# CAPITALIZATION/DISCOUNT

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3 Fundamentals, Techniques & Theory CAPITALIZATION/DISCOUNT RATES Once the analyst selects the appropriate type of earnings and estimates the amount of future earnings, an appropriate capitalization or discount rate must be determined. This rate is applied to the amount of estimated future earnings calculated. A capitalization rate is applied in a capitalization process to calculate value and a discount rate is applied in a discounting process to calculate value. For clarity, the rates are defined as follows: 1. Discount rate: A rate of return used to convert a series of future income amounts into their present value. 2. Capitalization rate: A divisor (or multiplier) used to convert a defined stream of income to a present indicated value. It is generally accepted in the valuation community that subtracting a company s expected longterm sustainable growth rate from its discount rate yields the capitalization rate. C. CAPITALIZATION VS. DISCOUNTING A distinction between the capitalization process and the discounting process is the utilization of a terminal value. Recall that the discounting process calculates the present value of a series of forecasted future benefits. Forecasts are made for a finite number of future periods. Thus, when valuing a company using a discounting process, the analyst must consider terminal values. The terminal value represents the value of a company in the terminal year of an earnings forecast, or what the company will be worth in x number of years. There are several methods of estimating terminal value, including price/earnings and other multiples. The most frequently used method is to capitalize terminal year earnings using an appropriate capitalization rate and then discount the results back to a present value. Recall that the capitalization rate is equal to the discount rate minus the projected growth rate. Thus, the discount rate and the capitalization rate are interchangeable only when there is no projected growth in the benefit stream. The following exhibits show the relationship between discounting and capitalizing future benefits under three future benefit assumptions: 1. The future benefit stream is linear and there is no growth. 2. The future benefit stream is linear but growing at a constant rate. 3. The future benefit stream reflects nonlinear growth by National Association of Certified Valuators and Analysts (NACVA). All rights reserved. Chapter Five 3 Used by Institute of Business Appraisers with permission of NACVA for limited purpose of collaborative training v1

4 CAPITALIZATION/DISCOUNT RATES Fundamentals, Techniques & Theory Assumptions Annual Benefits \$100,000 Discount Rate 20% Growth 0% Capitalization Rate 20% End of period discounting convention Linear Benefits Capitalization Rates vs. Discount Rates No Growth DISCOUNTING Year Annual Benefits \$100,000 \$100,000 \$100,000 \$100,000 \$100,000 Discount Factor Discounted Benefits 83,333 69,444 57,870 48,225 40,188 Year Annual Benefits \$100,000 \$100,000 \$100,000 \$100,000 \$100,000 Discount Factor Discounted Benefits 33,490 27,908 23,257 19,381 16,151 Year Annual Benefits \$100,000 \$100,000 \$100,000 \$100,000 \$100,000 Discount Factor Discounted Benefits 13,459 11,216 9,346 7,789 6,491 Year Annual Benefits \$100,000 \$100,000 \$100,000 \$100,000 \$100,000 Discount Factor Discounted Benefits 5,409 4,507 3,756 3,130 2,608 Year Annual Benefits \$100,000 \$100,000 \$100,000 \$100,000 \$100,000 Discount Factor Discounted Benefits 2,174 1,811 1,509 1,258 1,048 Year Annual Benefits \$100,000 \$100,000 \$100,000 \$100,000 \$100,000 Discount Factor Discounted Benefits Year Annual Benefits \$100,000 \$100,000 \$100,000 \$100,000 \$100,000 Discount Factor Discounted Benefits Sum of the Discounted Benefits (rounded) \$499,200 CAPITALIZING METHODOLOGY Annual Benefits \$100,000 Capitalization Rate 20.0% Capitalized Benefits \$500,000 Conclusion: This calculation demonstrates that, with no growth, the capitalization process produces the same result as the discounting process. 4 Chapter Five by National Association of Certified Valuators and Analysts (NACVA). All rights reserved v1 Used by Institute of Business Appraisers with permission of NACVA for limited purpose of collaborative training.

5 Fundamentals, Techniques & Theory CAPITALIZATION/DISCOUNT RATES Linear Benefits Capitalization Rates vs. Discount Rates With Growth Assumptions Annual Benefits \$100,000 Discount Rate 25% Growth 5% Capitalization Rate 20% End of period discounting convention DISCOUNTING Year Annual Benefits \$100,000 \$105,000 \$110,250 \$115,763 \$121,551 Discount Factor Discounted Benefits 80,000 67,200 56,448 47,417 39,830 Year Annual Benefits \$127,629 \$134,010 \$140,711 \$147,747 \$155,134 Discount Factor Discounted Benefits 33,457 28,104 23,607 19,830 16,657 Year Annual Benefits \$162,891 \$171,036 \$179,588 \$188,567 \$197,995 Discount Factor Discounted Benefits 13,992 11,754 9,873 8,293 6,966 Year Annual Benefits \$207,895 \$218,290 \$229,205 \$240,665 \$252,698 Discount Factor Discounted Benefits 5,852 4,915 4,129 3,468 2,913 Year Annual Benefits \$265,333 \$278,600 \$292,530 \$307,157 \$322,515 Discount Factor Discounted Benefits 2,447 2,056 1,727 1,451 1,218 Year Annual Benefits \$338,641 \$355,573 \$373,352 \$392,020 \$411,621 Discount Factor Discounted Benefits 1, Year Annual Benefits \$432,202 \$453,812 \$476,503 \$500,328 \$525,344 Discount Factor Discounted Benefits Sum of the Benefits (rounded) \$498,900 CAPITALIZING Annual Benefits \$100,000 Capitalization Rate 20% Capitalized Benefits \$500,000 Conclusion: This calculation demonstrates that, with linear growth, the capitalization process produces the same result as the discounting process by National Association of Certified Valuators and Analysts (NACVA). All rights reserved. Chapter Five 5 Used by Institute of Business Appraisers with permission of NACVA for limited purpose of collaborative training v1

6 CAPITALIZATION/DISCOUNT RATES Fundamentals, Techniques & Theory Nonlinear Growth Capitalization Rates vs. Discount Rates Assumptions Year 1 Year 2 Year 3 Year 4 Year 5 Base Benefits \$100,000 \$115,000 \$143,750 \$155,250 \$186,300 Growth 15.0% 25.0% 8.0% 20.0% 3.0% Annual Benefits \$115,000 \$143,750 \$155,250 \$ \$191,889 Discount Rate 25.0% 25.0% 25.0% 25.0% 25.0% Growth 15.0% 25.0% 8.0% 20.0% 3.0% Capitalization Rate 10.0% 0.0% 17.0% 5.0% 22.0% End of period discounting convention DISCOUNTING Year Annual Benefits \$ 115,000 \$ 143,750 \$ 155,250 \$ 186,300 \$ 191,889 Discount Factor Discounted Benefits \$ 92,000 \$ 92,000 \$ 79,488 \$ 76,308 \$ 62,878 Year Annual Benefits \$ 197,646 \$ 203,575 \$ 209,682 \$ 215,972 \$ 222,451 Discount Factor Discounted Benefits \$ 51,812 \$ 42,693 \$ 35,179 \$ 28,987 \$ 23,885 Year Annual Benefits \$ 229,125 \$ 235,999 \$ 243,079 \$ 250,371 \$ 257,882 Discount Factor Discounted Benefits \$ 19,682 \$ 16,218 \$ 13,363 \$ 11,011 \$ 9,073 Year Annual Benefits \$ 265,618 \$ 273,587 \$ 281,795 \$ 290,249 \$ 298,956 Discount Factor Discounted Benefits \$ 7,476 \$ 6,161 \$ 5,076 \$ 4,183 \$ 3,447 Year Annual Benefits \$ 307,925 \$ 317,163 \$ 326,678 \$ 336,478 \$ 346,572 Discount Factor Discounted Benefits \$ 2,840 \$ 2,340 \$ 1,928 \$ 1,589 \$ 1,309 Year Annual Benefits \$ 356,969 \$ 367,678 \$ 378,708 \$ 390,069 \$ 401,771 Discount Factor Discounted Benefits \$ 1,079 \$ 889 \$ 733 \$ 604 \$ 497 Year Annual Benefits \$ 413,824 \$ 426,239 \$ 439,026 \$ 452,197 \$ 465,763 Discount Factor Discounted Benefits \$ 410 \$ 338 \$ 278 \$ 229 \$ 189 Sum of the Discounted Benefits (rounded) \$ 696,172 TWO STAGE MODEL Year Terminal Annual Benefits \$ 115,000 \$ 143,750 \$ 155,250 \$ 186,300 \$ 191,889 \$ 197,646 Capitalization Rate 22.0% Capitalized Benefits \$ 898,391 Discount Factor Discounted Benefits \$ 92,000 \$ 92,000 \$ 79,488 \$ 76,308 \$ 62,878 \$ 294,385 Sum of discounted benefits and terminal value \$ 697,059 Conclusion: This calculation demonstrates that, with nonlinear growth, the capitalizing terminal value process produces the same result as the discounting process. 6 Chapter Five by National Association of Certified Valuators and Analysts (NACVA). All rights reserved v1 Used by Institute of Business Appraisers with permission of NACVA for limited purpose of collaborative training.

14 CAPITALIZATION/DISCOUNT RATES Fundamentals, Techniques & Theory F. CALCULATION OF CASH TO EARNINGS FACTOR When future earnings approximate future cash flows, no adjustment is necessary to convert the capitalization rate (or discount rate) for cash flows into a capitalization rate (or discount rate) for accrual earnings. However, when the analyst expects that future cash flows will not be consistent with future earnings, adjustment of the cash flow capitalization and discount rates is necessary. The following is one methodology used to determine the cash to earnings factor: Earnings Depr Working Capital CapX Debt Cash Flow Factor Prior Yr. \$ 3,948,781 \$ 248,626 \$ (395,000) \$ (529,336) \$ - \$ 3,273, % 2 nd Prior Yr. 2,010, ,669 (201,000) (13,130) - 2,106, % 3 rd Prior Yr. 5,499, ,066 (550,000) (227,431) - 5,039, % 4 th Prior Yr. 3,132, ,356 (313,000) (138,137) (45,103) 2,957, % 5 th Prior Yr. 1,641, ,768 (164,000) (286,059) 45,103 1,547, % 6 th Prior Yr. 837, ,189 (84,000) (317,588) (406,629) 320, % 7 th Prior Yr. 1,844, ,631 (184,000) (672,544) (128,955) 1,092, % \$18,915,651 \$2,032,305 \$(1,891,000) \$(2,184,225) \$(535,584) \$16,337, % Avgs \$ 2,700,000 \$ 290,000 \$ (270,000) \$ (310,000) \$ (80,000) \$ 2,330,000 After-tax net income capitalization rate for the current year (21.32%/86.37%) 24.69% After-tax net cash flow capitalization rate for the current year 21.32% Cash to earnings factor 3.37% Proof Earnings Cash Flow Benefit Stream \$ 2,700,000 \$ 2,330,000 Capitalization Rate 24.69% 21.32% Enterprise Value (rounded) \$10,900,000 \$10,900,000 In The Cost of Capital, Pratt advises: Another way of looking at cash flow would be to define it more broadly. Instead of considering only the cash flows investors actually receive, you might define net cash flows as those amounts that could be paid to equity investors without impeding a company s future growth. Of course these cash flows are not those paid to investors, but presumably, investors will ultimately realize the benefit of these amounts either through higher future dividends or, more likely, stock appreciation. Some analysts assume that over the long run, net (after-tax) income should be quite close to cash flow. Therefore, they assume that net income can be used as a proxy for net cash flow. This assumption should be questioned on a case-by-case basis. 14 Chapter Five by National Association of Certified Valuators and Analysts (NACVA). All rights reserved v1 Used by Institute of Business Appraisers with permission of NACVA for limited purpose of collaborative training.

15 Fundamentals, Techniques & Theory CAPITALIZATION/DISCOUNT RATES IV. CAPITAL ASSET PRICING MODEL (CAPM) CAPM, by definition, is an equilibrium asset pricing theory that shows that equilibrium rates of expected return on all risky assets are a function of their co-variance with the market portfolio. This method for determining a capitalization or discount rate is based on the theory that investors in risky assets require a rate of return above and beyond a risk free rate as compensation for bearing the risk associated with holding the investment. A. ASSUMPTIONS The assumptions underlying the capital asset pricing model are: 1. Investors are risk averse. 2. Rational investors seek to hold portfolios which are fully diversified. 3. All investors have identical investment holding periods. 4. All investors have the same expectations regarding expected rate of return, and how capitalization rates are generated. 5. There are no transaction costs. 6. There are no taxes. 7. The rate received from lending money is the same as the cost of borrowing. 8. The market has perfect diversity and liquidity so an investor can readily buy or sell any fractional interest. B. CALCULATION OF EXPECTED RETURN Expected return = Risk-free rate + Beta x Expected return Riskon a free market portfolio Rate Abbreviated, the variables and the equation appear as follows: ER i = R f + B (ER m R f ) The risk-free (R f ) rate is represented by the 20-year yield to maturity on US Government bonds. According to Ibbotson, the horizon of the chosen Treasury security should match the horizon of whatever is being valued. When valuing a business that is being treated as a going concern, the appropriate Treasury yield should be that of a long-term Treasury bond. The expected return on a market portfolio (R m ) is the actual return on the Standard and Poor s 500 (S&P 500) Index. The beta coefficient (B) is a key variable in the CAPM equation. In the standard CAPM calculation, it represents the co-variance of the rate of return on the subject security, with the rate of return on the market divided by the variance of the market. More simply, it is a measure of the volatility of the subject security as compared to the market. In order to fully understand beta, certain terms must be understood: 1. Variance is a measure of the squared deviation of the actual return of a security from its expected return. 2. Co-variance is a statistical measure of the interrelationship between two securities by National Association of Certified Valuators and Analysts (NACVA). All rights reserved. Chapter Five 15 Used by Institute of Business Appraisers with permission of NACVA for limited purpose of collaborative training v1

16 CAPITALIZATION/DISCOUNT RATES Fundamentals, Techniques & Theory In the standard calculation of CAPM, beta is computed using the return on investment (ROI) of the subject security. Since ROI is calculated using the price of stock, the analyst uses the standard CAPM very rarely. If the price of stock is known, is there a need for valuation? Some analysts alter the CAPM model by modifying certain variables. The risk-free rate (R f ) is represented by the intermediate term (five to 10 year) Treasury bond yield rate. Beta (B) is modified so that it represents the co-variance of the pre-tax return on equity (ROE) of the subject company, with the ROE of other specific companies or industry averages divided by the variance of the ROE of the industry. Finally, rather than using the expected return on a market portfolio as the ER m, it is represented by the average pre-tax ROE of the specific companies or the industry in which the subject company operates. 1. Calculation of Beta ( ) A beta 9 of 1.0 would indicate the subject company is no more or no less volatile than the industry. In this example the beta of indicates that the subject company is less volatile than the industry. As such, it would appear to be a better risk. Thus, a total risk-premium less than the industry would probably be appropriate for the company. Based on this analysis, it can be seen that the expected rate of return for a company should be positively related to its beta. 2. Security Market Line (SML) The expected return on a security with a beta of zero is the risk-free rate, since a zero beta indicates no relative risk. The expected return on a security with a beta of one is the expected return of the market, since a beta of one indicates that the security has the same relative risk as the market. A shortcoming of CAPM is the fact that it utilizes comparative information in its various forms. Since it may be extremely difficult to locate industry data, it may be difficult to use CAPM to develop a discount/capitalization rate. It is equally as difficult to find specific comparable company data for a closely held company. 3. Is CAPM a pre-tax or an after-tax method? The answer: it depends. CAPM describes the cost of equity for a given company, and is equal to the risk-free rate plus some amount to compensate for the risk involved in excess of the risk-free rate. Thus, there are several elements to CAPM coming from both sides of the tax equation. This risk-free rate is usually a government bond rate, which is pre-tax to the investor. The expected return on a market portfolio is generated from average returns of the market after corporate tax, usually comparing the return to that of the S&P 500. Beta is public market volatility, generated by stock transactions, which is after corporate tax (but again, pre-investor tax). These companies 10K forms do consider known tax liabilities in their bottom lines. However, this liability may not be the actual tax. In valuing a closely held company, beta is generally developed from comparable public companies or is calculated using the average pre-tax ROE (for equity capital) or ROI (for investment) of the specific company. ROI, as used to develop beta 10, is calculated as: (Ending Stock Price - Beginning Stock Price) + Dividends Beginning Stock Price 9 Historical beta research can be performed by KeyValueData. 10 or b often (but not always) indicate beta in financial equations. 16 Chapter Five by National Association of Certified Valuators and Analysts (NACVA). All rights reserved v1 Used by Institute of Business Appraisers with permission of NACVA for limited purpose of collaborative training.

20 CAPITALIZATION/DISCOUNT RATES Fundamentals, Techniques & Theory Companies that meet the criteria noted above are then divided evenly into twenty-five portfolios for each measure of size. Companies included in the high financial risk portfolio are shown as a separate line item in each of the size categories. C. DATA PRESENTATION The Duff & Phelps data are presented in a series of exhibits in Appendix X. Part I of the Report includes: Exhibits A-1 through A-8 Exhibits B-1 through B-8 ERP vs. company size (eight measures of size) Premiums over Capital Asset Pricing Model (CAPM) vs. company size (eight measures of size) Part II of the Report includes: Exhibits C-1 through C-8 Exhibits D-1 through D-3 Relation between size and company risk (eight measures of size) ERP vs. company risk (three measures of risk) The three company risk measures are as follows: D. DATA USE Operating margin (the lower the margin, the greater the risk) Coefficient of Variation in Operating Margin (the greater the coefficient of variation, the greater the risk) Coefficient of Variation in Return on Equity (the greater the coefficient of variation, the greater the risk) The ERPs developed by the Duff & Phelps data can be used to calculate a discount cost of equity using a build-up model (using the data reported in Exhibits A-1 through A-8) or the modified capital asset pricing model (MCAPM) (using the data reported in Exhibits B-1 through B-8). The Report suggests that the smoothed average premium is the most appropriate indicator for most of the portfolio groups. The smoothed premium refers to how the premium is determined. It can be calculated based on a regression analysis, with the average historical ERP as the dependent variable and the logarithm of the average sorting criteria as the independent variable. One benefit of the smoothed premium is if an analyst is estimating the required rate of return for a company significantly smaller than any of the companies found in the smallest of the 25 portfolios, it is appropriate to extrapolate the ERP using the slope and constant terms from the regression relationships used in deriving the smoothed premiums. 20 Chapter Five by National Association of Certified Valuators and Analysts (NACVA). All rights reserved v1 Used by Institute of Business Appraisers with permission of NACVA for limited purpose of collaborative training.

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