BKM CHAPTER 7
1. Which of the following factors reflect pure market risk for a given corporation?
a. Increased short-term interest rates
b. Fire in the corporate warehouse
c. Increased insurance costs
d. Death of the CEO, e. Increased labor costs)
(a) and (e) – The other three do not affect all participants in the economy.
2. When adding real estate to an asset allocation program that currently includes only
stocks, bonds, and cash alternatives (risk-free-money market investments), which of
the properties of real estate returns affect portfolio risk? Explain.
a. Standard Deviation
b. Expected Return
c. Correlation with the returns of other assets
(a) and (c). The portfolio risk (standard deviation) calculation now includes the variance
of real estate returns and correlation between real estate and stocks the correlation
between real estate and bonds. The correlation between real estate and money markets
will be zero.
Not (b) since the E(r) of real estate does not affect the portfolio’s risk.
Note: The question refers to correlation (ρij) between real estate and the other assets.
Since correlation is defined as the covariance between two assets’ returns divided by the
product of the standard deviations of the assets returns (ρij = σij/σiσj), the portfolio risk
formula can be stated in terms of either correlation or covariance.
3. Which of the following statements about the minimum variance portfolio of all risky
securities are valid? (Assume short sales are allowed.) Explain.
a. Its variance must be lower than those of all other securities or portfolios.
b. Its expected return can be lower than the risk-free rate.
c. It may be the optimal risky portfolio.
d. It must include all individual securities.
Note: In lectures, I referred to the minimum risk or minimum standard deviation portfolio.
Since the standard deviation is the square root of the variance, the portfolio with the
minimum standard deviation must also have the minimum variance so it is therefore the
minimum risk portfolio. All three terms describe the same portfolio.
(a) is valid. This is the definition of the minimum variance (aka minimum risk or
minimum standard deviation) portfolio.
(b) is not valid. The return of the minimum variance portfolio must exceed the return of
the risk free asset since the risk, while minimized, is still greater than risk-free (zero).
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(c) is not valid. It will not be the optimal risky portfolio because there exists another
portfolio, that when combined with the risk-free assets, will produce a larger CAL slope.
See your notes or Figure 7.13 on page 224. G is the Global minimum variance portfolio
and P is the optimal risky portfolio.
(d) is not valid. The minimum variance portfolio (or the optimal risky portfolio, or any
portfolio for that matter) may be formed using zero-weights for many assets.
The following data apply to Problems 4 through 10: A pension fund manager is
considering three mutual funds. The first is a stock fund, the second is a long-term
government and corporate bond fund, and the third is a T-bill money market fund that
yields a rate of 8%. The probability distribution of the risky funds is as follows:
E(r)
20%
12%
Stock Fund (S)
Bond Fund (B)
σ
30%
15%
The correlation between the funds is 0.10.
4. What are the investment proportions in the minimum-variance portfolio of the two
risky funds, and what is the expected value and standard deviation of its rate of
return?
The parameters of the opportunity set are:
E(rS) = 20%, E(rB) = 12%
σS = 30%, σB = 15%
ρAB = 0.10
From the standard deviations and the correlation coefficient we can generate the
“covariance matrix” (note that σSB = ρSB x σS X σB):
Bonds
1.00
0.10
Stocks
Bonds Stocks
Bonds
0.10
Bonds
.0225
.0045
1.00
Stocks
Stocks
.0045
.0900
00
For formula for the minimum variance weight is on page 213 in the text.
(I did not derive this in class.)
WSMin =
 B2   SB
0.0225  0.0045

 0.1739
2
2
 S   B  2 SB 0.0900  0.0225  (2  0.0045)
WSMin = 1  0.1739 = 0.8261
Expected Return Standard Deviation of the minimum variance portfolio are:
2
E(rMin) = (0.1739)(0.20) + (0.8261)(0.12) = 13.39%
σMin = [ w S2  S2  w 2B  2B  2 w S w B Cov (rS , rB )]1 / 2
= [(0.1739)2(0.30) 2 + (0.8261)2(0.15) 2 + 2(0.1739)(0.8261)(0.0045)]1/2 = 13.92%
5. Tabulate and draw the investment opportunity set of the two risky funds. Use
investment proportions for the stock fund of zero to 100% in increments of 20%.
WS
0%
20%
40%
60%
80%
100%
WB
100%
80%
60%
40%
20%
0%
E(r)
12.00%
13.60%
15.20%
16.80%
18.40%
20.00%

15.00%
13.94%
15.70%
19.53%
24.48%
30.00%
6. Draw a tangent from the risk-free rate to the opportunity set. What does your graph
show for the expected return and standard deviation of the optimal portfolio?
(Chart of everything later.)
7. Solve numerically for the proportions of each asset and for the expected return and
standard deviation of the optimal risky portfolio
The optimal risky portfolio is the portfolio along the frontier that maximizes the slope of
the CAL. The formula for the weight of an asset in the optimal risky portfolio is on page
217.
NOTE the change from E(r) – rf = E(R):
E(RS) = 0.20 – 0.08 = 0.12; E(RB) = 0.12 – 0.08 = 0.04
σS2 = 0.090; σB2 = 0.0225; σSB = 0.0045.
WS * 
E ( RS ) B2  E ( RB ) SB
E ( RS ) B2  E ( RB ) S2  [ E ( RS )  E (rB )] SB

[( 0.12 )  0.0225 ]  [( 0.04 )  0.0045 ]
 0.4516
(0.12  .0225 )  [0.04  0.09 ]  (0.12  .0.04 )  .0045 ]
WB* = 1  0.4516 = 0.5484
The mean and standard deviation of the optimal risky portfolio (P) are:
E(rP) = (0.4516)(0.20) + (0.5484)(0.12) = 15.61%
σP= [ w S2  S2  w 2B  2B  2 w S w B Cov (rS , rB )]1 / 2
= [(0.4516)2(0.30) 2 + (0.5484)2(0.15) 2 + 2(0.4616)(0.5484)(0.0045)]1/2 = 16.54%
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8. What is the reward-to-volatility ratio of the best feasible CAL?
The Slope of the CAL = E(rP) – rf)/σP. This is also reward-to-volatility ratio of the CAL.
The best feasible CAL is the CAL with highest reward-to-volatility ratio which is the
achievable CAL with the maximum slope.
The WS* equation above solves for the weight of the Stock fund in the risky portfolio P
that achieves the maximum slope. The “*” indicates an optimal (in this case maximum)
value.
Slope of the Optimal CAL =
E ( rp )  r f
p

0.1561 0.08
 0.4601
0.1654
9. You require that your combined portfolio yield an expected return of 14%, and that
it be efficient (which means it is on the best feasible CAL).
(a) What is the standard deviation of your portfolio?
Combined portfolios on the best feasible CAL are efficient because for any given
return, no other achievable allocation will have less risk (be to the left of the CAL).
Another way to say this is that combined portfolios on the best feasible CAL are
efficient because for any given risk, no other achievable allocation will have greater
return (be above the CAL).
All combined portfolios on the CAL will be combinations on P and the risk-free.
Given: E(rP) = 15.61%; σP= 16.54%; rf = 8%
Solve for the y that makes E(rC) = 14% then compute the σC= for that combined
portfolio:
E(rC) = rf + y[E(rP) – rf]  y = [E(rC) – rf]/[E(rP) – rf]
y = [0.14 – 0.08]/[0.1561 – 0.08] = 0.7884
σC= yσP = 0.7884(0.1654) = 13.04%
Note: This is NOT y* since this is an arbitrary combined portfolio and not the optimal
(utility maximizing) combined portfolio.
(b) What is the proportion of the combined portfolio C invested in the T-bill fund
and each of the two risky funds?
We need to compute the portion of C in stocks, bonds and the risk-free:
y = 78.84% (The portion of the combined portfolio in the risky stock-bond portfolio P)
WS* = 45.16% (The portion of P in the stock fund)
WB* = 54.84% (The portion of P in the bond fund)
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Stocks: (y)(WS*) = (0.7884)(0.4516) = 35.60%
Bonds: (y)(WB*) = (0.7884)(0.5484) = 43.24%
T-Bills: 1 – y = 1 – 0.7884 = 21.16%
Note the weights of the combined portfolio allocations sum to 1.
24%
Investment Opportunity Set
Optimal Risky
Portfolio (P)
20%
Risky Asset
Efficient Frontier
16%
E(r)
Optimal CAL
12%
MinimumRisk Portfolio
8%
4%
0%
0%
4%
8%
12%
16%
20%
24%
28%
32%
σ
10. If you were to use only the two risky funds, and still require an expected return of
14%, what would be the investment proportions of your portfolio? Compare its
standard deviation to that of the optimized portfolio in Problem 9. What do you
conclude?
E(rP) = WSE(rS) + (1 – WS)E(rB) = 14.00%
0.14 = WS(0.20) + (1 – WS)(0.12)
WS = 0.25
σP= [ w S2  S2  w 2B  2B  2 w S w B Cov (rS , rB )]1 / 2
= [(0.25)2(0.30) 2 + (0.75)2(0.15) 2 + 2(0.25)(0.75)(0.0045)]1/2 = 14.13%
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The 14% portfolio with Stocks and Bonds and the risk-free has a risk of 13.04% is on the
CAL. The 14% portfolio with just Stocks and Bonds has a risk of 14.13% and lies to the right
of the CAL. It is dominated by the portfolio on the CAL.
11. Stocks offer an expected rate of return of 18%, with a standard deviation of 22%.
Gold offers an expected return of 10% with a standard deviation of 30%.
(a) In light of the apparent inferiority of gold with respect to both mean return and
volatility, would anyone hold any gold? If so, demonstrate graphically why one
would do so.
Answer Part (b) first.
(b) Given the data above, re-answer (a) with the additional assumption that the
correlation coefficient between gold and stocks equals 1. Draw a graph
illustrating why one would or would not hold gold in one's portfolio. Could this
set of assumptions for expected returns, standard deviations, and correlation
represent an equilibrium for the security market?
The chart below shows the best available CAL when the correlation coefficient
between stocks and gold is one. Note the allocation curve between stocks and gold
(the red line) is a straight line (since ρ = 1). The Optimal (steepest sloped) CAL
passes through the 100% stock portfolio - so no gold is held. Contrast this with the
chart below when the correlation coefficient between stocks and gold is negative 0.50.
24%
P = 100% Stocks
20%
16%
E(r)
CAL
12%
ρ=1
8%
Gold
4%
0%
0%
4%
8% 12% 16% 20% 24% 28% 32%
σ
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The chart below shows the allocation curve between stocks and gold (the red line) for
a correlation coefficient between stocks and gold equal to negative 0.50. It is a
parabola-shaped line. Now holding some gold, an investor can achieve an optimal
risky portfolio P and an accompanying CAL (New CAL in blue) that has a steeper
slope than the CAL available with no gold (Old CAL in green).
24%
20%
Stocks
P
16%
E(r)
New
CAL
12%
Old
CAL
8%
Gold
4%
0%
0%
4%
8% 12% 16% 20% 24% 28% 32%
σ
12. Suppose that there are many stocks in the security market and that the
characteristics of stocks A and B are in the table below. The correlation coefficient is
-1. Suppose that it is possible to borrow at the risk-free rate, rf. What must be the
value of the risk-free rate? (Hint: Think about constructing a risk-free portfolio
from stocks A and B.)
Stock
A
B
E(r)
10%
15%
σ
5%
10%
The theory is that all risk-free portfolios must earn the risk-free rate. Since A and B are
perfectly negatively correlated, then the minimum risk portfolio containing these two
stocks will have zero risk and the return of this risk-free portfolio will earn the risk-free
rate.
Follow this procedure:
1) Calculate the weights of the minimum risk portfolio
2) Computer the return for this risk-free portfolio
3) Check to be sure it has zero risk
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1) The formula for the weight of Asset A in the minimum risk portfolio is:
WAMin = (σB2 – σAB)/( σA2 + σB2 – 2σAB)
Covariance = σAB = ρAB σA σA = (-1)(.05)(.10) = -0.005
WAMin = [0.102 – (-0.005)]/[0.052 + 0.102 – 2(-0.005)] = 0.6667
WBMin = 1 – 0.6667 = 0.3333
2) The formula for the return of the portfolio is:
E(rP) = WAE(rA) + WBE(rB) = 0.6667(0.10) + 0.3333(0.15) = 11.67%
3) Show that this 66.67%-33.33% has zero risk by calculating the risk of the portfolio:
σP = WA2σA2 + WA2σB2 + 2 WAWBσAB
= (0.6667)2(0.05)2 + (0.3333)2(0.15)2 + 2(0.6667)(0.3333)(-0.005) = 0
The conclusion is – in well function markets – the risk-free rate is 11.67%.
Note that there is a special case Minimum-Risk portfolio weight formula for the special
case in which ρ = -1. It is equation 7.12 on page 210. The WAMin formula above works
for any value of ρ. The formula above becomes equation 7.12 if you substitute ρAB σA σA
for the covariance term and plug in ρAB = -1.
14. The standard deviation of the portfolio is always equal to the weighted average of the
standard deviations of the assets in the portfolio. (True or false?)
False. The portfolio standard deviation equals the weighted average of the componentasset standard deviations only in the special case that all assets are perfectly positively
correlated.
Otherwise, as the formula for portfolio standard deviation shows, the portfolio standard
deviation is less than the weighted average of the component-asset standard deviations.
The portfolio variance is a weighted sum of the elements in the covariance matrix, with
the products of the portfolio proportions as weights.
16. Suppose that you have $1 million and the following two opportunities from which to
construct a portfolio: Risk-free asset earning 12% per year and a risky asset with
expected return of 30% per year and standard deviation of 40%. If you construct a
portfolio with a standard deviation of 30%, what is its expected rate of return?
y = σC/σP = 0.30/0.40 = 0.75
E(rC) = rf + y[E(rP) – rf] = 0.12 + 0.75[0.30 – 0.12] = 25.50%
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The following data are for Problems 17 through 19: The correlation coefficients between
pairs of stocks are as follows:
Corr(A,B) = .85
Corr(A,C) = .60
Corr(A,D) = .45
Each stock has an expected return of 8% and a standard deviation of 20%
17. If your entire portfolio is now composed of stock A and you can add some of only one
stock to your portfolio, would you choose (explain your choice):
Choose D since it has the lowest correlation with A and will reduce the portfolio’s risk by
the greatest amount.
Note that since all stocks have the same E(r), adding any of the other stocks will not
change the portfolio’s E(r).
18. Would the answer to Problem 17 change for more risk-averse or risk-tolerant
investors? Explain.
No, risk aversion is not a factor when choosing between risky assets. For all levels of risk
aversion, lower risk is better.
Extra Question: What if A = 0?
What if A = 0? This would be a “risk-neutral” investor. Risk neutral investors would not
care which portfolio they held since all portfolios have an expected return of 8% and
utility is not affected by risk.
Extra Question: What if A< 0?
If A < 0 then utility is increasing in risk. The person is called a “risk lover.”
In this case, a person would prefer adding no assets, since adding any asset that is not
perfectly positively correlated reduces risk. But if forced to add an asset, B would be
preferred since it would reduce risk the least.
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Additional Question:
(Note the following question’s similarity to Question 11.)
1. Consider Risky Asset A, Risky Asset B and the risk-free asset.
Given the points on the axes below representing risky assets A and B and the risk-free asset,
SKETCH and LABEL the following five items if -1 <  < 1:
(4) The optimal combined portfolio if y* = 0.5
(1) The risky asset opportunity set if -1 <  < 1
(5) The maximum obtainable utility curve
(2) The optimal risky portfolio if -1 <  < 1
(3) The optimal CAL
E(r)
Asset A
Asset B
Risk-Free

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2. Given the points on the axes below representing risky assets A and B and the risk-free asset,
SKETCH and LABEL the following five items if  = 1:
(4) The optimal combined portfolio if y* = 0.5
(1) The risky asset opportunity set if  = 1
(5) The maximum obtainable utility curve
(2) The optimal risky portfolio if  = 1
(3) The optimal CAL
E(r)
Asset A
Asset B
Risk-Free

CFA PROBLEMS
1.
a.
Restricting the portfolio to 20 stocks, rather than 40 to 50 stocks, will increase the
risk of the portfolio, but it is possible that the increase in risk will be minimal.
Suppose that, for instance, the 50 stocks in a universe have the same standard
deviation ( and the correlations between each pair are identical, with correlation
coefficient  Then, the covariance between each pair of stocks would be 2, and the
variance of an equally weighted portfolio would be:
1
n 1 2
 2P   2 

n
n
The effect of the reduction in n on the second term on the right-hand side would
be relatively small (since 49/50 is close to 19/20 and 2 is smaller than 2), but
the denominator of the first term would be 20 instead of 50. For example, if =
45% and = 0.2, then the standard deviation with 50 stocks would be 20.91%,
and would rise to 22.05% when only 20 stocks are held. Such an increase might
be acceptable if the expected return is increased sufficiently.
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b.
Hennessy could contain the increase in risk by making sure that he maintains
reasonable diversification among the 20 stocks that remain in his portfolio. This
entails maintaining a low correlation among the remaining stocks. For example, in
part (a), with = 0.2, the increase in portfolio risk was minimal. As a practical
matter, this means that Hennessy would have to spread his portfolio among many
industries; concentrating on just a few industries would result in higher
correlations among the included stocks.
2.
Risk reduction benefits from diversification are not a linear function of the number of
issues in the portfolio. Rather, the incremental benefits from additional diversification
are most important when you are least diversified. Restricting Hennesey to 10 instead of
20 issues would increase the risk of his portfolio by a greater amount than would a
reduction in the size of the portfolio from 30 to 20 stocks. In our example, restricting the
number of stocks to 10 will increase the standard deviation to 23.81%. The 1.76%
increase in standard deviation resulting from giving up 10 of 20 stocks is greater than
the 1.14% increase that results from giving up 30 of 50 stocks.
3.
The point is well taken because the committee should be concerned with the volatility of
the entire portfolio. Since Hennessy’s portfolio is only one of six well-diversified
portfolios and is smaller than the average, the concentration in fewer issues might have a
minimal effect on the diversification of the total fund. Hence, unleashing Hennessy to
do stock picking may be advantageous.
4.
d.
5.
6.
7.
8.
9.
c.
d.
b.
a.
c.
Portfolio Y cannot be efficient because it is dominated by another portfolio. For
example, Portfolio X has both higher expected return and lower standard
deviation.
10. Since we do not have any information about expected returns, we focus exclusively on
reducing variability. Stocks A and C have equal standard deviations, but the correlation
of Stock B with Stock C (0.10) is less than that of Stock A with Stock B (0.90).
Therefore, a portfolio comprised of Stocks B and C will have lower total risk than a
portfolio comprised of Stocks A and B.
11.
Fund D represents the single best addition to complement Stephenson's current portfolio,
given his selection criteria. First, Fund D’s expected return (14.0 percent) has the
potential to increase the portfolio’s return somewhat. Second, Fund D’s relatively low
correlation with his current portfolio (+0.65) indicates that Fund D will provide greater
diversification benefits than any of the other alternatives except Fund B. The result of
adding Fund D should be a portfolio with approximately the same expected return and
somewhat lower volatility compared to the original portfolio.
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The other three funds have shortcomings in terms of either expected return enhancement
or volatility reduction through diversification benefits. Fund A offers the potential for
increasing the portfolio’s return, but is too highly correlated to provide substantial
volatility reduction benefits through diversification. Fund B provides substantial
volatility reduction through diversification benefits, but is expected to generate a return
well below the current portfolio’s return. Fund C has the greatest potential to increase
the portfolio’s return, but is too highly correlated with the current portfolio to provide
substantial volatility reduction benefits through diversification.
12. a.
Subscript OP refers to the original portfolio, ABC to the new stock, and NP to
the new portfolio.
i. E(rNP) = wOP E(rOP ) + wABC E(rABC ) = (0.9  0.67) + (0.1  1.25) = 0.728%
ii. Cov = r  OP  ABC = 0.40  2.37  2.95 = 2.7966  2.80
iii. NP = [wOP2 OP2 + wABC2 ABC2 + 2 wOP wABC (CovOP , ABC)]1/2
= [(0.9 2  2.372) + (0.12  2.952) + (2  0.9  0.1  2.80)]1/2
= 2.2673%  2.27%
b.
Subscript OP refers to the original portfolio, GS to government securities, and NP
to the new portfolio.
i. E(rNP) = wOP E(rOP ) + wGS E(rGS ) = (0.9  0.67) + (0.1  0.042) = 0.645%
ii. Cov = r  OP  GS = 0  2.37  0 = 0
iii. NP = [wOP2 OP2 + wGS2 GS2 + 2 wOP wGS (CovOP , GS)]1/2
= [(0.9 2  2.372) + (0.12  0) + (2  0.9  0.1  0)]1/2
= 2.133%  2.13%
c.
Adding the risk-free government securities would result in a lower beta for the new
portfolio. The new portfolio beta will be a weighted average of the individual security
betas in the portfolio; the presence of the risk-free securities would lower that
weighted average.
d.
The comment is not correct. Although the respective standard deviations and expected
returns for the two securities under consideration are equal, the covariances between
each security and the original portfolio are unknown, making it impossible to draw the
conclusion stated. For instance, if the covariances are different, selecting one security
over the other may result in a lower standard deviation for the portfolio as a whole. In
such a case, that security would be the preferred investment, assuming all other factors
are equal.
e.
i. Grace clearly expressed the sentiment that the risk of loss was more important to
her than the opportunity for return. Using variance (or standard deviation) as a
measure of risk in her case has a serious limitation because standard deviation does
not distinguish between positive and negative price movements.
ii. Two alternative risk measures that could be used instead of variance are:
Range of returns, which considers the highest and lowest expected returns in the
future period, with a larger range being a sign of greater variability and therefore
of greater risk.
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Semivariance, which can be used to measure expected deviations of returns below
the mean, or some other benchmark, such as zero.
Either of these measures would potentially be superior to variance for Grace.
Range of returns would help to highlight the full spectrum of risk she is assuming,
especially the downside portion of the range about which she is so concerned.
Semivariance would also be effective, because it implicitly assumes that the
investor wants to minimize the likelihood of returns falling below some target rate;
in Grace’s case, the target rate would be set at zero (to protect against negative
returns).
13. a.
Systematic risk refers to fluctuations in asset prices caused by macroeconomic
factors that are common to all risky assets; hence systematic risk is often referred
to as market risk. Examples of systematic risk factors include the business cycle,
inflation, monetary policy and technological changes.
Firm-specific risk refers to fluctuations in asset prices caused by factors that are
independent of the market, such as industry characteristics or firm characteristics.
Examples of firm-specific risk factors include litigation, patents, management, and
financial leverage.
b.
Trudy should explain to the client that picking only the top five best ideas would
most likely result in the client holding a much more risky portfolio. The total risk
of a portfolio, or portfolio variance, is the combination of systematic risk and firmspecific risk.
The systematic component depends on the sensitivity of the individual assets to
market movements as measured by beta. Assuming the portfolio is well
diversified, the number of assets will not affect the systematic risk component of
portfolio variance. The portfolio beta depends on the individual security betas and
the portfolio weights of those securities.
On the other hand, the components of firm-specific risk (sometimes called
nonsystematic risk) are not perfectly positively correlated with each other and, as
more assets are added to the portfolio, those additional assets tend to reduce
portfolio risk. Hence, increasing the number of securities in a portfolio reduces
firm-specific risk. For example, a patent expiration for one company would not
affect the other securities in the portfolio. An increase in oil prices might hurt an
airline stock but aid an energy stock. As the number of randomly selected
securities increases, the total risk (variance) of the portfolio approaches its
systematic variance.
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