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

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.

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.

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%.

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.

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.

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.

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):