Discount and capitalization rates are both used to value businesses, particularly those that are closely held The terms are related, and the author explains the theories underlying their use and provides illustrations on how they are developed.

Discount and capitalization rates is probably the most important, difficult, and misunderstood area of valuing closely held businesses. Businesses are typically valued on the basis of income (see Idelle A. Howitt, "Valuing Closely Held Stock." The CPA Journal, September 1993, pp. 44-47, 91). Discount and cap rates arc critical to the final value estimate. However, there is much confusion about how to develop and use these rates in business valuations.

How Discount and Cap Rates Are Used

Discount and cap rates are used to convert some measure of income into an estimate of value. The measure of income can be various forms of cash flow or earnings. However, the discount or cap rate and the measure of income must be compatible, e.g., an after-tax discount rate should be applied to after-tax income.

Cap Rates. In the capitalization-of-income method of valuing a business, a cap rate is used to convert a single year income amount into a value estimate for the business as a whole. This method is appropriate when future income is expected to grow at a constant rate. Valuation theory requires that next year's income be capitalized, as the value of a business is based on expectations of future income. For example, assume a valuation date of December 31, 1993, estimated income for 1994 of $100,000, and a cap rate of 20%. The value of the business would be estimated at $500,000 as follows:

$100,000/.2 = $500,000.

Earnings multiples or capitalization factors are simply the reciprocal of cap rates. A 20% cap rate is the same as an earnings multiple of five times. In the above example, capitalizing the $100,000 earnings at 20% is the same as valuing the business at five times earnings.

Discount Rates. A discount rate is used in the discounted future income method of valuing a business. This method is appropriate when income is expected to grow at varying rates in future years. As shown in Exhibit 1, income is projected into the future for a number of years. The assumption in this example is that income will grow at 10% for the next five years then level off at five percent. At the end of year five, a terminal value is calculated by capitalizing year six income at 20% (why the cap rate is 20% is explained below). The income for each year and the terminal value are discounted to present value using a 25% discount rate. The present value amounts are added to arrive at an indicated value for the business.

Relationship Between Discount Rate and Cap Rate

The value of an asset is the present value of all future cash flows the asset is expected to provide. The future cash flows are discounted to present value using the required rate of return for the asset. The required rate of return depends on the riskiness of the asset. Greater risk requires a higher rate.

In financial theory there are three basic valuation models: no growth, constant growth, and variable growth. These models generally assume an unlimited asset life with a perpetual income stream.

The value of an asset using the no-growth model is simply the annual income divided by the required rate of return. As in our previous example, the value of an asset with a required rate of return of 20% and annual income of $100,000 is $500,000.

In the constant-growth model, the estimated long-term growth rate of future income is subtracted from the required rate of return. Using the same income and rate of return as above, assume that future income is expected to grow at 5% per year. The value of the asset is $666,667:

$100,000/(.2 - .05) = $666,667.

The discounted future income method illustrated in Exhibit 1 is an example of the variable-growth model. The discount rate of 25% is the required rate of return. The terminal value is calculated by using the constant-growth model to capitalize year six income. Income is expected to grow indefinitely at 5% after year five.

In valuation theory, a discount rate represents the required rate of return in an asset-valuation model. There is no difference between a discount rate and a cap rate when future income is not expected to grow. When future income is expected to grow at a constant rate, the cap rate is equal to the discount rate minus the expected future growth rate. The capitalization of income method and the discounted future income method will produce the same result where income is expected to grow at a constant rate if the cap rate is calculated correctly. Exhibit 2 shows how these two methods produce the same result where income is expected to grow at a constant 5% and the required rate of return (i.e., discount rate) is 25%. Note that the correct cap rate is 20% (25% - 5%).

Calculating the Discount Rate

A cap rate can be defined as a discount rate minus the expected long- term growth rate of future income. Therefore, to calculate a cap rate, one must first calculate a discount rate. According to current valuation theory, a discount rate is composed of two elements:

1. The current risk-free rate of return.

2. A risk premium appropriate to the asset in question.

This approach to discount rate determination is generally referred to as the buildup approach. The rate is built up by starting with the current risk-free rate and adding one or more components of risk appropriate to the business to be valued. The risk-free rate is determined by reference to U.S. government securities. The long-term Treasury bond is most commonly used. However, some appraisers use short- or intermediate-term government securities. The choice depends on the appraiser's assumption regarding the life expectancy of the business to be valued. Although there is some variation in technique and terminology, three basic methods are commonly used to calculate risk premiums.

Comparable Public Companies Method. This method is frequently referred to as the capital asset pricing model (CAPM) method. However, all three of the build-up methods are to some extent based on CAPM theory. CAPM is a basic theory that relates risk and return for any asset. It is based on the concept that the required rate of return for an asset is directly related to the riskiness of the asset. Greater risk requires a higher rate of return. CAPM measures risk in terms of the relative volatility of the asset price.

A risk premium can be developed from specific public company data if comparable public companies can be located. Using this approach the discount rate is calculated as follows:

DR = RFR + B(ER) + CS

where:

DR = discount rate RFR = risk-free rate B = beta ER = equity risk premium CS = company-specific risk factor

Beta is a measure of stock price volatility relative to the overall market. If a stock, say ABC Inc., tends to move up or down proportionately to the overall market, it has a beta of 1. If the stock movements are proportionately greater than the overall market, the stock has a beta greater than 1. If the stock movements are proportionately less, the stock has a beta less than 1. In CAPM theory, beta is a measure of risk. A stock with a beta of 1.5 is considered riskier than a stock with a beta of 1.

Beta factors for public companies can be found in stock market services such as Value Line Investment Survey (Value Line Publishing, Inc.). Business appraisers generally use the median beta for the comparable companies.

The equity risk premium is the additional return required over the risk- free rate for investing in the public stock market. It is calculated by subtracting the historical risk-free rate from the historical return on common stocks. This information can be found in Stocks, Bonds, Bills and Inflation (SBBI) by Ibbotson Associates which is published annually. SBBI uses the S & P 500 to measure returns on common stocks. When ER is multiplied by B, the equity risk factor is adjusted for the price volatility of the comparable companies. Thus, B(ER) should contain the risk factors associated with the comparable companies.

EXHIBIT1

USINGTHEDISCOUNTEDFUTUREINCOMEMETHODTOESTIMATETHEVALUEOFABUSINESS

25%Present

ForecastedValuePresent

IncomeFactorValue

Year1$100,0000.80000$80,000

Year2110,0000.6400070,400

Year3121,0000.5120061,952

Year4133,1000.4096054,518

Year5146,4100.3276847,976

Terminal

Value768,653*0.32768251,872

ValueEstimate$566,718

*TerminalValue:

Year5Income$146,410

GrowthFactorx1.05

Year6Income=$153,731

CapRate(.25-.05)/0.20

=$768,653

EXHIBIT2

RELATIONSHIPBETWEENDISCOUNTRATESANDCAPRATES

CAPRATE=DISCOUNTRATE-FUTUREGROWTH

(A)and(B)belowproducethesameresultifthecaprateiscalculated

correctly

(A)discountedfutureincomemethodassumingconstant5%growthofincomeand

discountrateof25%

25%Present

ForecastedValuePresent

IncomeFactorValue

Year1$100,0000.80000$80,000

Year2105,0000.6400067,200

Year3110,2500.5120056,448

Year4115,7630.4096047,416

Year5121,5510.3276839,830

Terminal

Value638,141*0.32768209,106

ValueEstimate$500,000

*TerminalValue:

Year5Income$121,551

GrowthFactorx1.05

Year6Income=$127,628

CapRate(.25-.05)/0.20

=$638,141

(B)Capitalizationofincomemethodassuminga5%growthrateand25%discount

rate

Discountrate-Futuregrowthrate=Caprate

Year1income/caprate=Valueestimate

$100,000/(.25-.05)=$500,000

EXHIBIT3

USINGCOMPARABLEPUBLICCOMPANIESTOCALCULATEADISCOUNTRATE

Beta

Factor

COMPCO11.50

COMPCO21.30

COMPCO31.60

COMPCO41.40

COMPCO51.70

MedianBeta(B)1.50

Long-termtreasurybond

yieldatvaluationdate(RFR)0.07

Equityriskpremiumper

1993SBBI(ER):

Historicalreturnon

commonstocks0.124

Historicalreturnonlongterm

govtbonds-0.052

0.072

Company-specificriskfactor(CS)0.07

Discountrate=RFR+B(ER)+CS

=.07+1.5(.072)+.07

=.248

Roundedto.25

EXHIBIT4

USINGSMALLPUBLICCOMPANIESTOCALCULATEADISCOUNTRATE

Long-termtreasurybondyield

atvaluationdate(RFR)0.07

Equityriskpremiumper

Exhibit3(ER)0.072

Smallstockriskfactor

per1993SBBI(SR):

Historicalreturnon

smallcompanystocks0.176

Historicalreturnon

commonstocks-0.124

0.052

Company-specificriskfactor(CS)0.06

Discountrate=RFR+ER+SR+CS

=.07+.072+.052+.06

=.254

Roundedto.25

The company-pecific risk factor represents any additional risk not associated with the comparable public companies. This factor must be estimated by the appraiser. However, it should be based on his or her analysis of the company as well as his or her subjective judgment. The company-specific risk factor should consider the following at a minimum, relative to the comparable public companies:

* Size of the company

* Depth of management

* Financial structure

* Product diversification

* Reliance on major customers

* Reliance on major suppliers

* Geographical diversity of customers

* Length of time in business

* Pattern of earnings

* Economic factors of special concern to the company

Exhibit 3 illustrates how a discount rate would be calculated using the comparable public companies method.

Small Public Companies Mehod. This method is very similar to the comparable public companies method. It may be used when no comparable public companies can be found. With this method, the discount rate is calculated as follows:

DR = RFR + ER + SR + CS

SR is an additional risk premium associated with small public companies. It is calculated by subtracting the historical return on common stocks from the historical return on small company stocks. Returns on small stocks can also be found in SBBI.

The company-specific risk factor would be determined as above except the comparison would be to small public companies in general. Therefore, an industry risk element would also be considered. Exhibit 4 illustrates how a discount rate would be calculated using this method.

Risk Premium Scale Method. This method was developed by James H. Schilt, a well-known business appraiser. Based on his experience as a business appraiser and financial analyst, Schilt developed a risk premium scale for closely held businesses.

As shown in Exhibit 5, the scale has five categories. Each category has a range of rates. A business would be positioned on the scale according to the category descriptions. The appropriate rate would then be selected from the category range based on the appraiser's analysis of the company and his judgment.

The risk premium selected from the scale is added to the risk-free rate to form the discount rate. Therefore, all risk factors over the risk- free rate are embodied in this risk premium.

Matching the Discount Rate to the Income Stream

The measure of income to be discounted or capitalized depends on how the discount rate is calculated. Schilt clearly states his risk premium scale is intended to be used with pretax net income.

For the other two methods, net cash flow is the theoretically correct income stream. The discount rate in both methods is based on net cash returns in the public stock market. In practice, however, net income (after tax) is frequently used to discount multiple years and is generally used when a single year is capitalized. Some appraisers feel the discount rate should be adjusted when net income is used because net income will generally be greater than net cash flow.

For a growing business, net cash flow is typically projected as follows:

Net income + Depreciation and amortization - Cash needed for asset replacement - Increase in working capital needed as the business grows + Increase in debt to finance business growth.

Cash needed for asset replacement will generally exceed depreciation and amortization. Therefore, when net income is used as the income stream, an upward adjustment should be made to the discount or cap rate. This adjustment should be proportionate to the ratio of net income to net cash flow. For example, assume net income is $60,000 and net cash flow is $50,000. Also assume the net cash flow discount rate is 20%. The net income discount rate would be 24% as follows:

$60,000/$50,000 = 1.2 20% x 1.2 = 24%

Other appraisers feel the difference between net cash flow and net income, in the long run, is usually small and can generally be ignored or included in the company-specific risk factor.

Net of Debt Assumption

The methodology discussed above for calculating discount rates assumes only equity capital is being valued. This net-of-debt approach is the most common approach to valuing closely held businesses. It assumes the firm's long-term debt will remain in place. If the firm has no long-term debt, this assumption is not an issue and the above methodology will apply.

There are cases, however, where the net-of-debt assumption is not appropriate. For example, the valuation may contemplate a specific buyer who will not rely on long-term debt. In such cases, a TABULAR DATA OMITTED debt-free approach should be used. In the debt-free approach, the firm's weighted average cost of capital (WACC) should be used as the discount rate. The WACC is calculated by taking a weighted average of the firm's cost of equity capital and cost of debt. The cost of equity capital is the discount rate calculated using one of the build-up methods discussed above. The cost of debt is the after-tax rate of interest on the firm's debt. For example, assume the following:

Discountrateperbuild-up

method25%

Interestrateondebt10%

Marginaltaxrate40%

Debttoequityratio1to2

The after-tax cost of debt is .1(1 - .4) or 6%. The WACC is 18.7% calculated as follows:

Weighted

CostWeightCost

Equity.252/3.167

Debt.061/3.020

WACC.187

Control Premium and Marketability Discount

Discount and cap rates based on data from the public stock market result in valuations considered to be marketable minority interest. Stock market data are based on transactions for minority interests trading in a ready market. If a controlling interest is being valued, the final value should be adjusted for a control premium and a marketability discount. A controlling interest in a business is generally worth more than a minority interest. A readily marketable business is generally worth more than one that is not readily marketable. These adjustments are made to the final value estimate for the business, and discount and cap rates are not adjusted for these factors. If a minority interest is being valued, a premium for control is not necessary.

Of the three methods discussed above for calculating discount rates, the first two are based on public stock market data. The risk premium scale method is not based on the public stock market and thus control premiums and marketability discounts are not needed. However, if a minority interest is being valued, a discount for lack of control is needed with the risk premium scale method.

APPENDIX 1: USING IBBOTSON ASSOCIATES DATA

Stocks, Bonds, Bills and Inflation (SBBI) has become a standard reference source for business appraisers. It has been published annually since 1983. The 1993 edition contains 219 pages.

The book contains tables, charts and narrative regarding capital returns and inflation in the U.S. since 1926. According to the authors, the intended audience is anyone serious about investments. They specifically mention practitioners and scholars in finance, economics and business; portfolio strategists; and security analysts. Chief financial officers are also mentioned. Business appraisers are not mentioned. Thus, SBBI is not written specifically for business appraisers.

Return data are presented for the following securities:

* Common Stocks

* Small Company Stocks

* Long-term Corporate Bonds

* Long-term Government Bonds

* Intermediate-term Government Bonds

* U.S. Treasury Bills

Data are also presented for the Consumer Price Index.

In SBBI, annual returns on stocks are composed of capital appreciation returns, income returns and reinvestment returns. For example, in 1992 the total annual return on common stocks was 7.67%. Appreciation was 4.46%, income was 3.03% and the return for reinvestment was .18%. Income refers to dividend payments and reinvestment refers to reinvesting the dividends during the year.

Returns on common stocks are calculated using the S & P 500 index. Returns on small company stocks are calculated using the DFA Small Company Fund, a mutual fund that invests in over 2,000 small public companies.

Returns are shown for each year since 1926. Cumulative returns are also shown for all holding periods since 1926. For example, the annual return on common stocks from the beginning of 1982 to the end of 1992 was 16.7%.

A business appraiser should not take an uninformed or mechanical approach in using SBBI. He or she should have some understanding of how the data series are constructed and resolve several issues that do not have clear cut answers.

What Time Period to Use?

A business appraiser must decide what time period to use in determining the rate of return on public company stocks. The SBBI authors favor using the period from 1926 to present because that period includes all of the types of events that affect stock prices. It includes wars, depression and inflationary periods. However, a more recent period may be a better prediction of the future. Using the past twenty years, for example, would include several economic cycles and be more representative of the current economy. An argument could also be made for using the post war period of 1946 to present to eliminate the great depression and World War II.

Arithmetic or Geometric Mean?

SBBI shows rate of return data based on both arithmetic and geometric means. The appraiser must decide which mean to use. The arithmetic mean is a simple average of the rates of return for each year. The geometric mean is based on compounding and is generally less than the arithmetic mean. For example, the 1926-1992 annual rate of return for common stocks was 12.4% based on an arithmetic mean and 10.3% based on a geometric mean. The authors recommend using the arithmetic mean because investors tend to use arithmetic means in forming their expectations of future returns. However, most financial decision models are based on compounding.

Is the Small Company Risk Premium Valid?

The small company stock series presents another issue for the appraiser to resolve. For the 1926-1992 period the small stock return exceeded the return on the common stock series. In terms of arithmetic means, the small stock return was 17.6% compared to 12.4% for common stocks. Business appraisers commonly use this 5.2% difference as a small stock risk premium in developing their discount rates. However, small stock returns do not exceed common stock returns for all past periods. For example, for the 1982-1992 period, the rate of return on small stock was 13% compared to 16.7% for common stocks.

Thus, SBBI should be used carefully. The appraiser should understand the numbers being extracted from this source and be prepared to explain how the business valuation issues discussed above were resolved.

APPENDIX 2: WHERE TO FIND INFORMATION ON DISCOUNT AND CAP RATES

Literature on Discount and Cap Rates

Fishman, Pratt, Griffith and Wilson, Guide to Business Valuations (Practitioners Publishing Company, 1993), Chapter 5

Shannon Pratt, Valuing a Business: the Analysis and Appraisal of Closely Held Companies 2nd ed. (Dow Jones-Irwin, 1989), Chapter 8

Gregory A. Gilbert, "Discount Rates and Capitalization Rates - Where are We?" Business Valuation Review, 1990, pp. 108-113

William P. Dukes and Oswald D. Bowlin, "A Comparison of Valuation Techniques for Closely Held Firms," Business Valuation Review, June 1993, pp. 80-91

CAPM Theory

Any book on financial management or corporate finance. For example: George E. Pinches, Essentials of Financial Management, 3rd ed. (Harper and Rowe, 1990), Chapter 7

Yields on Government securities

Wall Street Journal (Dow Jones Co.), published daily

U.S. Financial Data (Federal Reserve Bank of St. Louis), published weekly

Federal Reserve Bulletin (Board of Governors of the Federal Reserve System), published monthly

Beta Factors

Value Line Investment Survey (Value Line Publishing, Inc.)

Equity Risk Premiums

Stocks, Bonds, Bills and Inflation, 1993 Yearbook: Market Results From 1926 - 1992 (Ibbotson Associates), published annually

Randy Swad, PhD, CPA, is professor of accounting at California State University, Fullerton.