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CHAPTER 13

RETURN, RISK, AND THE SECURITY MARKET LINE

KEY CONCEPTS AND SKILLS

Know how to calculate expected returns

Understand the impact of diversification

Understand the systematic risk principle

Understand the security market line

Understand the risk-return trade-off

Be able to use the Capital Asset Pricing Model

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CHAPTER OUTLINE

Expected Returns and Variances

Portfolios

Announcements, Surprises, and Expected Returns

Risk: Systematic and Unsystematic

Diversification and Portfolio Risk

Systematic Risk and Beta

The Security Market Line

The SML and the Cost of Capital: A Preview

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Lecture Tip: You may find it useful to emphasize the economic foundations of the material in this chapter. Specifically, we assume:
-Investor rationality: Investors are assumed to prefer more money to less and less risk to more, all else equal. The result of this assumption is that the ex ante risk-return trade-off will be upward sloping.

-As risk-averse return-seekers, investors will take actions consistent with the rationality assumptions. They will require higher returns to invest in riskier assets and are willing to accept lower returns on less risky assets.

-Similarly, they will seek to reduce risk while attaining the desired level of return, or increase return without exceeding the maximum acceptable level of risk.

EXPECTED RETURNS

Expected returns are based on the probabilities of possible outcomes

In this context, “expected” means average if the process is repeated many times

The “expected” return does not even have to be a possible return

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Use the following example to illustrate the mathematical nature of expected returns:

Consider a game where you toss a fair coin: If it is Heads, then student A pays student B $1. If it is Tails, then student B pays student A $1. Most students will remember from their statistics that the expected value is $0 (=.5(1) + .5(-1)). That means that if the game is played over and over then each student should expect to break-even. However, if the game is only played once, then one student will win $1 and one will lose $1.

EXAMPLE: EXPECTED RETURNS

State Probability C T___

Boom 0.3 0.15 0.25

Normal 0.5 0.10 0.20

Recession ??? 0.02 0.01

RC = .3(15) + .5(10) + .2(2) = 9.9%
RT = .3(25) + .5(20) + .2(1) = 17.7%

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Suppose you have predicted the following returns for stocks C and T in three possible states of the economy. What are the expected returns?

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What is the probability of a recession? 1- 0.3 - 0.5 = 0.2

If the risk-free rate is 4.15%, what is the risk premium?

Stock C: 9.9 – 4.15 = 5.75%

Stock T: 17.7 – 4.15 = 13.55%

VARIANCE AND STANDARD DEVIATION

Variance and standard deviation measure the volatility of returns

Using unequal probabilities for the entire range of possibilities

Weighted average of squared deviations

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It’s important to point out that these formulas are for populations, unlike the formulas in chapter 12 that were for samples (dividing by n-1 instead of n). Further, the probabilities that are used account for the division.

Remind the students that standard deviation is the square root of the variance.

EXAMPLE: VARIANCE AND STANDARD DEVIATION

Consider the previous example. What are the variance and standard deviation for each stock?

Stock C

2 = .3(0.15-0.099)2 + .5(0.10-0.099)2 + .2(0.02-0.099)2 = 0.002029

 = 4.50%

Stock T

2 = .3(0.25-0.177)2 + .5(0.20-0.177)2 + .2(0.01-0.177)2 = 0.007441

 = 8.63%

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It is helpful to remind students that the standard deviation (but not the variance) is expressed in the same units as the original data, which is a percentage return in our example.

ANOTHER EXAMPLE

Consider the following information:

State Probability ABC, Inc. Return

Boom .25 0.15

Normal .50 0.08

Slowdown .15 0.04

Recession .10 -0.03

What is the expected return?

What is the variance?

What is the standard deviation?

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E(R) = .25(0.15) + .5(0.08) + .15(0.04) + .1(-0.03) = 8.05%

Variance = .25(.15-0.0805)2 + .5(0.08-0.0805)2 + .15(0.04-0.0805)2 + .1(-0.03-0.0805)2 = 0.00267475

Standard Deviation = 5.17%

PORTFOLIOS

A portfolio is a collection of assets

An asset’s risk and return are important in how they affect the risk and return of the portfolio

The risk-return trade-off for a portfolio is measured by the portfolio expected return and standard deviation, just as with individual assets

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Lecture Tip: Each individual has their own level of risk tolerance. Some people are just naturally more inclined to take risk, and they will not require the same level of compensation as others for doing so. Our risk preferences also change through time. We may be willing to take more risk when we are young and without a spouse or kids. But, once we start a family, our risk tolerance may drop.

EXAMPLE: PORTFOLIO WEIGHTS

Suppose you have $15,000 to invest and you have purchased securities in the following amounts. What are your portfolio weights in each security?

$2000 of C

$3000 of KO

$4000 of INTC

$6000 of BP

C: 2/15 = .133

KO: 3/15 = .2

INTC: 4/15 = .267

BP: 6/15 = .4

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C – Citigroup

KO – Coca-Cola

INTC – Intel

BP – BP

Show the students that the sum of the weights = 1

PORTFOLIO EXPECTED RETURNS

The expected return of a portfolio is the weighted average of the expected returns of the respective assets in the portfolio

You can also find the expected return by finding the portfolio return in each possible state and computing the expected value as we did with individual securities

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EXAMPLE: EXPECTED PORTFOLIO RETURNS

Consider the portfolio weights computed previously. If the individual stocks have the following expected returns, what is the expected return for the portfolio?

C: 19.69%

KO: 5.25%

INTC: 16.65%

BP: 18.24%

E(RP) = .133(19.69%) + .2(5.25%) + .267(16.65%) + .4(18.24%) = 15.41%

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PORTFOLIO VARIANCE

Compute the portfolio return for each state:
RP = w1R1 + w2R2 + … + wmRm

Compute the expected portfolio return using the same formula as for an individual asset

Compute the portfolio variance and standard deviation using the same formulas as for an individual asset

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EXAMPLE: PORTFOLIO VARIANCE

Consider the following information on returns and probabilities:

Invest 50% of your money in Asset A

State Probability A B Portfolio

Boom .4 30% -5% 12.5%

Bust .6 -10% 25% 7.5%

What are the expected return and standard deviation for each asset?

What are the expected return and standard deviation for the portfolio?

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If A and B are your only choices, what percent are you investing in Asset B? 50%

Asset A: E(RA) = .4(30) + .6(-10) = 6%

Variance(A) = .4(30-6)2 + .6(-10-6)2 = 384

Std. Dev.(A) = 19.6%

Asset B: E(RB) = .4(-5) + .6(25) = 13%

Variance(B) = .4(-5-13)2 + .6(25-13)2 = 216

Std. Dev.(B) = 14.7%

Portfolio (solutions to portfolio return in each state appear with mouse click after last question)

Portfolio return in boom = .5(30) + .5(-5) = 12.5

Portfolio return in bust = .5(-10) + .5(25) = 7.5

Expected return = .4(12.5) + .6(7.5) = 9.5 or

Expected return = .5(6) + .5(13) = 9.5

Variance of portfolio = .4(12.5-9.5)2 + .6(7.5-9.5)2 = 6

Standard deviation = 2.45%

Note that the variance is NOT equal to .5(384) + .5(216) = 300 and

Standard deviation is NOT equal to .5(19.6) + .5(14.7) = 17.17%

What would the expected return and standard deviation for the portfolio be if we invested 3/7 of our money in A and 4/7 in B? Portfolio return = 10% and standard deviation = 0

Portfolio variance using covariances:

COV(A,B) = .4(30-6)(-5-13) + .6(-10-6)(25-13) = -288

Variance of portfolio = (.5)2(384) + (.5)2(216) + 2(.5)(.5)(-288) = 6

Standard deviation = 2.45%

ANOTHER EXAMPLE

Consider the following information on returns and probabilities:

State Probability X Z

Boom .25 15% 10%

Normal .60 10% 9%

Recession .15 5% 10%

What are the expected return and standard deviation for a portfolio with an investment of $6,000 in asset X and $4,000 in asset Z?

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Portfolio return in Boom: .6(15) + .4(10) = 13%

Portfolio return in Normal: .6(10) + .4(9) = 9.6%

Portfolio return in Recession: .6(5) + .4(10) = 7%

Expected return = .25(13) + .6(9.6) + .15(7) = 10.06%

Variance = .25(13-10.06)2 + .6(9.6-10.06)2 + .15(7-10.06)2 = 3.6924

Standard deviation = 1.92%

Compare to return on X of 10.5% and standard deviation of 3.12%

And return on Z of 9.4% and standard deviation of .49%

Using covariances:

COV(X,Z) = .25(15-10.5)(10-9.4) + .6(10-10.5)(9-9.4) + .15(5-10.5)(10-9.4) = .3

Portfolio variance = (.6*3.12)2 + (.4*.49)2 + 2(.6)(.4)(.3) = 3.6868

Portfolio standard deviation = 1.92% (difference in variance due to rounding)

Lecture Tip: Here are a few tips to pass along to students suffering from “statistics overload”:

-The distribution is just the picture of all possible outcomes

-The mean return is the central point of the distribution

-The standard deviation is the average deviation from the mean

-Assuming investor rationality (two-parameter utility functions), the mean is a proxy for expected

return and the standard deviation is a proxy for total risk.

EXPECTED VS. UNEXPECTED RETURNS

Realized returns are generally not equal to expected returns

There is the expected component and the unexpected component

At any point in time, the unexpected return can be either positive or negative

Over time, the average of the unexpected component is zero

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ANNOUNCEMENTS AND NEWS

Announcements and news contain both an expected component and a surprise component

It is the surprise component that affects a stock’s price and therefore its return

This is very obvious when we watch how stock prices move when an unexpected announcement is made or earnings are different than anticipated

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Lecture Tip: It is easy to see the effect of unexpected news on stock prices and returns. Consider the following two cases: (1) On November 17, 2004 it was announced that K-Mart would acquire Sears in an $11 billion deal. Sears’ stock price jumped from a closing price of $45.20 on November 16 to a closing price of $52.99 (a 7.79% increase) and K-Mart’s stock price jumped from $101.22 on November 16 to a closing price of $109.00 on November 17 (a 7.69% increase). Both stocks traded even higher during the day. Why the jump in price? Unexpected news, of course. (2) On November 18, 2004, Williams-Sonoma cut its sales and earnings estimates for the fourth quarter of 2004 and its share price dropped by 6%. There are plenty of other examples where unexpected news causes a change in price and expected returns.

EFFICIENT MARKETS

Efficient markets are a result of investors trading on the unexpected portion of announcements

The easier it is to trade on surprises, the more efficient markets should be

Efficient markets involve random price changes because we cannot predict surprises

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SYSTEMATIC RISK

Risk factors that affect a large number of assets

Also known as non-diversifiable risk or market risk

Includes such things as changes in GDP, inflation, interest rates, etc.

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UNSYSTEMATIC RISK

Risk factors that affect a limited number of assets

Also known as unique risk and asset-specific risk

Includes such things as labor strikes, part shortages, etc.

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Lecture Tip: You can expand the discussion of the difference between systematic and unsystematic risk by using the example of a strike by employees. Students will generally agree that this is unique or unsystematic risk for one company. However, what if the UAW stages the strike against the entire auto industry. Will this action impact other industries or the entire economy? If the answer to this question is yes, then this becomes a systematic risk factor. The important point is that it is not the event that determines whether it is systematic or unsystematic risk; it is the impact of the event.

RETURNS

Total Return = expected return + unexpected return

Unexpected return = systematic portion + unsystematic portion

Therefore, total return can be expressed as follows: Total Return = expected return + systematic portion + unsystematic portion

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DIVERSIFICATION

Portfolio diversification is the investment in several different asset classes or sectors

Diversification is not just holding a lot of assets

For example, if you own 50 Internet stocks, you are not diversified

However, if you own 50 stocks that span 20 different industries, then you are diversified

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Video Note: “Portfolio Management” looks at the value of diversification.

TABLE 13.7

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THE PRINCIPLE OF DIVERSIFICATION

Diversification can substantially reduce the variability of returns without an equivalent reduction in expected returns

This reduction in risk arises because worse than expected returns from one asset are offset by better than expected returns from another

However, there is a minimum level of risk that cannot be diversified away and that is the systematic portion

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A discussion of the potential benefits of international investing may be helpful at this point.

FIGURE 13.1

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DIVERSIFIABLE RISK

The risk that can be eliminated by combining assets into a portfolio

Often considered the same as unsystematic, unique or asset-specific risk

If we hold only one asset, or assets in the same industry, then we are exposing ourselves to risk that we could diversify away

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TOTAL RISK

Total risk = systematic risk + unsystematic risk

The standard deviation of returns is a measure of total risk

For well-diversified portfolios, unsystematic risk is very small

Consequently, the total risk for a diversified portfolio is essentially equivalent to the systematic risk

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SYSTEMATIC RISK PRINCIPLE

There is a reward for bearing risk

There is not a reward for bearing risk unnecessarily

The expected return on a risky asset depends only on that asset’s

systematic risk since unsystematic risk can be diversified away

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A discussion of diversification via mutual funds and ETFs may add to the students’ understanding.

MEASURING SYSTEMATIC RISK

How do we measure systematic risk?

We use the beta coefficient

What does beta tell us?

A beta of 1 implies the asset has the same systematic risk as the overall market

A beta < 1 implies the asset has less systematic risk than the overall market

A beta > 1 implies the asset has more systematic risk than the overall market

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Lecture Tip: Remember that the cost of equity depends on both the firm’s business risk and its financial risk. So, all else equal, borrowing money will increase a firm’s equity beta because it increases the volatility of earnings. Robert Hamada derived the following equation to reflect the relationship between levered and unlevered betas (excluding tax effects):
L = U(1 + D/E)

where:

L = equity beta of a levered firm;

U = equity beta of an unlevered firm;

D/E = debt-to-equity ratio

TABLE 13.8 – SELECTED BETAS

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www: Click on the web surfer icon to go to the stock screener on Yahoo! Finance where you can do searches for various betas. Students are often interested to see the range of betas that are out there.

Lecture Tip: Students sometimes wonder just how high a stock’s beta can get. In earlier years, one would say that, while the average beta for all stocks must be 1.0, the range of possible values for any given beta is from - to +.

Today, the Internet provides another way of addressing the question. Go to the Yahoo! Finance stock screener by clicking the web surfer icon. This site allows you to search many financial markets by fundamental criteria. For example, as of September 8, 2012, a search for stocks with betas of at least 2.00 turns up 1,041 stocks.

TOTAL VS. SYSTEMATIC RISK

Consider the following information:

Standard Deviation Beta

Security C 20% 1.25

Security K 30% 0.95

Which security has more total risk?

Which security has more systematic risk?

Which security should have the higher expected return?

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Security K has the higher total risk

Security C has the higher systematic risk

Security C should have the higher expected return

WORK THE WEB EXAMPLE

Many sites provide betas for companies

Yahoo Finance provides beta, plus a lot of other information under its Key Statistics link

Click on the web surfer to go to Yahoo Finance

Enter a ticker symbol and get a basic quote

Click on Key Statistics

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EXAMPLE: PORTFOLIO BETAS

Consider the previous example with the following four securities

Security Weight Beta

C .133 2.685

KO .2 0.195

INTC .267 2.161

BP .4 2.434

What is the portfolio beta?

.133(2.685) + .2(.195) + .267(2.161) + .4(2.434) = 1.947

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Which security has the highest systematic risk? C

Which security has the lowest systematic risk? KO

Is the systematic risk of the portfolio more or less than the market? more

BETA AND THE RISK PREMIUM

Remember that the risk premium = expected return – risk-free rate

The higher the beta, the greater the risk premium should be

Can we define the relationship between the risk premium and beta so that we can estimate the expected return?

YES!

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EXAMPLE: PORTFOLIO EXPECTED RETURNS AND BETAS

E(RA)

A

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Based on the example in the book:

Point out that there is a linear relationship between beta and expected return. Ask if the students remember the form of the equation for a line.

Y = mx + b

E(R) = slope (Beta) + y-intercept

The y-intercept is = the risk-free rate, so all we need is the slope

Lecture Tip: The example in the book illustrates a greater than 100% investment in asset A. This means that the investor has borrowed money on margin (technically at the risk-free rate) and used that money to purchase additional shares of asset A. This can increase the potential returns, but it also increases the risk.

REWARD-TO-RISK RATIO: DEFINITION AND EXAMPLE

The reward-to-risk ratio is the slope of the line illustrated in the previous example

Slope = (E(RA) – Rf) / (A – 0)

Reward-to-risk ratio for previous example =
(20 – 8) / (1.6 – 0) = 7.5

What if an asset has a reward-to-risk ratio of 8 (implying that the asset plots above the line)?

What if an asset has a reward-to-risk ratio of 7 (implying that the asset plots below the line)?

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Ask students if they remember how to compute the slope of a line: rise / run

If the reward-to-risk ratio = 8, then investors will want to buy the asset. This will drive the price up and the expected return down (remember time value of money and valuation). When will the flurry of trading stop? When the reward-to-risk ratio reaches 7.5.

If the reward-to-risk ratio = 7, then investors will want to sell the asset. This will drive the price down and the expected return up. When will the flurry of trading stop? When the reward-to-risk ratio reaches 7.5.

MARKET EQUILIBRIUM

In equilibrium, all assets and portfolios must have the same reward-to-risk ratio, and they all must equal the reward-to-risk ratio for the market

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SECURITY MARKET LINE

The security market line (SML) is the representation of market equilibrium

The slope of the SML is the reward-to-risk ratio: (E(RM) – Rf) / M

But since the beta for the market is always equal to one, the slope can be rewritten

Slope = E(RM) – Rf = market risk premium

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Based on the discussion earlier, we now have all the components of the line:

E(R) = [E(RM) – Rf] + Rf

Lecture Tip: Although the realized market risk premium has on average been approximately 8.5%, the historical average should not be confused with the anticipated risk premium for any particular future period. There is abundant evidence that the realized market return has varied greatly over time. The historical average value should be treated accordingly. On the other hand, there is currently no universally accepted means of coming up with a good ex ante estimate of the market risk premium, so the historical average might be as good a guess as any. In the late 1990’s, there was evidence that the risk premium had been shrinking. In fact, Alan Greenspan was concerned with the reduction in the risk premium because he was afraid that investors had lost sight of how risky stocks actually are. Investors had a wake-up call in late 2000 and 2001 (and again in 2008 and 2009).

THE CAPITAL ASSET PRICING MODEL (CAPM)

The capital asset pricing model defines the relationship between risk and return

E(RA) = Rf + A(E(RM) – Rf)

If we know an asset’s systematic risk, we can use the CAPM to determine its expected return

This is true whether we are talking about financial assets or physical assets

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FACTORS AFFECTING
EXPECTED RETURN

Pure time value of money: measured by the risk-free rate

Reward for bearing systematic risk: measured by the market risk premium

Amount of systematic risk: measured by beta

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EXAMPLE - CAPM

Consider the betas for each of the assets given earlier. If the risk-free rate is 4.15% and the market risk premium is 8.5%, what is the expected return for each?

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Security	Beta	Expected Return
C	2.685	4.15 + 2.685(8.5) = 26.97%
KO	0.195	4.15 + 0.195(8.5) = 5.81%
INTC	2.161	4.15 + 2.161(8.5) = 22.52%
BP	2.434	4.15 + 2.434(8.5) = 24.84%

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Lecture Tip: Students should remember that in an efficient market, security investments have an NPV = 0, on average. However, the NPV does not imply that a company’s investments in new projects must have an NPV of zero. Firms attempt to invest in projects with a positive NPV, and those that are consistently successful will trade at higher prices, all else equal. The ability to generate positive NPV projects reflects the fundamental differences in physical asset markets and financial asset markets. Physical asset markets are generally less efficient than financial asset markets, and cash flows to physical assets are often owner dependent.

FIGURE 13.4

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QUICK QUIZ

How do you compute the expected return and standard deviation for an individual asset? For a portfolio?

What is the difference between systematic and unsystematic risk?

What type of risk is relevant for determining the expected return?

Consider an asset with a beta of 1.2, a risk-free rate of 5%, and a market return of 13%.

What is the reward-to-risk ratio in equilibrium?

What is the expected return on the asset?

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Reward-to-risk ratio = 13 – 5 = 8%

Expected return = 5 + 1.2(8) = 14.6%

COMPREHENSIVE PROBLEM

The risk free rate is 4%, and the required return on the market is 12%.

What is the required return on an asset with a beta of 1.5?

What is the reward/risk ratio?

What is the required return on a portfolio consisting of 40% of the asset above and the rest in an asset with an average amount of systematic risk?

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R = .04 + 1.5 x (.12 - .04) = .16

The reward/risk ratio is 8%

R = (.4 x .16) + (.6 x .12) = .136

CHAPTER 13

END OF CHAPTER

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