Chapter 11

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Chapter 11Systematic Risk and the Equity Risk Premium

Copyright © 2011 Pearson Australia (a division of Pearson Australia Group Ltd) –9781442502000 / Berk/DeMarzo/Harford / Fundamentals of Corporate Finance / 1st edition

While for large portfolios investors should expect to experience higher returns for higher risk, the same does not hold true for individual shares.

Shares have both unsystematic, diversifiable risk and systematic, undiversifiable risk—only the systematic risk is rewarded with higher expected returns.

With no reward for bearing unsystematic risk, rational investors should choose to diversify.

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11.1 The Expected Return of a Portfolio


Portfolio weights We can describe a portfolio by its portfolio weights,

which are the fractions of the individual investment in the portfolio:

These portfolio weights add up to 100% (that is, w1 + w2 + … + wN = 100%), so that they represent the way we have divided our money between the different individual investments in the portfolio.

(Eq. 11.1)

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FORMULA!



Portfolio returns The return on a portfolio, Rp, is the weighted

average of the returns on the investments in the portfolio, where the weights correspond to portfolio weights:

(Eq. 11.2)

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FORMULA!



Problem: Suppose you invest $100,000 and buy 4,000 shares of

Qantas at $10 per share ($40,000) and 1,500 shares of Woolworths at $40 per share ($60,000).

If Qantas’ shares goes up to $12 each and Woolworths’ shares falls to $38 each and neither paid dividends, what is the new value of the portfolio?

What return did the portfolio earn? Show that Eq. 11.2 is true by calculating the individual returns of the shares and multiplying them by their weights.

If you don’t buy or sell any shares after the price change, what are the new portfolio weights?

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Example 11.1 Calculating Portfolio Returns (pp.338-9)


Solution:Plan: Your portfolio:

4,000 shares of Qantas: $10 $12 ($2 capital gain) 1,500 shares of Woolworths: $40 $38 ($2 capital

loss) To calculate the return on your portfolio, calculate its

value using the new prices and compare it to the original $100,000 investment.

To confirm that Eq. 11.2 is true, calculate the return on each share individually using Eq. 10.1 from Chapter 10, multiply those returns by their original weights in the portfolio, and compare your answer to the return you just calculated for the portfolio as a whole. 6




Execute: The new value of your Qantas shares is 4,000 × $12 =

$48,000 and the new value of your Woolworths shares is 1,500 × $38 = $57,000.

So, the new value of your portfolio is $48,000 + $57,000 = $105,000, for a gain of $5,000 or a 5% return on your initial $100,000 investment.

Since neither share paid any dividends, we calculate their returns simply as the capital gain or loss divided by the purchase price.

Return on Qantas’ shares was $2/$10 = 20%, and return on Woolworths’ shares was –$2/$40 = –5%.

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Execute (cont’d): The initial portfolio weights were 40,000/$100,000

= 40% for Qantas and $60,000/$100,000 = 60% for Woolworths, so we can also calculate the return of the portfolio from Eq.11.2 as:

After the price change, the new portfolio weights are:

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Evaluate: The $3,000 loss on your investment in Woolworths

was offset by the $8,000 gain in your investment in Qantas, for a total gain of $5,000 or 5%.

The same result comes from giving a 40% weight to the 20% return on Qantas and a 60% weight to the –5% loss on Woolworths; you have a total net return of 5%.

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The expected return of a portfolio is simply the weighted average of the expected returns of the investments within it, using the portfolio weights:

We started by stating that you can describe a portfolio by its weights.

These weights are used in calculating both a portfolio’s return and expected return.

Table 11.1 summarises these concepts.

(Eq. 11.3)

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FORMULA!



Table 11.1 Summary of Portfolio Concepts

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Example 11.2 Portfolio Expected Return (p.340)

Problem: Suppose you invest $10,000 in Ford (F) shares,

and $30,000 in Luis International (L) shares. You expect a return of 10% for Ford, and 16% for

Luis. What is the expected return for your portfolio?

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Solution:Plan:

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Execute:

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Evaluate: The importance of each share for the expected

return of the overall portfolio is determined by the relative amount of money you have invested in it.

Most (75%) of your money is invested in Luis, so the overall expected return of the portfolio is much closer to Luis’ expected return than it is to Ford’s.

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11.2 The Volatility of a Portfolio Investors in a company care not only about the

return, but also about the risk of their portfolios. When we combine shares in a portfolio, some of

their risk is eliminated through diversification. The amount of risk that will remain depends upon

the degree to which the shares share common risk.

The volatility of a portfolio is the total risk, measured as standard deviation, of the portfolio.

In this section we describe the tools to quantify the degree to which two shares share risk and to determine the volatility of a portfolio.

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Diversifying Risks Let’s begin with a simple example of how risk

changes when shares are combined in a portfolio. Table 11.2 shows returns for three hypothetical

shares, along with their average returns and volatilities.

Note that while the three shares have the same volatility and average return, the pattern of returns differs.

In years when the airline shares performed well, the oil shares tended to do poorly (see 2004–05), and when the airlines did poorly, the oil shares tended to do well (2007–08).

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11.2 The Volatility of a Portfolio


Table 11.2 Returns for Three Shares and Portfolios of Pairs of Shares

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Table 11.2 also shows the returns for two portfolios of the shares.

The first portfolio is an equal investment in the two airlines, North Air and West Air.

The second portfolio is an equal investment in West Air and Tassie Oil—bottom rows display the average return and volatility for each share and portfolio of shares.

Note that the 10% average return of both portfolios is equal to the 10% average return of the shares.

However, as Figure 11.1 illustrates, their volatilities (standard deviations)—12.1% for portfolio 1 and 5.1% for portfolio 2—are very different from the 13.4% volatility for the individual shares and from each other.

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Figure 11.1 Volatility of Airline and Oil Portfolios

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Diversifying risks This example demonstrates two important things: First, by combining shares into a portfolio, we

reduce risk through diversification. Because the shares do not move identically,

some of the risk is averaged out in a portfolio.

As a result, both portfolios have lower risk than the individual shares.

Second, the amount of risk that is eliminated in a portfolio depends upon the degree to which the shares face common risks and move together.

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Measuring co-movement: Correlation Correlation: a barometer of the degree to which

the returns share common risk, calculated as the covariance of the returns divided by the standard deviation of each return.

The closer the correlation is to +1, the more the returns tend to move together as a result of common risk.

When the correlation equals 0, the returns are uncorrelated (no tendency to move together or opposite of one another).

Finally, the closer the correlation is to –1, the more the returns tend to move in opposite directions.

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Figure 11.2 Correlation

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When will share returns be highly correlated with each other?

Share returns will tend to move together if they are affected similarly by economic events.

Thus, shares in the same industry tend to have more highly correlated returns than shares in different industries.

This tendency is illustrated in Table 11.4, which shows the volatility (standard deviation) of individual share returns and the correlation between them for several common shares.

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Table 11.3 Annual Volatilities and Correlations for Selected Shares (Based on Monthly Returns, 2004–08)

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When combining shares into a portfolio, unless the shares all have a perfect positive correlation of +1 with each other, the risk of the portfolio will be lower than the weighted average volatility of the individual shares (Figure 11.1).

Contrast this fact with a portfolio’s expected return: The expected return of a portfolio is equal to the

weighted average expected return of its shares, but the volatility of a portfolio is less than the weighted average volatility.

As a result, it’s clear that we can eliminate some volatility by diversifying.

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The volatility of a large portfolio As we add more shares to our portfolio, the

diversifiable firm-specific risk for each share matters less and less—only common risk still matters.

The benefit of diversification is most dramatic initially—the decrease in volatility going from one to two shares is much larger than the decrease going from 100 to 101 shares.

In an equally weighted portfolio, the same amount of money is invested in each share.

Even for a very large portfolio, however, we cannot eliminate all of the risk—the systematic risk remains.

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Figure 11.4 Volatility of an Equally Weighted Portfolio versus the Number of Shares

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11.3 Measuring Systematic Risk Our goal is to understand the impact of risk on the

firm’s capital providers. By understanding how they view risk, we can

quantify the relation between risk and required return to produce a discount rate for our present value calculations.

To recap:1. The amount of a share’s risk that is diversified

away depends on the portfolio that you put it in.

2. If you build a large enough portfolio, you can diversify away all unsystematic risk, but you will be left with systematic risk.

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Role of the market portfolio Because every security is owned by someone, the

sum of all investors’ portfolios must equal the portfolio of all risky securities available in the market.

But everyone wants to hold the most diversified portfolio, so for supply and demand to balance, everyone must hold the market portfolio.

The market portfolio is the portfolio of all risky investments, held in proportion to their value.

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11.3 Measuring Systematic Risk


Market risk and beta We can measure a share’s systematic risk by

estimating the share’s sensitivity to the market portfolio, which we refer to as its beta (β):

A share’s beta (β) is the percentage change in its return that we expect for each 1% change in the market’s return.

There are many data sources that provide estimates of beta based on historical data.

Typically, these data sources estimate betas using two to five years of weekly or monthly returns and, in Australia, use the All Ordinaries Index as the market portfolio.

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Table 11.5 Average Betas for Shares by Industry (Based on Monthly Data for 5 Years to June 2009)

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The beta of the overall market portfolio is 1, so you can think of a beta of 1 as representing average exposure to systematic risk.

However, as the table demonstrates, many industries and companies have betas much higher or lower than 1.

The differences in betas by industry are related to the sensitivity of each industry’s profits to the general health of the economy.

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Example 11.4 Total Risk versus Systematic Risk (pp.352-3)

Problem: Suppose that, in the coming year, you expect

Qantas shares to have a standard deviation of 30% and a beta of 1.2, and Woolworths’ shares to have a standard deviation of 41% and a beta of 0.6.

Which share carries more total risk? Which has more systematic risk?

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Execute: Total risk is measured by standard deviation,

therefore, Woolworths’ shares have more total risk.

Systematic risk is measured by beta. Qantas has a higher beta, and so has more

systematic risk.

Evaluate: A share can have high total risk, but if a lot of it is

diversifiable, it can still have low or average systematic risk.

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Example 11.4 Total Risk versus Systematic Risk (pp.352-3)


11.4 Putting It All Together: The Capital Asset Pricing Model One of our goals in this chapter is to calculate the

cost of equity capital for a company listed on the stock exchange, which is the best available expected return offered in the market on an investment of comparable risk and term.

Thus, in order to calculate the cost of capital, we need to know the relation between the risk of the company and its expected return.

In this section, we put all the pieces together to build a model for determining the expected return of any investment.

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CAPM equation relating risk to expected return

Intuitively, the expected return on any investment includes two components: A baseline risk-free rate of return that we would

demand to compensate for inflation and the time value of money, even if there were no risk of losing our money.

A risk premium that varies with the amount of systematic risk in the investment.

Expected return = Risk-free rate + Risk premium for systematic risk

37FORMULA!

11.4 Putting It All Together: The Capital Asset Pricing Model


We devoted the last section to measuring systematic risk.

Beta is our measure of the amount of systematic risk in an investment:

Expected return for investment i = (Risk-free rate) + βi x [Risk premium per unit of systematic risk]

FORMULA!



Capital asset pricing model The equation for the expected return of an

investment:

(Eq. 11.6)

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FORMULA!



The CAPM simply says that the return we should expect on any investment is equal to the risk-free rate of return plus a risk premium proportional to the amount of systematic risk in the investment.

Specifically, the risk premium of an investment is equal to the market risk premium multiplied by the amount of systematic (market) risk present in the investment, measured by its beta with the market (βi).

Because investors will not invest in this security unless they can expect at least the return given in Eq. 11.6, we also call this return the investment’s required return.

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Example 11.5 Calculating the Expected Return for a Share (p.356)

Problem: Suppose the risk-free return is 5% and you

measure the market risk premium to be 7%. Qantas has a beta of 1.33. According to the CAPM, what is its expected

return?

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Solution: Plan: We can use Eq. 11.6 to calculate the expected

return according to the CAPM. For that equation, we will need the market risk

premium, the risk-free return and the share’s beta.

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Execute:

Evaluate: Because of Qantas’ beta of 1.33, investors will

require a risk premium of 9.31% over the risk-free rate for investments in its shares to compensate for the systematic risk of Qantas shares.

This leads to a total expected return of 14.31%.

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Can there be shares that have a negative beta? While the vast majority of shares have a positive

beta, it is possible to have returns that co-vary negatively with the market.

Firms that provide goods or services that are in greater demand in economic contractions than in booms fit this description.

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Example 11.6 A Negative Beta Share (p.357)

Problem: Suppose the shares of Bankruptcy Auction

Services Limited (BAS) have a negative beta of –0.30.

How does its required return compare to the risk-free rate, according to the CAPM?

Does your result make sense?

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Solution:Plan: We can use the CAPM equation (Eq. 11.6) to

calculate the expected return of this negative beta share just like we would a positive beta share.

We don’t have the risk-free rate or the market risk premium, but the problem doesn’t ask us for the exact expected return, just whether or not it will be more or less than the risk-free rate.

Using Eq. 11.6, we can answer that question.

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Execute: Because the expected return of the market is

higher than the risk-free rate, Eq. 11.6 implies that the expected return of BAS will be below the risk-free rate.

As long as the market risk premium is positive, then the second term in Eq. 11.6 will have to be negative if the beta is negative.

For example, if the risk-free rate is 5% and the market risk premium is 7%:

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Evaluate: This result seems odd—why would investors be willing

to accept a 2.2% expected return on this share when they can invest in a safe investment and earn 5%?

The answer is that a savvy investor will not hold BAS alone; instead, the investor will hold it in combination with other securities as part of a well-diversified portfolio.

These other securities will tend to rise and fall with the market. But because BAS has a negative beta, its correlation with the market is negative, which means that BAS tends to perform well when the rest of the market is doing poorly.

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The CAPM and portfolios We can apply CAPM to portfolios as well. Therefore, the expected return of a portfolio should

correspond to the portfolio’s beta. We calculate the beta of a portfolio made up of

securities each with weight wi as follows:

That is, the beta of a portfolio is the weighted average beta of the securities in the portfolio.

(Eq. 11.7)

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FORMULA!

(Eq. 11.8)


The big picture The CAPM marks the culmination of our examination of

how investors in capital markets trade-off risk and return, providing a powerful and widely used tool to quantify the return that should accompany a particular amount of systematic risk.

The Valuation Principle tells us to use this cost of capital to discount the future expected cash flows of the firm to arrive at the value of the firm.

Thus, the cost of capital is an essential input to analyse investment opportunities and so knowing this overall cost of capital is critical to a company’s success at creating value for its investors.

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Chapter 11