Chapter 06 – Efficient Diversification
CHAPTER 06
EFFICIENT DIVERSIFICATION
1. So long as the correlation coefficient is below 1.0, the portfolio will benefit from
3. a and b will have the same impact of increasing the Sharpe ratio from .40 to .45.
4. The expected return of the portfolio will be impacted if the asset allocation is changed.
Since the expected return of the portfolio is the first item in the numerator of the Sharpe
5. Total variance = Systematic variance + Residual variance = β2 Var(rM) + Var(e)
When β = 1.5 and σ(e) = .3, variance = 1.52 × .22 + .32 = .18. In the other scenarios:
sMs(e) b
TOTAL
Variance
Corr Coeff
0.2 0.3 1.65 0.1989 0.7399
a. Both will have the same impact. Total variance will increase from .18 to .1989.
6. a. Without doing any math, the severe recession is worse and the boom is better.
Thus, there appears to be a higher variance, yet the mean is probably the same
since the spread is equally large on both the high and low side. The mean return,
however, should be higher since there is higher probability given to the higher
returns.
Chapter 06 – Efficient Diversification
b. Calculation of mean return and variance for the stock fund:
(A) (B) (C) (D) (E) (F) (G)
Col. B
Col. B
×
Col. C
Col. F
Severe recession 0.05 –40 –2.0 –51.2 2621.44 131.07
Mild recession 0.25 –14 –3.5 –25.2 635.04 158.76
Scenario
Deviation
from
Expected
Return
Squared
Deviation
Rate of
Return
Probability
c. Calculation of covariance:
(A) (B) (C) (D) (E) (F)
Col. C Col. B
Stock Bond
Fund Fund Col. D Col. E
Normal growth 0.40 5.8 3 17.4 6.96
Deviation from
Mean Return
Scenario
Probability
7. a. One would expect variance to increase because the probabilities of the extreme
outcomes are now higher.
Chapter 06 – Efficient Diversification
b. Calculation of mean return and variance for the stock fund:
Col. B Col. B
Col. C Col. F
Severe recession 0.10 –0.37 –0.037 –0.465 0.2162 0.0216
Scenario
Probability
Stock
Rate of
Return
Deviation
from
Expected
Return
Squared
Deviation
c. Calculation of covariance
Col. C Col. B
Stock Bond
Fund Fund Col. D Col. E
Severe recession 0.1 –0.465 -0.122 0.05673 0.00567
Mild recession 0.2 –0.205 0.119 –0.024395 –0.0049
Deviation from
Mean Return
Scenario
Probability
8. The parameters of the opportunity set are:
From the standard deviations and the correlation coefficient we generate the covariance
matrix [note that Cov(rS, rB) = SB]:
The minimum-variance portfolio proportions are:
Chapter 06 – Efficient Diversification
wMin(S) = B2 － Cov(rS, rB)
S2 + B2 － 2Cov(rS, rB) = 529 － 110.4
1,024 + 529 － (2 ×110.4) = .3142
The mean and standard deviation of the minimum variance portfolio are:
E(rMin) = ( .3142 15%) + ( .6858 9%) = 10.89%
% in stocks % in bonds Exp. Return Std dev.
Sharpe Ratio
0.20 0.80 0.10 0.20 0.23
0.40 0.60 0.11 0.20 0.29
0.6466 0.3534 0.1288 0.233382 0.3162 Tangency Portfolio
1.00 0.00 0.15 0.32 0.30
9.
15
20
Investment Opportunity Set
Chapter 06 – Efficient Diversification
10. The Sharpe ratio of the optimal CAL is:
11.
a. The equation for the CAL is:
b. The mean of the complete portfolio as a function of the proportion invested in
the risky portfolio (y) is:
E(rC) = (l − y)rf + yE(rP) = rf + y[E(rP) − rf] = 5.5 + y(12.88 − 5.5)
To prevent rounding error, we use the spreadsheet with the calculation of the
previous parts of the problem to compute the proportion in each asset in the
complete portfolio:
18
20
Investment Opportunity Set
Chapter 06 – Efficient Diversification
Proportion of stocks in complete portfolio
W(s) = W(risky portfolio)*% in stock of the risky portfolio
12. Using only the stock and bond funds to achieve a mean of 12%, we solve:
The efficient portfolio with a mean of 12% has a standard deviation of only 20.61%.
13.
a. Although it appears that gold is dominated by stocks, gold can still be an
attractive diversification asset. If the correlation between gold and stocks is
sufficiently low, gold will be held as a component in the optimal portfolio.
b. If gold had a perfectly positive correlation with stocks, gold would not be a part
of efficient portfolios. The set of risk/return combinations of stocks and gold
0%
2%
4%
12%
0% 10% 20% 30%
Corr = 1
Standard
Deviation
Return
Chapter 06 – Efficient Diversification
14. Since Stock A and Stock B are perfectly negatively correlated, a risk-free portfolio can
be created and the rate of return for this portfolio in equilibrium will always be the risk-
free rate. To find the proportions of this portfolio [with wA invested in Stock A and wB
= (1 –wA ) invested in Stock B], set the standard deviation equal to zero. With perfect
negative correlation, the portfolio standard deviation reduces to:
P = ABS[wAA −wBB]
15. Since these are annual rates and the risk-free rate was quite variable during the sample
period of the recent 20 years, the analysis has to be conducted with continuously
compounded rates in excess of T-bill rates. Notice that to obtain cc rates we must
Chapter 06 – Efficient Diversification
Annual returns from Table 2 Continuously compounded rates Excess returns
Year
Large
Stock
Long-
Term T–
Bonds
T-Bills
Large
Stock
Long-
Term T–
Bonds
T-Bills
Large
Stock
Long-
Term T–
Bonds
1991 30.66 18.39 5.60 26.74 16.88 5.45 21.29 11.43
1993 9.87 15.48 2.90 9.41 14.39 2.86 6.55 11.53
1995 37.71 31.67 5.60 32.00 27.51 5.45 26.55 22.06
1997 33.17 15.08 5.25 28.65 14.05 5.12 23.53 8.93
1999 21.04 –8.74 4.69 19.10 –9.15 4.58 14.51 –13.73
2003 28.69 2.38 1.02 25.22 2.35 1.01 24.21 1.34
2005 4.91 6.50 2.98 4.79 6.30 2.94 1.86 3.36
2007 3.53 10.25 4.67 3.47 9.76 4.56 –1.10 5.20
SD 19.64 8.88
Corr(stocks,bonds) 0.13
Stocks Bonds Mean SD
0.0 14.06 8.88
0.2 0.8 3.95 8.55
0.4 0.6 3.85 10.05
0.6 0.4 3.74 12.74
0.8 0.2 3.63 16.04
1.0 03.53 19.64
Min–Var 0.1338 0.8662 3.99 8.44
Weights in
Portfolio
The bond portfolio is less risky as represented by its lower standard deviation. Yet, as
the portfolio table shows, mixing .87% of bonds with 13% stocks would have produced
16. If the lending and borrowing rates are equal and there are no other constraints on
portfolio choice, then the optimal risky portfolios of all investors will be identical.
17. No, it is not possible to get such a diagram. Even if the correlation between A and B
18. In the special case that all assets are perfectly positively correlated, the portfolio
standard deviation is equal to the weighted average of the component-asset standard
19. The probability distribution is:
Probability
Rate of Return
.7
100%
.3
-50%
20. The expected rate of return on the stock will change by beta times the unanticipated
change in the market return: 1.2 ( .08 – .10) = –2.4%
21.
a. The risk of the diversified portfolio consists primarily of systematic risk. Beta
measures systematic risk, which is the slope of the security characteristic line (SCL).
b. The undiversified investor is exposed primarily to firm-specific risk. Stock A
has higher firm-specific risk because the deviations of the observations from the
22. Using “Regression” command from Excel’s Data Analysis menu, we can run a
regression of Ford’s excess returns against those of S&P 500, and obtain the following
data. The Beta of Ford is .87.
Chapter 06 – Efficient Diversification
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.37
R Square 0.14
Coefficients
Standard
Error t Stat P-value
23. A scatter plot results in the following diagram. The slope of the regression line is 2.0
and intercept is 1.0.
-2
0
1
2
-1 -0.5 0 0.5 1
24. a. Regression output produces the following:
Alpha Beta E(r) – rf VAR SD
S&P -0.6123 16.2541 4.0316
Chapter 06 – Efficient Diversification
d. We use Equation 6.16 to compute wG
O
Then, plug wG
O into Equation 6.17 to compute the optimal position of Google in
the optimal risky portfolio and the weight in the market index:
wG
∗ = – .6279 / [1 + (– .6279 (1 – 1.3916)] = –50.40%
SUMMARY OUTPUT: Regression of Google on S&P 500 (excess returns)
Regression Statistics
Multiple R 0.4391
R Square 0.1928
Coefficients
Standard
Error
t Stat P-value
25.
a.
Strategy:
Three-In
One-In
Third-In-Three
Risk Premium
R + R + R = 3R
0 + 0 + R = R
3* 1/3 R = R
Chapter 06 – Efficient Diversification
b. The One-In strategy has the lowest Sharpe ratio
c. The Third-In-Three provides the highest price of risk.
f. The Third-In-Three investment’s price of risk is exactly three times higher than that
of the Three-In.
CFA 1 Answer:
CFA 2 Answer:
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
CFA 3
Answer:
Chapter 06 – Efficient Diversification
i. E(rNP) = wOP E(rOP ) + wABC E(rABC ) = ( .9 .67) + ( .1 1.25) = .7280%
b. Subscript OP refers to the original portfolio, GS to government securities, and
NP to the new portfolio.
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
expected returns for the two securities under consideration are identical, the
correlation coefficients between each security and the original portfolio are
e. 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
CFA 4 Answer:
Chapter 06 – Efficient Diversification
CFA 5
Answer:
Risk reduction benefits from diversification are not a linear function of the number of
CFA 6
Answer:
The point is well taken because the committee should be concerned with the volatility
CFA 7
Answer:
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
b. Trudy should explain to the client that picking only the five best ideas would
most likely result in the client holding a much more risky portfolio. The total
Chapter 06 – Efficient Diversification