Discussion Issues and Derivations

Riskfree Rate

  1. Estimating the riskfree rate when the government is not default free
    We have implicitly assumed in our discussion of risk free rates that the government is a default-free entity and that it issues long term bonds. There are a number of economies where one or both of these assumptions can be challenged. In some emerging market countries, where governments in the past have failed to meet their promised obligations, the government is not viewed as default free. There are many other markets where the government does not issue long term bonds, and the best that one can obtain is a short term government rate.
    There are three solutions to this problem. One is to bypass it entirely by doing the analysis in a different currency (such as the U.S. dollar) where a riskfree rate is easy to obtain. The other is to find the rate at which the largest and safest corporations in that country can borrow long term at in the local currency and reduce that rate by a small default premium (say 20 or 30 basis points) to arrive at a long term riskfree rate. The third solution exists only if there are long term forward contracts on the local currency.Since interest rate parity drives forward contract pricing, the long term local currency rate can be obtained from the price of the forward contract and the long term interest rate on the foreign currency.
  2. Real versus Nominal Riskfree Rates
    Real riskfree rates do not include a premium for expected inflation and should be used if the cash flows are estimated using a similar premise. In practice, it is a good idea to steer away from nominal cash flows and discount rates when inflation hits double digits. One solution is to use a different currency which is more stable; in high-inflation economies, it is common to do investment analyses and valuations in U.S. dollars. The other is to use real cash flows and discount rates.
    Obtaining real riskfree rates can be trivial if a inflation-protected government bond trades in the market. In the United States, for instance, the rate on inflation-linked bonds that were introduced in 1997 , is the real riskfree rate. Unfortunately, this option is generally unavailable in those high-inflation economies where the need to use real riskfree rates is the greatest. In those markets the real risk free rate has to be estimated indirectly. We would propose that the real risk free rate be set equal the the expected long term real growth rate of that economy. While this rate may be about 3% for the U.S. economy, it is likely to be higher for other economies such as Brazil and China.

Risk Premium

  1. Geometric versus Arithmetic Risk Premiums: Which is better?
    The conventional wisdom is that the arithmetic mean is the better estimate. This is true if
    (1) you consider each year to be a period (and the CAPM to be a one-period model)
    (2) annual returns in the stock and bond markets are serially uncorrelated
    As we move to longer time horizons, and as returns become more serially correlated (and empirical evidence suggests that they are), it is far better to use the geometric risk premium. In particular, when we use the risk premium to estimate the cost of equity to discount a cash flow in ten years, the single period in the CAPM is really ten years, and the appropriate returns are defined in geometric terms.
    In summary, the arithmetic mean is more appropriate to use if you are using the Treasury bill rate as your riskfree rate, have a short time horizon and want to estimate expected returns over that horizon.
    The geometric mean is more appropriate if you are using the Treasury bond rate as your riskfree rate, have a long time horizon and want to estimate the expected return over that long time horizon.
  2. The Mechanics of Implied Equity Premiums
    This approach assumes that the market, overall, is correctly priced. Consider, for instance, a very simple valuation model for stocks:
    Value =
    This is essentially the present value of dividends growing at a constant rate forever (see the time value appendix to chapter 5). Three of the four inputs in this model can be obtained externally - the current level of the market (value), the expected dividends next period and the expected growth rate in earnings and dividends in the long term. The only "unknown" is then the required return on equity; when we solve for it, we get an implied expected return on stocks. Subtracting out the riskfree rate will yield an implied equity risk premium.
    To illustrate, assume that the current level of the S&P 500 Index is 900, the expected dividend yield on the index is 2% and the expected growth rate in earnings and dividends in the long term is 7%. Solving for the required return on equity yields the following:
    900 = (.02*900) /(r - .07)
    Solving for r,
    r = (18+63)/900 = 9%
    If the current riskfree rate is 6%, this will yield a premium of 3%.
    The advantage of this approach is that it is market-driven and current, and does not require any historical data. It is, however, bounded by whether the model used for the valuation is the right one and the availability and reliability of the inputs to that model. For instance, in the above example, some might take issue with the use of dividends and the assumption of "constant" growth. Finally, it is based upon the assumption that the market is correctly priced.
  3. Estimating equity risk premium for an emerging market or a market with limited history?
    The risk premium should be a function of the volatility in the underlying economy and the political risk associated with that particular market. Other things remaining equal, we would expect markets which are riskier than the U.S. to have larger premiums than the U.S., especially looking forward. While no direct measure of this risk may exist, most countries are rated by ratings agencies based at least partially on these criteria. The advantages of these ratings are that they can be associated with default premia, that allow us to quantify the effect on the risk premium. The use of country bond ratings to estimate equity risk premiums for these countries may be disquieting to some. In defense, it should be noted that there is a high correlation between country bond premiums and country equity returns. Furthermore, many of the factors considered by the ratings agencies in analysing country bond risk are also factors in evaluating country equity risk.
    It is true that the premium that emerges from looking at a country's rating reflects the default risk on its debt and that the equity premium is likely to be higher, especially for near-term horizons. The default premium from the rating can be scaled upwards, using the relative volatility of the stock and the bond markets in that country. To illustrate, assume that Brazil has a BB rating, and that the default premium is 2%. In addition, assume that the annualized standard deviation in the Brazilian stock market is 34.3%, while the standard deviation in Brazilian government bond prices is 10.9%. The risk premium for Brazil can then be estimated to be:
    Equity Risk Premium for Brazil = US Risk Premium + Default Premium based on Rating (Equity Standard Deviation/ Bond Standard Deviation) = 5.5% + 2% (34.3%/10.9%) = 11.8%. This premium should be adjusted down as the time horizon lengthens.
    An alternative approach exists when all of the country risk can be hedged away using market-traded instruments (options, futures and forwards). The annual percentage cost of hedging away the risk can be added on to the base country premium to arrive at the risk premium for a foreign market. For instance, assume that you are a U.S. investor, with a base premium of 5.5% for domestic investments, and that you can buy insurance against country-specific risk for 2% a year. The total premium used for for this country will then be 7.5%.


  1. Setting Regression Parameters for Beta Estimates: How far back? Daily, weekly or monthly data?
    There are two estimaton decisions the analyst must make in setting up the regression described above. The first concerns the length of the estimation period. Most estimates of betas, including those by Value Line and Standard and Poors, use five years of data, while Bloomberg uses two years of data. The trade-off is simple: A longer estimation period provides more data, but the firm itself might have changed in its risk characteristics over the time period. For instance, using data from 1985 to 1994 to estimate betas for Microsoft might increase the amount of data available, but it will lead to a beta estimate that is much higher than the true beta, since Microsoft was a smaller and riskier firm in 1985 than it was in 1994.
    The second estimation issue relates to the return interval. Returns on stocks are available on an annual, monthly, weekly, daily and even on a intra-day basis. Using daily or intra-day returns will increase the number of observations in the regression, but it exposes the estimation process to a significant bias in beta estimates related to non-trading. For instance, the betas estimated for small firms, which are more likely to suffer from non-trading, are biased downwards when daily returns are used. Using weekly or monthly returns can reduce the non-trading bias significantly. To illustrate, the beta for America Online, a small information services firm, was 1.20 using daily returns from 1990 and 1994, while it was 1.80 using monthly returns. The latter is a much more reliable estimate of the firm’s beta.
  2. Why do beta estimates vary across services?
    It is not uncommon to find very different beta estimates reported for a firm by different services at the same point in time. There are several reasons for these differences –
    1. The services might not be looking at the same historical time period. Value Line and S&P, for instance, use 5-year estimates, while Bloomberg, in its default calculation, uses a 2-year estimate.
    2. The services also often using different return interevals to estimate betas. Bloomberg and Value Line use weekly returns to get their beta estimates while S&P uses monthly returns; there are services that even use daily returns.
    3. The adjustments made to regression betas vary widely across the services. Bloomberg employs the simple adjustment towards one for all the betas that it estimates, whereas a service like Barra adjusts betas using a variety of fundamental information about the firm, such as its dividend yield.
    While these beta differences are troubling, note that the beta estimates delivered by each of these services comes with a standard error, and it is very likely that all of the betas reported for a firm fall within the range of the standard errors from the regressions.
  3. What is the right market index to use in estimating betas?
    In most cases, analysts are faced with a mind boggling array of choices among indices when it comes to estimating betas. Some analysts use only the local index, but others are willing to experiment. One common practice is to use the index that is most appropriate for the investor who is looking at the stock. Thus, if the analysis is being done for a U.S. investor, the S&P 500 index is used. This is generally not appropriate. By this rationale, an investor who owns only two stocks should use an index composed of only those stocks to estimate betas.
    The right index to use in analysis should be determined by who the marginal investor in Aracruz is - a good indicator is to look at the largest holders of stock in the company and the markets where the trading volume is heaviest. If the marginal investor is, in fact, a Brazilian investor, it is reasonable to use a well-constructed Brazilian index. If the marginal investor is a global investor, a more relevant measure of risk may emerge by using the global index. Over time, you would expect global investors to displace local investors as the marginal investors, because they will perceive far less of the risk as market risk and thus pay a higher price for the same security. Thus, one of the ironies of our notion of risk is that Aracruz will be less risky to an overseas investor who has a global portfolio than to a Brazilian investor with all of his or her wealth in Brazilian assets.
  4. Historical Betas versus Expected Betas: Reversion and Financial Fundamentals
    In much of corporate finance and valuation, our interest is in the beta looking forward and not the beta looking back. A regression beta, even if well estimated, reflects the firm as it existed over the period of the regression in terms of business and financial risk. If the firm has changed on either dimension, it can be argued that the beta looking forward will be different from the historical beta. There are three ways in which we can make this adjustment.
    One simplistic way of adjusting historical betas is to assume that betas will move towards one in the long term and adjust beta estimates towards one.
    A more accurate way of estimating forward looking betas is to estimate them from the bottom up, based upon the current business mix of the firm and estimating a weighted average of the sector betas. This can then be levered up using the current or expected financial leverage of the firm.
    A third approach is to relate betas to observable financial characteristics (such as the size of the firm, its dividend yield and debt ratio) through statistical analysis (such as a regression). The firm's specific financial characteristics will then yield a predicted beta for the firm.
  5. Betas and Leverage: Derivation and Extensions
    To estimate the relationship between leverage and betas, let us begin with the assumption that debt bears no market risk (which is consistent with studies that have found that default risk is non-systematic). Debt creates a tax benefit which is reflected on the asset side of the balance sheet (In market value terms):
    Assets Liabilities
    Assets A B(unlev) Debt D 0
    Tax Benefit of Debt tD 0 Equity E B(levered)
    Betas are weighted averages,
    B(unlev) (E + D - tD)/(D+E) = B(levered)(E/(D+E))
    Solve for B(levered),
    B(levered) = B(unlev) (E + D - tD)/E= B(unlev) (1 + (1-t)D/E)

    If debt has a beta (B(Debt))
    B(unlev) (E + D - tD)/(D+E) + B(Debt)) (tD/(D+E)) = B(levered) (E/(D+E)) + B(Debt) D/(D+E)
    B(levered)= B(unlev) (1 + (1-t)D/E) - B(Debt) (1-t) [D/(D+E)]
  6. Accounting Betas
    Changes in earnings at a division or a firm, on a quarterly or annual basis, can be regressed against changes in earnings for the market, in the same periods, to arrive at an estimate of a "market beta" to use in the CAPM. While the approach has some intuitive appeal, it suffers from three potential pitfalls. First, accounting earnings tend to be smoothed out relative to the underlying value of the company, resulting in betas that are "biased down", especially for risky firms, or "biased up", for safer firms. In other words, betas are likely to be closer to one for all firms using accounting data. Second, accounting earnings can be influenced by non-operating factors, such as changes in depreciation or inventory methods, and by allocations of corporate expenses at the divisional level. Finally, accounting earnings are measured, at most, once every quarter, and often only once every year, resulting in regressions with few observations and not much power.
  7. Estimating Bottom-up Betas
    The bottom-up beta for a firm, asset or project can be estimated as follows.
    1. Identify the business or businesses that make up the firm, asset or project.
    2. Estimate the unlevered beta(s) for the business or businesses that the firm is involved in.
    3. To calculate the unlevered beta for the firm, take a weighted average of the unlevered betas, using the market values of the different businesses that the firm is involved in. If the market value is not available, use a reasonable proxy such as operating income or revenues.
    4. Calculate the leverage for the firm, using market values if available. If not, use the target leverage specified by the management of the firm or industry-typical debt ratios.
    5. Estimate the levered beta for the firm (and each of its businesses) using the unlevered beta from step 3 and the leverage from step 4.

Cost of Debt

  1. What is debt?
    Some new securities, at first sight, are difficult to categorize as either debt or equity. To check where on the spectrum between straight debt and straight equity these securities fall, answer the following questions:
    1. Are the payments on the securities contractual or residual?
    - If contractual, it is closer to debt
    - If residual, it is closer to equity
    2. Are the payments tax deductible?
    - If yes, it is closer to debt
    - If no, if is closer to equity
    3. Do the cash flows on the security have a high priority or a low priority if the firm is in financial trouble?
    - If it has high priority, it is closer to debt.
    - If it has low priority, it is closer to equity.
    4. Does the security have a fixed life?
    - If yes, it is closer to debt
    - If no, it is closer to equity
    5. Does the owner of the security get a share of the control of management of the firm?
    - If no, it is closer to debt.
    - If yes, if is closer to equity
  2. Which is a better estimate of the cost of debt - the rating-based interest rate or the yield to maturity on an issued bond?
    The yield to maturity on an issued bond has the advantage of being a market-determined rate. It will be skewed by any special features that the bond may have and the degree to which the bond is secured, relative to other debt. For instance, if the bonds issued by a firm have first priority on the assets, the yield to maturity on these bonds will be lower than the company's true cost of debt, which should represent the cost of the entire debt pool. That is why using the ratings and estimating a cost of debt based on the rating may provide a better estimate of debt.
  3. Estimating Synthetic Ratings
    To estimate synthetic ratings we use the financial characteristics of the firm under question. As an example, if a firm has an interest coverage ratio of 5.2, and other firms with similar interest coverage ratios have an A rating, this firm's synthetic rating is A.


  1. Gross Debt or Net Debt
    Many analysts prefer to subtract the cash from the gross debt to arrive at a net debt figure. If net debt is used, it is important that we be consistent in both our estimation of unlevered beta and in our measure of the debt ratio.
    The unlevered beta that should be used for the firm should then be just the unlevered beta without the cash. When this beta is levered, it will be levered using the net debt ratio, and this estimate of the levered beta will be slightly different from the estimate using gross debt. The difference arises because netting out debt makes the assumption that the tax benefit from having debt is exactly offset by the tax effects of earning interest on cash. It is generally not a good idea to net debt if the debt is very risky (since netting is predicated on the asusmption that debt and cash are both riskless) or if the interest rate earned on cash is substantially lower than the interest rate paid on debt.
  2. Estimating market value of debt when debt is not traded
    The market value of debt is usually more difficult to obtain directly since very few firms have all of their debt in the form of bonds outstanding trading in the market. Many of them have non-traded debt, such as bank debt, which is specified in book value terms but not market value terms. A simple way to convert book value debt into market value debt is to treat the entire debt on the books as one coupon bond, with a coupon set equal to the interest expenses on all of the debt and the maturity set equal to the face-value weighted average maturity of the debt, and to then value this coupon bond at the current cost of debt for the company. Thus, the market value of $ 1billion in debt, with interest expenses of $ 60 million and a maturity of 6 years, when the current cost of debt is 7.5% can be estimated as follows:
    Estimated Market Value of Debt =
  3. Dealing with Operating Leases
    Operating leases are treated as expenses in the income statement. Given that they are commitments to make payments in the future, it is better to take the present value of the operating leases at the unsecured cost of debt and to treat that amount is debt.
  4. Estimating market value of equity
    The market value of equity is generally the number of shares outstanding times the current stock price. If there is more than one class of shares outstanding, the market values of all of these securities should be aggregated and treated as equity. Finally, if there other equity claims in the firm - warrants and conversion options in other securities - these should also be valued and added on to the value of the the equity in the firm.
  5. Dealing with Preferred Stock
    Preferred stock shares some of the characteristics of debt - the preferred dividend is pre-specified at the time of the issue and is paid out before common dividend -- and some of the characteristics of equity - the payments of preferred dividend are not tax deductible. If preferred stock is viewed as perpetual, the cost of preferred stock can be written as follows:
    kps = Preferred Dividend per share/ Market Price per preferred share