Chapter 18 – Portfolio Performance Evaluation
CHAPTER 18
PORTFOLIO PERFORMANCE EVALUATION
1. The dollar-weighted average will be the internal rate of return between the initial and
final value of the account, including additions and withdrawals. Using Excel’s XIRR
function, utilizing the given dates and values, the dollar-weighted average return is as
follows:
Date
Account
January 1
$148,000.00
February 1
$2,500.00
March 1
$4,000.00
April 1
$1,500.00
May 1
$13,460.00
June 1
-$23,000.00
July 1
$3,000.00
August 1
$198,000.00
26.43%
=XIRR(C13:C20,B13:B20)
2. 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):
()
αρ
σσ
Pf PM
PP
E r r S
−=+
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.
3. The IRR (i.e., the dollar-weighted return) cannot 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
Chapter 18 – Portfolio Performance Evaluation
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either the geometric or the arithmetic average because more money is invested at the
higher rates than at the lower rates. On the other hand, if withdrawals gradually reduce
the amount invested as the rate of return increases, then the IRR is less than each of the
other averages. (Similar scenarios are illustrated with numerical examples in the text,
where the IRR is shown to be less than the geometric average, and in Concept Check 1,
where the IRR is greater than the geometric average.)
a. Possibly. Alpha alone does not determine which portfolio has a larger Sharpe
ratio. Sharpe measure is the primary factor, since it tells us the real return per
4. a. In any given year the expected value of the portfolio is the starting value. We
will assume “fall” means the portfolio value goes to zero. Therefore—
Starting Value $1,000
Ending Value = $1,000 (.5 x 2+.5 x 0) = $1,000 → Expected Return of 0%
2.
a. Arithmetic average: ̅rABC = 10%; ̅rXYZ = 10%
b. Dispersion: σABC = 7.07%; σXYZ = 13.91%
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consent of McGraw-Hill Education.
rXYZ = (1.30 × 1.12 × 1.18 × 1.00 × 0.90)1/5 – 1 = 0.0911 = 9.11%
Despite the fact that the two stocks have the same arithmetic average, the
geometric average for XYZ is less than the geometric average for ABC. The
reason for this result is the fact that the greater variance of XYZ drives the
geometric average further below the arithmetic average.
d. Your expected rate of return would be the arithmetic average, or 10%.
e. Even though the dispersion is greater, your expected rate of return would
still be the arithmetic average, or 10%.
6. a. Time-weighted average returns are based on year-by-year rates of return:
Year
Return = (Capital gains + Dividend)/Price
2016 − 2017
[($120 – $100) + $4]/$100 = 24.00%
2017 – 2018
[($90 – $120) + $4]/$120 = –21.67%
2018 − 2019
[($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%
b.
Date
Cash
Flow
Explanation
1/1/13
–$300
Purchase of three shares at $100 each
1/1/14
–$228
Purchase of two shares at $120 less dividend income on three shares held
1/1/15
$110
Dividends on five shares plus sale of one share at $90
1/1/16
$416
Dividends on four shares plus sale of four shares at $100 each
Chapter 18 – Portfolio Performance Evaluation
−228
7.
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 =
(1.1111 × 1.0 × 1.0)1/3 – 1 = 0.0357 = 3.57%
b. Time-weighted arithmetic average rate of return = (11.11% + 0 + 0)/3 = 3.70%
8. a. The alphas for the two portfolios are:
αA = 12% – [5% + 0.7 × (13% – 5%)] = 1.4%
αB = 16% – [5% + 1.4 × (13% – 5%)] = –0.2%
Ideally, you would want to take a long position in Portfolio A and a short position
in Portfolio B.
b. If you will hold only one of the two portfolios, then the Sharpe measure is the
9.
Chapter 18 – Portfolio Performance Evaluation
a.
Stock A
Stock B
(i)
Alpha = regression intercept
1.0%
2.0%
(ii)
Information ratio =
α
σ(e )
P
P
0.0971
0.1047
(iii)
*Sharpe measure =
σ
Pf
P
rr−
0.4907
0.3373
(iv)
†Treynor measure =
β
Pf
P
rr−
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
parameters rM = 14% and rf = 6%. The standard deviation of each stock’s returns is
given in the problem.
† The beta to use for the Treynor measure is the slope coefficient of the regression
equation presented in the problem.
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.
10. 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.
Timing ability is indicated by the curvature of the plotted line. Lines that become
We can therefore classify performance for the four managers as follows:
Selection
Ability
Timing Ability
Chapter 18 – Portfolio Performance Evaluation
A.
Bad
Good
B.
Good
Good
C.
Good
Bad
D.
Bad
Bad
11. 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%
c. Asset Allocation:
(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%
12. a. Manager: (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
Chapter 18 – Portfolio Performance Evaluation
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.
0.08
0.30%
2.4%
Japan
0.00
0.10
0.0
U.S.
−0.04
0.40
−1.6
Germany
−0.07
0.20
−1.4
Contribution of stock selection:
−0.6%
13. 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
14. The use of universes of managers to evaluate relative investment performance does, to
some extent, overcome statistical problems, as long as those manager groups can be
made sufficiently homogeneous with respect to style.
15. a. The manager’s alpha is 10% – [6% + 0.5 × (14% – 6%)] = 0
b. From Black-Jensen-Scholes and others, we know that, on average, portfolios with
16. a. The most likely reason for a difference in ranking is due to the absence of
diversification in Fund A. The Sharpe ratio measures excess return per unit of total
17. The within sector selection calculates the return according to security selection. This is
done by summing the weight of the security in the portfolio multiplied by the return of
the security in the portfolio minus the return of the security in the benchmark:
Large Cap Sector: 0.6 (.17-.16)= 0.6%
Mid Cap Sector: 0.15 (.24–.26) –0.3%
Small Cap Sector: 0.25 (.20-.18)= 0.5%
Total Within-Sector Selection = 0.6% – 0.3% 0.5% 0.8%
=
+=
18. Primo Return
0.6 17% 0.15 24% 0.25 20% 18.8%= + + =
To isolate the impact of Primo’s pure sector allocation decision relative to the
benchmark, multiply the weight difference between Primo and the benchmark portfolio in
each sector by the benchmark sector returns:
To isolate the impact of Primo’s pure security selection decisions relative to the
benchmark, multiply the return differences between Primo and the benchmark for each
sector by Primo’s weightings:
19. Because the passively managed fund is mimicking the benchmark, the
2
R
of the
20. a. The euro appreciated while the pound depreciated. Primo had a greater stake in the
euro-denominated assets relative to the benchmark, resulting in a positive currency
allocation effect. British stocks outperformed Dutch stocks resulting in a negative
Chapter 18 – Portfolio Performance Evaluation
Copyright © 2018 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written
with the British investments. Since they had a greater proportion invested in Dutch
stocks relative to the benchmark, we assume that they had a positive security
allocation effect in total. However, this cannot be known for certain with this
information. It is the best choice, however.
21. a.
Miranda S&P
.102 .02 .225 .02
.2216 .5568
σ .37 .44
Pf
P
rr SS
−− − −
→ = = = =
b. To compute
2
M
measure, blend the Miranda Fund with a position in T-bills such
that the adjusted portfolio has the same volatility as the market index. Using the
data, the position in the Miranda Fund should be .44/.37 = 1.1892 and the position
in T-bills should be 1 – 1.1892 = -0.1892 (assuming borrowing at the risk-free rate).
The adjusted return is:
*(1.1892) 10.2% (.1892) 2% .1175 11.75%
P
r= − = =
Calculate the difference in the adjusted Miranda Fund return and the benchmark:
*
211.75% ( 22.50%) 34.25%
M
P
M r r= − = − − =
[Note: The adjusted Miranda Fund is now 59.46% equity and 40.54% cash.]
c.
Miranda S&P
.102 .02 .225 .02
.0745 .245
β 1.10 1.00
Pf
P
rr TT
−− − −
→ = = = = −
d.
22. This exercise is left to the student; answers will vary.
CFA 1. a. Manager A
Strength. Although Manager A’s one-year total return was somewhat below the
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
α [ β ( )]
0.102 [0.02 1.10 ( 0.225 0.02)]
.3515 35.15%
P P f P M f
r r r r= − + −
= − + − −
==
Chapter 18 – Portfolio Performance Evaluation
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consent of McGraw-Hill Education.
currency return compared to a –5.2 percent currency return on the international
index. Based on this outcome, Manager B’s strength appears to be expertise in the
currency selection area.
Weakness. Manager B had a marked shortfall in local market return. Therefore,
Manager B appears to be weak in security/market selection ability.
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
2. Recommendation: Another strategy would be to combine the portfolios of
Manager A and Manager B, with Manager A making country exposure and
CFA 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 to
adopt a long-term horizon. The board specifically instructed the investment
manager to give priority to long-term results.
b. The sample of pension funds had a much larger share invested in equities than did
c. Alpine’s alpha measures its risk-adjusted performance compared to the market:
d. Note that the last five years, and particularly the most recent year, have been bad
for bonds, the asset class that Alpine had been encouraged to hold. Within this
Chapter 18 – Portfolio Performance Evaluation
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consent of McGraw-Hill Education.
has been superior on a risk-adjusted basis. Its overall disappointing returns were
due to a heavy asset allocation weighting towards bonds which was the board’s,
not Alpine’s, choice.
e. A trustee may not care about the time-weighted return, but that return is more
indicative of the manager’s performance. After all, the manager has no control over
the cash inflows and outflows of the fund.
CFA 3. a. Method I does nothing to separately identify the effects of market timing
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. But
such net purchases can be motivated by withdrawals from or contributions to the
fund rather than the manager’s decisions. (Note that this is an open-ended mutual
fund.) Therefore, it is inappropriate to evaluate the manager based on whether net
purchases turn out to be reliable bullish or bearish signals.
17 8 8.182
1.1
−=
(.24 .08) 0.888
.18
−=
CFA 6. a. Treynor measures
(10 6) (12 6)
Portfolio X: 6.67 S&P 500: 6.00
0.6 1.0
−−
==
Sharpe measures
(.10 .06) (.12 .06)
Portfolio X: 0.222 S&P 500: 0.462
0.18 .13
−−
==
Chapter 18 – Portfolio Performance Evaluation
Portfolio X outperforms the market based on the Treynor measure, but
underperforms based on the Sharpe measure.
b. The two measures of performance are in conflict because they use different
CFA 7. Geometric average = (1.15 × 0.90)1/2 – 1 = 0.0173 = 1.73%
CFA 11. Time-weighted average return =
1/2
(1.15 1.1) 1 12.47% − =
[The arithmetic mean is:
15% 10% 12.5%
2
+=
]
To compute dollar-weighted rate of return, cash flows are:
Dollar-weighted rate of return = 11.71% (Solve for IRR in financial calculator).
CFA 12. 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.
Chapter 18 – Portfolio Performance Evaluation
• The chosen index may not represent the entire universe of securities. For example,
the S&P 500 Index represents 65 to 70 percent of U.S. equity market
capitalization.
• The chosen index (e.g., the S&P 500) may have a large capitalization bias.
• 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.
• The normal portfolio must be continually updated, requiring substantial resources.
Median of the manager universe:
• It can be difficult to identify a universe of managers appropriate for the
investment style of the plan’s managers.
• A manager universe may exhibit survivorship bias; managers who have gone out of
business are removed from the universe, resulting in a performance measure that
overstates the actual performance had those managers been included.
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 =
σ
Pf
P
rr−
The Treynor measure is calculated by dividing the portfolio risk premium (i.e.,
actual portfolio return minus the risk-free return) by the portfolio beta:
β
Pf
P
rr−
Jensen’s alpha is calculated by subtracting the market risk premium, adjusted
for risk by the portfolio’s beta, from the actual portfolio excess return (risk
premium). It can be described as the difference in return earned by the portfolio
Chapter 18 – Portfolio Performance Evaluation
compared to the return implied by the Capital Asset Pricing Model or Security
Market Line:
α [ β ( )]
P P f P M f
r r r r= − + −
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
relevant risk is systematic risk, and it measures excess return per unit of
systematic risk. Jensen’s alpha assumes that the relevant risk is systematic risk,
and it measures excess return at a given level of systematic risk.
CFA 13. i. 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. Incorrect. Valid benchmarks are unambiguous and can be replicated. The median
manager benchmark is ambiguous because the weights of the individual securities
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.
CFA 14. a. Sharpe ratio =
σ
Pf
P
rr−
22.1% 5.0% 24.2% 5.0%
: 1.02 : 0.95
16.8% 20.2%
Williamson Joyner
SS
−−
==
β
P
Pf
rr−
1.2 0.8
Williamson Joyner
Chapter 18 – Portfolio Performance Evaluation
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