## Cost of Equity

The cost of equity is the rate of return that investors require to make an equity investment in a firm. All of the risk and return models described in the previous chapter need a riskfree rate and a risk premium (in the CAPM) or premiums (in the APM and multi-factor models). We will begin by discussing those common inputs before we turn our attention to the estimation of risk parameters.

### I. Riskfree Rate

Most risk and return models in finance start off with an asset that is defined as risk free and use the expected return on that asset as the risk free rate. The expected returns on risky investments are then measured relative to the risk free rate, with the risk creating an expected risk premium that is added on to the risk free rate.

### Requirements for an asset to be riskfree

We defined a riskfree asset as one where the investor knows the expected returns with certainty. Consequently, for an investment to be riskfree, i.e., to have an actual return be equal to the expected return, two conditions have to be met -

• There has to be no default risk, which generally implies that the security has to be issued by a government. Note, though, that not all governments are default free and the presence of government or sovereign default risk can make it very difficult to estimate riskfree rates in some currencies.

• There can be no uncertainty about reinvestment rates, which implies that there are no intermediate cash flows. To illustrate this point, assume that you are trying to estimate the expected return over a five-year period and that you want a risk free rate. A six-month treasury bill rate, while default free, will not be risk free, because there is the reinvestment risk of not knowing what the treasury bill rate will be in six months. Even a 5-year treasury bond is not risk free, since the coupons on the bond will be reinvested at rates that cannot be predicted today. The risk free rate for a five-year time horizon has to be the expected return on a default-free (government) five-year zero coupon bond.

This clearly has painful implications for anyone doing corporate financial analysis, where expected returns often have to be estimated for periods ranging from multiple years. A purist's view of risk free rates would then require different risk free rates for each period and different expected returns. As a practical compromise, however, it is worth noting that the present value effect of using risk free rates that vary from year to year tends to be small for most well behaved1 term structures. In these cases, we could use a duration matching strategy, where the duration of the default-free security used as the risk free asset is matched up to the duration2 of the cash flows in the analysis. If, however, there

1 By well behaved term structures, I would include a normal upwardly sloping yield curve, where long term rates are at most 2-3% higher than short term rates.

2 In investment analysis, where we look at projects, these durations are usually between 3 and 10 years. In valuation, the durations tend to be much longer, since firms are assumed to have infinite lives. The duration are very large differences, in either direction, between short term and long term rates, it does pay to stick with year-specific risk free rates in computing expected returns.

Cash Flows and Risk free Rates: The Consistency Principle

The risk free rate used to come up with expected returns should be measured consistently with how the cash flows are measured. If the cashflows are nominal, the riskfree rate should be in the same currency in which the cashflows are estimated. This also implies that it is not where a project or firm is domiciled that determines the choice of a risk free rate, but the currency in which the cash flows on the project or firm are estimated. Thus, Disney can analyze a proposed project in Mexico in dollars, using a dollar discount rate, or in pesos, using a peso discount rate. For the former, it would use the US treasury bond rate as the riskfree rate but for the latter, it would need a peso riskfree rate.

Under conditions of high and unstable inflation, valuation is often done in real terms. Effectively, this means that cash flows are estimated using real growth rates and without allowing for the growth that comes from price inflation. To be consistent, the discount rates used in these cases have to be real discount rates. To get a real expected rate of return, we need to start with a real risk free rate. While government bills and bonds offer returns that are risk free in nominal terms, they are not risk free in real terms, since expected inflation can be volatile. The standard approach of subtracting an expected inflation rate from the nominal interest rate to arrive at a real risk free rate provides at best an estimate of the real risk free rate. Until recently, there were few traded default-free securities that could be used to estimate real risk free rates; but the introduction of inflation-indexed treasuries has filled this void. An inflation-indexed treasury security does not offer a guaranteed nominal return to buyers, but instead provides a guaranteed real return. In early 2004, for example, the inflation indexed US 10-year treasury bond rate was only 1.6%, much lower than the nominal 10-year bond rate of 4%.

in these cases is often well in excess of ten years and increases with the expected growth potential of the firm.

4.1. ^: What is the right riskfree rate?

The correct risk free rate to use in the capital asset pricing model a. is the short term government security rate b. is the long term government security rate c. can be either, depending upon whether the prediction is short term or long term.

Our discussion, hitherto, has been predicated on the assumption that governments do not default, at least on local borrowing. There are many emerging market economies where this assumption might not be viewed as reasonable. Governments in these markets are perceived as capable of defaulting even on local borrowing. When this is coupled with the fact that many governments do not borrow long term locally, there are scenarios where obtaining a l risk free rate in the local currency, especially for the long term, becomes difficult. In these cases, there are compromises that give us reasonable estimates of the risk free rate.

• Look at the largest and safest firms in that market and use the rate that they pay on their long-term borrowings in the local currency as a base. Given that these firms, in spite of their size and stability, still have default risk, you would use a rate that is marginally lower3 than the corporate borrowing rate.

• If there are long term dollar-denominated forward contracts on the currency, you can use interest rate parity and the treasury bond rate (or riskless rate in any other base currency) to arrive at an estimate of the local borrowing rate.4

• You could adjust the local currency government borrowing rate by the estimated default spread on the bond to arrive at a riskless local currency rate. The default

3 Reducing the corporate borrowing rate by 1% (which is the typical default spread on highly rated corporate bonds in the U.S) to get a riskless rate yields reasonable estimates.

4 For instance, if the current spot rate is 38.10 Thai Baht per US dollar, the ten-year forward rate is 61.36 Baht per dollar and the current ten-year US treasury bond rate is 5%, the ten-year Thai risk free rate (in nominal Baht) can be estimated as follows.

In Practice: What if there is no default-free rate?

3 Reducing the corporate borrowing rate by 1% (which is the typical default spread on highly rated corporate bonds in the U.S) to get a riskless rate yields reasonable estimates.

4 For instance, if the current spot rate is 38.10 Thai Baht per US dollar, the ten-year forward rate is 61.36 Baht per dollar and the current ten-year US treasury bond rate is 5%, the ten-year Thai risk free rate (in nominal Baht) can be estimated as follows.

Solving for the Thai interest rate yields a ten-year risk free rate of 10.12%.

spread on the government bond can be estimated using the local currency ratings5 that are available for many countries. For instance, assume that the Brazilian government bond rate (in nominal Brazilian Reals (BR)) is 14% and that the local currency rating assigned to the Brazilian government is BB+. If the default spread for BB+ rated bonds is 5%, the riskless Brazilian real rate would be 9%. Riskless BR rate = Brazil Government Bond rate - Default Spread = 14% -5% = 9%

### II. Risk premium

The risk premium(s) is clearly a significant input in all of the asset pricing models. In the following section, we will begin by examining the fundamental determinants of risk premiums and then look at practical approaches to estimating these premiums.

What is the risk premium supposed to measure?

The risk premium in the capital asset pricing model measures the extra return that would be demanded by investors for shifting their money from a riskless investment to an average risk investment. It should be a function of two variables:

1. Risk Aversion of Investors: As investors become more risk averse, they should demand a larger premium for shifting from the riskless asset. While of some of this risk aversion may be inborn, some of it is also a function of economic prosperity (when the economy is doing well, investors tend to be much more willing to take risk) and recent experiences in the market (risk premiums tend to surge after large market drops).

2. Riskiness of the Average Risk Investment: As the riskiness of the average risk investment increases, so should the premium. This will depend upon what firms are actually traded in the market, their economic fundamentals and how good they are at managing risk. For instance, the premium should be lower in markets where only the largest and most stable firms trade in the market.

5 Ratings agencies generally assign different ratings for local currency borrowings and dollar borrowing, with higher ratings for the former and lower ratings for the latter.

Since each investor in a market is likely to have a different assessment of an acceptable premium, the premium will be a weighted average of these individual premiums, where the weights will be based upon the wealth the investor brings to the market. Put more directly, what Warren Buffett, with his substantial wealth, thinks is an acceptable premium will be weighted in far more into market prices than what you or I might think about the same measure.

In the arbitrage pricing model and the multi-factor models, the risk premiums used for individual factors are similar wealth-weighted averages of the premiums that individual investors would demand for each factor separately.

Assume that stocks are the only risky assets and that you are offered two investment options:

• A riskless investment (say a Government Security), on which you can make 4%

• A mutual fund of all stocks, on which the returns are uncertain

How much of an expected return would you demand to shift your money from the riskless asset to the mutual fund?

e. Between 10-12%

Your answer to this question should provide you with a measure of your risk premium. (For instance, if your answer is 6%, your premium is 2%.)

### Estimating Risk Premiums

There are three ways of estimating the risk premium in the capital asset pricing model - large investors can be surveyed about their expectations for the future, the actual premiums earned over a past period can be obtained from historical data and the implied premium can be extracted from current market data. The premium can be estimated only from historical data in the arbitrage pricing model and the multi-factor models.

### 1. Survey Premiums

Since the premium is a weighted average of the premiums demanded by individual investors, one approach to estimating this premium is to survey investors about their expectations for the future. It is clearly impractical to survey all investors; therefore, most surveys focus on portfolio managers who carry the most weight in the process. Morningstar regularly survey individual investors about the return they expect to earn, investing in stocks. Merrill Lynch does the same with equity portfolio managers and reports the results on its web site. While numbers do emerge from these surveys, very few practitioners actually use these survey premiums. There are three reasons for this reticence:

— There are no constraints on reasonability; individual money managers could provide expected returns that are lower than the riskfree rate, for instance.

— Survey premiums are extremely volatile; the survey premiums can change dramatically, largely as a function of recent market movements.

— Survey premiums tend to be short term; even the longest surveys do not go beyond one year.

^ 4.3: Do risk premiums change?

In the previous question, you were asked how much of a premium you would demand for investing in a portfolio of stocks as opposed to a riskless asset. Assume that the market dropped by 20% last week, and you were asked the same question today. Would your premium a. be higher?

c. be unchanged?

### 2. Historical Premiums

The most common approach to estimating the risk premium(s) used in financial asset pricing models is to base it on historical data. In the arbitrage pricing model and multi- factor models, the raw data on which the premiums are based is historical data on asset prices over very long time periods. In the CAPM, the premium is defined as the difference between average returns on stocks and average returns on risk-free securities over an extended period of history.

### Basics

In most cases, this approach is composed of the following steps. It begins by defining a time period for the estimation, which can range to as far back as 1871 for U.S. data. It then requires the calculation of the average returns on a stock index and average returns on a riskless security over the period. Finally, it calculates the difference between the returns on stocks and the riskless return and uses it as a risk premium looking forward. In doing so, we implicitly assume that

1. 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.)

2. It assumes that the average riskiness of the "risky" portfolio (stock index) has not changed in a systematic way across time.

### Estimation Issues

While users of risk and return models may have developed a consensus that historical premium is, in fact, the best estimate of the risk premium looking forward, there are surprisingly large differences in the actual premiums we observe being used in practice. For instance, the risk premium estimated in the US markets by different investment banks, consultants and corporations range from 4% at the lower end to 12% at the upper end. Given that we almost all use the same database of historical returns, provided by Ibbotson Associates6, summarizing data from 1926, these differences may seem surprising. There are, however, three reasons for the divergence in risk premiums. • Time Period Used: While there are many who use all the data going back to 1926, there are almost as many using data over shorter time periods, such as fifty, twenty or even ten years to come up with historical risk premiums. The rationale presented by those who use shorter periods is that the risk aversion of the average investor is likely to change over time and that using a shorter and more recent time period provides a

6 See "Stocks, Bonds, Bills and Inflation", an annual edition that reports on the annual returns on stocks, treasury bonds and bills, as well as inflation rates from 1926 to the present. (http://www.ibbotson.com)

more updated estimate. This has to be offset against a cost associated with using shorter time periods, which is the greater noise in the risk premium estimate. In fact, given the annual standard deviation in stock prices7 between 1928 and 2002 of 20%, the standard error8 associated with the risk premium estimate can be estimated as follows for different estimation periods in Table 4.1.

Estimation Period |
Standard Error of Risk Premium Estimate |

5 years |

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