Aswath Damodaran, Applied Corporate Finance, Study notes of Strategic Management

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Applied Corporate Finance

Aswath Damodaran

Stern School of Business, NYU

adamodar@stern.nyu.edu

Fourth Edition

Forthcoming in 2014

CHAPTER 4

RISK MEASUREMENT AND HURDLE RATES IN PRACTICE

In the last chapter, we presented the argument that the expected return on an

equity investment should be a function of the market or non-diversifiable risk embedded

in that investment. Here we turn our attention to how best to estimate the parameters of

market risk in each of the models described in the previous chapter—the capital asset

pricing model, the arbitrage-pricing model, and the multifactor model. We will present

three alternative approaches for measuring the market risk in an investment; the first is to

use historical data on market prices for the firm considering the project, the second is to

use the market risk parameters estimated for other firms that re in the same business as

the project being analyzed, and the third is to use accounting earnings or revenues to

estimate the parameters.

In addition to estimating market risk, we will also discuss how best to estimate a

riskless rate and a risk premium (in the CAPM) or risk premiums (in the APM and

multifactor models) to convert the risk measures into expected returns. We will present a

similar argument for bringing default risk into a cost of debt and then bring the discussion

to fruition by combining both the cost of equity and debt to estimate a cost of capital,

which will become the minimum acceptable hurdle rate for an investment.

Cost of Equity

The cost of equity is the rate of return that investors require to invest in the equity

of a firm. All of the risk and return models described in the previous chapter need a risk-

free rate and a risk premium (in the CAPM) or premiums (in the APM and multifactor

models). We begin by discussing those common inputs before turning attention to the

estimation of risk parameters.

I. Risk-Free 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

are very large differences in either direction between short-term and long-term rates, it

does pay to use year-specific risk-free rates in computing expected returns.

Estimating Risk Free Rates

The risk-free rate used to come up with expected returns should be measured

consistently with how the cash flows are measured. If the cash flows are nominal, the

risk-free rate should be in the same currency in which the cash flows are estimated. This

also implies that it is not where a project or firm is located 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 Germany in US dollars, using

a dollar discount rate, or in Euros, using a Euro discount rate.

Risk free rates: Default Free Governments

If you assume that governments are default free, the simplest measure of a risk

free rate is the interest rate on a market-traded long-term government bond. But how we

do make this judgment on the default risk in the government? The easiest way is to use

the sovereign rating for the government and to assume that any sovereign that is Aaa

rated is default free, though it comes at the cost of having to trust the ratings agencies to

be right in their assessment. Using this principle, we estimate risk free rates in ten

currencies in November 2013 in figure 4.1, where the issuing governments are Aaa rated

by Moody’s, and assuming that this rating implies no default risk.

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

of the firm.

Note that if these are truly default free rates, the key reason for differences in risk free

rates across currencies is expected inflation. The risk free rate in Australian dollars is

higher than the risk free rate in Swiss Francs, because expected inflation is higher in

Australia than in Switzerland.

A Real Risk free 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. Although government bills and

bonds offer returns that are risk-free in nominal terms, they are not risk-free in real terms,

because 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 only an estimate of the real risk-free rate.

Using this approach, we are able to derive risk free rates in a host of currencies in Figure

4.2, where the issuing government is perceived as having default risk:

The risk free rates vary widely across these emerging market currencies for the same

reason that they do across developed market currencies in Figure 4.1, i.e., because of

differences in expected inflation.

Risk free Rate: No Local Currency Government Bonds

The starting point in the preceding analyses of risk free rates, for governments with or

without default risk, is a long-term government bond in the local currency, with a market-

set interest rate. There are some currencies, though, where the government either does not

issue long term bonds in the local currency or those bonds are not market-traded. There

are two choices for an analyst facing this problem:

a. Build up approach: The risk free rate in a currency is composed of two components:

an expected inflation rate and an expected real interest rate. If you have an estimate of

expected inflation in a currency, you can build up a risk free rate in that currency by

adding a real interest rate to it. The latter can be estimated either by using the

inflation indexed treasury bond (TIPs) rate or set equal to expected long term real

growth in the economy. Thus, if the expected inflation rate in Vietnamese Dong is

9.5% and the ten-year TIPs rate is 0.50%, the risk free rate in Vietnamese Dong is

b. Differential Inflation approach: In a variant, you can start with the risk free rate in US

dollars or Euros and add the differential between expected inflation in the currency in

question and expected inflation in the US dollar to estimate a risk free rate in the local

currency. For example, if the ten-year US treasury bond rate is 2.5% and the expected

inflation rate in Peruvian Sul is 3% higher than the expected inflation rate in the US,

the Peruvian Sul is 5.5%.

The key conclusion, though, is that currencies matter in analysis because they have

different expectations of inflation embedded in them. As long as you are consistent about

assuming the same expected inflation rate in both your cash flows and your discount rate,

it matters little what currency you do your analysis in.

4.1. What Is the Right Risk-Free Rate?

The correct risk-free rate to use in the CAPM

a. is the short term government security rate.

b. is the long term government security rate.

c. can be either, depending on whether the prediction is short-term or long-term.

Illustration 4.1: Estimating Riskfree Rates

The companies that we are analyzing in this book include two US companies,

(Disney and Bookscape), a Brazilian company (Vale), an Indian company (Tata Motors),

a Chinese company (Baidu) and a German bank (Deutsche Bank). We estimated riskfree

rates in five currencies, as well as in real terms, on November 5, 2013, and will use these

riskfree rates for the rest of the book:

a. In US dollars (2.75%): The ten-year US treasury bond rate was 2. 7 5%. While

concerns about the credit worthiness of the US government have increased in recent

years, we will use 2 .5% as the riskfree rate in any dollar based computation.

b. In Euros (1.75%): For a Euro riskfree rate, we looked at ten-year Euro denominated

government bonds and noted that at least 12 different European governments have

accept these CDS spreads are measures of the sovereign default spreads, the default

spread for Brazil would be 2.59%. Note that the quirks in the CDS market, where there is

counter party risk and other frictions, result in no country having a CDS spread of zero.

One modification that you could use is to net out the CDS spread of the country with the

lowest spread (Norway has a spread of 0.32% in November 2013) from the rest to get a

more robust version of the default spread to use in estimating risk free rates.

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 CAPM measures the extra return that would be

demanded by investors for shifting their money from a riskless investment to the market

portfolio or risky investments, on average. 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. Although some of this

risk aversion may be inherent, 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 on what firms are

actually traded in the market, their economic fundamentals, and how involved they

are in managing risk.

Because each investor in a market is likely to have a different assessment of an

acceptable equity risk premium, the premium will be a weighted average of these

individual premiums, where the weights will be based on 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 APM and the multifactor 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.

4.2 What Is Your Risk Premium?

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 percent
• 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?

a. Less than 4 percent

b. Between 4 and 6 percent

c. Between 6 and 8 percent

d. Between 8 and10 percent

e. Between 10 and 12 percent

f. More than 12 percent

Your answer to this question should provide you with a measure of your risk premium.

(For instance, if your answer is 6 percent, your premium is 2 percent.)

Estimating Risk Premiums

There are three ways of estimating the risk premium in the CAPM: 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 APM and the multi-factor models.

1. Survey Premiums

Because the premium is a weighted average of the premiums demanded by

individual investors, one approach to estimating this premium is to survey investors about

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 APM and multifactor models,

the premiums are based on historical data on asset prices over very long time periods

which are used to extract factor-specific risk premiums. 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, the difference between the average

returns on stocks and the riskless return it is defined as the risk premium looking forward.

In doing this, we implicitly assume the following:

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. The average riskiness of the “risky” portfolio (stock index) has not changed in a

systematic way across time.

Estimation Issues

Users of risk and return models may have developed a consensus that the historical

premium is in fact the best estimate of the risk premium looking forward, but there are

surprisingly large differences in the actual premiums used in practice. For instance, the

risk premium estimated in the U.S. markets by different investment banks, consultants,

and corporations range from 4 percent at the lower end to 12 percent at the upper end.

Given that they almost all use the same database of historical returns, provided by

Ibbotson Associates,^9 summarizing data from 1926, these differences may seem

surprising. There are, however, three reasons for the divergence in risk premiums.

9 See “Stocks, Bonds, Bills and Inflation,” an annual publication that reports on the annual returns on

stocks, Treasury bonds and bills, and inflation rates from 1926 to the present. Available online at

www.ibbotson.com.

• Time Period Used : Although there are some who use all of the Ibbotson which goes

back to 1926, there are 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 using a shorter and more recent time period

provides a more updated estimate. This has to be offset against a cost associated with

using shorter time periods, which is the greater estimation error in the risk premium

estimate. In fact, given the annual standard deviation in stock returns between 1928

and 20 13 of 20 percent,^10 the standard error associated with the risk premium estimate

can be estimated as follows for different estimation periods in Table 4.2.^11

Table 4.2 Standard Errors in Risk Premium Estimates

Estimation Period Standard Error of Risk Premium Estimate

5 years 20/√5 = 8.94%

10 years 20/√10 = 6.32%

25 years 20/√25 = 4.00%

50 years 20/√50 = 2.83%

Note that to get reasonable standard errors, we need very long time periods of

historical returns. Conversely, the standard errors from ten- and twenty-year estimates

are likely to be almost as large or larger than the actual risk premiums estimated. This

cost of using shorter time periods seems, in our view, to overwhelm any advantages

associated with getting a more updated premium.

• Choice of Risk-Free Security : The Ibbotson database reports returns on both Treasury

bills and bonds and the risk premium for stocks can be estimated relative to each.

Given that short term rates have been lower than long term rates in the United States

for most of the past seven decades, the risk premium is larger when estimated relative

to shorter-term government securities (such as Treasury bills). The risk-free rate

chosen in computing the premium has to be consistent with the risk-free rate used to

(^10) For the historical data on stock returns, bond returns, and bill returns, check under Updated Data at

www.damodaran.com..

arithmetic as opposed to geometric averages. The effect of these choices is summarized

in Table 4.3, which uses returns from 1928 to 20 12.^14

Table 4.3 Historical Risk Premiums (%) for the United States, 1928- 2012

Stocks – Treasury Bills Stocks – Treasury Bonds

Arithmetic Geometric Arithmetic Geometric

1928 – 2012 7.65%

(2.20%)

Note that even with only three slices of history considered, the premiums range from

1.72% to 7.65%, depending upon the choices made. If we take the earlier discussion

about the “right choices” to heart, and use a long-term geometric average premium over

the long-term rate as the risk premium to use in valuation and corporate finance, the

equity risk premium that we would use would be 4.20%. The numbers in brackets below

the arithmetic average premiums are the standard errors in the estimates and note that

even the estimate over the longest period (1928-2012) comes with significant standard

error and that the ten-year estimate is almost useless given the standard error.

Historical Premiums in Other Markets

Although historical data on stock returns is easily available and accessible in the

United States, it is much more difficult to get for foreign markets. The most detailed look

at these returns estimated the returns you would have earned on twenty equity markets

between 1900 and 2012 and compared these returns with those you would have earned

investing in bonds.^15 Table 4.4 presents the risk premiums—that is, the additional

returns—earned by investing in equity over short term and long-term government bonds

over that period in each of the fourteen markets.

Table 4.4 Equity Risk Premiums by Country

Stocks minus Short term Governments Stocks minus Long term Governments

(^14) The raw data on Treasury bill rates, Treasury bond rates, and stock returns was obtained from the Federal

Reserve data archives maintained by the Fed in St. Louis.

(^15) Dimson, E.,, P Marsh and M Staunton, 2002, Triumph of the Optimists: 101 Years of Global Investment

Returns, Princeton University Press, NJ and Global Investment Returns Yearbook, 2006, ABN

AMRO/London Business School.

Country

Geometric Mean

Arithmetic Mean

Standard Error

Standard Deviation

Geometric Mean

Arithmetic Mean

Standard Error

Standard Deviation

Australia 6.6% 8.1% 1.7% 17.6% 5.6% 7.5% 1.9% 19.9%

Austria 5.6% 10.5% 3.6% 37.7% 2.8% 22.1% 14.7% 154.8%

Belgium 2.7% 5.2% 2.3% 24.0% 2.3% 4.3% 2.0% 21.0%

Canada 4.1% 5.5% 1.6% 17.1% 3.4% 5.0% 1.7% 18.3%

Denmark 2.8% 4.6% 1.9% 20.5% 1.8% 3.3% 1.6% 17.5%

Finland 5.8% 9.3% 2.8% 30.0% 5.3% 8.9% 2.8% 30.1%

France 5.9% 8.6% 2.3% 24.4% 3.0% 5.3% 2.1% 22.8%

Germany 5.9% 9.8% 3.0% 31.7% 5.2% 8.6% 2.7% 28.4%

Ireland 3.2% 5.4% 2.0% 21.3% 2.6% 4.6% 1.9% 19.8%

Italy 5.6% 9.5% 3.0% 31.8% 3.4% 6.8% 2.8% 29.5%

Japan 5.7% 8.9% 2.6% 27.6% 4.8% 8.9% 3.1% 32.7%

Netherlands 4.2% 6.4% 2.1% 22.7% 3.3% 5.6% 2.1% 22.2%

New

Zealand

4.2% 5.8% 1.7% 18.3% 3.7% 5.3% 1.7% 18.1%

Norway 2.9% 5.8% 2.5% 26.3% 2.2% 5.2% 2.6% 27.8%

South

Africa

6.3% 8 .3% 2.1% 21.9% 5.4% 7.1% 1.8% 19.5%

Spain 3.1% 5.3% 2.0% 21.7% 2.1% 4.1% 1.9% 20.7%

Sweden 3.6% 5.7% 1.9% 20.6% 2.9% 5.1% 2.0% 20.8%

Switzerland 3.4% 5.1% 1.8% 18.8% 2.0% 3.5% 1.7% 17.6%

U.K. 4.3% 6.0% 1.9% 19.8% 3.7% 5.0% 1.6% 17.1%

U.S. 5.3% 7.2% 1.8% 19.6% 4.2% 6.2% 1.9% 20.5%

Europe 3.3% 5.1% 1.8% 19.3% 3.4% 4.8% 1.5% 16.3%

World-ex

U.S.

3.5% 5.1% 1.8% 18.6% 3.0% 4.1% 1.4% 14.7%

World 4.1% 5.5% 1.6% 17.0% 3.2% 4.4% 1.4% 15.3%

Note that the risk premiums, averaged across the markets, are lower than risk premiums

in the United States. For instance, the geometric average risk premium for stocks over

long-term government bonds, across the non-US markets, is only 3.0%, lower than the

4.2% for the US markets. The results are similar for the arithmetic average premium,

with the average premium of 3.5% across markets being lower than the 5.3% for the

United States. In effect, the difference in returns captures the survivorship bias, implying

that using historical risk premiums based only on US data will results in numbers that are

too high for the future. Note that the “noise” problem persists, even with averaging across

20 markets and over 112 years. The standard error in the global equity risk premium

India’s default spread of 2.00% (based on the rating) to the US equity risk premium

and similar adjustments would be made for Brazil and China.

ERP for India = 4.20 % (US ERP) + 2.25% (Default spread for India) = 6.45%

ERP for Brazil = 4.20% (US ERP) + 2.00% (Default spread for Brazil) = 6.20%

ERP for China = 4.20% (US ERP) + 0.80% (Default spread for China) = 5.00%

The cost of equity for a company that operates in these markets would then be based

on these larger equity risk premiums, scaled up or down for individual companies,

based on their betas.

Cost of equity = Risk-free rate + Beta * (U.S. ERP + Country Bond Default Spread)

In a variation on this approach, some analysts prefer to add the default spread

separately (and thus not scale the value to the beta):

Cost of equity = Risk-free rate + Beta * (U.S. ERP) + Country Bond Default Spread

One reason that we adjusted the government bond rate for the default spread is to

prevent the double counting that will occur, if you don’t make that adjustment, since

both the risk free rate and the equity risk premium will then have the default spreads

embedded in them.

2. Relative Standard Deviation : There are some analysts who believe that the equity

risk premiums of markets should reflect the differences in equity risk, as measured by

the volatilities of these markets. A conventional measure of equity risk is the standard

deviation in stock prices; higher standard deviations are generally associated with

more risk. If you scale the standard deviation of one market against another, you

obtain a measure of relative risk.

Relative Standard Deviation (^) Country X =

Standard Deviation (^) Country X

Standard Deviation (^) US

This relative standard deviation when multiplied by the premium used for U.S. stocks

should yield a measure of the total risk premium for any market.

Equity risk premiumCountry X = Risk PremiumU.S. * Relative Standard deviationCountry X

Assume for the moment that you are using a mature market premium for the United

States of 4.20 percent and the annual standard deviation of U.S. stocks is 15 percent.

The annualized standard deviation in the Brazilian equity index is 21 percent,

18

yielding a total risk premium for Brazil:

Equity Risk Premium Brazil

The country risk premium can be isolated as follows:

Country Risk PremiumBrazil = 5.88% – 4.20% = 1.68%

A similar approach could be used for India and China, yielding the following:

Equity Risk PremiumIndia = 4. 20 %*

Equity Risk PremiumChina = 4. 20 %*

Although this approach has intuitive appeal, there are problems with using standard

deviations computed in markets with widely different market structures and liquidity.

There are very risky emerging markets that have low standard deviations for their

equity markets because the markets are illiquid. This approach will understate the

equity risk premiums in those markets.

3. Default Spreads + Relative Standard Deviations : The country default spreads that

come with country ratings provide an important first step, but still only measure the

premium for default risk. Intuitively, we would expect the country equity risk

premium to be larger than the country default risk spread since equities are riskier

than bonds. To address the issue of how much higher, we look at the volatility of the

equity market in a country relative to the volatility of the country bond used to

estimate the default spread. This yields the following estimate for the country equity

risk premium.

Country Risk Premium = Country Default Spread *

σEquity

σ (^) Country Bond

To illustrate, consider the case of Brazil. As noted earlier, the dollar-denominated

bonds issued by the Brazilian government trade with a default spread of 2 percent

18 Both the U.S. and Brazilian standard deviations were computed using weekly returns for two years from

the beginning of 2002 to the end of 2003. You could use daily standard deviations to make the same

judgments, but they tend to have much more estimation error in them.