# I. Estimating Discount Rates

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1 I. Estimating Discount Rates DCF Valuation Aswath Damodaran 1

2 Estimating Inputs: Discount Rates Critical ingredient in discounted cashflow valuation. Errors in estimating the discount rate or mismatching cashflows and discount rates can lead to serious errors in valuation. At an intuitive level, the discount rate used should be consistent with both the riskiness and the type of cashflow being discounted. Equity versus Firm: If the cash flows being discounted are cash flows to equity, the appropriate discount rate is a cost of equity. If the cash flows are cash flows to the firm, the appropriate discount rate is the cost of capital. Currency: The currency in which the cash flows are estimated should also be the currency in which the discount rate is estimated. Nominal versus Real: If the cash flows being discounted are nominal cash flows (i.e., reflect expected inflation), the discount rate should be nominal Aswath Damodaran 2

3 Cost of Equity The cost of equity should be higher for riskier investments and lower for safer investments While risk is usually defined in terms of the variance of actual returns around an expected return, risk and return models in finance assume that the risk that should be rewarded (and thus built into the discount rate) in valuation should be the risk perceived by the marginal investor in the investment Most risk and return models in finance also assume that the marginal investor is well diversified, and that the only risk that he or she perceives in an investment is risk that cannot be diversified away (I.e, market or non-diversifiable risk) Aswath Damodaran 3

4 The Cost of Equity: Competing Models Model Expected Return Inputs Needed CAPM E(R) = R f + β (R m - R f ) Riskfree Rate Beta relative to market portfolio Market Risk Premium APM E(R) = R f + Σ j=1 β j (R j - R f ) Riskfree Rate; # of Factors; Betas relative to each factor Factor risk premiums Multi E(R) = R f + Σ j=1,,n β j (R j - R f ) Riskfree Rate; Macro factors factor Betas relative to macro factors Macro economic risk premiums Proxy E(R) = a + Σ j=1..n b j Y j Proxies Regression coefficients Aswath Damodaran 4

5 The CAPM: Cost of Equity Consider the standard approach to estimating cost of equity: Cost of Equity = R f + Equity Beta * (E(R m ) - R f ) where, R f = Riskfree rate E(R m ) = Expected Return on the Market Index (Diversified Portfolio) In practice, Short term government security rates are used as risk free rates Historical risk premiums are used for the risk premium Betas are estimated by regressing stock returns against market returns Aswath Damodaran 5

6 Short term Governments are not riskfree On a riskfree asset, the actual return is equal to the expected return. Therefore, there is no variance around the expected return. For an investment to be riskfree, then, it has to have No default risk No reinvestment risk Thus, the riskfree rates in valuation will depend upon when the cash flow is expected to occur and will vary across time A simpler approach is to match the duration of the analysis (generally long term) to the duration of the riskfree rate (also long term) In emerging markets, there are two problems: The government might not be viewed as riskfree (Brazil, Indonesia) There might be no market-based long term government rate (China) Aswath Damodaran 6

7 Estimating a Riskfree Rate Estimate a range for the riskfree rate in local terms: Approach 1: Subtract default spread from local government bond rate: Government bond rate in local currency terms - Default spread for Government in local currency Approach 2: Use forward rates and the riskless rate in an index currency (say Euros or dollars) to estimate the riskless rate in the local currency. Do the analysis in real terms (rather than nominal terms) using a real riskfree rate, which can be obtained in one of two ways from an inflation-indexed government bond, if one exists set equal, approximately, to the long term real growth rate of the economy in which the valuation is being done. Do the analysis in another more stable currency, say US dollars. Aswath Damodaran 7

8 A Simple Test You are valuing Ambev, a Brazilian company, in U.S. dollars and are attempting to estimate a riskfree rate to use in the analysis. The riskfree rate that you should use is The interest rate on a Brazilian Real denominated long term Government bond The interest rate on a US \$ denominated Brazilian long term bond (C- Bond) The interest rate on a US \$ denominated Brazilian Brady bond (which is partially backed by the US Government) The interest rate on a US treasury bond Aswath Damodaran 8

9 Everyone uses historical premiums, but.. The historical premium is the premium that stocks have historically earned over riskless securities. Practitioners never seem to agree on the premium; it is sensitive to How far back you go in history Whether you use T.bill rates or T.Bond rates Whether you use geometric or arithmetic averages. For instance, looking at the US: Arithmetic average Geometric Average Historical Period T.Bills T.Bonds T.Bills T.Bonds % 6.84% 6.21% 5.17% % 4.68% 4.74% 3.90% % 6.90% 9.44% 6.17% Aswath Damodaran 9

10 If you choose to use historical premiums. Go back as far as you can. A risk premium comes with a standard error. Given the annual standard deviation in stock prices is about 25%, the standard error in a historical premium estimated over 25 years is roughly: Standard Error in Premium = 25%/ 25 = 25%/5 = 5% Be consistent in your use of the riskfree rate. Since we argued for long term bond rates, the premium should be the one over T.Bonds Use the geometric risk premium. It is closer to how investors think about risk premiums over long periods. Aswath Damodaran 10

11 Country Risk Premiums Historical risk premiums are almost impossible to estimate with any precision in markets with limited history - this is true not just of emerging markets but also of many Western European markets. For such markets, we can estimate a modified historical premium beginning with the U.S. premium as the base: Relative Equity Market approach: The country risk premium is based upon the volatility of the market in question relative to U.S market. Country risk premium = Risk Premium US * σ Country Equity / σ US Equity Country Bond approach: In this approach, the country risk premium is based upon the default spread of the bond issued by the country. Country risk premium = Risk Premium US + Country bond default spread Combined approach: In this approach, the country risk premium incorporates both the country bond spread and equity market volatility. Aswath Damodaran 11

12 Step 1: Assessing Country Risk Using Country Ratings: Latin America - March 2001 Country Rating Typical Spread Market Spread Argentina B Bolivia B Brazil B Colombia Ba Ecuador Caa Guatemala Ba Honduras B Mexico Baa Paraguay B Peru Ba Uruguay Baa Venezuela B Aswath Damodaran 12

13 Step 2: From Bond Default Spreads to Equity Spreads Country ratings measure default risk. While default risk premiums and equity risk premiums are highly correlated, one would expect equity spreads to be higher than debt spreads. One way to adjust the country spread upwards is to use information from the US market. In the US, the equity risk premium has been roughly twice the default spread on junk bonds. Another is to multiply the bond spread by the relative volatility of stock and bond prices in that market. For example, Standard Deviation in Bovespa (Equity) = 32.6% Standard Deviation in Brazil C-Bond = 17.1% Adjusted Equity Spread = 5.37% (32.6/17.1%) = 10.24% Ratings agencies make mistakes. They are often late in recognizing and building in risk. Aswath Damodaran 13

14 Another Example: Assessing Country Risk Using Currency Ratings: Western Europe Country Rating Typical Spread Actual Spread Austria Aaa 0 Belgium Aaa 0 Denmark Aaa 0 Finland Aaa 0 France Aaa 0 Germany Aaa 0 Greece A Ireland AA Italy Aa Netherlands Aaa 0 Norway Aaa 0 Portugal A Spain Aa Sweden Aa Switzerland Aaa 0 Aswath Damodaran 14

15 Greek Country Risk Premium Country ratings measure default risk. While default risk premiums and equity risk premiums are highly correlated, one would expect equity spreads to be higher than debt spreads. One way to adjust the country spread upwards is to use information from the US market. In the US, the equity risk premium has been roughly twice the default spread on junk bonds. Another is to multiply the bond spread by the relative volatility of stock and bond prices in that market. For example, Standard Deviation in Greek ASE(Equity) = 40.5% Standard Deviation in Greek GDr Bond = 26.1% Adjusted Equity Spread = 0.95% (40.5%/26.1%) = 1.59% Ratings agencies make mistakes. They are often late in recognizing and building in risk. Aswath Damodaran 15

16 From Country Spreads to Corporate Risk premiums Approach 1: Assume that every company in the country is equally exposed to country risk. In this case, E(Return) = Riskfree Rate + Country Spread + Beta (US premium) Implicitly, this is what you are assuming when you use the local Government s dollar borrowing rate as your riskfree rate. Approach 2: Assume that a company s exposure to country risk is similar to its exposure to other market risk. E(Return) = Riskfree Rate + Beta (US premium + Country Spread) Approach 3: Treat country risk as a separate risk factor and allow firms to have different exposures to country risk (perhaps based upon the proportion of their revenues come from non-domestic sales) E(Return)=Riskfree Rate+ β (US premium) + λ (Country Spread) Aswath Damodaran 16

17 Estimating Company Exposure to Country Risk Different companies should be exposed to different degrees to country risk. For instance, a Brazilian firm that generates the bulk of its revenues in the United States should be less exposed to country risk in Brazil than one that generates all its business within Brazil. The factor λ measures the relative exposure of a firm to country risk. One simplistic solution would be to do the following: λ = % of revenues domestically firm / % of revenues domestically avg firm For instance, if a firm gets 35% of its revenues domestically while the average firm in that market gets 70% of its revenues domestically λ = 35%/ 70 % = 0.5 There are two implications A company s risk exposure is determined by where it does business and not by where it is located Firms might be able to actively manage their country risk exposures Aswath Damodaran 17

18 Estimating E(Return) for Embraer Assume that the beta for Embraer is 0.88, and that the riskfree rate used is 4.5%. (Real Riskfree Rate) Approach 1: Assume that every company in the country is equally exposed to country risk. In this case, E(Return) =4.5% % (5.51%) = 19.59% Approach 2: Assume that a company s exposure to country risk is similar to its exposure to other market risk. E(Return) = 4.5% (5.51% %) = 18.36% Approach 3: Treat country risk as a separate risk factor and allow firms to have different exposures to country risk (perhaps based upon the proportion of their revenues come from non-domestic sales) E(Return)= 4.5% (5.51%) (10.24%) = 14.47% Embraer is less exposed to country risk than the typical Brazilian firm since much of its business is overseas. Aswath Damodaran 18

19 Implied Equity Premiums If we use a basic discounted cash flow model, we can estimate the implied risk premium from the current level of stock prices. For instance, if stock prices are determined by a variation of the simple Gordon Growth Model: Value = Expected Dividends next year/ (Required Returns on Stocks - Expected Growth Rate) Dividends can be extended to included expected stock buybacks and a high growth period. Plugging in the current level of the index, the dividends on the index and expected growth rate will yield a implied expected return on stocks. Subtracting out the riskfree rate will yield the implied premium. This model can be extended to allow for two stages of growth - an initial period where the entire market will have earnings growth greater than that of the economy, and then a stable growth period. Aswath Damodaran 19

20 Estimating Implied Premium for U.S. Market: Jan 1, 2002 Level of the index = 1148 Treasury bond rate = 5.05% Expected Growth rate in earnings (next 5 years) = 10.3% (Consensus estimate for S&P 500) Expected growth rate after year 5 = 5.05% Dividends + stock buybacks = 2.74% of index (Current year) Year 1 Year 2 Year 3 Year 4 Year 5 Expected Dividends = \$34.72 \$38.30 \$42.24 \$46.59 \$ Stock Buybacks Expected dividends + buybacks in year 6 = (1.0505) = \$ = 34.72/(1+r) /(1+r) /(1+r) /(1+r) 4 + (51.39+(54.73/(r-.0505))/(1+r) 5 Solving for r, r = 8.67%. (Only way to do this is trial and error) Implied risk premium = 8.67% % = 3.62% Aswath Damodaran 20

21 Implied Premium for US Equity Market 7.00% 6.00% 5.00% Implied Premium 4.00% 3.00% 2.00% 1.00% 0.00% Year Aswath Damodaran 21

22 Implied Premium for Brazilian Market: March 1, 2001 Level of the Index = Dividends on the Index = 4.40% of (Used weighted yield) Other parameters Riskfree Rate = 4.5% (real riskfree rate) Expected Growth Next 5 years = 13.5% (Used expected real growth rate in Earnings) After year 5 = 4.5% (real growth rate in long term) Solving for the expected return: Expected return on Equity = 11.16% Implied Equity premium = 11.16% -4. 5% = 6.66% Aswath Damodaran 22

23 The Effect of Using Implied Equity Premiums on Value Embraer s value per share (using historical premium + country risk adjustment) = BR Embraer s value per share (using implied equity premium of 6.66%) = BR Embraer s stock price (at the time of the valuation) = BR Aswath Damodaran 23

24 Estimating Beta The standard procedure for estimating betas is to regress stock returns (R j ) against market returns (R m ) - R j = a + b R m where a is the intercept and b is the slope of the regression. The slope of the regression corresponds to the beta of the stock, and measures the riskiness of the stock. This beta has three problems: It has high standard error It reflects the firm s business mix over the period of the regression, not the current mix It reflects the firm s average financial leverage over the period rather than the current leverage. Aswath Damodaran 24

25 Beta Estimation: The Noise Problem Aswath Damodaran 25

26 Beta Estimation: The Index Effect Aswath Damodaran 26

27 Determinants of Betas Product or Service: The beta value for a firm depends upon the sensitivity of the demand for its products and services and of its costs to macroeconomic factors that affect the overall market. Cyclical companies have higher betas than non-cyclical firms Firms which sell more discretionary products will have higher betas than firms that sell less discretionary products Operating Leverage: The greater the proportion of fixed costs in the cost structure of a business, the higher the beta will be of that business. This is because higher fixed costs increase your exposure to all risk, including market risk. Financial Leverage: The more debt a firm takes on, the higher the beta will be of the equity in that business. Debt creates a fixed cost, interest expenses, that increases exposure to market risk. Aswath Damodaran 27

28 Equity Betas and Leverage The beta of equity alone can be written as a function of the unlevered beta and the debt-equity ratio β L = β u (1+ ((1-t)D/E)) where β L = Levered or Equity Beta β u = Unlevered Beta (Asset Beta) t = Corporate marginal tax rate D = Market Value of Debt E = Market Value of Equity While this beta is estimated on the assumption that debt carries no market risk (and has a beta of zero), you can have a modified version: β L = β u (1+ ((1-t)D/E)) - β debt (1-t) (D/E) Aswath Damodaran 28

29 Solutions to the Regression Beta Problem Modify the regression beta by changing the index used to estimate the beta adjusting the regression beta estimate, by bringing in information about the fundamentals of the company Estimate the beta for the firm using the standard deviation in stock prices instead of a regression against an index. accounting earnings or revenues, which are less noisy than market prices. Estimate the beta for the firm from the bottom up without employing the regression technique. This will require understanding the business mix of the firm estimating the financial leverage of the firm Use an alternative measure of market risk that does not need a regression. Aswath Damodaran 29

30 Bottom-up Betas The bottom up beta can be estimated by : Taking a weighted (by sales or operating income) average of the unlevered betas of the different businesses a firm is in. j =k j =1 j Operating Income j Operating Income Firm (The unlevered beta of a business can be estimated by looking at other firms in the same business) Lever up using the firm s debt/equity ratio levered = [ ] unlevered 1+ (1 tax rate) (Current Debt/Equity Ratio) The bottom up beta will give you a better estimate of the true beta when It has lower standard error (SE average = SE firm / n (n = number of firms) It reflects the firm s current business mix and financial leverage It can be estimated for divisions and private firms. Aswath Damodaran 30

31 Bottom-up Beta: Firm in Multiple Businesses Boeing in 1998 Segment Estimated Value Unlevered Beta Segment Weight Commercial Aircraft 30, % Defense 12, % Estimated Value = Revenues of division * Enterprise Value/Sales Business Unlevered Beta of firm = 0.91 (.7039) (.2961) = 0.88 Levered Beta Calculation Market Value of Equity = \$ 33,401 Market Value of Debt = \$8,143 Market Debt/Equity Ratio = 24.38% Tax Rate = 35% Levered Beta for Boeing = 0.88 (1 + (1 -.35) (.2438)) = 1.02 Aswath Damodaran 31

32 Siderar s Bottom-up Beta Siderar is an Argentine steel company. Business Unlevered D/E Ratio Levered Beta Beta Steel % 0.71 Proportion of operating income from steel = 100% Levered Beta for Siderar= 0.71 Aswath Damodaran 32

33 Comparable Firms? Can an unlevered beta estimated using U.S. steel companies be used to estimate the beta for an Argentine steel company? Yes No Aswath Damodaran 33

34 The Cost of Equity: A Recap Preferably, a bottom-up beta, based upon other firms in the business, and firm s own financial leverage Cost of Equity = Riskfree Rate + Beta * (Risk Premium) Has to be in the same currency as cash flows, and defined in same terms (real or nominal) as the cash flows Historical Premium 1. Mature Equity Market Premium: Average premium earned by stocks over T.Bonds in U.S. 2. Country risk premium = Country Default Spread* ( σequity/σcountry bond) or Implied Premium Based on how equity market is priced today and a simple valuation model Aswath Damodaran 34

35 Estimating the Cost of Debt The cost of debt is the rate at which you can borrow at currently, It will reflect not only your default risk but also the level of interest rates in the market. The two most widely used approaches to estimating cost of debt are: Looking up the yield to maturity on a straight bond outstanding from the firm. The limitation of this approach is that very few firms have long term straight bonds that are liquid and widely traded Looking up the rating for the firm and estimating a default spread based upon the rating. While this approach is more robust, different bonds from the same firm can have different ratings. You have to use a median rating for the firm When in trouble (either because you have no ratings or multiple ratings for a firm), estimate a synthetic rating for your firm and the cost of debt based upon that rating. Aswath Damodaran 35

36 Estimating Synthetic Ratings The rating for a firm can be estimated using the financial characteristics of the firm. In its simplest form, the rating can be estimated from the interest coverage ratio Interest Coverage Ratio = EBIT / Interest Expenses For Siderar, in 1999, for instance Interest Coverage Ratio = 161/48 = 3.33 Based upon the relationship between interest coverage ratios and ratings, we would estimate a rating of A- for Siderar. With a default spread of 1.25% (given the rating of A-) For Titan s interest coverage ratio, we used the interest expenses and EBIT from Interest Coverage Ratio = 55,467/ 4028= Aswath Damodaran 36

37 Interest Coverage Ratios, Ratings and Default Spreads If Coverage Ratio is Estimated Bond Rating Default Spread(1/99) Default Spread(1/01) > 8.50 AAA 0.20% 0.75% AA 0.50% 1.00% A+ 0.80% 1.50% A 1.00% 1.80% A 1.25% 2.00% BBB 1.50% 2.25% BB 2.00% 3.50% B+ 2.50% 4.75% B 3.25% 6.50% B 4.25% 8.00% CCC 5.00% 10.00% CC 6.00% 11.50% C 7.50% 12.70% < 0.20 D 10.00% 15.00% Aswath Damodaran 37

38 Cost of Debt computations Companies in countries with low bond ratings and high default risk might bear the burden of country default risk For Siderar, the rating estimated of A- yields a cost of debt as follows: Pre-tax Cost of Debt in 1999 = US T.Bond rate + Country default spread + Company Default Spread = 6% % % = 12.50% The synthetic rating for Titan is AAA. The default spread in 2001 is 0.75%. Pre-tax Cost of Debt = Riskfree Rate + Company Default Spread+ Country Spread = 5.10% % %= 6.80% Aswath Damodaran 38

39 Synthetic Ratings: Some Caveats The relationship between interest coverage ratios and ratings, developed using US companies, tends to travel well, as long as we are analyzing large manufacturing firms in markets with interest rates close to the US interest rate They are more problematic when looking at smaller companies in markets with higher interest rates than the US. Aswath Damodaran 39

40 Weights for the Cost of Capital Computation The weights used to compute the cost of capital should be the market value weights for debt and equity. There is an element of circularity that is introduced into every valuation by doing this, since the values that we attach to the firm and equity at the end of the analysis are different from the values we gave them at the beginning. As a general rule, the debt that you should subtract from firm value to arrive at the value of equity should be the same debt that you used to compute the cost of capital. Aswath Damodaran 40

41 Estimating Cost of Capital: Titan Cements Equity Cost of Equity = 5.10% (4%+1.59%) = 10.47% Market Value of Equity = 739,217 million GDr (78.7%) Debt Company default spread Mature market premium Cost of debt = 5.10% % +0.95%= 6.80% Country default spread Market Value of Debt = 199,766 million GDr (21.3 %) Greek country premium Cost of Capital Cost of Capital = % (.787) % ( ) (0.213)) = 9.33 % Aswath Damodaran 41

42 Titan Cement: Book Value Weights Titan Cement has a book value of equity of 135,857 million GDR and a book value of debt of 200,000 million GDR. Estimate the cost of capital using book value weights instead of market value weights. Aswath Damodaran 42

43 Estimating A U.S. Dollar Cost of Capital: Siderar - An Argentine Steel Company Equity Cost of Equity = 6.00% (4% %) = 16.32% Market Value of Equity = 3.20* = 995 million (94.37%) Debt Cost of debt = 6.00% % (Country default) +1.25% (Company default) = 12.5% Market Value of Debt = 59 Mil (5.63%) Cost of Capital Mature Market Premium Cost of Capital = % (.9437) % ( ) (.0563)) = % (.9437) % (.0563)) = % Country Risk Premium for Argentina Aswath Damodaran 43

44 Converting a Dollar Cost of Capital into a Peso cost of capital Approach 1: Use a peso riskfree rate in all of the calculations above. For instance, if the peso riskfree rate was 10%, the cost of capital would be computed as follows: Cost of Equity = 10.00% (4% %) = 20.32% Cost of Debt = = 10.00% % (Country default) +1.25% (Company default) = 16.5% (This assumes the peso riskfree rate has no country risk premium embedded in it.) Approach 2: Use the differential inflation rate to estimate the cost of capital. For instance, if the inflation rate in pesos is 7% and the inflation rate in the U.S. is 3% Cost of capital= (1 + Cost of Capital \$ ) 1+ Inflation Peso 1+ Inflation \$ = (1.07/1.03) = > 20.37% Aswath Damodaran 44

45 Dealing with Hybrids and Preferred Stock When dealing with hybrids (convertible bonds, for instance), break the security down into debt and equity and allocate the amounts accordingly. Thus, if a firm has \$ 125 million in convertible debt outstanding, break the \$125 million into straight debt and conversion option components. The conversion option is equity. When dealing with preferred stock, it is better to keep it as a separate component. The cost of preferred stock is the preferred dividend yield. (As a rule of thumb, if the preferred stock is less than 5% of the outstanding market value of the firm, lumping it in with debt will make no significant impact on your valuation). Aswath Damodaran 45

46 Recapping the Cost of Capital Cost of borrowing should be based upon (1) synthetic or actual bond rating (2) default spread Cost of Borrowing = Riskfree rate + Default spread Marginal tax rate, reflecting tax benefits of debt Cost of Capital = Cost of Equity (Equity/(Debt + Equity)) + Cost of Borrowing (1-t) (Debt/(Debt + Equity)) Cost of equity based upon bottom-up beta Weights should be market value weights Aswath Damodaran 46

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