PowerPointto accompany
Chapter 11
Systematic Risk
and the Equity
Risk Premium
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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.
2
11.1 The Expected Return of a Portfolio
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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)
3
FORMULA!
11.1 The Expected Return of a Portfolio
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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)
4
FORMULA!
11.1 The Expected Return of a Portfolio
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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?
5
Example 11.1 Calculating Portfolio Returns (pp.338-9)
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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
Example 11.1 Calculating Portfolio Returns (pp.338-9)
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Example 11.1 Calculating Portfolio Returns (pp.338-9)
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%.
7
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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:
8
Example 11.1 Calculating Portfolio Returns (pp.338-9)
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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%.
9
Example 11.1 Calculating Portfolio Returns (pp.338-9)
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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)
10
FORMULA!
11.1 The Expected Return of a Portfolio
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Table 11.1 Summary of Portfolio Concepts
11
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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?
12
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Solution:
Plan:
You have a total of $40,000 invested:
$10,000/$40,000 = 25% in Ford: E[RF]=10%
$30,000/$40,000 = 75% in Luis: E[RL]=16%
Using Eq.11.3, calculate the expected return on your whole portfolio by weighting the expected returns of the stocks in your portfolio by their portfolio weights.
13
Example 11.2 Portfolio Expected Return (p.340)
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Execute:
The expected return on your portfolio is:
14
Example 11.2 Portfolio Expected Return (p.340)
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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.
15
Example 11.2 Portfolio Expected Return (p.340)
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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.
16
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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).
17
11.2 The Volatility of a Portfolio
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Table 11.2 Returns for Three Shares and Portfolios of Pairs of Shares
18
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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.
19
11.2 The Volatility of a Portfolio
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Figure 11.1 Volatility of Airline and Oil Portfolios
20
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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.
21
11.2 The Volatility of a Portfolio
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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.
22
11.2 The Volatility of a Portfolio
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Figure 11.2 Correlation
23
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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.
24
11.2 The Volatility of a Portfolio
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Table 11.3 Annual Volatilities and Correlations for Selected Shares (Based on Monthly Returns, 2004–08)
25
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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.
26
11.2 The Volatility of a Portfolio
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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.
27
11.2 The Volatility of a Portfolio
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Figure 11.4 Volatility of an Equally Weighted Portfolio versus the Number of Shares
28
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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.
29
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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.
30
11.3 Measuring Systematic Risk
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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.
31
11.3 Measuring Systematic Risk
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Table 11.5 Average Betas for Shares by Industry (Based on Monthly Data for 5 Years to June 2009)
32
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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.
33
11.3 Measuring Systematic Risk
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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?
34
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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.
35
Example 11.4 Total Risk versus Systematic Risk (pp.352-3)
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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.
36
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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
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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!
11.4 Putting It All Together: The Capital Asset Pricing Model
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Capital asset pricing model
The equation for the expected return of an
investment:
(Eq. 11.6)
39
FORMULA!
11.4 Putting It All Together: The Capital Asset Pricing Model
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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.
40
11.4 Putting It All Together: The Capital Asset Pricing Model
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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?
41
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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.
42
Example 11.5 Calculating the Expected Return for a Share (p.356)
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Execute:
Using Eq. 11.6:
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%.
43
Example 11.5 Calculating the Expected Return for a Share (p.356)
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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.
44
11.4 Putting It All Together: The Capital Asset Pricing Model
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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?
45
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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.
46
Example 11.6 A Negative Beta Share (p.357)
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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%:
E[RBAS] = 5% – 0.30(7%) = 2.9%.
47
Example 11.6 A Negative Beta Share (p.357)
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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.48
Example 11.6 A Negative Beta Share (p.357)
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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)
49
11.4 Putting It All Together: The Capital Asset Pricing Model
FORMULA!
(Eq. 11.8)
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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.
50
11.4 Putting It All Together: The Capital Asset Pricing Model