ACTIVE MANAGEMENT AND PERFORMANCE MEASUREMENT
1. The dollar-weighted average return or IRR for the cash flow given, based on daily
compounding, is .05553 percent or an annual rate of 22.46 percent. (Solved
through spreadsheet by using XIRR function, and assuming 365 days in a year.)
2. As established in the following result from the text, the Sharpe ratio depends on
both alpha for the portfoli () and the correlation between the portfolio and the
market index (ρ):
E(r P r f P 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 averages. A number of scenarios can be developed to illustrate this
conclusion. For example, consider a scenario where the rate of return each period
consistently increases over several time periods. If the amount invested also
increases each period, and then all of the proceeds are withdrawn at the end of
several periods, the IRR is greater than 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.)
4. It is not necessarily wise to shift resources to timing at the expense of security
selection. There is also tremendous potential value in security analysis. The
decision on whether to shift resources has to be made on the basis of the macro,
compared to the micro, forecasting ability of the portfolio management team.
5. a. Arithmetic average: r ABC = 10% r XYZ = 10%
21-1 b. Dispersion: ABC = 7.07%, XYZ = 13.91%. XYZ has greater dispersion.
(We used 5 degrees of freedom to calculate standard deviations.)
c. Geometric average:
rABC = (1.2 1.12 1.14 1.03 1.01)5– 1 = .0977 = 9.77%
rXYZ = (1.3 1.12 1.18 1.0 .90)– 1 = .0911 = 9.11%
Despite the equal arithmetic averages, XYZ has a lower geometric average.
The reason is that the greater variance of XYZ drives the geometric average
further below the arithmetic average.
d. In terms of “forward-looking” statistics, the arithmetic average is the better
estimate of expected return. Therefore, if the data reflect the probabilities of
future returns, 10 percent is the expected return of both stocks.
6. a. Time-weighted average returns are based on year-by-year rates of return.
Year Return [(Capital gains + Dividend)/Price]
2010–2011 [($120 – $100) + $4]/$100 = 24.00%
2011–2012 [($90 – $120) + $4]/$120 = –21.67%
2012–2013 [($100 – $90) + $4]/$90 = 15.56%
Arithmetic mean: (24% – 21.67% + 15.56%)/3 = 5.96%
Geometric mean: (1.24 .7833 1.1556) 1/3– 1 = .0392 = 3.92%
Date Cash Flow Explanation
1/1/2010 –300 Purchase of three shares at $100 each.
1/1/2011 –228 Purchase of two shares at $120 less
dividend income on three shares held.
1/1/2012 110 Dividends on five shares plus sale of one
share at $90.
1/1/2013 416 Dividends on four shares plus sale of
four shares at $100 each.
21-2 Dollar-weighted return = Internal rate of return = –.1606%.
7. Time Cash Flow ($) 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 = (1.11111.0 1.0)1/3– 1 =
.0357 = 3.57%.
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%. (You can find this
using a financial calculator by setting n = 3, PV = (–)270, FV = 0, PMT =
100, and solving for the interest rate.) The IRR exceeds the other averages
because the investment fund was the largest when the highest return occurred.
21-3 8. a. E(r) (%)
Portfolio A 12 12 .7
Portfolio B 16 31 1.4
Market index 13 18 1.0
Risk-free asset 5 0 0.0
The alphas for the two portfolios are:
A= 12% – [5% + 0.7(13% – 5%)] = 1.4%
B= 16% – [5% + 1.4(13% – 5%)] = – 0.2%
Ideally, you would want to take a long position in A and a short position in B.
b. If you will hold only one of the two portfolios, then the Sharpe measure is the
12 – 5
SA= 12 = .583
16 – 5
SB= 31 = .355
Portfolio A is preferred using the Sharpe criterion.
9. a. Stock A Stock B
i. Alpha is the intercept of the regression 1% 2%
ii. Appraisal ratio = /(e ) .0971 .1047
iii. Sharpe ratio* = (r – r )/ .4907 .3373
p f p
iv. Treynor measure** = (r – r )/ 8.833% 10.500%
p f p
*To compute the Sharpe ratio, note that for each stockp r f r can be
computed from the right-hand side of the regression equation, using the
assumed parameters r M 14% and r = f%. 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, then Sharpe’s measure is the one to use. A’s
is higher, so it is preferred.
ii. If the stock is mixed with the index fund, the contribution to the overall
Sharpe measure is determined by the appraisal ratio; therefore, B is
21-4 iii. If it is one of many stocks, then Treynor’s measure counts, and B is
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
steeper as you move to the right of the graph show good timing ability. The
steeper slope shows that the manager maintained higher portfolio sensitivity to
market swings (i.e., a higher beta) in periods when the market performed well.
This ability to choose more market-sensitive securities in anticipation of market
upturns is the essence of good timing. In contrast, a declining slope as you move
to the right means that the portfolio was more sensitive to the market when the
market did poorly and less sensitive when the market did well. This indicates poor
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
11. a. Sharpe ratio = (r P r )f P
Williamson Capital: Sharpe ratio = (22.1% 5.0%)/16.8% = 1.02
Joyner Asset Management: Sharpe ratio = (24.2% 5.0%)/20.2% = 0.95
Treynor measure = (r P r f/ P
Williamson Capital: Treynor measure = (22.1% 5.0%)/1.2 = 14.25
Joyner Asset Management: Treynor measure = (24.2% 5.0%)/0.8 = 24.00
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 1.02, respectively), so Joyner must be less diversified than Williamson.
The Treynor measure indicates that Joyner has a higher return per unit of
21-5 systematic risk than does Williamson, while the Sharpe ratio indicates that
Joyner has a lower return per unit of total risk than does Williamson.
12. Support. A manager could be a better performer in one type of circumstance. 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
performance over an entire cycle. Also, to the extent that observing a manager
over an entire cycle increases the number of observations, it would improve the
reliability of the measurement.
Contradict. If we adequately control for exposure to the market (i.e., adjust for
beta), then market performance should not affect the relative performance of
individual managers. It is therefore not necessary to wait for an entire market
cycle to pass before you evaluate a manager.
13. It does, to some degree, if those manager groups can be made sufficiently
homogeneous with respect to style.
14. a. The manager’s alpha is 10 – [6 + .5(14 – 6)] = 0.
b. From Black-Jensen-Scholes and others, we know that, on average, portfolios
with low beta have had positive alphas. (The slope of the empirical security
market line is shallower than predicted by the CAPM—see Chapter 7.)
Therefore, given the manager’s low beta, performance could be subpar despite
the estimated alpha of zero.
15. a. Bogey: (0.60 2.5%) + (0.30 1.2%) + (0.10 0.5%) = 1.91%
Actual: (0.70 2.0%) + (0.20 1.0%) + (0.10 0.5%) = 1.65%
b. Security selection:
(1) (2) (3) = (1) × (2)
Differential Manager’s Contribution
Market Market (Manager Portfolio to
– Index) Weight 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:
21-6 (1) (2) (3) = (1) × (2)
Excess weight Index Contribution
Market (Manager – to
Benchmark) Return 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%
Security selection –0.39%
Asset allocation 0.13%
Excess performance –0.26%
16. Time-weighted average return = (1.151.1) 1/1 12.47%
(The arithmetic mean is: 2 12.5% .)
To compute dollar-weighted rate of return, cash flows are:
CF 0 −$500,000
CF 1 −$500,000
CF 2 ($500,000 × 1.15 × 1.10) + ($500,000 × 1.10) = $1,182,500
Dollar-weighted rate of return = 11.71%
17. 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 risk, while the Treynor ratio measures excess return per unit of
systematic risk. Since fund A performed well on the Treynor measure and so
poorly on the Sharpe measure, it seems that the fund carries a greater amount
of unsystematic risk, meaning it is not well diversified and systematic risk is
not the relevant risk measure.
18. 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