1. Estimating the Discount Rate
P.V. Viswanath
Based on Damodaran’s Corporate Finance

2. Inputs required to use the CAPM
According to the CAPM, the required rate of return on an asset will be:
Required ROR = Rf + b (E(Rm) - Rf)
The inputs required to estimate the required ROR are:
(a) the current risk-free rate
(b) the expected market risk premium (the premium expected for investing in risky assets over the riskless asset)
(c) the beta of the asset being analyzed.
Summarizes the inputs. Note that we are replacing the last component (E(Rm-Rf) with the expected risk premium..
P.V. Viswanath

3. The Riskfree Rate
The riskfree rate is the rate on a zero coupon government bond matching the time horizon of the cash flow being analyzed.
Theoretically, this means using different riskfree rates for each cash flow - the 1 year zero coupon rate for the cash flow in year 1, the 2-year zero coupon rate for the cash flow in year
Practically, if there is substantial uncertainty about expected cash flows, it is enough to use a single riskfree rate for all flows.
Using a long term government rate (even on a coupon bond) as the riskfree rate on all of the cash flows in a long term analysis will yield a close approximation of the true value.
For short term analysis, it is appropriate to use a short term government security rate as the riskfree rate.
From a present value standpoint, using different riskfree rates for each cash flow may be overkill, except in those cases where your interest rates are very different for different time horizons (a very upward sloping or downward sloping yield curve)
P.V. Viswanath

4. Measurement of the risk premium
The risk premium is the premium that investors demand for investing in an average risk investment, relative to the riskfree rate.
As a general proposition, this premium should be
greater than zero
increase with the risk aversion of the investors in that market
increase with the riskiness of the “average” risk investment
Implicit here are two questions - Which investor’s risk premium? What is the average risk investment?
P.V. Viswanath

5. The Historical Premium Approach
This is the default approach used by most to arrive at the premium to use in the model
In most cases, this approach does the following
it defines a time period for the estimation (1926-Present, 1962-Present....)
it calculates average returns on a stock index during the period
it calculates average returns on a riskless security over the period
it calculates the difference between the two
and uses it as a premium looking forward
The limitations of this approach are:
it assumes that the risk aversion of investors has not changed in a systematic way across time. (The risk aversion may change from year to year, but it reverts back to historical averages)
it assumes that the riskiness of the “risky” portfolio (stock index) has not changed in a systematic way across time.
This is the basic approach used by almost every large investment bank and consulting firm.
P.V. Viswanath

6. Historical Average Premiums for the United States
Many practitioners use the Ibbotson data base and estimate historical premiums. There are three reasons for why the premium estimated may differ:
1. How far back you go (My personal bias is to go back as far as possible. Stock prices are so noisy that you need very long time periods to get reasonable estimates)
2. Whether you use T.Bill or T.Bond rates ( You have to be consistent. Since I will be using the T.Bond rate as my riskfree rate, I will use the premium over that rate)
3. Whether you use arithmetic or geometric means (If returns were uncorrelated over time, and you were asked to estimate a 1-year premium, the arithmetic mean would be used. Since returns are negatively correlated over time, and we are estimating premiums over longer holding periods, it makes more sense to use the compounded return, which gives us the geometric average)
Thus, I should be using 6.60% as my premium.
In practice, a risk premium of about 5.5% is used. However, depending upon perceptions of investor risk preferences at a given time, this number can be moved upwards or downwards.
P.V. Viswanath

7. Estimating Beta
The standard procedure for estimating betas is to regress stock returns (Rj) against market returns (Rm) -
Rj = a + b Rm
where a is the intercept and b is the slope of the regression.
Often five years of monthly data is used to estimate these parameters.
The slope of the regression corresponds to the beta of the stock, and measures the riskiness of the stock.
Betas reflect not just the volatility of the underlying investment but also how it moves with the market:
Beta (Slope) = Correlationjm (sj / sm)
Note that sj can be high but beta can be low (because the asset is not very highly correlated with the market)
P.V. Viswanath

8. Estimating Performance
The intercept of the regression provides a simple measure of performance during the period of the regression, relative to the capital asset pricing model.
Rj = Rf bj (Rm - Rf)
= Rf (1-bj) + b Rm Capital Asset Pricing Model
Rj = aj + bj Rm Regression Equation
If aj > Rf (1-bj) ..Stock did better than expected during reg period
aj = Rf (1-bj) ..Stock did as well as expected during regr period
aj < Rf (1-bj) ..Stock did worse than expected during reg period
Jensen's alpha, a measure of stock performance, is measure as aj - Rf (1-b)
Jensen’s alpha can also be computed by estimating the expected return during the period of the regression, using the actual return on the market during the period, the riskfree rate during the period and the estimated beta, and then comparing it to the actual return over the period.
Algebraically, you should get the same answer.
P.V. Viswanath

9. Example: Estimating Expected Returns
Boeing’s Beta was estimate on December 31, 1998 to be 0.96
Riskfree Rate = 5.00% (Long term Government Bond rate)
Risk Premium = 5.50% (Approximate historical premium)
Expected Return = 5.00% (5.50%) = 10.31%
Note that this expected return would have been different if we had decided to use a different historical premium or the implied premium.
As a potential investor in Boeing, what does this expected return of 10.31% tell you?
This is the return that I can expect to make in the long term on Boeing, if the stock is correctly priced and the CAPM is the right model for risk,
This is the return that I need to make on Boeing in the long term to break even on my investment in the stock
P.V. Viswanath

10. How managers use this expected return
Managers at Boeing
need to make at least 10.31% as a return for their equity investors to break even.
this is the hurdle rate for projects, when the investment is analyzed from an equity standpoint
In other words, Boeing’s cost of equity is 10.31%.
What is the cost of not delivering this cost of equity?
The cost of equity is what equity investors in your company view as their required return.
The cost of not delivering this return is more unhappy stockholders, a lower stock price, and if you are a manager, maybe your job.
Going back to the corporate governance section, if stockholders have little or no control over managers, managers are less likely to view this as the cost of equity.
P.V. Viswanath

11. Fundamental Determinants of Betas
Type of Business: Firms in more cyclical businesses or that sell products that are more discretionary to their customers will have higher betas than firms that are in non-cyclical businesses or sell products that are necessities or staples.
Operating Leverage: Firms with greater fixed costs (as a proportion of total costs) will have higher betas than firms will lower fixed costs (as a proportion of total costs)
Financial Leverage: Firms that borrow more (higher debt, relative to equity) will have higher equity betas than firms that borrow less.
Betas do not come from regressions. They come from what firms do - what business they are in, what their cost structure is and how much debt they carry.
P.V. Viswanath

12. Equity Betas and Leverage
The beta of equity alone can be written as a function of the unlevered beta and the debt-equity ratio
L = u (1+ (1-t)D/E)
where
L = Levered or Equity Beta
u = Unlevered Beta
t = Corporate marginal tax rate
D = Market Value of Debt
E = Market Value of Equity
The unlevered beta measures the riskiness of the business that a firm is in and is often called an asset beta.
This is based upon two assumptions
Debt bears no market risk (which is consistent with studies that have found that default risk is non-systematic)
Debt creates a tax benefit
Assets Liabilities
Assets A (bu) Debt D (bD =0)
Tax Benefits tD (bD=0) Equity E (bL)
Betas are weighted averages,
Bu (E + D - tD)/(D+E) = bL(E/(D+E))
Solve for bL,
bL = Bu (E + D - tD)/E= Bu (1 + (1-t)D/E)
If debt has a beta (bD)
Bu (E + D - tD)/(D+E) + bD tD/(D+E) = bL(E/(D+E)) + bD D/(D+E)
bL = Bu (1 + (1-t)D/E) - bD ((1-t) D/E))
P.V. Viswanath

13. Effects of leverage on betas: Boeing
The regression beta for Boeing is This beta is a levered beta (because it is based on stock prices, which reflect leverage) and the leverage implicit in the beta estimate is the average market debt equity ratio during the period of the regression (1993 to 1998)
The average debt equity ratio during this period was 17.88%.
The unlevered beta for Boeing can then be estimated:(using a marginal tax rate of 35%)
= Current Beta / (1 + (1 - tax rate) (Average Debt/Equity))
= 0.96 / ( 1 + ( ) (0.1788)) = 0.86
Note that betas reflect the average leverage over the period and not the current leverage of the firms. Firms whose leverage has changed over the period will have regression betas that are different from their true betas.
P.V. Viswanath

14. Boeing : Beta and Leverage
Debt to Debt/Equity Beta Effect Capital Ratio of Leverage
0.00% 0.00%
10.00% 11.11%
20.00% 25.00%
30.00% 42.86%
40.00% 66.67%
50.00% %
60.00% %
70.00% %
80.00% %
90.00% %
Since equity investors bear all of the non-diversifiable risk, the beta of Disney’s equity will increase as the leverage increases.
P.V. Viswanath

15. Betas are weighted Averages
The beta of a portfolio is always the market-value weighted average of the betas of the individual investments in that portfolio.
Thus,
the beta of a mutual fund is the weighted average of the betas of the stocks and other investment in that portfolio
the beta of a firm after a merger is the market-value weighted average of the betas of the companies involved in the merger.
Betas are always weighted averages - where the weights are based upon market value. This is because betas measure risk relative to a market index.
P.V. Viswanath

16. Firm Betas versus divisional Betas
Firm Betas as weighted averages: The beta of a firm is the weighted average of the betas of its individual projects.
At a broader level of aggregation, the beta of a firm is the weighted average of the betas of its individual division.
The same principle applies to a firm. To the degree that the firm is in multiple businesses, its beta reflects all of these businesses.
P.V. Viswanath

17. Bottom-up versus Top-down Beta
The top-down beta for a firm comes from a regression
The bottom up beta can be estimated by doing the following:
Find out the businesses that a firm operates in
Find the unlevered betas of other firms in these businesses
Take a weighted (by sales or operating income) average of these unlevered betas
Lever up using the firm’s debt/equity ratio
The bottom up beta will give you a better estimate of the true beta when
the standard error of the beta from the regression is high (and) the beta for a firm is very different from the average for the business
the firm has reorganized or restructured itself substantially during the period of the regression
when a firm is not traded
Bottom-up betas build up to the beta from the fundamentals, rather than trusting the regression.
The standard error of an average beta for a sector, is smaller by a factor of √n, where n is the number of firms in the sector. Thus, if there are 25 firms in a sector, the standard error of the average is 1/5 the average standard error.
P.V. Viswanath

18. The Home Depot’s Comparable Firms
Note that
Comparable firms carries an element of subjective judgment. You could look at this list and argue that these firms are not truly comparable to the Home Depot. They might be smaller and riskier than the Home Depot. They might carry more or less operating leverage.
P.V. Viswanath

19. Estimating The Home Depot’s Bottom-up Beta
Average Beta of comparable firms = 0.93
D/E ratio of comparable firms = ( )/16,232 = 14.01%
Unlevered Beta for comparable firms = 0.93/(1+(1-.35)(.1401))
= 0.86
If the Home Depot’s D/E ratio is 20%, our bottom-up estimate of Home Depot’s beta is 0.86[1+(1-.35)(.2)] =
I averaged first and then unlevered the beta for comparable firms.
Alternatively, you could have unlevered each firm’s beta and then averaged the unlevered betas. The reason I do not do this is because the individual firms’s betas might be misestimated. Conceptually, however, there is no problem with doing this. (The answer is not that different; the average of the unlevered betas is 0.88)
P.V. Viswanath

20. From Cost of Equity to Cost of Capital
The cost of capital is a composite cost to the firm of raising financing to fund its projects.
In addition to equity, firms can raise capital from debt.
If the firm has bonds outstanding, and the bonds are traded, the yield to maturity on a long-term, straight (no special features) bond can be used as the interest rate.
If the firm is rated, use the rating and a typical default spread on bonds with that rating to estimate the cost of debt.
If the firm is not rated,
and it has recently borrowed long term from a bank, use the interest rate on the borrowing or
estimate a synthetic rating for the company, and use the synthetic rating to arrive at a default spread and a cost of debt
Capital is more than just equity. It also includes other financing sources, including debt.
P.V. Viswanath

21. Estimating Market Value Weights
Market Value of Equity should include the following
Market Value of Shares outstanding
Market Value of Warrants outstanding
Market Value of Conversion Option in Convertible Bonds
Market Value of Debt is more difficult to estimate because few firms have only publicly traded debt. There are two solutions:
Assume book value of debt is equal to market value
Estimate the market value of debt from the book value; for Boeing, the book value of debt is $6,972 million, the interest expense on the debt is $ 453 million, the average maturity of the debt is years and the pre-tax cost of debt is 5.50%.
Estimated MV of Boeing Debt =
The market value of debt is estimated by considering all debt as if it were one large coupon bond.
The average maturity of debt can be obtained from the 10-K.
P.V. Viswanath

22. Estimating Cost of Capital: Boeing
Equity
Cost of Equity = 5% (5.5%) = 10.58%
Market Value of Equity = $32.60 Billion
Equity/(Debt+Equity ) = 82%
Debt
After-tax Cost of debt = 5.50% (1-.35) = 3.58%
Market Value of Debt = $ 8.2 Billion
Debt/(Debt +Equity) = 18%
Cost of Capital = 10.58%(.80)+3.58%(.20) = 9.17%
Summarizes the inputs from the last 90 pages.
The financing choice becomes simpler if the sources of capital can be boiled down to debt and equity. Consequently, we have condensed all of the debt -short as well as long term debt- into one figure and attached the long term cost of debt to it. (We are implicitly assuming that the rolled-over cost of short term debt is equal to the long term cost of debt)
Special cases:
Hybrid Securities: If you have convertible debt, it is best to break down the convertible debt into debt and equity components.
Preferred Stock: This has to be treated as a third component of capital, with a cost set equal to the preferred dividend yield (without a tax benefit)
P.V. Viswanath

23. Choosing a Hurdle Rate
Either the cost of equity or the cost of capital can be used as a hurdle rate, depending upon whether the returns measured are to equity investors or to all claimholders on the firm (capital)
If returns are measured to equity investors, the appropriate hurdle rate is the cost of equity.
If returns are measured to capital (or the firm), the appropriate hurdle rate is the cost of capital.
While the cost of equity and capital can be very different numbers, they can both be used as hurdle rates, as long as the returns and cash flows are defined consistently.
P.V. Viswanath

24. Back to First Principles
Invest in projects that yield a return greater than the minimum acceptable hurdle rate.
The hurdle rate should be higher for riskier projects and reflect the financing mix used - owners’ funds (equity) or borrowed money (debt)
Returns on projects should be measured based on cash flows generated and the timing of these cash flows; they should also consider both positive and negative side effects of these projects.
Choose a financing mix that minimizes the hurdle rate and matches the assets being financed.
If there are not enough investments that earn the hurdle rate, return the cash to stockholders.
The form of returns - dividends and stock buybacks - will depend upon the stockholders’ characteristics.
P.V. Viswanath