Chapter 24 – Portfolio Performance Evaluation
CHAPTER 24: PORTFOLIO PERFORMANCE EVALUATION
PROBLEM SETS
1. As established in the following result from the text, the Sharpe ratio depends on both
alpha for the portfolio ( P) and the correlation between the portfolio and the market
index (ρ):
M
P
P
P
fP S
rrE
+=
−)(
Specifically, this result demonstrates that a lower correlation with the market index
reduces the Sharpe ratio. Hence, if alpha is not sufficiently large, the portfolio is inferior
to the index. Another way to think about this conclusion is to note that, even for a
portfolio with a positive alpha, if its diversifiable risk is sufficiently large, thereby
reducing the correlation with the market index, this can result in a lower Sharpe ratio.
2. The IRR (i.e., the dollar-weighted return) can not be ranked relative to either the
geometric average return (i.e., the time-weighted return) or the arithmetic average
return. Under some conditions, the IRR is greater than each of the other two averages,
and similarly, under other conditions, the IRR can also be less than each of the other
3. It is not necessarily wise to shift resources to timing at the expense of security selection.
24-2
4. a. Arithmetic average:
%10rABC =
;
%10rXYZ =
b. Dispersion: ABC = 7.07%; XYZ = 13.91%
c. Geometric average:
rABC = (1.20 1.12 1.14 1.03 1.01)1/5 – 1 = 0.0977 = 9.77%
d. In terms of “forward looking” statistics, the arithmetic average is the
better estimate of expected rate of return. Therefore, if the data reflect the
5. a. Time-weighted average returns are based on year-by-year rates of return:
Year
Return = (capital gains + dividend)/price
2005 − 2006
[($120 – $100) + $4]/$100 = 24.00%
2006 − 2007
[($90 – $120) + $4]/$120 = –21.67%
2007 − 2008
[($100 – $90) + $4]/$90 = 15.56%
Arithmetic mean: (24% – 21.67% + 15.56%)/3 = 5.96%
Geometric mean: (1.24 0.7833 1.1556)1/3 – 1 = 0.0392 = 3.92%
Chapter 24 – Portfolio Performance Evaluation
24-3
b.
Date
Explanation
1/1/05
Purchase of three shares at $100 each
1/1/06
Purchase of two shares at $120 less dividend income on three shares held
1/1/07
Dividends on five shares plus sale of one share at $90
1/1/08
Dividends on four shares plus sale of four shares at $100 each
416
110
−228
−300
Dollar-weighted return = Internal rate of return = –0.1607%
6.
Time
Cash flow
Holding period return
0
3(–$90) = –$270
1
$100
(100–90)/90 = 11.11%
2
$100
0%
3
$100
0%
a. Time-weighted geometric average rate of return =
b. Time-weighted arithmetic average rate of return = (11.11% + 0 + 0)/3 = 3.70%
The arithmetic average is always greater than or equal to the geometric average;
the greater the dispersion, the greater the difference.
c. Dollar-weighted average rate of return = IRR = 5.46%
Chapter 24 – Portfolio Performance Evaluation
7. a. The alphas for the two portfolios are:
A = 12% – [5% + 0.7(13% – 5%)] = 1.4%
b. If you will hold only one of the two portfolios, then the Sharpe measure is the
appropriate criterion:
583.0
12
512
SA=
−
=
355.0
31
516
SB=
−
=
Using the Sharpe criterion, Portfolio A is the preferred portfolio.
8.
a.
Stock A
Stock B
(i)
Alpha = regression intercept
1.0%
2.0%
(ii)
Information ratio = P /(eP)
0.0971
0.1047
(iii)
*Sharpe measure = (rP – rf)/P
0.4907
0.3373
(iv)
**Treynor measure = (rP – rf )/P
8.833
10.500
* To compute the Sharpe measure, note that for each stock, (rP – rf ) can be
computed from the right-hand side of the regression equation, using the assumed
b. (i) If this is the only risky asset held by the investor, then Sharpe’s measure is the
appropriate measure. Since the Sharpe measure is higher for Stock A, then A is the
best choice.
24-5
9. We need to distinguish between market timing and security selection abilities. The
intercept of the scatter diagram is a measure of stock selection ability. If the
manager tends to have a positive excess return even when the market’s performance
is merely “neutral” (i.e., has zero excess return), then we conclude that the manager
has on average made good stock picks. Stock selection must be the source of the
positive excess returns.
We can therefore classify performance for the four managers as follows:
Selection
Ability
Timing Ability
A.
Bad
Good
B.
Good
Good
C.
Good
Bad
D.
Bad
Bad
10. a. Bogey: (0.60 2.5%) + (0.30 1.2%) + (0.10 0.5%) = 1.91%
b. Security Selection:
(1)
(2)
(3) = (1) (2)
Market
Differential return
within market
(Manager – index)
Manager’s
portfolio weight
Contribution to
performance
Equity
–0.5%
0.70
−0.35%
Bonds
–0.2%
0.20
–0.04%
Cash
0.0%
0.10
0.00%
Contribution of security selection:
−0.39%
24-6
(1)
(2)
(3) = (1) (2)
Market
Excess weight
(Manager – benchmark)
Index
Return
Contribution to
performance
Equity
0.10%
2.5%
0.25%
Bonds
–0.10%
1.2%
–0.12%
Cash
0.00%
0.5%
0.00%
Contribution of asset allocation:
0.13%
Summary:
Security selection –0.39%
11. a. Manager return: (0.30 20) + (0.10 15) + (0.40 10) + (0.20 5) = 12.50%
b. Added value from country allocation:
(1)
(2)
(3) = (1) (2)
Country
Excess weight
(Manager – benchmark)
Index Return
minus bogey
Contribution to
performance
U.K.
0.15%
−1.8%
−0.27%
Japan
–0.20%
1.2%
–0.24%
U.S.
−0.05%
0.2%
−0.01%
Germany
0.10%
−1.8%
−0.18%
Contribution of country allocation:
−0.70%
c. Added value from stock selection:
(1)
(2)
(3) = (1) (2)
Country
Differential return
within country
(Manager – Index)
Manager’s
country weight
Contribution to
performance
U.K.
8%
0.30%
2.4%
Japan
0%
0.10%
0.0%
U.S.
−4%
0.40%
−1.6%
Germany
−7%
0.20%
−1.4%
Contribution of stock selection:
−0.6%
Summary:
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12. Support: A manager could be a better performer in one type of circumstance than in
another. For example, a manager who does no timing, but simply maintains a high beta,
will do better in up markets and worse in down markets. Therefore, we should observe
13. The use of universes of managers to evaluate relative investment performance does, to
b. From Black-Jensen-Scholes and others, we know that, on average, portfolios with
CFA PROBLEMS
1. a. Manager A
Strength. Although Manager A’s one-year total return was somewhat below the
international index return (–6.0 percent versus –5.0 percent), this manager
apparently has some country/security return expertise. This large local market
Chapter 24 – Portfolio Performance Evaluation
24-8
Manager B
Strength. Manager B’s total return exceeded that of the index, with a marked
positive increment apparent in the currency return. Manager B had a –1.0 percent
currency return compared to a –5.2 percent currency return on the international
b. The following strategies would enable the fund to take advantage of the
strengths of each of the two managers while minimizing their weaknesses.
1. Recommendation: One strategy would be to direct Manager A to make no
currency bets relative to the international index and to direct Manager B to
make only currency decisions, and no active country or security selection
bets.
2. Recommendation: Another strategy would be to combine the portfolios of
Manager A and Manager B, with Manager A making country exposure and
2. a. Indeed, the one year results were terrible, but one year is a poor statistical base
from which to draw inferences. Moreover, the board of trustees had directed Karl
b. The sample of pension funds had a much larger share invested in equities than did
Alpine. Equities performed much better than bonds. Yet the trustees told Alpine to
c. Alpine’s alpha measures its risk-adjusted performance compared to the market:
Chapter 24 – Portfolio Performance Evaluation
d. Note that the last 5 years, and particularly the most recent year, have been bad
for bonds, the asset class that Alpine had been encouraged to hold. Within this
e. A trustee may not care about the time-weighted return, but that return is more
3. a. Method I does nothing to separately identify the effects of market timing and
b. Method II is not perfect, but is the best of the three techniques. It at least attempts
to focus on market timing by examining the returns for portfolios constructed from
bond market indexes using actual weights in various indexes versus year-average
c. Method III uses net purchases of bonds as a signal of bond manager optimism.
24–11
14. a. Each of these benchmarks has several deficiencies, as described below.
Market index:
• A market index may exhibit survivorship bias. Firms that have gone out of
business are removed from the index, resulting in a performance measure that
overstates actual performance had the failed firms been included.
• The chosen index may not be investable. There may be securities in the index
that cannot be held in the portfolio.
Benchmark normal portfolio:
• This is the most difficult performance measurement method to develop and
calculate.
• Comparison with a manager universe does not take into account the risk taken in
the portfolio.
• The median of a manager universe does not represent an “investable” portfolio;
that is, a portfolio manager may not be able to invest in the median manager
portfolio.
Chapter 24 – Portfolio Performance Evaluation
24–12
b. i. The Sharpe ratio is calculated by dividing the portfolio risk premium (i.e.,
actual portfolio return minus the risk-free return) by the portfolio standard
deviation:
Sharpe ratio = (rP – rf)/P
ii. The Sharpe ratio assumes that the relevant risk is total risk, and it measures
excess return per unit of total risk. The Treynor measure assumes that the
15. i. The statement is incorrect. Valid benchmarks are unbiased. Median manager
benchmarks, however, are subject to significant survivorship bias, which results in
several drawbacks, including the following:
• The performance of median manager benchmarks is biased upwards.
ii. The statement is incorrect. Valid benchmarks are unambiguous and able to be
replicated. The median manager benchmark, however, is ambiguous because the
weights of the individual securities in the benchmark are not known. The
iii. The statement is correct. The median manager benchmark may be inappropriate
because the median manager universe encompasses many investment styles and,
therefore, may not be consistent with a given manager’s style.
Chapter 24 – Portfolio Performance Evaluation
16. a. Sharpe ratio = (rP – rf)/P
Williamson Capital: Sharpe ratio = (22.1% − 5.0%)/16.8% = 1.02
b. The difference in the rankings of Williamson and Joyner results directly from the
difference in diversification of the portfolios. Joyner has a higher Treynor measure
(24.00) and a lower Sharpe ratio (0.95) than does Williamson (14.25 and 1.202,