales of August we want to predict for September, the 3-month

moving average requires that we add August being the latest and drop off May being the

earliest so that the September value becomes an average of August, July, and June. The

predicted sales for October would be an average of September, August, and July while June

is dropped off, and so on. The average would move forward throughout the series, hence the

name of the “moving” average. We can obtain the general formula for the moving average

this way. Let us call the actual data of Sales A where t is the current time and can reflect the

number of observations such that if we are in July, A would be A , and the previous month

is June and can be denoted by A which is also A = A = A The month after July is

August which can be denoted by A or A = A = A . So the forecast for August is an

average of July, June, and May.

or

t

t 7

6 t–1 7–1 6.

8 t+1 7+1 8

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The denominator 3 is the number of periods (p) that would constitute the elements of the

average according to the forecaster, and it could be any number the forecaster chooses. This

would make the formula more general as

So, if we are in November (11) and want to calculate a 5-month forecast for December (12),

the forecasted value for December would be

The last observation is A and it is equal to the last term in the general formula

Column no. 3 of Table 9.12 shows the actual sales of customized computers as a local store

during the 16 quarters of 2008–2011. The three-quarter forecasts are

197

198

7

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shown in column no. 4 starting at the fourth observation, 184.3, as the first average of

observations 1, 2, and 3.

Next on the column is 195.6 as the forecast value for sales in the first quarter of 2009. It is

obtained by averaging only the previous three observations and dropping off the earliest.

Column no. 7 of the table shows the forecasts using a five-quarter moving average. For

example, the predicted value of sales for the third quarter in 2011 (176.8) is obtained by

averaging out the previous five actual values.

Table 9.12 Three-Quarter and Five-Quarter Forecasts of Customized Computers

198

199

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As to why and when the forecast uses a specific value of P, it depends on the forecaster's

design, objectives, and justifications of the forecast. Generally speaking, the larger the P or

the number of observations constituting the average, the smoother the forecasts and the

more effective the impact on dealing with randomness that would be spread out over more

observations, each of which would get less weight as P increases. If we graph this time

series in its three columns of estimates as they are shown in Table 9.13, it would be clearer

that the forecast values obtained by using five-quarter moving average produced a

smoother curve (c) as compared to the

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other curves (Figure 9.8). The actual data (curve A), and the three-quarter moving average

forecasts (curve B). It is a visual confirmation to the notion that a greater value of P would

result in ironing out most of the randomness in the time series data.

Table 9.13 Three-quarter and 5-quarter Forecasts of Customized Computers

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200

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Figure 9.8 Smoothness of Five-Quarter Versus Three-Quarter Forecasts

The Weighted Moving Average

The simple moving average method assumes uniformity among the observations of time

series data in terms of their impact on forming the predicted value. It is, in fact, one of the

shortcomings that counteracts the simplicity of the model. As a response to such an

important pitfall, forecasters came up with the idea of weighing the importance of the

observations that would form the average. The weighted moving average, therefore, allows

the forecasters to assign certain weights to each observation. The weights are supposed to be

consistent with the external influences on the average. The forecasters may consider giving

more or less importance to a specific observation or term, according to their knowledge and

discretion. All assigned weights may or may not total to one (1).

The forecast value for the next period (F ) will be calculated as

where w = w , w ,……, w are the number of weights that are assigned to as many observations as available for the average. The 3-month moving average, has a P = 3 and,

therefore, has three weights (w , w , w ) assigned to each of the three actual observations,

A , A , A .

t+1

i 1 2 p

1 2 3

1 2 3

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Example

Suppose that the forecaster assigned the following weights to the four quarters of 2011:

0.15, 0.25, 0.37, 0.23. What would be the four-quarter predicted sales for the first quarter of

2012? Use the actual sales in the previous table.

Since the summation of weights is equal to 1 (0.15 + 0.25 + 0.37 + 0.23), we can skip dividing

by Σw and just calculate the numerator part of the equation

Example

What if the weights are 0.18, 0.24, 0.33, 0.20?

In this case, the summation of the weights is 0.95, which would have to be the denominator.

Example

Suppose that at the end of the first quarter of 2012 the sales turn out to be 164. Use the four-

quarter weighted moving average to predict the sales in the second quarter of 2012

assuming the same weights of the last example.

Since we use the four-quarter moving average and we have a new actual sales of the first

quarter of 2012, we should drop off the earliest quarter in the group which is the first

quarter of 2011 (154). So, the four quarters forming the average now are 2012.1, 2011.4,

2011.3, and 2012.2.

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Exponential Smoothing

Exponential smoothing model is another way to respond to the uniformity assumption of

the moving average method, which treats all time periods in the series equally. This method

assumes that the most recent past is more predictive of the future value of the forecast than

the distant past. This assumption prompted the introduction of a constant (α) to signify the weight of the immediate past while the earlier data take a (1–α) weight. In this case, α is assigned by the forecaster's discretion and the extent of his belief in the increasing impact of

the most recent data on the prediction. The value of α is between 0 and 1 but frequently has been given values between 0.10 and 0.35.

So, the exponential model calculates the value of the forecast for the next period (F ) as a

weighted average of the actual observation in the current period (A ) and

t+1

t

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the forecast value for the same period (F ), where α is assigned to the current actual and (1– α) is assigned to the smoothed forecast (F ).

Logically, if we write this equation for the current period (t), we get

and if we substitute (2) into (1), we get

and if we substitute for F as

we get

and if we keep substituting for the forecast of the past periods F , F and earlier, we will

realize that we have an equation of an exponentially weighted moving average with its

weights forming a geometric progression

which illustrates that any value of α would produce a decreasing values of weights. For example, if α is 0.40, the rest of the weights would be calculated as in Table 9.14.

This shows that the weights get smaller and smaller as we go back in the past. It dropped

here to 1% at the eighth term back. It is a confirmation that assigning a higher α would place a greater importance on the most recent past as a predictor of the future forecast, and

logically placing less importance on the earlier observations. However, greater value of α produces less smoothing, so for a smoother line a smaller α would help.

Table 9.14 Value of Weights as Geometric Progression

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t

t–1

t–2 t–3

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If we slightly rearrange the original format of the exponential smoothing model,

we can get the most practical format to use for calculations.

We use this equation to produce forecasts in the F and F columns of Table 9.15, using two

values of α, 0.20 and 0.45. For the first forecast of the first quarter of 2008, we use the actual observation (170) for A and the general average of the actual (173.9) for F , but for the rest

of forecasts, F would be the previous F. For example, to predict the sales for the second

Table 9.15 Exponential Smoothing Forecasts Using Two Values of Alpha

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1 2

t t

t

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quarter of 2010, using an α of 0.20, we use 188 for A and 170.9 for F as they are the current data at the time to predict for the following quarter.

t t

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9.11 Barometric Forecasting

Due to the continuous and significant overlap between micro and macro level of economic

activities, firm managers have traditionally realized the need to be aware of the directions

of the major macroeconomic variables when they forecast for their own variables at the

micro level. For example, predicting the sales or profits for a specific product is inextricably

connected to consumer demand, income, employment level, inflation, and so on of many

aggregate economic variables. Generally, any business activity at the firm level can be

connected to the general conditions of the economy. Economists have noticed the significant

impact that some economic variables can have on the rest of the economy, and the acquired

importance in the prediction of many other variables at both macro and micro levels of

economics. Researchers at the National Bureau of Economic research (NBER) have

considered some major economic variables as “indicators,” and have classified them into

three groups.

1. The leading indicators are those variables which change first, prompting other variables

to follow in their changes.

2. The lagging indicators are those variables whose changes tend to follow the changes of

others.

3. The coincidental indicators are those variables whose changes just coincide with the

changes of other variable.

Based on this categorization, and on the fact that in practice certain variables tend to move

ahead of others, gave the idea that the changes in the leading economic indicators (LEI) can

be used to predict the changes in others which follow. Figure 9.9

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shows a typical set of time series for these three groups. We can see that the turning points

of peak and trough is happening first in the leading indicators followed after a period of

time by the lagged indicators while the coincidental move almost in tandem with the leading

group.

Figure 9.9 Major Groups of Economic Indicators

The time during which the leading change precedes the following change is called the lead

time. It varies from cycle to cycle and it is different in the peak and trough cases. In Figure

9.9, we can see that it is taking the lagging indicator a shorter time to follow the peak of the

leading as compared to the time it takes to follow the trough. This variability is a classic

characteristic of the business cycle. Recorded data on reference dates of all peaks and

troughs of the United States business cycle since 1854 shows a great deal of variability,

especially in the lead time and how long each phase lasts. For example, in 1990–1991 the

trough lead time was 2 months and the peak lead time was 6 months. In 2001, the trough

lead time was 8 and the peak lead time was 14 months.

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Once again, the main idea of the barometric forecasting is to utilize the LEI as a predictor or

barometer for short-term changes in a set of time series data that exhibits a good correlation

of their changes over time. A typical example to illustrate the plausibility of this idea is the

causal relationship among some variables. The changes in the number of building permits

issued in the entire economy can serve as a predictor of the activity of the construction

sector. An increase in the new consumer orders can indicate an increase ahead in the

production, employment, income, and so on. The other typical example is the fluctuation of

the stock market indices such as the Dow Jones and Nasdaq and how they are used as

predictors of the state of the economy.

Economic indicators data are published monthly in Business Cycle Indicators that is issued

by The Conference Board. There are more than 300 major indicators but the short list of the

most common include 21, 10 in the leading category, 7 in the lagging, and 4 in the

coincidental, as they are shown in Table 9.16.

Barometric forecasting depends on the composite indices that are developed out of each

group of these indicators. Each composite index is, in fact, a weighted average of the

components of the group to signify the direction of movement in the whole group. Since

some of the components move up and some move down, a so called diffusion index has

been developed to represent the collective movement in the group. A diffusion index value

of 100 means that all of the components in the group are increasing, a value of a means all

of them are decreasing, and any other percentage would refer to the increasing aspect. A

60% means 6 out of 10 components are going up.

One of the most popular composite measures is the LEI. It has been developed by the US

Bureau of Economic Analysis. Typically, the LEI can signal the march toward recession or

the way to recovery. One of the known criteria is that three consecutive months of decline is

a strong signal for a recession, and three consecutive quarters of decline is a confirmation to

be in a recession. As for the diffusion index, history has shown that a value of about 50

reflects growth in the economic activity and under 50 reflects a downturn. Despite the fact

that the leading indicators have correctly predicted all recessions which occurred since

1948, it also has predicted some recessions that did not occur. This is a matter of the extent

of its accuracy

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that may have become part of its shortcomings in forecasting. That is its inability to measure

the magnitude of the change, and being restricted to identifying only the direction of the

change. Despite all the shortcomings of the economic and business indicators, they remain

an important tool in the prediction of short-term changes in the general economic activity,

and the turning points in business cycles. Their prediction can be highly useful, especially in

conjunction with other types of forecasting techniques.

Table 9.16 Typical Set of the Three Major Indicators: Leading, Lagging and Coincidental

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9.12 Testing Forecasting Accuracy

As it was mentioned before, the main objective of the forecasting process is to produce

forecasts that are as close as possible to the actual data. By this logic, we can define any

forecast error (E ) as the difference between the actual value (A) and the forecasted value

(F):

f

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Furthermore, this logic would dictate that the accuracy of our forecast would be measured

by how close the projected value is to the actual value, or how small the difference (E ) is

between them. So, the smaller the forecast error, the more accurate the forecasted value, the

more reliable the forecasting model, and the more sound the forecasting process.

The difference or the error term is typically due to many reasons related to the design of

model, its specifications, the type of data and the way of collecting them, and to the process

of data analysis. The major reasons that produce the error are the following:

the omitted variables

the equation misspecification

the random fluctuations

the economic misinterpretations

the explanatory variables identification and reliance

There are typical tests of the forecast goodness which targets the value of error in order to

check if it is at its minimum. Among these accuracy checks are:

The RMSE Check

Considering the previous method of the moving average, we can run a simple test to check

which of the two procedures the three-quarter or five-quarter forecasts is better. That is to

say which procedure, would produce a forecast value closer to the actual. This test uses the

squared forecast errors, and calculates and compares what is called the root-mean-square

error (RMSE). The measure would be that the smaller the RMSE, the better the procedure.

where n is the total number of observations and P is the number of terms constituting the

average. The end of columns 6 and 9 of Table 9.12 calculates the summation of the squared

errors in both procedures. Plugging in the values 7,762.15 and 6,456.04 in the formula above

reveals that:

For the three-quarter forecast, the RMSE is

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and for the five-quarter forecast, the RMSE is

Since the five-quarter procedure has a relatively less RMSE (24.22 < 24.44), it would be a

little better than the three-quarter procedure. So, if we want to predict for

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the first quarter of 2012, we will use the five-quarter procedure that produces 167.4 instead

of using the three-quarter procedure that produces 163.

We can use the RMSE in the exponential smoothing method too. Summarizing the results of

Table 9.15 in Table 9.17, we can check which α value produces a better forecast. For the forecasts produced with α = 0.20, we get

Table 9.17 Two Forecasts to Calculate the Root-Mean-Square Error (RMSE)

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and for the forecasts produced with α = 0.45, we get

Since using α of 0.20 produced a smaller RMSE (22.6 < 24.7), it means that α value of 0.22 is more appropriate than α value of 0.45 to bring the forecasts closer to their values.

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Figure 9.10 Smoothness of High vs. Low Alpha Values

This concludes that if we want to predict the sales of the first quarter of 2012, we should rely

on the estimate of (166.08) instead of the estimate of (157.9) as the first one uses an α of 0.20 and the second uses an α of 0.45.

or

Graphing the time series in its three forms, the actual, the forecasts using an α of 0.20, and the forecasts using an α of 0.45, as they are in Figure 9.10 shows that a smaller α (0.20) produces a smoother line.

The MAD Check

MAD stands for mean absolute deviation. It is another method, similar to the RMSE, to test

for the appropriateness and accuracy of the forecasting techniques, especially in terms of

how close the forecasted values are to the actual data. Just like the RMSE, the MAD test is

particularly useful when conducting short range forecasts such as those performed by the

moving average and the exponential smoothing. It is a simple measure of the overall

forecast error, which depends on the absolute value of the error terms, as opposed to

squaring them as it is in the RMSE calculations. The absolute value is obtained by just

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ignoring the signs of the deviation between the actual and predicted values. It is to say that

we consider all of them positive as they are squeezed in between the two vertical lines.

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Table 9.18 shows the absolute values of the deviations between the actual sales and four

different forecasts as we calculated them earlier. At the end of the table, MAD is calculated

for every forecast method:

The three-quarter forecast:

The four-quarter forecast:

The exponential forecast when α = 0.20:

The exponential forecast when α = 0.45:

Table 9.18 Mean Absolute Deviation (MAD) for Four Forecasts

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These results are very similar to what we got earlier by the RMSE tests. They confirm again

that the least forecast error (21.4) was found in the exponential smoothing technique using

an α value of 0.20. This would be a reason to consider this particular method as the best among the four methods used on this specific

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time series. The MAD results also confirmed that using the four-quarter was better than the

three-quarter moving average, and using an α of 0.20 was better than an α of 0.45.

Table 9.19 Forecast Check by Mean Absolute Percent Error (MAPE) Method

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The MAPE Check

This is the third accuracy test for forecasting estimation. MAPE stands for mean absolute

percent error. In this method, the forecast error is expressed as a proportion of the actual

value. The summation of the absolute values of those proportions is divided by the number

of observations and multiplied by 100 to get the MAPE value.

Table 9.19 shows the data related to the exponential forecasting using the two values of α, 0.20 and 0.45. We created columns 6 and 9 by taking the absolute value of the ratio of the

forecast errors (in column 5 and 8) and the actual values (in column 3). The summations of

columns 6 and 9 gave us the numerators in the MAPE formula above.

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– For the exponential forecast when α = 0.20:

– For the exponential forecast when α = 0.45:

This test indicates that the forecasts using an α value of 0.20 are more accurate than the forecasts using an α value of 0.45. This also confirms the results obtained by the previous checks.

Finally, it can be said that forecasting accuracy and reliability would basically refer to the

predictive consistency and effectiveness of the forecasting process, and to its capacity to

produce projected estimates that are significantly close to the real data that will be realized

later.

Practically, forecasting accuracy proved to exhibit an inverse relationship with the time

horizon of the forecast. The shorter the past period of time the forecaster uses and the

shorter the future period he would predict for, the more accurate the forecast. In other

words, the recent past is much more predictive for the near future. Accuracy is often related

to more careful and sophisticated procedures which normally does not come cheap. It

should, however, make managers aware that the more desire and enthusiasm they have to

get accurate forecasts, the more understanding and willingness is required of them to bear

the necessary cost.

9.13 Summary

Pro forma statement and financial forecasting is the topic of this chapter. Pro forma

statement was defined and two examples were given: pro forma balance sheet and pro

forma income-expense statement. Also the sales ratio method to construct a pro forma

statement was explained. After that, the discussion was moved to financial forecasting.

Forecasting process was defined and its role and importance, and its steps were explained.

Different types of forecasting models were addressed. In the qualitative models we saw the

opinions and polls, and survey and market research. In the quantitative models we

discussed the structural and non-structural models, and we articulated the time series

analysis as a representative of the non-structural models. To explain the variations in the

economic and financial data, we defined and graphically illustrated the secular trend,

seasonal variation, cyclical fluctuations, and random changes. After that, we moved to a

more technical discussion to show how to fit a forecasting model, with examples and graphs.

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To adjust for seasonality, we discussed three methods: the simple average of errors, the

actual/forecast ratio, and the dummy variables method. Other models of forecasting were

addressed such as the smoothing models, where we explained with examples the simple

moving average, the weighted moving average, and the exponential smoothing method. Also

discussed was the barometric model of forecasting, followed by the ways to test the accuracy

of the forecasting methods. In this regard, we explained the RMSE, MAD, and finally, the

MAPE.

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