Learning goals. At the end of this chapter, you should be able to: (1) properly construct a present value (PV) model; (2) understand the need for homogeneous measures when building PV models; and (3) describe PV model dimensions that require homogeneous measures.

Learning objectives. To achieve your learning goals, you should complete the following objectives:

• Define PV models and describe their uses.
• Learn how to compare challenging and defending investments.
• Learn how to convert a challenging investment’s future earnings and costs to their value in the present.
• Learn how to represent the cost of sacrificing a defending investment by using its internal rate of return (IRR).
• Learn about PV model dimensions that require homogeneous measures to accurately and to consistently compare investments.

Introduction

Firm managers either continue their commitment to an existing investment called a defender or disinvest in the defender and commit to a new investment called a challenger. The financial manager’s assignment is to analyze and to rank defenders and challengers. What adds complexity to the ranking process is that defenders and challengers are sometimes measured in different units. These differences in measures between challenging and defending investments may result in unstable and inconsistent rankings when more than one ranking method is used. This chapter intends to describe present value (PV) models that consistently and accurately rank defending and challenging investments using homogeneous (same) measures.

What is a Present Value (PV) Model?

A PV model is a mathematical expression that represents the value of future cash flow in present dollars. The present value of future cash flow earned by a challenging investment exchanged at the discount rate equal to the defender’s IRR is called net present value (NPV). The NPV of the future cash flow of an investment discounted by its own IRR is zero.

To rank uniquely a defending and challenging investment requires that we reduce their current and future earning to a single number in the same period. To make this point, that ranking investments uniquely requires a one-dimensional measure, assume that we determine the winner and loser of a sporting event by several different measures. For example, suppose that the winner of the Super Bowl football game depended on the following measures: points earned, yards gained, yards earned on the ground divided by the yards earned passing, yards penalized, injuries sustained, and the number of persons viewing the contest. Most football sports fans would agree that the success measures just described matter—but we will never determine who wins and who loses with such multidimensional measures unless in some rare event one team dominated in all dimensions. So, we must decide which measure matters most and in the case of football—the measure that matters most is points earned.

To avoid the indecisiveness of a multi-dimension metric when ranking investments, we convert future cash flow to the present creating a one-dimensional number that allows us to rank investments uniquely.

The Need for Homogeneous Measures

The idea that future cash flow can be valued by its worth in the present period may be one of the most important and pervasive concepts in financial management. It is the basis for ranking physical investments and for valuing bonds, stocks, insurance, pension funds, housing, land, cars—and almost anything that has more than one period of economic life generating cash flow.

We might have hoped that converting investments’ future cash flow to their value in the present would produce one agreed on method for valuing investments. It has not—because PV models have several dimensions and PV models have only resolved differences in measurement for one dimension—time. Other dimensions such as investment size, term, loan terms, taxes, measures of return, liquidity, and risk also require that we measure them using the same—homogeneous—measures before we can rank investments consistently and accurately.

To make the point that a lack of homogeneous measures may create ranking conflicts, compare ranking investments to a horse race. We organize horse races so that other factors besides the horses’ speed do not influence the horse race outcome. For example, we expect that only horses enter the race. We require that the horses all begin the race at the same time and place. We expect that all horses will run the same distance—that implies that not only has the starting point been determined but the finish line as well. Finally, we expect that everyone agree that the criterion for ranking horses is the time interval between when each horse starts and finishes the race.

Ranking investments is, of course, not a horse race. Yet, the process of ranking investments according to their earnings and horses according to their speed have many elements in common. First, we assume that we are comparing investments of the same size (only horses run the race). We also assume that we are comparing investments over the same time-period, term, (the length of the race is the same for all horses). Finally, we assume that the criterion used to rank investments (present value) would provide the same rankings for each investment just as the time required to run the race would be used to rank the speed of the horses.

In the remainder of this chapter and in the following chapters, we focus on several PV model dimensions that must be measured the same, homogeneously, so that our investment rankings will be consistent and accurate.

Homogeneous Time Measures

The first and most important homogeneous measure requires that investment cash flow from defenders and challengers be measured in present dollars. Creating homogeneous time measures by discounting future cash to flow to their value in the present is in essence what PV models achieve. They convert future cash flow to equivalent values in the present. But how do we find the present value of future cash flow? The key is to recognize that prices, including the price of future dollars valued in the present, are ratios. Ratios tell us the per unit value of what is in the denominator. For the ratio A/B, the ratio tells us the number of units of A for one unit of B.

To illustrate, consider the price of a tank of gas. To calculate the price per gallon of gasoline, we divide the money paid for a tank of gas (A) by the gallons of gas purchased (B). The ratio tell us the per unit price of a gallon of gasoline. If the ratio (money paid)/(gallons of gas purchased) were ($20) / (10 gallons) = $2.00, we would say that the price of a one gallon of gasoline is $2.00.

Exchanging future dollars for a present dollar. Now consider a special kind of ratio that describes the rate at which we exchange present dollars (A) and future dollars (B) between time-periods. We exchange present and future dollars every time we lend or borrow money, make or liquidate an investment, or invest in or withdraw money from a retirement account. To illustrate, suppose I borrow (lend) present dollars V₀ (A) now and in one year from now, I repay (receive) future dollars R₁ (B) where the subscript indicates the end of the time-period in which the cash flow occurs. The result of the division (A/B) is equal to (1 + r) where r is expressed as a decimal because the dollar units in the ratio cancel. Thus, the ratio of A/B or future dollars to present dollar is equals to 1 plus r percent:

(7.1)  

Using our convention for describing ratios, we would describe equation (7.1) as follows: (1 + r) future dollars can be exchanged for one present dollar or that a present dollar is compounded to its future value at the rate of r percent.

To illustrate, suppose V₀ equals $100 and R₁ equals $110. In this example the ratio of future to present dollars is: $110/$100 = 110%, and we could say that one present dollar can be exchanged for $1.10 future dollars. Sometimes we simply say that one dollar in the present can be compounded to 1.10 future dollars.

Were we to describe the ratio of future dollars for present dollars, we would say $1.00 future dollar can be exchange for 1/(1 + r) present dollars.

(7.2)  

To illustrate, suppose V₀ equals $100 and R₁ equals $110. In this example the ratio of present to future dollars is $100/$110 = $0.91 and we could say that one future dollar can be exchanged for $0.91 present dollars. Sometimes we say that one dollar in the future can be discounted to 0.91 present dollars.

At this point, it may be useful to explain the use of the variable r. It appears frequently in PV models that involve comparing money across time. In its most general use, it is a percent or a rate. It could be the market rate of interest. It could be the rate of interest earned on one’s assets or equity. Or it could be the rate charged on loans, of which there are several. We will try to make it clear which rate r refers to and in some cases add a superscript to r to clarify. For example, the market rate of interest is defined as rm. If we want to refer to r in a particular period, we would subscript it with the appropriate time-period. The rate r in the t^th period could be written as r^t.

Exchanging present dollars for a future dollar over more than one period. If we know the exchange rate between present and future dollars, then it is a small step to convert present dollars to their equivalent in the future or to convert future dollars to their equivalent in the present. We simply multiply by the appropriate ratio. To convert present dollars to their future value we multiply by (1 + r).

(7.3)  

To convert future dollars to their value in the present we multiply by 1/(1 + r).

(7.4)  

This last ratio, the present value of future cash flow, is of particular interest. We find this ratio to be of special interest because, as was mentioned earlier, we live and make decisions in the present. Thus, converting future cash flows to their equivalent present value makes it possible to evaluate capital budgets with future consequences in terms of present dollars.

To summarize, using numbers from our previous example where r = 10% or .1, the future value of $100 is $100(1.1) = $110. The present value of $110 future dollars is $110[1/1.1)] = $100.

Completing the gasoline purchase analogy, if the price of gasoline were $2 per gallon, we might ask how much will 10 gallons of gasoline cost? Ten gallons times $2 is equal to $20. Or we might ask how many gallons of gasoline can I purchase for $20? Twenty dollars divided by $2 is equal to 10 gallons.

To illustrate the importance of calculating the present value of future cash flow, suppose that a “down-on-her-luck friend” approaches you. She explains that her wealthy aunt has promised her $100 in one year, but she needs the money now. She asks you what you would offer in present dollars in return for her future $100 dollars. You quickly calculate (assuming an exchange rate of 110%), and report that the present value of $100 future dollars (received in one period from now) is $90.91. Or, we might say that the discounted present value of $100 future dollars is $90.91.

Just as we found the present value of one period in future dollars, we can also find the present value of future cash flows received in two or more periods in the future. Returning to our earlier example, suppose this same down-on-her-luck friend offers you R₁ dollars in one period and R₂ dollars two periods in the future and asks how many present dollars would you offer for the exchange if the exchange rate is 1 + r.

We can find the present value of R₂ in two steps. First, convert R₂ to its equivalent value in period-one dollars by using the ratio [R₂/(1 + r)]. The important point is that the exchange rate is assumed to be (1 + r) between any two periods, including between periods one and two. Next, discount period one dollars to their equivalent in the present:

(7.5)  

We would follow a similar procedure to find the present value of dollars received in three periods from the present. Suppose you were offered $100 for the next three periods. What is the present value of these future cash flows if the exchange rate were 110 percent between any two periods? The answer is:

(7.6)  

In the remainder of this book, models which evaluate financial strategies by converting future cash flows to their equivalent in the present are referred to as PV models.

Homoegenous Measures and Cash Flow

The second homogeneous measure required when building a PV model is to represent economic activities of the firm by their cash flow. In our horse race analogy, the cash flow principle is equivalent to letting only horses run the race. Some activities of the firm we characterize as noncash flow such as appreciation (depreciation) of assets, increases in inventories of unsold goods, and increases in accounts receivable and payable. But these events are not included in PV models because they do not produce cash flow.

One justification for the cash flow principle is that, at some point, we expect all economic activities to generate cash flow. At some point, we expect inventories to be liquidated and generate cash. At some point, we expect accounts receivable and accounts payable to be settled for cash. At some point we expect long-term assets that have appreciated (depreciated) to be sold and the difference between their acquisition and sale price to capture the noncash flow of depreciation (appreciation). Thus, in effect, we do count all economic activities of the firm by recognizing only cash flow; however, we count them only when they create cash flow.

Defenders and Challengers and Homogeneous Measures

The defender’s opportunity cost. An opportunity cost is a benefit, profit, or value of something that must be given up to acquire or achieve something else. A PV model compares the returns from a defender that must be sacrificed to acquire the challenger with the returns from the challenger. Both the opportunity costs of the defender and the returns from the challenger must be measured in homogeneous units.

To represent what we sacrifice by liquidating the defender to purchase the challenger, we find the defender’s internal rate of return (IRR) r that equates the defender’s discounted future earnings to its liquidated value in the present. We represent this trade-off in equation (7.7). In this equation, R_t t = 1, …, n are the defender’s cash flow in period t if it were not liquidated. The variable S_n is the defender’s liquidation value assuming it would be held for n periods. And, V₀ is the present value of the defender’s future cash flow and liquidation value exchanged for present dollars at rate r:

(7.7)  

The discount rate r is the opportunity cost of the sacrificing the defender to acquire the challenger measures as a percent. Another way to describe the defender’s opportunity cost is to describe it as the exchange rate between the defender’s present and future dollars.

Prices are not opportunity costs. The price of a good is the amount of money, or money equivalent, paid to obtain it. The amount of money or money equivalent paid to obtain the good represents the direct cost of what is given up to obtain something desired or to avoid something disliked. However, there may be other costs to acquire a good—or to avoid a bad—besides the price paid. For example, one might consider the cost of attending a movie to be the price of the ticket. However, transportation costs to and from the theater could add to the actual value of what must be exchanged to attend the movie. And there may be other costs of attending the movie such as lost earnings as a result of missing work.

Suppose the movie was playing at the same time that the moviegoer ate his or her prepaid dinner meal. Now the cost of the movie includes not only the price of the ticket plus transportation costs and lost wages but also the value of the skipped meal. The cost of the ticket, transportation costs, lost wages, and the skipped meal together represent the opportunity cost of attending the movie, which is different from the price of the movie ticket.

Opportunity costs and perfect capital markets. The capital (or financial) market is where people trade today’s present dollars for future dollars and vice versa. In a perfect capital market, dollars trade between adjacent time-periods at the (same) market rate of interest r^m. For a perfect capital market to exist, the following conditions are required: no barriers to entry; no participant can influence the price; transactions are costless to complete; relevant information about the market is widely and freely available; products and services are homogeneous; no distorting taxes exist; and investment possibilities are continuously divisible. Finally, the firm’s opportunity cost of capital is the same regardless of the size or purpose of the amount being borrowed or lent.

Though markets for some financial investments are considered highly efficient, they are not perfect. Rates of return on savings rarely equal the rate paid to borrow funds. Moreover, rates of return on investments typically depend on the size and economic lifetime of the investments. Hence, in the real world, investors face imperfect capital markets. We allow for imperfections in capital markets in our PV models by allowing the rate of return on an investment being considered for adoption, a challenger, to differ from the rate of return on an investment that must be sacrificed to adopt the challenger, a defender

Defenders may include investments that must be liquidated, investments that must be foregone, or credit reserves (unused borrowing capacity) that must be exchanged for debt funds. Thus, an interest rate on a loan equals the opportunity cost of capital on a defender only if the credit reserve used up has no value. In imperfect markets, the opportunity cost of the defender and the market rates of interest on loans are rarely equal. Since PV models focus on retaining the defender or acquiring the challenger, deciding between them requires that they be measured in the same homogeneous measures. Indeed, as we will show in later chapters, one reason why PV models may rank challengers and defenders inconsistently is because they fail to measure them in homogeneous measures. This point is so important that we will discuss it in more detail in later chapters. The focus on homogeneous measures leads us to consider other principles that will guide us when constructing PV models.

Homogeneous Rates of Return Measures

Another homogeneous measure is the rate of return measures used in PV models. The homogeneity of returns measure requires that if the return to the challenger is measured as a return to equity invested, then the discount rate that measures the defender’s opportunity cost must also measure the defender’s return on equity invested. If the return to the challenger is measured as a return on the investment, then the discount rate that measures the defender’s opportunity cost must also measure the defender’s return on the investment. The return on the investment (ROI) corresponds to the return on assets (ROA) when describing a firm. To avoid confusion we will describe the rate of return on equity in the investment or in the firm as ROE. And we will describe the return to the firm’s assets or investment as ROA.

So how do we build PV models that homogeneously measure returns on assets, ROAs, and returns on equity, ROEs? We begin by finding ROAs. When finding the ROA for the defender, we ignore cash flow that includes borrowing or lending activities. The advantage of this approach is that it measures the rate of return on assets independent of the returns from the loan used to finance the investment. We write the defender’s cash flow and its corresponding IRR that measures ROA as:

(7.8)  

Where V₀ is the initial investment, R₁ is net cash flow earned in the first period, and S₁ is the liquidation value of the investment after one period. Then we write the ROA as:

(7.9)  

Interpreted, equation (7.9) equates ROA to cash returns R₁ plus the cash value of capital appreciation (depreciation) equal to the difference between the liquidation value and the beginning value of the asset all divided by the beginning value of the asset. It measures the rate of returns on all of the firm’s assets including returns generated by human capital, manufactured capital, social capital, natural capital, and financial capital.

Why would we want to exclude earnings from debt capital while including earnings from other forms of capital? To do so would overstate or understate the earnings generated by the other forms of capital. This is the argument for accounting for cash flow associated with debt capital, an approach consistent with the calculation of the return on equity. To measure the rate of return on the firm’s equity, we pay the cost of debt used to finance the firm’s assets—netting out the contributions of debt capital. We calculate the firm’s rate of return on its equity (ROE).

To include cash flow associated with debt capital, assume that an asset is acquired using debt D₀ plus equity E₀ whose sum is equal to V₀. At the beginning of the project, the firm supplies equity capital to purchase the investment creating a negative cash flow of E0 dollars in the beginning period. Then it receives D₀ from the lender and pays D₀ to the seller to complete the purchase of the investment. The receiving and paying debt capital D₀ at the beginning of the period cancel out making net cash flow from debt capital at the beginning of the investment zero. At the end of the period, the investment repays debt capital D₀ plus interest iD₀ to the lender. Thus at the end of the period, cash flows of D cancel each other out. Thus, we can express these cash flow as:

(7.10)  

And we can write the ROE as follows:

(7.11)  

We are interested in the relationship between ROA and ROE. To determine that relationship, we first solve for R₁ in equation (7.9) and find that it equals:

(7.12)  

Then we make the substitution for R₁ in equation (7.11) and obtain the result:

(7.13)  

What equation (7.13) reveals is that ROE > ROA as long as the firm earns a positive return on its debt capital (ROA > i) confirming an earlier result.

The cash flow principle requires that careful distinction be made between a cash transaction and a noncash transaction. Sometimes the distinction is not always clear. For example, an asset’s book value depreciation does not itself generate a cash flow. Depreciation expenses of an investment do, however, generate a tax shield that creates a cash flow in the form of reduced tax payment. Thus, we include the cash flow associated with tax savings resulting from depreciation of an asset but not the depreciation.

Increased inventories of unsold goods do not create a cash flow. However, when the inventory is liquidated at the end of the period, it is converted to cash and enters in the present value calculations.

Total Costs and Returns Measures

All cash costs and cash returns associated with an investment should be included when determining an investment’s present value. Consider how the total costs and returns principle is applied in several practical situations. Whenever low-interest loans or preferential tax treatments are tied to the ownership of a durable, these concessions will influence the present value of the investment. To ignore these benefits (costs) would lead to an under or over evaluation of the worth of the investment.

Sometimes an investment such as land has more than one source of returns. Mineral deposits, potential recreational use, and urbanization pressures may create expected returns over and above those associated with agricultural use. Pollution standards may impose costs in addition to those normally experienced. All these expected cash costs and returns that influence the value of the durable should be included in the PV model.

Other Homogeneous Measures

We require that cash flow associated with both the challenger and defender be measured in homogeneous units. Therefore, we have discussed the requirement that their returns both be measured in the present time period and that only cash flow be included when measuring economic activities of the defender and challenger. There are at least four other homogeneous measure requirement that are so important that we devote a chapter to each one.

Homogeneous size measures (Chapter 9) require that initial and periodic investment sizes for challengers and defenders must be equal.

Homogeneous term measures (Chapter 10) require that the defenders and challengers experience economic activity over equal terms.

Homogeneous tax-rate measures (Chapter 11) require that we account for differences in effective tax rates for defenders and challengers that depend on cash flow patterns and capital gains (loses).

Homogeneous profit measures (Chapter 12) require that we calculate rates of return in PV models and AIS following the same (homogeneous) methods.

Homogeneous investment measures (Chapter 13) recognize that PV models may be constructed for an incremental or a stand-alone investment, and homogeneity of investment types requires that both the defender and challenger are either incremental or stand-alone investments.

Homogeneous liquidity measures (Chapter 14) recognize that firms and investments can be differentiated by their liquidity and requires that the defender and challengers are measured using consistent liquidity measures.

Homogeneous risk measures (Chapter 15) recognize that risk tolerance is different for individuals. Therefore, to measure risk in homogeneous units, we must account for influence of risk attitudes of certainty equivalent representations of risky investments.

If the cash flow for a prospective challenger is measured in after-tax nominal values with no risk over n years with an initial investment of V₀ dollars, then the defender whose ROA or ROE is used as the discount rate when calculating the challenger’s net present value (NPV) should be measured in similar units—in after-tax nominal values with no risk over n years with an initial investment of V₀ dollars.

Summary and Conclusions

The discount rate in a PV model represents the rate of return earned by a defender. It is the rate of return sacrificed by disinvesting in the defender to acquire the challenger. The defender’s rate of return measures either its ROA or ROE. Choosing the appropriate discount rate for PV models describing a challenger is perhaps the most difficult task of investment analysis. The choice essentially involves the identification of challengers and opportunity costs associated with defenders.

The relevant costs in economic models have always been opportunity costs. These costs reflect what is given up when an alternative is chosen. This chapter has examined how the calculation of opportunity costs is influenced by time.

This chapter also discussed homogeneous measures that should direct the construction of PV models. They should measure cash flow in the present. We should count only cash flow when finding present values for challengers and defenders. And we should measure the defender’s opportunity cost of capital and the challenger’s cash flow as either ROA or ROE. We will devote entire chapters on special homogeneous measures such as homogeneous sizes, homogeneous terms, homogeneous tax rates, homogeneous investment types, homogeneous rates of return, homogeneous liquidity measures, and homogeneous risk measures.

Our goal is that by using homogeneous measures to construct PV models we can obtain consistent, stable, and accurate investment ranking.  Knowing that one’s investment rankings are stable, accurate, and consistent is essential for the success of a financial manager whose goal is to maximize the present value of the firm’s investments.

Questions

1. Underlying present value (PV) analysis is the fundamental concept that a dollar today is not valued the same as a dollar received in the future. List several reasons that might explain why a dollar today may not be valued the same as a dollar in the future.
2. The idea that the value of a good today is not the same as the value of the good in the future is a universal concept. Suppose you were offered a 2018 Ford F-150 pickup truck for delivery today or the same truck in exactly the same condition delivered in 2023. What would you offer for the truck today versus what you would offer for the identical truck delivered in 2023? If the prices you would offer are different, please explain why?
3. Building on the concept of present and future dollars, suppose that you invest $100 today. Then assume that one year later your investment returns $110 in cash.
   1. What is the present value of the future earnings?
   2. What is the future value of $100?
   3. What is the ratio that converts dollars received one period in the future to their present value?
   4. What is the ratio that converts present dollars to their value one period in the future?
   5. Interpret the ratios: $100/$110 and $110/$100 in terms of the problem described above. How would you use the two ratios? (Hint: what ratio would convert a present dollar to a future dollar and what ratio would convert a future dollar to its worth in the present?)
4. Suppose you are considering a ten-day spring break vacation. Transportation, lodging, and meals will cost about $2,300. Explain the difference between the cost of $2,300 and your opportunity cost of the vacation?
5. Suppose that you invest $100 today. Then assume that one year later your investment returns $110 in cash. Assume that your opportunity cost of capital is 12%.
   1. What is the present value of the future earnings?
   2. What is the future value of $100?
   3. What is the ratio that converts dollars received one period in the future to their present value?
   4. What is the ratio that converts present dollars to their value one period in the future?
   5. Compare the ratios: $100/$110 and $110/$100 in terms of the problem described in problem 3 and the ratios 1/1.12 and 1.12/1 described in this problem. What is the main difference between the problems and what difference does it make?
6. In a PV model, two investments are being compared, a defender and a challenger. Briefly distinguish between a defending and a challenging investment. Suppose you are considering replacing an aging orchard with new trees. What would be the challenging investment and what would be the defending investment? What consideration would you include when comparing a challenger and a defender? What are the possible decisions you might take in this investment problem?
7. In practice, financial managers face more complicated investment problems than deciding between a single defender and a single challenger. Describe how you might solve an investment problem with more than one challenger? One challenger but more than one defender? When choosing between multiple defenders, what would be the appropriate criterion for the preferred defender? How would you analyze a problem consisting of multiple challengers and defenders? What principle from micro economic theory might guide you in preparing your answer?
8. A financial manager believes that a defending investment can earn $500 for the next six years. He also believe that his defending investment can be sold today for $2,500. What is the opportunity cost of sacrificing the defender to invest in a challenger. In other words find the IRR of the defending investment.
9. Reconsider the defender in question 8 and suppose that the defending investment is financed by a $2,000 that does not require principle payment—only interest payments at the rate of 4% per year.  Now find the rate of return on equity invested in the defender. Compare your results with those obtained in question 8 and explain the differences if they exist.