Major Money Market Rates

The different securities shown in Table 7.1 offer different interest rates. However, a few money market rates, such as those shown in Table 7.3 and Exhibit 7.1, are cited very often and serve as benchmarks for other rates in the marketplace.

Fisher Equation

One of the first great American economists, Irving Fisher, clearly delineated the relationship between nominal and real interest rates.⁵ This relationship has been termed the Fisher effect.⁶ Equation 7.3 shows that the nominal rate of interest is equal to the real rate of return and the expected rate of inflation.

(7.3)

where:

• r_n = nominal rate of interest
• r_r = real rate of interest
• p = inflation rate

If we rearrange Equation 7.3, we get Equation 7.4:

(7.4)

There are three components to the transformed Fisher equation. The first is the real return that an investor expects to receive on an investment. The second component of the nominal rate reflects inflationary expectations. The third component shows that the rate of return itself reflects the reduced buying power of the rate that is actually earned. This last component is usually small and therefore is often dropped to result in a short form. This simplified Fisher equation is shown in Equation 7.5:

(7.5)

EXAMPLE USING THE FISHER EQUATION

Let us assume that the real rate of interest is 2%. Let us further assume that the inflation rate is 2.5%. The Fisher equation 7.3 shows us that the nominal rate is:

(7.6)

We can see that the simple Fisher equation indicates a nominal rate of 4.5, which is the simple sum of the (expected) real rate of 2% plus the (expected) rate of inflation of 2.5%. In the low‐interest‐rate environment that has prevailed after the subprime crisis, this issue, while important, is somewhat less important than what it would be in a more normal‐interest‐rate environment.

Additional Comment of Nominal Versus Real Rates in Business Interruption Analysis

It is important to know the difference between nominal and real rates when one is applying and analyzing different interest rates. However, analysts and experts in litigation need to recognize that the rates with which they often deal – rates that are quoted in the marketplace – are quoted in nominal terms. Applying the Fisher equation allows one to gain a sense of how much of the quoted rate is due to price expectations. This, however, may not play a direct role in a particular case.

Comment on Ex‐Ante and Ex‐Post Rates

It is useful to be aware of the difference between ex‐ante rate and ex‐post rates. As an example, interest rates are based upon ex‐ante expectations. The lender may quote a rate that includes the sum of the real rate of return that they seek plus a rate that reflects their expectation of inflation. For example, if a lender seeks a 3% rate of return after inflationary effects, and if it expects inflation over the period of the loan to be 2.5%, then it may quote an interest rate on the loan of 5.5%. However, this reflects the expectation that over the period of the loan inflation will be 2.5%. Let’s assume the loan is a one‐year loan. If, instead of inflation being 2.5%, it is (coincidentally) 5.5%, then the real return is 0% (5.5% – 5.5%).

Relationship Between Risk and Return

The higher the level of risk of a security, the higher its rate of return. To understand this, consider two securities that are the same in all respects including their rates of return but that differ in their risk. Security Y has more risk than security X. Sellers seeking to sell both security X and Y will find that buyers are unwilling to buy Y – it offers only the same return as X but requires buyers to assume more risk than X. The only way that sellers will be able to sell Y is to offer a higher rate of return than X. How much higher depends on the risk that buyers perceive in Y. The result of this risk‐return trade‐off process is a ranking of securities in the marketplace; securities with higher risk have to offer higher rates of return in order to compete for the available funds of investors.

Table 7.5 shows several average rates of return for securities that vary in their risk levels. Common stocks have higher risk than corporate bonds, which, in turn, have higher risk than Treasuries. Table 7.5 illustrates how these securities offer higher rates of return in accordance to the risk that the market perceives in these securities. Exhibit 7.5 shows the general risk‐return trade‐off for different broad categories of securities.

Sources of Rate of Return Data

The most frequently cited source of rate of return data is the annual volume Stocks, Bills, Bonds and Inflation published by Morningstar.⁷ It is a volume that every damages expert should own. The book is an outgrowth of a seminal study published by Roger Ibbotson and Rex Sinquefeld in 1976.⁸ The study quantified the historical rates of return on various securities and became the start of the database that is published in the Stocks, Bills, Bonds and Inflation yearbook. The book provides a wide variety of rate of returns statistics as well as tables that allow the expert to look up the average rate of return for many broad categories of securities from 1926 through the last full year. The security categories covered by Morningstar are Treasury bills, intermediate‐term Treasuries, long‐term Treasuries, corporate bonds, large‐capitalization common stocks, and small‐capitalization common stocks. The book also includes many statistics on the variability of securities, helping to explain their risk and rate of return.

Calculating Rates of Return

Experts need to be aware of how the rates of return they use are computed as well as the difference between the rates quoted on a security and its rate of return. A one‐period rate of return is defined as the income derived during the period divided by the purchase price. For a security such as a stock purchased at the start of the period, this could be the sum of any change in the price of the security plus any dividends received divided by the purchase price (see Equation 7.7).

(7.7)

For an interest‐paying bond, the one‐period rate of return is similarly computed, as shown in Equation 7.8.

(7.8)

Data sources such as Stocks, Bills, Bonds and Inflation show various rates of return for large categories of securities of large‐capitalization stocks. Such returns can be readily computed using the value of market indices, such as the Standard & Poor’s (S&P) 500. Periodic returns can be computed in a similar fashion as shown in Equation 7.9.

(7.9)

Computing Average Rates of Return over Historical Periods

There are two ways that a historical average rate of return can be computed: the arithmetic and the geometric mean. In Chapter 5 we discussed the difference between the arithmetic and geometric means. In this chapter we revisit the topic in the context of rate of return.

The arithmetic mean annual rate of return is the simple average of the annual rates of return over the historical period. This is expressed in Equation 7.10.

(7.10)

where:

• r_a = the arithmetic rate of return
• r_t = the rate of return generated over the period tn = the number of periods used for computing the average

The geometric mean can be expressed as shown in Equation 7.11.

(7.11)

where:

• r_g = the geometric mean

The geometric mean can also be expressed using the beginning and ending period values as shown in Equation 7.12.

(7.12)

where:

• r_g = the geometric mean
• Y_n = the value at the end of period n
• Y₀ = the value at time 0
• n = the number of periods over which we are computing the geometric mean

Under most conditions, the geometric mean generally is less than the arithmetic mean.⁹ In cases where returns are constant, the geometric mean and the arithmetic mean are equal.

One way to explain the difference between the geometric and arithmetic means is that the geometric mean is a backward‐looking measure that reflects the change in value over more than one time period. However, when we are focusing on single periods, the arithmetic mean is more appropriate. The arithmetic mean is the appropriate measure to use for forecasting and discounting, as it better measures “a typical performance over single periods and serves as the correct rate for forecasting, discounting, and estimating the costs of capital.”¹⁰

Another way to compare the geometric and arithmetic means is to note that “the geometric average tells you what you actually earned per year on average, compounded annually. The arithmetic average tells you what you earned in a typical year. You should use whatever one answers the question you want answered.¹¹

Term Structure of Interest Rates

Another important relationship involving rates that may be used to discount future losses or to bring historical loss amounts to present value terms is what is known as the term structure of interest rates. This refers to the relationship between rates of securities at different maturities and of a similar credit quality.¹² The more common circumstance is for longer‐term securities, such as long‐term Treasuries, to have a higher yield than their shorter‐term equivalents. Various theories attempt to explain the common upward‐sloping nature of this relationship, which is sometimes graphically depicted as a yield curve.¹³ One of the more often‐cited theories is the expectation theory, which states that long‐term rates are a function of current short‐term rates and expected future short‐term rates. When future short‐term rates are expected to rise, then long‐term rates, being partially a function of future rates, have to be higher than short‐term rates. Sometimes, however, we can have a downward‐sloping yield curve. This is called an inverted yield curve. Some analysts believe that a flat or downward‐sloping yield curve is a predictor of recession. For experts in litigation, the yield curve enters the picture when they have to decide whether to use long‐term rates or short‐term rates to discount future losses to present value. While this topic is discussed later in this chapter, at this point we will just state that many experts believe that one should roughly match the term of the securities being used as a guide to the discount rate, with the term or the loss stream being discounted.

In recent years, the historical term structure of interest rates has been disrupted. Expansionary monetary policy in the post‐Great Recession world has left interest rates are very low levels. Even as the economy continued to grow for a prolonged economic expansion, the Federal Reserve kept rates low. These low rates applied to not only short term but also to long term rates. This created uncertainty in markets and analysts had to judge whether such unusual interest rates would persist. This uncertainty was compounded by the Coronavirus crisis.

In a world of flat yield curves and repressed yields, the normal term structure premium has declined. This may affect the selection of long term rates – rates which are often used to discount longer term profits projections.

Selection of Appropriate Prejudgment Rate

There are differing views within economics and finance as to the appropriate rate of return to apply to historical damages to bring them to trial date terms. At a minimum, past losses should be converted to present value terms using a risk‐free rate, such as the rate on U.S. Treasury bills.¹⁴ Some believe that the defendant’s debt rate should be used as the prejudgment rate of return based on the theory that once the defendant engaged in its wrongful actions, it was, in effect, in debt to the plaintiff – as it “owed” the plaintiff its lost profits.¹⁵ The use of the defendant’s debt rate, which is higher than the risk‐free rate, results in a higher present value of losses. This choice of prejudgment returns, however, is not generally accepted within this field. The use of the risk‐free rate, or the defendant’s debt rate, will not fully compensate the plaintiff for its opportunity costs. If the plaintiff had received the lost profits at the time it would have earned them “but for” the actions of the defendant, it would have either reinvested them in the business and/or distributed a component to shareholders. The use of the lower rates just described may not compensate the plaintiff for the lost investment opportunities that it could have enjoyed. Another guide to use in the determination of the prejudgment rate is the plaintiff’s cost of capital. This is the expected rate of return that the market requires in order to attract funds to a particular usage or investment.¹⁶ This is the weighted average of each of the individual costs of capital for each of the components in the company’s capital structure. For example, if the firm derived half its capital from borrowing and half from equity investments, each weight is 0.50. The rate of return on equity would be higher than the debt rate due to the increased risk associated with equity investments. Let us assume that the debt rate is 10% and the equity rate is 15%. The weighted cost of capital would be as shown in Equation 7.13.

(7.13)

where:

• w_d = the percent of debt in the capital structure
• r_d = the rate paid on debt
• w_e = the percent of equity in the capital structure
• r_e = the rate of return on equity

Based on the parameters of the above example, the cost of capital would be:

When the firm has a more varied capital structure, with different layers of debt and other forms of capital, such as preferred stock, the cost of capital can be expressed in the weighted average form shown in Equation 7.14.

(7.14)

Equation 7.14 simply means that the cost of capital is the weighted average of the costs of the components of the firm’s capital mix.

For a profitable firm, the return provided by the cost of capital may be lower than the actual rate of return the firm earns in a given year (depending on how you define this rate of return). The cost of capital, however, is a rate of return that is sufficient for a company to meet its interest obligations and repay its debt principal, while also allowing for an equity return that is consistent with this type of business. It is important to note that the payment of a rate of return equal to the cost of capital is not a “break‐even” return. It is a return that is equal to the return contracted for by new nonequity capital providers; it enables new equity holders to receive the return they expected at the time they made their investment.

In computing damages for a division of a diversified firm that operates in many lines of business and has several divisions, it may be more appropriate to use the divisional cost of capital. When that is not easily defined, the economist can look at the costs of capital for companies that are similar to the division in question. The individual components of the capital mix of a firm and their respective costs are discussed next.

Cost of Debt

The rate on debt is more straightforward than the equity rate. The debt rate is stipulated when the firm enters into borrowing agreements. For example, if the firm always borrows at the prime rate, this may be a good estimate of the debt rate. The situation becomes only slightly more complicated when the firm has more than one source of debt.

If the company has borrowed from banks at varying rates, a weighted average can be taken for a historical period, such as five years. The weights used are the amounts borrowed. When the company has borrowed in securities markets by issuing corporate bonds, however, the rate on these securities needs to be added to the debt rate calculation in the same weighted manner as before.

It is the norm in corporate finance to express the debt rate on an after‐tax cost basis.¹⁷ This reflects the tax benefits that are derived through the tax deductibility of interest payments. On an after‐tax basis, the cost of debt can be expressed as shown in Equation 7.15.

(7.15)

where:

• t = the firm’s tax rate
• r_d = the before‐tax debt rate
• r^t_d = the after‐tax debt rate

For public debt in the form of corporate bonds, these rates can be further adjusted to reflect the flotation costs.

Preferred Stock

Preferred stock typically pays a constant dividend and is more similar to debt than common stock. The cost of preferred stock can be expressed as shown in Equation 7.16.

(7.16)

where:

• d_p = the preferred stock dividend
• P_net = the net proceeds of the preferred stock offering after deducting flotation costs

Rate on Equity

The determination of the appropriate rate of return on equity is less straightforward. This is the rate of return that is expected by the firm’s equity investors. For publicly held companies, the historical return on equity is a readily available statistic. It tends to be higher for firms that are riskier. For newer firms, however, the rate of return on equity provided by securities markets may not be as useful. In such instances, it is valuable to consider the equity rates of proxy companies. Because these companies are in the same industry, are of similar size, and have similar risk characteristics as the injured company, the rate of return on equity for such companies can be used as a proxy for the injured company’s equity rate. The rate of return on equity may be different depending on whether the equity is derived from internal sources, such as retained earnings, or external ones. The cost of internal equity capital is often represented by the Gordon model, as shown in Equation 7.17.¹⁸

(7.17)

where:

• k_e = the cost of internal equity
• d₁ = the next period’s dividend
• P₀ = the stock price at time 0
• g = the growth rate of dividends

Using another approach, the appropriate risk‐adjusted return relationship can be approximated using the capital asset pricing model, or CAPM. This model generates betas to weight the difference between the market rate of return and the risk‐free rate. Betas are derived from a regression analysis of the historical movements of the individual firm’s stock return (r_i) and the excess market return (r_m – r_f). Betas reflect the variability of the stock’s return that can be explained by market movements. This is referred to as systematic risk. Systematic risk is that component of total risk that cannot be reduced through diversification. Each firm has its own beta, which is used to adjust the excess return (r_m – r_f) and reflect the firm’s unique risk characteristics. Using this model, the equity rate can be derived as shown in Equation 7.18.

(7.18)

where:

• k_e = the cost of equity
• r_f = the risk‐free rate
• β_i = the beta for company i
• r_m = the market rate of return

The capital asset pricing model, and betas in particular, have been the subject of criticism.¹⁹ The strength of the hypothesized positive, linear relationship between betas and security returns was challenged when Fama and French showed that the inclusion of other variables, such as size, significantly reduces the explanatory power of betas. They have expanded the usual CAPM into a three‐factor model, which includes other explanatory variables, such as firm size. Their results follow some other challenges to the capital asset pricing model. However, the CAPM has certainly not been discarded and continues to be a valuable tool in corporate finance.

Betas for many companies can be found in a variety of places – many of them free. Examples include various investment publications, such as Value Line Investment Survey, include betas in the collection of financial statistics they publish on the companies covered in their publication.²⁰ Another source is Yahoo Finance.

For privately held companies, the rate of return on equity is more difficult to establish. These statistics are not as readily available as they are for publicly held firms. The problem is even greater for the smaller private companies, which, since they are not encumbered by the requirement to meet expected returns for stockholders, tend to have higher “costs” than they otherwise would. The rates of return on equity for these businesses, therefore, are not comparable to those provided by securities markets. The rates for similar public firms can be used after the addition of a risk premium in order to account for the increased risk and illiquidity associated with many private firms. The expert needs to determine the appropriateness of this approach on a case‐by‐case basis.

Sources of Cost of Capital Data

The cost of capital for a company can be measured directly by determining the components in the company’s capital mix and the relevant rate that the company pays for each one of them. Using these data, weights for each component can be computed and applied to the relevant rates to arrive at a weighted average cost of capital. These data may be acquired directly from the company or from public filings that the company has generated.

Another source of cost of capital data is the Cost of Capital Quarterly yearbook.²¹ This data source includes capital costs, such as the cost of equity, and is organized by Standard Industrial Classification (SIC) code. Capital costs are available for approximately 300 different SIC codes. The data are organized by industry category rather than by company.

Morningstar/Ibbotson Associates culls these data from the Compustat data files. This is a data source published by Standard & Poor’s that has financial data, including financial statements, for a large number of public companies.

Cost of Capital as an Element of Damages

In one District of Columbia case, a plaintiff appealed a district court’s award of the costs of capital, which was applied to damages resulting from a breach of contract.²² The Court of Appeals indicated that such an award is appropriate in a breach of contract claim but not in a negligence claim. In its ruling, the court stated:

It remains unclear whether pre‐judgment interest (interest from the time of the tort to the date of court judgment) is available in a negligence action. Nonetheless, … in addition to finding Williams negligent, the District Court had found that Straight has breached its contract with Smoot. Further, because Williams agreed to indemnify Straight in full, the court did not error by including the costs of capital in the damage award assessed against Williams – whether or not the court allows a cost of capital award in a negligence action.

Cost of Capital in Public Utility Environment

Cost of capital is a fundamental concept in the public utility rate‐making process. Many hearings in which economic experts testified on the costs of capital have been held before various rate‐making commissions. The commissions typically meet to decide whether a given utility may receive an increase in its rates to reflect higher costs as well as a return on invested capital. Courts ruling on related issues have endorsed the use of a rate of return that is sufficient to allow a company to maintain its credit and attract capital.²³ This rate has been found to be one that affords a rate of return similar to other investments that have comparable risk levels.

A type of lawsuit that occurs in the construction industry but also elsewhere is a suit that is the result of a plaintiff’s loss. Construction contractors are often requested by purchasers of their services, such as other companies or governmental entities, to post a bond. The bond guarantees their performance and protects the purchasers from a variety of failures on the part of the contractor. Such bonds may be purchased by a surety. In making their assessment of the contractor’s bonding capacity, they tend to look at a limited number of financial variables. Paramount among them is the liquidity of the contractor as measured by the size of its working capital. Working capital is the difference between current assets and current liabilities. This is explained in the next excerpts from S.C. Anderson v. Bank of America.²⁴

Anderson claimed consequential damages, consisting of lost profits, based upon an alleged impairment of bonding capacity.

A contractor must furnish a performance bond if it is to be awarded a public works construction project. A bid bond is a document issued by a bonding company which is attached by the contractor to its bid. The bond represents to the project owner that the contractor is bondable, and if the contractor is awarded the job, the surety company will issue a performance bond covering the work.

A surety, such as Travelers, calculates a contractor’s bonding capacity by assessing, among other things, the contractor’s working capital. Working capital consists of current assets less allowed current liabilities. By mid‐1986, Anderson’s financial statements disclosed that the CPII and TAII receivables had not been paid for several months. This impacted Anderson’s working capital and, in August, 1986, resulted in a reduction in Anderson’s bonding capacity from an aggregate exposure of $10 million to an aggregate exposure of $5 million.

There are several ways to measure the damages that a plaintiff may suffer as a result of a loss of bonding capacity. One way is to measure the reduced amount of business the plaintiff experienced due to the lower bonding capacity. The lower capacity may prevent him from taking on projects beyond a certain combined level. Defendants should explore whether the loss of the bonding capacity is truly due to their actions or is the result of other factors, such as mismanagement by the contractor. Defendants should also be mindful that a lower total business volume does not necessarily imply lower overall profitability. Some companies can generate more total profits on a smaller business volume. Their pursuit of growth, however, may cause them to take on more business and dilute their overall profitability. It can also happen that the plaintiff is prevented from bidding larger jobs due to a lower bonding capacity but is able to bid a larger number of smaller jobs – some of which may not have the same bonding requirement of larger jobs. The defendant should compute the profitability margin by job for the plaintiff in order to assess the relative profit contribution of large versus small jobs.

Another factor that should be explored by defendants in cases over loss of bonding capacity is the cost of securing alternative bonding capacity. If alternative bonding capacity can be acquired at a higher price, then this difference in price may be one measure of damages instead of a lost profits analysis. If that is the case, then the issue simplifies into a higher costs of capital analysis as opposed to a lost profits exercise.

Computation of Prejudgment Interest in Patent Infringement Cases

The courts have not presented a unified position on how to compute the prejudgment return in patent infringement cases. Several rates have been found acceptable, ranging from money market rates to the prevailing prime rate during the infringement period.²⁵ However, there have been cases in which a plaintiff was able to demonstrate that rates in excess of the prime rate, which were the rates that it had to pay to obtain capital to stay solvent during the infringement period, were appropriate. In patent cases, prejudgment interest is applied to what are known as base damages. Base damages are the amount of damages that are awarded. These usually are a measure of lost profits or a reasonable royalty.

Risk Adjustment of Past Losses

Past losses are usually not known with certainty – they have to be estimated. Given this uncertainty, their estimation should reflect some risk adjustment process. The uncertainty associated with past losses, however, is markedly different from that of future losses. There are a number of important deterministic variables that influence the level of sales and profits that a firm can receive. These include factors such as the overall level of demand in the economy, the performance characteristics of particular types of products, consumer preferences, actions of competitors, and others. In making an ex‐post projection, some of these factors are already known. Indeed, the methodological framework presented in this book attempts to explicitly take into account such factors in the loss estimation process. Given the more complete nature of the information set that applies to past losses compared to future losses, there is a very different level of uncertainty associated with them. For future losses, many of the deterministic variables are unknown; for past losses, they are known historical values. To the extent that the historical loss estimation process already took the variation in the relevant risk factors into account, the past losses may already be risk‐adjusted through the use of the information set that is available as of the date of the analysis. If the expert explicitly attempted to do this in the estimation process, then no further risk adjustment of past estimated losses is necessary. If not, then some accommodation for such uncertainty needs to be made. The risk adjustment process for past losses presents a fertile area for cross‐examination when the expert has ignored this issue in estimating losses.

Build‐up Method

One of the more often cited methods of adding a risk premium to the combined to arrive at a risk‐adjusted rate uses the process shown in Equation 7.19.

(7.19)

where:

• R=risk‐adjusted return
• R_f=the risk‐free rate. This is usually based on the rate on long‐term U.S. Treasuries.
• R_m=the equity risk premium (ERP). This is the risk premium in excess of the risk‐free rate that is associated with equity investments. Specifically, it represents the expected return on a diversified portfolio of stocks in excess of the expected yield on government securities.
• R_s=the size premium. This is a premium based on the size of the company being evaluated. The smaller the company, the greater the risk premium. Much of the data on size premiums are based on market capitalization, which applies to public companies as opposed to closely held entities.
• R_fs=the firm‐specific risk. This is an additional premium that is based on the firm‐specific risk that each company possesses. It is often a function of more subjective factors, such as the expert’s judgment of the risk associated with the company and the specific monetary stream being evaluated.

Equity Risk Premium

The equity risk premium is additional return above the risk‐free rate, which is associated with equity‐based risks. This is shown in Equation 7.20.

(7.20)

where:

• RP = the equity risk premium
• Rm = the expected return of the stock market. This is usually measured by the S&P500 although some would argue that while the market is often measured in financial research using the S&P500, a broader based measure might be more useful.
• Rf = the expected risk‐free rate

For an extended time period, many experts simply use a 7.1% equity risk premium derived from Morningstar data. However, over time that premium may change as a function of market conditions, which affect the returns in equity markets as well as the changes in the risk‐free rate of return.

In valuation analysis, including analysis of the value of damages in litigated matters, the equity risk premium is an expected value. However, these expectations are, in part, based upon historical values such as the historical rates of return in equity markets. When historical values are higher than more current rates of return, one may conclude that there is a basis to believe that the equity risk premium can be too high.³⁶

Researchers have long known that using historical data on realized returns does not necessarily tell what the expected returns (ex‐ante), as opposed to ex‐post returns, were in the historical time period.³⁷ Some have tried to estimate such expected premiums. Arnott and Bernstein arrive at a 2.4% geometric average, which translates into a 4.5% arithmetic average – well below what was, in the past, an often‐used equity risk premium of approximately 7% from Morningstar.³⁸ Others, such as Fama and French, have used fundamental factors, such as expected dividend yield and expected earnings growth, to arrive at the conclusion that expected equity returns could not have been as high as historical realized returns.³⁹ They also found expected returns lower than realized returns. However, Roger Ibbotson, one of the founders of the data source that is now marketed by Morningstar, and Peng Chen responded to some of this critical research with a series of studies that attempted to trace expected returns to a variety of fundamental factors such as earnings, dividends, inflation, return on equity, performance of the economy, and other factors.⁴⁰ They arrived at a 6.1% arithmetic expected premium.

When assessing the expected risk premium, it is useful to consider that the risk that the risk premium seeks to reflect is the risk that the actual returns from an equity investment will be different from the expected value and, instead, will vary. So, variability, as measured by the variance around an expected value, can be judged two ways: (a) past data that reflects historical variation and (b) estimates of how that variability may differ in the future.

Using historical data, the equity risk premium will vary based upon the estimation time period. The Duff & Phelps estimates for the 20‐year period 1996–2015 revealed an arithmetic average of 5.28% and a geometric average of 3.38%.⁴¹ Over the 30‐year time period 1986–2015 they derived an arithmetic average of 6.07% and a geometric average of 4.38%.

Another way to get a sense of the expected equity risk premium is to survey financial managers. Graham and Harvey analyzed the surveys of chief financial officers (CFOs) that have been conducted quarterly from June 2000 through December 2017.⁴² The risk premium was defined as the difference between the expected 10‐year return on the S&P500 and the 10‐year Treasury bond. Admittedly, their response rate was low – in the 5%–8% range. Nonetheless, they got an average of 351 responses in each quarterly survey. In the December survey that got a premium of 4.41%, which was above the historical average of 3.64%.

Using CAPM with Morningstar Data

An alternative to directly using the Morningstar data is to employ the capital asset pricing model and the betas it provides to account for the equity risk premium. If one does so, one needs to recognize that some of the size effect may be accounted for in the betas. This requires a beta‐adjusted size premium to be used instead of the unadjusted size premiums that are otherwise utilized. Industry effects should also be incorporated into a correct beta. However, a firm‐specific premium may still need to be added. While theoretically useful, this process of arriving at a risk‐adjusted discount rate requires that the correct beta be selected. If one is evaluating a closely held company, a ready‐made beta may not exist; then one has to make a judgment about the extent to which betas from other publicly held companies apply.

Duff & Phelps Research on the Size Premium

Roger Grabowski and David King have conducted research on the relationship between size and risk. They and others have pointed out that ranking companies by market capitalization may not be as relevant a benchmark – especially when trying to arrive at risk premiums for closely held companies. They note that market value is a function of more than just a company’s size; it is also a function of the discount rate that is used to value the company by the market. In addition, market capitalization is just one way to judge a company’s size. Examples of other methods include revenues, profits, and assets. Grabowski and King provide different measures of size by which to gauge the size premium.⁴³ The expert can then compute an average size premium that is a function of the eight different measures of size that relate to the subject company.

Risk‐Adjusting the “Numerator” as Opposed to the “Denominator”

Most of the discussion of risk‐adjusting forecasted future lost profits is focused on making sure that the discount rate fully accounts for the risk of the forecasted value. One other way that risk can be accounted for is to “discount” the forecasted values that appear in the numerator. This is sometimes done in capital budgeting under what is known as the certainty equivalence method. Discounting using a fully adjusted discount rate in effect reduces the forecasted values to arrive at ones that are more certain. A similar effect can be derived from reducing the forecast to arrive at more conservative values, thereby attempting to account for the fact that the future values are uncertain. Experts need to be aware that if they have accounted for risk by reducing the “numerator,” they would not want to fully account for risk again by using it for the full risk premium.

One of the benefits of using risk premiums embedded into the discount rate is that they are usually derived from more objective, market‐based data. Adjustments by the expert to the numerator may be more ad hoc and subjective. If the experts base these adjustments on a considered study of the subject company and the forecast itself, however, then the resulting adjustment may be reliable. Each case is different, and the process needs to be decided on a case‐by‐case basis.

Discounting with Nominal Versus Real Rates

There has been much debate in the forensic economics literature regarding whether discounting should be done using nominal or real interest rates.⁴⁴ Much of this debate has centered around personal injury litigation. However, many of the same principles apply to the measurement of commercial damages. Therefore, it is instructive to explore this debate and the court’s position on it.

One of the simplest ways of projecting future earnings and then discounting them to the present is simply to assume that the rate of earnings growth is approximately equal to the interest rate used to discount the future earnings. The process of discounting then becomes quite simple: one simply needs to take the length of the loss period and multiply it by the annual loss in the first year of the loss. The result is an inflation‐adjusted and discounted loss. This method is referred to as the total offset method. The courts have acknowledged the mathematical validity of using a net discount rate, one that deducts the inflation rate from the discount rate so that discounting is done using a real or inflation‐adjusted rate. However, the courts have not accepted the simplistic total offset method.⁴⁵ Given that it can be shown that interest rates have been generally above the rate of inflation, it is difficult to justify using the total offset method.

The decision of whether to use a nominal or a real discount rate depends on how the inflation adjustment is done in the projection. If revenues and lost profits are projected using an annual inflation adjustment, then a nominal interest rate should be used to discount the projected amounts to present value terms. Only when the projection is not adjusted for the effects of inflation would it make sense to use a real rate in the discounting process. Given that the methodology described in this book specifically incorporates inflation and growth into the projection process, nominal rates rather than real rates are used for discounting.

Court Position on the Appropriate Risk Premium

The process of discounting to present value has been well received and accepted by the courts, and there is abundant legal precedent to confirm this.⁴⁶ In fact, the Delaware Chancery Court has characterized the discount cash flow method of valuation, an application of discounting that is discussed in Chapter 8, as the preferred method of valuation.⁴⁷ Furthermore, Robert Dunn has observed that in the few instances in which the courts have refused to recognize that only discounted profits should be allowed, the defendant had failed to introduce proof as to what such discounted values would be.⁴⁸ Thus, at a minimum, a defendant should be mindful of the need to prove this issue to the court.

We have explored the reasoning behind all of the components that should be incorporated in the risk premium so as to arrive at a discount rate that fully reflects the risk of the projected future lost profits stream. Some courts, though, have been reluctant to accept a discount rate that incorporates all of this risk. For example, in American List Corp. v. U.S. News & World Report, the trial court accepted an 18% discount rate based on the perceived risk associated with the plaintiff’s ability to fulfill the contract in the future.⁴⁹ However, the New York Appeals Court rejected the 18% discount rate as too high and remanded for a recomputation of this discount rate. In stating its reasoning, the court asserted:

Defendant argues that in discounting the total amount due under the contract to its present value, the court may factor in the risk that the nonrepudiating party will be unable to perform the contract in the future. Such a rule, however, does violence to the settled principles of the doctrine of anticipatory breach because it would require the nonrepudiating party to prove its ability to perform in the future, despite the fact the doctrine is intended to operate to relieve the nonrepudiating party from that very performance.⁵⁰

One court was reluctant to accept a higher risk premium. However, this should not preclude the expert from applying standard financial principles to construct a risk‐adjusted discount rate that fully reflects the risk of the projected stream of lost profits.

Defense Objections for Plaintiff’s Discount Rate Assumptions

A defendant may believe that the plaintiff’s lost profits projection is too high because the discount rate he is using is too low, thereby causing the future projected lost profits to be insufficiently discounted. This may be the case if the risk premium incorporated into the discount rate is too small. This situation can come about due to an overly optimistic assessment by the plaintiff with regard to the projected likelihood of future profitability.

If the defendant disagrees with the plaintiff’s discount rate assumptions, it must put forward its reasoned objections at trial; it cannot rely on a posttrial appeal process to air objections. This position was clearly articulated by the U.S. Court of Appeals for the Fifth Circuit in Lehrman v. Gulf Oil Corp.⁵¹ Here the court said if the defense does not put forward its position on plaintiff’s discount rate at trial, it cannot raise an issue with the plaintiff’s assumptions on appeal.

Gulf also claims that the jury should have been instructed to award only the present value of lost future profits, and we agree that this would be the better practice. But we do not find this issue raised in Gulf’s objection to the court’s charge.

The defense can challenge the plaintiff’s discount rate assumptions either through cross‐examination or through its own damages expert. One of the problems with trying to do this by cross‐examination is that counsel has to try to get the plaintiff’s expert to concede that his rate is too low or to try to get a jury to understand that it is too low. The latter may be difficult without a more complete explanation of the reasoning, which may be best provided by testimony produced by the defendant through its own damages expert. Such testimony would address the expected variability of the projected stream of lost profits and would try to relate this to the variability of returns on securities in the marketplace as well as other proxies.

Process of Capitalization and the Loss of an Indefinite Stream of Future Profitability

Assuming it is legally established that the plaintiff should be compensated for the loss of a stream of future profitability of indefinite length, it is possible for the expert to value such a stream using the process of capitalization. Applied to commercial damages analysis, this process values a continuous stream of projected future profits. This valuation is done by dividing the growth‐adjusted capitalization rate into the next period’s projected lost profits. The growth adjustment refers to the projected rate of growth of the profit stream. If, for example, the selected discount rate is 15% and the annual profit stream is projected to grow at a 5% rate, then the growth‐adjusted capitalization rate equals 10%. Mathematically, dividing by 0.10 is the same as multiplying by 10. Similarly, using a growth‐adjusted capitalization rate of 20% is equivalent to multiplying by 5. For each rate that is used as a divisor, there is an equivalent multiplier that is implied by the divisor. The capitalization process is shown in Equation 7.21.

(7.21)

where:

• k_i = the capitalization rate for firm i prior to growth adjustment
• g = the growth rate of future annual earnings

Difference Between a Capitalization Rate and a Discount Rate

When one forecasts specific future monetary amounts, such as cash flows, and then uses a discount rate to convert each of these amounts to present value terms, it is known as discounting. The rate used to convert the future amounts to present value terms is the discount rate. Continuous monetary streams that are growing at a certain positive rate or not growing at all are called perpetuities. Computing the present value of such a stream is called capitalization. The interest rate used for this present value conversion of a perpetuity is called a capitalization rate. Sometimes these terms are used interchangeably, but such usage is generally incorrect.

Using Capitalization in a Business Interruption Loss Analysis

This book discusses two ways of converting future loss amounts to present value terms. The first requires projecting specific lost profits into the future and then discounting them to present value terms using the relevant discount rate. The second is to stipulate a lost profits value for a forthcoming year and to assume that this earnings stream would continue indefinitely. If this occurs, the present value of such a stream can be computed by capitalizing it or, stated another way, by dividing it by the relevant capitalization rate. While the latter method sometimes is used to value businesses, especially closely held businesses, it is not generally used as the sole method of computing the present value of a lost profits stream. One of the reasons for this is the implied length of the earnings stream. In order to use capitalization, one has to assume that the lost profits stream would have continued indefinitely. A court has to accept the length of this earnings stream. If a court will only accept a more limited loss period, then one must project specific annual losses for the relevant loss period and then discount them accordingly.

Sometimes the capitalization process may be used in conjunction with discounting. For example, assume that a business interruption has occurred two years prior to the trial date and the expert has projected losses for an additional three years into the future. The question may arise: What about losses after that third year? Would the plaintiff have been able to generate profits after the third year? Is it reasonable to conclude that the loss period would extend into the future indefinitely? If there is a case‐specific reason why losses should be projected for three more years, such as that was the remaining length of a contract, then those losses may be projected separately. If, however, the plaintiff argues that it has lost an income stream of indefinite length, then this value can be shown by capitalizing the losses.

Unless there are case‐specific reasons for extending losses for a period of indefinite length, the capitalization analysis may not be relevant. It is a calculation that may not necessarily be relevant depending on the facts of the case and the relevant law. Experts should make their retaining attorneys understand the nature of a capitalization calculation. This calculation assumes that the losses will go on for an infinite time period. This sounds like a calculation that results in an extremely high value solely due to its infinite length. In reality, however, the present value process that is implicit in the capitalization computation converts values far into the future into very low values. However, some of the pragmatic issues that have to be dealt with are the legal responsibility of the defendant for losses of such an infinite length in light of the plaintiff’s obligation to mitigate its losses as time goes by. These issues may make the capitalization of losses calculation irrelevant to a case.

Statistical Stability of Capitalization Rates

One issue that affects the reliability of capitalization rates is the statistical stability of these rates. The issue of statistical stability is discussed in Chapter 5, where the concept of stationarity was introduced. Nonstationary statistical series lack certain properties that affect the reliability of forecasts upon which they are based. This issue is relevant to capitalization rates if such rates are based on historical data, as is often the case. Bowles and Lewis investigated the time series properties of capitalization rates with an eye toward assessing the reliability of such rates as they are used in a litigation context.⁵² They wanted to determine if the historical mean of a capitalization rate series was covariance stationary. If it was, they concluded that it would be appropriate for present value calculations. They did this research using aggregated macroeconomic data, where g was the quarterly change in after‐tax profits in the economy and r was the rate of return of the S&P500. They applied time series tests to these two series and a combined capitalization rate series based on the difference between r and g. Their research allowed them to reject a null hypothesis of a unit root for all of these series, leading them to conclude that capitalization rates are stationary and therefore useful to present value calculations.

The Bowles and Lewis research is useful, but readers must bear in mind that it is based on broad‐based macroeconomic data. The capitalization rate that an expert uses in a particular case, however, would presumably be based on a firm‐specific g, which might not have the same statistical properties.