# Risk and Return: Estimating Cost of Capital

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1 Lecture: IX 1 Risk and Return: Estimating Cost o Capital The process: Estimate parameters or the risk-return model. Estimate cost o equity. Estimate cost o capital using capital structure (leverage) inormation. The cost o equity can be estimated using the Dividend Growth Model (studied earlier), Capital Asset Pricing Model (CAPM). E( R ) = R + β E( R ) R i i [ ] M Inputs to the CAPM are: The current risk-ree rate, The expected return on the market index, and The beta o the asset being analyzed. Hence the equation is actually estimated as ollows: E( R i ) = R + βi[ Market Risk Premium] Estimation issues: What is the correct risk-ree rate to use in the model? How should we measure the risk premium to be used in calculating the expected return on the market index? How should we estimate beta?

2 Lecture: IX 2 Estimating Risk-ree Rates Two approaches: Use a short-term Govt. security rate (usually the 3- month T-bill rate). Use a long-term Govt. bond rate (usually the 30-yr bond rate). Which one should be used? Match-up the horizon o the project or asset being analyzed with the maturity o the risk-ree asset. Managers looking at long-term projects should use the long-term Govt. bond rate. I the investment horizon is short (under 1 year), then use the short-term T-bill rate. Example: Pepsi Cola Corp. has a beta o What is their cost o equity, i the expected return on the market is 13% (3-month T-bill rate is 5%, 30-yr T-bond rate is 6.4%) Using the short-term rate: Cost o equity = 5% (13%-5%) = 14.28% Using the long-term rate: Cost o equity = 6.4% +1.16(13%-6.4%)=14.06% Will the cost o equity rom a long-term perspective always be lower than that rom a short-term perspective? Why or why not?

3 Lecture: IX 3 Estimating Risk Premium Deined as the dierence between average returns on stocks and average returns on risk-ree securities over the measurement period. Generally based on historical data. Two issues: How long should the measurement period be? Should arithmetic or geometric averages be used to compute the risk premium? Length o the measurement period: In practice, people use at least 10 years o data. Should use the longest possible period, i there are no trends in the premium. Much o the data on US stocks is available rom 1926 onwards. Oten, data rom 1926 till now is used. Arithmetic or Geometric averages? Arithmetic mean is the average o the annual returns or the period under consideration. Geometric mean is the compounded annual return over the same period.

4 Lecture: IX 4 Example: Year Price Return % % Arithmetic average return = [100%+(-40%)]/2 = 30% Geometric average return = (2x0.6) - 1 = (9.54%) There can be dramatic dierences in premiums based on the averaging method! Arithmetic mean is argued as being more consistent with the mean-variance ramework o CAPM and a better predictor o premiums in the next period. Geometric mean accounts or compounding, and is argued to be a better predictor o the average premium in the long run. Geometric mean generally yields lower premium estimates. Since expected returns are compounded over long periods o time, the geometric mean provided a better estimate o the risk premium. In the US, the premium has been about 3.82% rom European markets have had lower premiums, while Britain has had higher (6.25%).

5 Lecture: IX 5 What determines the size o the risk premium? More volatile economies have higher risk premiums (e.g., emerging markets, with high-growth high-risk economies - like South America, Russia). Political risk and instability leads to higher premiums - various rating agencies publish these surveys (e.g. Iraq would have high premiums!) Market structure aects risks in stocks - or economies where listed companies are large, diversiied and stable (e.g., Germany and Switzerland), risk premiums are lower. In the US and UK, many smaller and riskier companies are also listed, thereby increasing the premium or investing in stocks.

6 Lecture: IX 6 Estimating Beta The conceptual way: Cov( Ri, RM ) Previously, beta was deined by β i = 2 σ ( RM ) Using historical returns or a market index and the stock being analyzed, we can estimate beta. Example: A stock had the ollowing returns over the last 5 years, as compared to the return on the S&P 500 index. What is an estimate o the stock s beta? Year Home Depot s return S&P 500 return 1-15% -10% 2 3% 15% 3 12% 8% 4 58% 30% 5 44% 22% here, s.d.(r M ) = 13.62% (i.e., Var(R M ) = ) Cov(R i, R M ) = hence, β = / = 1.8

7 Lecture: IX 7 Estimating beta the real-world way: CAPM can be written as a one-actor model: R = R + β R R i = R [ ] M ( 1 β ) + β R M Ri = a + brm This is a linear regression o stock returns (R i ) against market returns (R M ). The slope o this regression is the beta o the stock. The intercept o this regression provides a simple measure o the perormance o the stock relative to CAPM, during the regression period: I a > R (1 β ), stock did better than expected a = R a < R (1 β ), (1 β ), stock did as well as expected stock did worse than expected The dierence between a and R (1-β) is called Jensen s alpha; it provides a measure o whether the asset underor out-perormed the market on a risk-adjusted basis. The R-squared (R 2 )o this regression provides an estimate o the proportion o risk that can be attributed to market wide actors (systematic risk) - the balance (1- R 2 ) can be attributed to irm-speciic risk (unsystematic risk). Is high R-squared good? As an analyst, would you recommend investors with limited unds to buy high R-squared stocks?

8 Lecture: IX 8 Example: Estimating beta or Intel ( ) We can compute monthly returns to a stockholder in Intel as ollows: priceintel, j priceint el, j 1 + dividends j Stock returnintel, j = price Intel, j 1 Monthly returns on the market index (S&P 500)are given by: index j index j 1 market returnintel, j = + Dividend Yield j index Regress the monthly time series o Intel s stock returns on the market s return. The slope o this regression comes to 1.39, which is Intel s beta, during The intercept o this regression is 2.09%. Since the returns are monthly, the risk-ree rate on a monthly basis averaged 0.4% during We can, thereore, compute the Jensen s alpha, to measure Intel s perormance relative to the market: Jensen ' s α = Intercept R (1 β ) = 2.09% 0.4%(1 1.39) = 2.25% Hence, Intel perormed 2.25% better than expected, based on CAPM, on a monthly basis ( ). This results in an annualized excess return o 30.6%. The R-squared o the regression was 22.9%, implying that 22.9% o the risk in Intel comes rom market-wide sources, and the balance (77.1%) comes rom irmspeciic components (this component is diversiiable, hence unrewarded in CAPM). j 1

9 Lecture: IX 9 Estimation issues in the beta regression: Length o estimation period (Value Line and S&P use 5 years o data, Bloomberg uses 2 years) - longer period provides more data, but the irm s risk characteristics may not remain stable over longer periods. Return Interval - Using daily or intraday data increases observations, but induces bias due to non-trading days (i stock is not traded requently, the returns on nontraded days would be zero, thereby biasing the beta downwards). E.g., or America Online ( ), the beta is 1.8 using monthly returns, but 1.2 using daily returns (which one is more reliable, and why?). Choice o market index - standard practice is to estimate betas relative to the index o the market in which the stock trades (US stocks relative to NYSE Composite, British stocks relative to the FTSE, Japanese stocks relative to the Nikkei, etc.). But it may not be appropriate or international or cross-border investors. Statistical issues - whether betas should be adjusted to relect the likelihood o estimation errors and biases. These techniques are most useul when daily returns are used; less useul or longer return intervals. When the betas o stocks listed on overseas markets are estimated against the NYSE Composite instead o their local indices, are the betas likely to increase or decrease? Which beta would you use and why?

10 Lecture: IX 10 The Determinants o Beta The cyclical nature o business - the more sensitive a business is to market conditions, higher is its beta. E.g., housing irms have higher betas than ood processing companies. Degree o Operating Leverage - ratio o %change in operating proits to %change in sales (deines the relationship between ixed costs and total costs - high ixed costs implies high Operating Leverage). High Operating Leverage implies a higher variability in earnings, hence higher beta or the irm. Degree o Financial Leverage (debt/equity ratio) - higher debt implies higher obligated payments, - hence, in bad times, the income goes down more; in good times, the income goes up more. - so more debt increases the variance in net income, increasing the equity beta (what we estimate using stock returns) o the irm. D E β irm = βdebt + βequity D + E D + E D 1 βequity = + β irm, as βdebt 0 E - β irm is called the unlevered beta o the irm (β U ), i.e., the beta o the irm without any debt. - β equity is the levered beta or equity in the irm (β L ). - In the presence o taxes, the levered beta is given by: β = β 1+ (1 t)( D / E) L U [ ]

11 Lecture: IX 11 The properties o beta: Levered betas are always greater than unlevered betas in the presence o inancial leverage (debt). I a irm has multiple divisions/businesses, its beta will be the weighted average o the betas o each business line, with the weights based on the market value o each. European companies have stricter labor laws than US companies, making it more diicult or them to lay o employees during economic downturns. How should this aect their betas? Example: Boeing has a beta o 0.95, a debt/equity ratio o 5%, and a tax rate o 34%. How would their beta change i their debt ratio went up to 25%? How does it aect their cost o equity, i the T- bond rate is 6.5% and the market risk premium is 5.5%? present cost o equity = 6.5% (5.5%) = 11.73% unlevered beta = 0.95/[1+(1-0.34)(0.05)] = levered beta at 25% debt ratio = 0.912[1+(1-0.34)(0.25)] = 1.07 Equity cost at 25% debt ratio = 6.5% (5.5%) = 12.39% What can you iner rom this?

12 Lecture: IX 12 Weighted Average Cost o Capital (WACC) The weighted average o costs o dierent components o inancing (debt, equity, and hybrids). E D WACC = ke + kd ( 1 t) D + E D + E The debt-equity proportional weights must be estimated using the market values o equity and debt, not the book values (cost o capital measures the cost o issuing securities to inance projects, and these securities are issued at market values, not book values!). k e is the cost o equity. k d is the pre-tax cost o debt, while k d (1-t) is the ater-tax cost o debt. The cost o debt is the current cost to the irm o borrowing unds to inance projects. It is not - the coupon rate on the outstanding bonds, nor - the rate at which the company borrowed in the past. The cost o debt is the Yield-to-Maturity on its debt. Example: In March 1995, Pepsi Cola Corp. had a cost o equity o 13.33%, a cost o debt o 8% (pretax), and 34% tax rate. Its equity had a book value o \$7.05 billion and a market value o \$32 billion. The book value o debt was \$9.75 billion, while the market value o debt was \$10 billion. What was its WACC? WACC = 13.33% + 8%(1 34%) = 11.41% (using book value weights gives a totally distorted WACC o 8.66%!)

13 Lecture: IX 13 A Comprehensive Example o Risk, Return and Costs o Financing - Home Depot (again!) By regressing monthly returns or Home Depot on the S&P 500 index returns, over , we get a beta o 1.38 (hence Home Depot stock is riskier than the average market). The beta estimates rom dierent estimation services are dierent - Value Line reports 1.30 or Home Depot - why? (Value Line uses weekly returns, and statistically adjusts beta or long-term biases). In Jan 95, 30-yr T-bond rate was 7.5%. Using a historical risk premium o 5.5%, the expected return is: Expected return = 7.5% (5.5%) = 15.09% (In other words, the cost o equity or Home Depot was 15.09% in Jan 95) The intercept on the beta regression was 2.19%. Using an average risk-ree rate o 6.5% during , Jensen s α is 2.39% per month (32.82% excess annual return). The average Jensen s α or the industry was -0.02% per month. (This suggests that Home Depot s superior perormance was due entirely to irm-speciic actors) R-squared was 33.76%. Hence, 33.76% o the risk in Home Depot s stock comes rom market-wide actors. (This is understandable, since Home Depot s business is home improvement, which will suer during economic recessions)

14 Lecture: IX 14 Home Depot had an average debt-equity ratio o 4%, so their unlevered beta comes to 1.34 (using 34% tax rate). (So currently, bulk o their risk is due to business risk, not inancial leverage - i they increase leverage to 50%, beta would go above 2). Home Depot had a pre-tax cost o debt o 8.5%, and market values o debt and equity o \$900 million and \$ billion (stock price x # o shares), respectively. Using a 34% tax rate, their WACC comes to 14.70%. (This is the appropriate benchmark to use or evaluating their projects).

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