RISK AND RETURN: PART II
In Chapter 2 we presented the key elements of adding a risk-free asset to the set of
risk and return analysis. There we saw that investment opportunities. This leads all
much of the risk inherent in a stock can be investors to hold the same well-diversified
eliminated by diversification, so rational portfolio of risky assets, and then to account
investors should hold portfolios of stocks for differing degrees of risk aversion by
rather than just one stock. We also introduced combining the risky portfolio in different
the Capital Asset Pricing Model (CAPM), proportions with the risk-free asset.
which links risk and required rates of return, Additionally, we show how betas are
using a stock’s beta coefficient as the relevant actually calculated, and we discuss two
measure of risk. alternative views of the risk/return
In this chapter, we extend the Chapter 2 relationship, the Arbitrage Pricing Theory
material by presenting an in-depth treatment (APT) and the Fama-French 3-Factor
of portfolio concepts and the CAPM. We Model.
continue the discussion of risk and return by
The riskiness of a portfolio, because it is assumed to be a single asset held in isolation, is
measured by the standard deviation of its return distribution. This equation is exactly the
same as the one for the standard deviation of a single asset, except that here the asset is a
portfolio of assets (for example, a mutual fund).
∑ (k )
Portfolio standard deviation = σ p = ˆ 2
− k p Pi .
Two key concepts in portfolio analysis are covariance and the correlation coefficient.
Covariance is a measure of the general movement relationship between two variables. It
combines the variance or volatility of a stock’s returns with the tendency of those returns to
move up or down at the same time other stocks move up or down. The following equation
defines the covariance (Cov) between Stocks A and B:
Covariance = Cov (AB) = ∑ (k Ai - k A ) (k Bi - k B )Pi .
If the returns move together, the terms in parentheses will both be positive or both
be negative, hence the product of the two terms will be positive, while if the returns
move counter to one another, the products will tend to be negative.
Cov(AB) will be large and positive if two assets have large standard deviations and
tend to move together; it will be large and negative for two high σ assets which
move counter to one another; and it will be small if the two assets’ returns move
randomly, rather than up or down with one another, or if either of the assets has a
small standard deviation.
The correlation coefficient also measures the degree of comovement between two variables,
but its values are limited to the range from -1.0 (perfect negative correlation) to +1.0 (perfect
positive correlation). The relationship between covariance and the correlation coefficient
can be expressed as
Correlation coefficient(AB) = r AB = .
σA σ B
The correlation coefficient standardizes the covariance.
The sign of the correlation coefficient is the same as the sign of the covariance, so a
positive sign means that the variables move together, a negative sign indicates that
they move in opposite directions, and if r is close to zero, they move independently
of one another.
Under the assumption that the distributions of returns on the individual securities are normal,
the following equation can be used to determine the riskiness of a two-asset portfolio:
Portfolio SD = σp = w 2 σ 2 + (1 − w A ) 2 σ 2 + 2 w A (1 − w A ) rAB σ A σ B .
A A B
Here wA is the fraction of the portfolio invested in Security A, so (1 – w A) is the
fraction invested in Security B.
One important use of portfolio risk concepts is to select efficient portfolios. An efficient
portfolio provides the highest expected return for any degree of risk, or the lowest degree
of risk for any expected return.
While the riskiness of a multi-asset portfolio usually decreases as the number of stocks
increase, the portfolio’s risk depends on the degree of correlation among the stocks.
In general, the average correlation between two stocks is +0.5 to +0.7, and hence holding
stocks in portfolios will reduce, but not eliminate, risk.
The attainable, or feasible, set of portfolios represents all portfolios that can be constructed
from a given set of stocks.
The optimal portfolio is found by determining the efficient set of portfolios and then
choosing from the efficient set the single portfolio that is best for the individual investor.
The efficient set of portfolios is also called the efficient frontier.
Portfolios to the left of the efficient set are not possible because they lie outside the
Portfolios to the right of the boundary line (interior portfolios) are inefficient
because some other portfolio would provide either a higher return with the same
degree of risk or lower risk for the same rate of return.
An indifference curve (or risk/return trade-off function) reflects an individual investor’s
attitude towards risk. The optimal portfolio for each investor is found at the tangency point
between the efficient set of portfolios and one of the investor’s indifference curves. This
tangency point marks the highest level of satisfaction the investor can attain.
The Capital Asset Pricing Model (CAPM) specifies the relationship between risk and
required rates of return on assets when they are held in well-diversified portfolios.
As in all financial theories, a number of assumptions were made in the development of the
CAPM. These assumptions are:
All investors focus on a single holding period, and seek to maximize the expected
utility of their terminal wealth.
All investors can borrow or lend an unlimited amount at a given risk-free rate of
Investors have homogeneous expectations.
All assets are perfectly divisible and perfectly liquid.
There are no transactions costs.
There are no taxes.
All investors are price takers.
The quantities of all assets are given and fixed.
Theoretical extensions in finance literature have relaxed some of the assumptions, and in
general these extensions have led to conclusions that are reasonably consistent with the basic
theory. However, even the extensions contain assumptions that are both strong and
unrealistic, so the validity of the model can only be established through empirical tests.
The Capital Market Line (CML) specifies a linear relationship between expected return
and risk for efficient portfolios.
The equation of the CML may be expressed as follows:
k M − k RF
k p = k RF +
ˆ σp .
Here k p is the expected (and in equilibrium, required) rate of return on an efficient portfolio,
kRF is the rate of interest on risk-free securities, k is the return on the market portfolio, σM
is the standard deviation of the market portfolio, and σp is the standard deviation of the
efficient portfolio in question.
In words, the expected rate of return on any efficient portfolio (that is, any portfolio on the
CML) is equal to the riskless rate plus a risk premium, and the risk premium is equal to
( k M - k RF) / σM multiplied by the portfolio’s standard deviation, σp.
The slope of the CML reflects the aggregate attitude of investors toward risk.
An efficient portfolio is one that is well diversified, hence all of its unsystematic risk has
been eliminated and its only remaining risk is market risk. Therefore, unlike individual
stocks, the riskiness of an efficient portfolio is measured by its standard deviation, σp.
The relevant measure of risk for use in the Security Market Line (SML) equation is the
stock’s beta coefficient, which measures the volatility of a stock relative to that of a
portfolio containing all stocks.
Beta is estimated by plotting historical returns on a particular stock versus returns on a
market index. The slope of the regression line, or characteristic line, is the stock’s beta
coefficient. The statistical equation for beta is:
Cov k j , k M ) r jM σ j σ M σj
bj = = = r jM
M σ 2
bj = the slope, or beta coefficient, for Stock j.
kj = the historical (realized) rate of return on Stock j.
kM = the historical (realized) rate of return on the market.
r jM = the correlation between Stock j and the market.
σj = the standard deviation of Stock j.
σM = the standard deviation of the market.
A stock’s beta, hence its market risk, depends on its correlation with the stock market
as a whole, rj M, and its own variability, σj, relative to the variability of the market, σM.
We assume that the historical relationship between Stock j and the market as a
whole, as given by its characteristic line, will continue into the future.
Besides general market movements, each firm also faces events that are unique to it
and independent of the general economic climate. This component of total risk is
the stock’s diversifiable, or company-specific, risk, and rational investors will
eliminate its effects by holding diversified portfolios of stocks.
An individual stock tends to move with the market as economic conditions change.
This component of total risk is the stock’s market, or non-diversifiable, risk. Even
well-diversified portfolios contain some market risk.
Total risk (variance) equals market risk plus diversifiable risk.
Total = Market + Diversifiable
risk risk risk
2 2 2 2
σ j = b j σM + σe j .
If all the points plotted on the characteristic line, then all of the stock’s total risk would
be market risk. On the other hand, if the points are widely scattered about the
regression line, much of the stock’s total risk would be diversifiable.
If the stock market never fluctuated, then stocks would have no market risk.
Beta is the measure of relative market risk, but the stock’s actual risk depends on
both its beta (market risk), and on the volatility of the market.
The diversifiable risk can and should be eliminated by diversification, so the
relevant risk is market risk, not total risk.
A stock’s risk premium, (k M − k RF) b j , depends only on its market risk, not its total
Since the CAPM depends on some unrealistic assumptions, it must be tested empirically to
determine if it gives accurate estimates of k i.
Betas are generally calculated for some past period, and the assumption is made that the
relative volatility of a stock will remain constant in the future. However, conditions may
change and alter a stock’s future volatility, which is the item of real concern to investors.
The CAPM should use expected (future) data, yet only historical data are generally available.
Studies indicate that the CAPM is a better concept for structuring investment portfolios than
it is for purposes of estimating the cost of capital for individual securities.
Studies of the CAPM based on the slope of the SML have generally showed a significant
positive relationship between realized returns and systematic risk and that the relationship
between risk and return appears to be linear.
According to the CAPM, high-beta stocks should provide higher returns than low-beta
stocks. However, the Fama-French study revealed no relationship between historical betas
and historical returns—low-beta stocks provided about the same returns as high-beta stocks.
The CAPM is extremely appealing at an intellectual level.
The CAPM framework is clearly a useful way to think about the riskiness of assets.
Thus, as a conceptual model, the CAPM is of truly fundamental importance.
Estimates of ki found through the use of the CAPM are subject to potentially large errors,
because we do not know precisely how to measure any of the inputs required to implement
Because the CAPM is logical in the sense that it represents the way risk-averse people ought
to behave, the model is a useful conceptual tool.
It is appropriate to think about many financial problems in a CAPM framework. However, it
is equally important to recognize the limitations of the CAPM when using it in practice.
The CAPM assumes that required rates of return depend on only one risk factor, the
stock’s beta coefficient, but required returns may be a function of several risk factors. An
approach called the Arbitrage Pricing Theory (APT) can include any number of risk
ˆ ( ˆ ) ( ˆ )
k i = k i + F1 − F1 b i 1 + + Fj − Fj b ij + e i ,
k i = the realized rate of return on Stock i.
k i = the expected rate of return on Stock i.
Fj = the realized value of economic Factor j.
Fj = the expected value of Factor j.
b ij = the sensitivity of Stock i to economic Factor j.
e i = the effect of unique events on the realized return of Stock i.
This equation shows that the realized return on any stock is equal to the stock’s expected
return, increases or decreases which depend on unexpected changes in fundamental
economic factors times the sensitivity of the stock to these changes, and a random term
which reflects changes unique to the firm or industry.
Theoretically, one could construct a portfolio such that (1) the portfolio was riskless and (2)
the net investment in it was zero. Such a zero investment portfolio must have a zero
expected return, or else arbitrage operations would occur that would cause the prices of the
underlying assets to change until the portfolio’s expected return was zero.
The end result is APT:
k i = k RF + ( k 1 − k RF ) b i1 + + ( k j − k RF ) b ij ,
where kj is the required rate of return on a portfolio that is sensitive only to the jth
economic factor (bj = 1.0) and has zero sensitivity to all other factors.
The primary theoretical advantage of the APT is that it permits several economic factors to
influence individual stock returns, whereas the CAPM assumes that the impact of all factors,
except those unique to the firm, can be captured in a single measure, the volatility of the
stock with respect to the market portfolio.
The APT also requires fewer assumptions than the CAPM and hence is a more
The APT does not assume that all investors hold the market portfolio, a CAPM
requirement that clearly is not met in practice.
However, the APT does not identify the relevant factors, nor does it even tell us how many
factors should appear in the model. The APT is in an early stage of development, and there
are still many unanswered questions.
The Fama-French 3-Factor Model is a multi-factor model, like the APT, except it specifies
three specific factors: (1) a market factor, like CAPM; (2) a size factor, based on the
difference in returns between a portfolio with Small sized firms and a portfolio with Big
sized firms (SMB); and (3) a factor based on the difference in returns between a portfolio
with High ratios of Book value/Market value of equity (B/M ratios) and a portfolio with
Low B/M ratios. The model is shown below:
_ _ _ _ _ _
k i − k RF = ai + bi k M − k RF + ci k SMB + di k HML + ei ,
ki = historical (realized) rate of return on Stock i.
k RF = historical (realized) rate of return on the risk free rate.
kM = historical (realized) rate of return on the market.
k SMB = historical (realized) rate of return on the small size portfolio
minus the big size portfolio.
k HML = historical (realized) rate of return on the high B/M portfolio
minus the low B/M portfolio.
ai = vertical axis intercept term for Stock i.
bi, ci and di = slope coefficients for Stock i.
ei = random error, reflecting the difference between the actual return
on Stock i in a given period and the return as predicted by the
Using the Fama-French 3-factor model, the expected return is:
k i = k RF + a i + b i k M − k RF + c i k SMB + d i k HML ,
where kSMB and kHML are the expected returns on the Small Minus Big and High Minus
1. __________ is a measure of the general movement relationship between two variables,
while the __________ __________ also measures the degree of comovement between
two stocks but its values are limited from -1.0 to +1.0.
2. A(n) ____________ ____________ is that portfolio which provides the highest expected
return for any given degree of risk, or the lowest degree of risk for any expected return.
3. The ___________, or ____________, set of portfolios represents all portfolios that can
be constructed from a given set of stocks.
4. The ___________ ___________ is found by determining the efficient set of portfolios
and then choosing from the efficient set the single portfolio that is best for the individual
5. The efficient set of portfolios is also called the ________ __________.
6. A(n) ______________ ____________ (or risk/return trade-off function) reflects an
individual investor’s attitude towards risk.
7. The __________ ________ __________ ________ specifies the relationship between
risk and required rates of return on assets when they are held in well-diversified
8. The _________ _______ _____ specifies a linear relationship between expected return
and risk for efficient portfolios.
9. The relevant measure of risk for use in the Security Market Line (SML) equation is the
stock’s ______ ______________, which measures the volatility of a stock relative to that
of a portfolio containing all stocks.
10. Beta is estimated by plotting historical returns on a particular stock versus returns on a
market index. The slope of the regression line, or _________________ _______, is the
stock’s beta coefficient.
11. Besides general market movements, each firm also faces events that are unique to it and
independent of the general economic climate. This component of total risk is the stock’s
________________, or __________-___________, risk, and rational investors will
eliminate its effects by holding diversified portfolios of stocks.
12. The relevant risk is _________ risk, not total risk.
13. Studies indicate that the CAPM is a better concept for structuring _____________
_____________ than it is for purposes of estimating the cost of capital for individual
14. The __________ __________ __________ uses several risk factors for determining the
required return on a stock.
15. The most commonly used continuous distribution is the _________ _______________,
which is symmetric about the expected value and its tails extend out to plus and minus
16. The _____________ _____________ standardizes the covariance.
17. The sign of the correlation coefficient is the ______ as the sign of the covariance. A(n)
__________ sign means that the variables move together, a(n) __________ sign indicates
that they move in opposite directions, and if r is close to zero, they move
_______________ of one another.
18. Portfolios to the ______ of the efficient set are not possible because they lie outside the
attainable set, while portfolios to the _______ of the boundary line are inefficient
because some other portfolio would provide either a higher return with the same degree
of risk or lower risk for the same rate of return.
19. The _______ of the CML reflects the aggregate attitude of investors toward risk.
20. An individual stock tends to move with the market as economic conditions change. This
component of total risk is the stock’s ________, or _____-_______________ risk.
21. _______ risk equals market risk plus diversifiable risk.
22. A stock’s risk premium depends only on its ________ risk, not its total risk.
23. The standard deviation of a portfolio is not the weighted average of the standard
deviations of the individual stocks in the portfolio.
a. True b. False
24. If the correlation coefficient between two stocks is +1.0, risk can be completely
a. True b. False
25. Total risk is relevant only for assets held in isolation.
a. True b. False
26. The Arbitrage Pricing Theory identifies the relevant factors for determining the required
a. True b. False
27. Which of the following statements is most correct?
a. It is difficult to interpret the magnitude of the correlation coefficient, so a related
statistic, the covariance, is often used to measure the degree of comovement between
b. The sign of the correlation coefficient is the same as the sign of the covariance, so
a positive sign means that the variables move together, a negative sign indicates that
they move in opposite directions, and if the correlation coefficient is close to zero,
they move independently of one another.
c. Attainable portfolios are defined as those portfolios which provide the highest
expected return for any degree of risk, or the lowest degree of risk for any expected
d. The efficient set of portfolios is also called the efficient frontier.
e. Statements b and d are both correct.
28. Which of the following statements is most correct?
a. The Capital Market Line (CML) specifies a curvilinear relationship between
expected return and risk, whereas the Security Market Line (SML) specifies a linear
relationship between expected return and risk.
b. An efficient portfolio is one that is well diversified, so like individual stocks, all
the riskiness of an efficient portfolio is measured by its standard deviation, σp.
c. A stock’s beta coefficient is the y-intercept of its characteristic line.
d. Empirical tests of the stability of beta coefficients have indicated that the betas of
individual stocks are stable, hence that past betas for individual securities are good
estimators of their future risk, while betas of portfolios of ten or more randomly
selected stocks are not stable, hence that past portfolio betas are not good estimators
of future portfolio volatility.
e. All of the above statements are false.
1. You are evaluating two potential investment opportunities: Stocks A and B. The
expected rate of return on Stock A is 15.8% and its standard deviation is 2.8%. Stock
B’s expected rate of return is 20.5% with a standard deviation of 3.5%. What is the
coefficient of variation (CV) for Stocks A and B, respectively?
a. 0.12; 0.12 b. 0.18; 0.12 c. 0.18; 0.17 d. 0.25; 0.17 e. 0.25; 0.25
2. Refer to Self-Test Problem 1. Assume that Stocks A and B have a correlation coefficient
of 0.65. What is the covariance between Stocks A and B?
a. 3.89 b. 4.35 c. 5.12 d. 6.37 e. 7.19
3. Refer to Self-Test Problem 1. Assume that Stocks A and B have a covariance of -5.4.
What is the correlation coefficient between Stocks A and B?
a. -0.55 b. -0.70 c. -0.85 d. -0.92 e. -1.00
4. Given the information below, calculate the betas for Stocks A and B.
Year Stock A Stock B Market
1 -5% 10% -10%
2 10 20 10
3 25 30 30
(Hint: Think rise over run.)
a. 1.0; 0.5 b. 0.75; 0.5 c. 0.75; 1.0 d. 0.5; 0.5 e. 0.75; 0.25
(The following data apply to the next two Self-Test Problems.)
You are given the following information:
Year Stock N Market
1 -5% 10%
2 -8 15
3 7 -10
The risk-free rate is equal to 7 percent, and the market required return is equal to 10
5. What is Stock N’s beta coefficient?
a. 1.00 b. -0.50 c. 0.60 d. -0.75 e. -0.60
6. What is Stock N’s required rate of return?
a. 6.40% b. 5.20% c. 8.80% d. 5.90% e. 7.00%
7. Stock Y and the Market had the following rates of return during the last 4 years. What is
Stock Y’s beta? (Hint: You will need a financial calculator to calculate the beta
1995 10.0% 10.0%
1996 16.0 13.5
1997 -7.5 -4.0
1998 0.0 5.5
a. 1.25 b. 0.75 c. 1.00 d. 1.34 e. 1.57
8. Stock Y, Stock Z, and the Market had the following rates of return during the last 4
Y Z Market
1995 10.0% 10.0% 10.0%
1996 16.0 11.5 13.5
1997 -7.5 1.0 -4.0
1998 0.0 6.0 5.5
The expected future return on the market is 15 percent, the real risk-free rate is 3.75
percent, and the expected inflation rate is a constant 5 percent. If the market risk
premium rises by 3 percentage points, what will be the change in the required rate of
return of the riskier stock?
a. 4.01% b. 3.67% c. 4.88% d. 3.23% e. 4.66%
ANSWERS TO SELF-TEST QUESTIONS
1. Covariance; correlation coefficient 13. investment portfolios; securities
2. efficient portfolio 14. Arbitrage Pricing Theory
3. attainable; feasible 15. normal distribution
4. optimal portfolio 16. correlation coefficient
5. efficient frontier 17. same; positive; negative;
6. indifference curve independently
7. Capital Asset Pricing Model 18. left; right
8. Capital Market Line 19. slope
9. beta coefficient 20. market; non-diversifiable
10. characteristic line 21. Total
11. diversifiable; company-specific 22. market
23. a. The standard deviation of a portfolio depends on the correlations among the
stocks as well as their individual standard deviations. The calculation for the
portfolio standard deviation is:
∑ (k )
σp = ˆ 2
− k p Pi .
24. b. A correlation coefficient of -1.0 is required to combine two stocks into a riskless
25. a. When assets are combined into portfolios, then the relevant risk is an asset’s
market risk, which is the contribution of the asset to the riskiness of the portfolio.
26. b. The APT does not identify the relevant factors beforehand, nor does it even tell
how many factors should appear in the model.
27. e. Statement a is false. It is difficult to interpret the magnitude of the covariance, so
the correlation coefficient is used to measure comovement between two variables.
Statement c is false; this statement is true only for efficient portfolios. Both
statements b and d are true, so statement e is the correct choice.
28. e. Statement a is false. The CML specifies the relationship between risk and return
for efficient portfolios, while the SML specifies the relationship between risk and
return for individual securities. Statement b is false. The standard deviation of an
individual stock should not be used to measure the stock’s riskiness because some of
its risk as reflected in the standard deviation can be eliminated by diversification.
Statement c is false; beta is the slope of the characteristic line. Statement d is false;
just the reverse is true. Betas of portfolios of 10 or more randomly selected stocks
have been shown to be stable, while the betas of individual securities have been
shown to be unstable. Consequently, statement e is the correct choice.
SOLUTIONS TO SELF-TEST PROBLEMS
1. c. Stock A: CV = 2.8%/15.8% = 0.18. Stock B: CV = 3.5%/20.5% = 0.17. Since
the CV for Project A is higher, it has more risk per unit of expected return.
2. d. Cov(AB) = 0.65 (2.8) (3.5) = 6.37.
3. a. rAB = -5.4/[(2.8)(3.5)] = -0.55.
4. b. Stock A: bA = Rise/Run = [10 – (-5)]/[10 – (-10)] = 15/20 = 0.75.
Stock B: bB = Rise/Run = [20 – 10]/[10 – (-10)] = 10/20 = 0.50.
This problem can also be worked using most financial calculators having
5. e. bN = Rise/Run = [-8 ) – (-5)]/(15 – 10) = -3/5 = -0.60.
Again, this problem can also be worked using most financial calculators having
6. b. kN = 7% + (10% – 7%)(-0.60) = 7% + (-1.80%) = 5.20%.
7. d. Use the regression feature of the calculator. Enter data for the market and Stock
Y, and then find BetaY = 1.3374 rounded to 1.34.
8. a. We know kM = 15%; k* = 3.75%; IP = 5%.
Original RPM = kM – kRF = 15% – (3.75% + 5%) = 6.25%.
RPM increases by 3% to 9.25%.
Find the change in k = ∆k for the riskier stock.
First, find the betas for the two stocks. Enter data in the regression register, then
find bY = 1.3374 and bZ = 0.6161.
Y is the riskier stock. Originally, its required return was k Y = 8.75% +
6.25%(1.3374) = 17.11%. When RPM increases by 3 percent, k Y = 8.75% + (6.25%
+ 3%)(1.3374) = 21.12%. Difference = 21.12% – 17.11% = 4.01%.