Solutions for Chapter 18: Questions and Problems
CHAPTER 18
EVALUATION OF PORTFOLIO PERFORMANCE
Answers to Questions
1.
The two major factors would be: (1) attempt to derive risk-adjusted returns that exceed a
naive buy-and-hold policy and (2) completely diversify - i.e., eliminate all unsystematic
risk from the portfolio. A portfolio manager can do one or both of two things to derive
superior risk-adjusted returns. The first is to have superior timing regarding market
cycles and adjust your portfolio accordingly. Alternatively, one can consistently select
undervalued stocks. As long as you do not make major mistakes with the rest of the
portfolio, these actions should result in superior risk-adjusted returns.
2.
Treynor (1965) divided a fund’s excess return (return less risk-free rate) by its beta. For a
fund not completely diversified, Treynor’s “T” value will understate risk and overstate
performance. Sharpe (1966) divided a fund’s excess return by its standard deviation.
Sharpe’s “S” value will produce evaluations very similar to Treynor’s for funds that are
well diversified. Jensen (1968) measures performance as the difference between a fund’s
actual and required returns. Since the latter return is based on the CAPM and a fund’s
beta, Jensen makes the same implicit assumptions as Treynor - namely, that funds are
completely diversified. The information ratio (IR) measures a portfolio’s average return
in excess of that of a benchmark, divided by the standard deviation of this excess return.
Like Sharpe, it can be used when the fund is not necessarily well-diversified.
3.
For portfolios with R2 values noticeably less than 1.0, it would make sense to compute
both measures. Differences in the rankings generated by the two measures would suggest
less-than-complete diversification by some funds - specifically, those that were ranked
higher by Treynor than by Sharpe.
4.
Jensen’s alpha () is found from the equation Rjt – RFRt = j + j[Rmt – RFRt] +ejt. The aj
indicates whether a manager has superior (j > 0) or inferior (j < 0) ability in market
timing or stock selection, or both. As suggested above, Jensen defines superior (inferior)
performance as a positive (negative) difference between a manager’s actual return and his
CAPM-based required return. For poorly diversified funds, Jensen’s rankings would
more closely resemble Treynor’s. For well-diversified funds, Jensen’s rankings would
follow those of both Treynor and Sharpe. By replacing the CAPM with the APT,
differences between funds’ actual and required returns (or “alphas”) could provide fresh
evaluations of funds.
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Solutions for Chapter 18: Questions and Problems
5.
The Information Ratio (IR) is calculated by dividing the average return on the
portfolio less a benchmark return by the standard deviation of the excess return. The IR
can be viewed as a benefit-cost ratio in that the standard deviation of return can be
viewed as a cost associated in the sense that it measures the unsystematic risk taken on
by active management. Thus IR is a cost-benefit ratio that assesses the quality of the
investor’s information deflated by unsystematic risk generated by the investment
process.
6.
The difference by which a manager’s overall actual return beats his/her overall
benchmark return is termed the total value-added return and decomposes into an
allocation effect and a selection effect. The former effect measures differences in weights
assigned by the actual and benchmark portfolios to stocks, bonds and cash times the
respective differences between market-specific benchmark returns and the overall
benchmark return. The latter effect focuses on the market-specific actual returns less the
corresponding market-specific benchmark returns times the weights assigned to each
market by the actual portfolio. Of course, the foregoing analysis implicitly assumes that
the actual and benchmark market-specific portfolios (e.g., stocks) are risk-equivalent. If
this is not true the analysis would not be valid.
7.
When measuring the performance of an equity portfolio manager, overall returns can be
related to a common total risk or systematic risk. Factors influencing the returns achieved
by the bond portfolio manager are more complex. In order to evaluate performance based
on a common risk measure (i.e., market index), four components must be considered that
differentiate the individual portfolio from the market index. These components include:
(1) a policy effect, (2) a rate anticipation effect, (3) an analysis effect, and (4) a trading
effect. Decision variables involved include the impact of duration decisions, anticipation
of sector/quality factors, and the impact of individual bond selection.
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Solutions for Chapter 18: Questions and Problems
CHAPTER 18
Answers to Problems
1(a).
.15  .07
0.05
.20  .07

.10
.10  .07

.03
.17  .07

.06
.13  .07

.04
SP 
SQ
SR
SS
Market
.08
 1.60
.05
.13

 1.30
.10
.03

 1.00
.03
.10

 1.67
.06
.06

 1.50
.04

1(b).
TP 
.15  .07 .08

 .0800
1.00
1.00
TQ 
.20  .07 .13

 .0867
1.50
1.50
TR 
.10  .07 .03

 .0500
.60
.60
TS 
.17  .07 .10

 .0909
1.10
1.10
Market 
.13  .07 .06

 .0600
1.00
1.00
Sharpe
Treynor
P
Q
R
S
Market
1(c).
2
4
5
1
3
3
2
5
1
4
It is apparent from the rankings above that Portfolio Q was poorly diversified since
Treynor ranked it #2 and Sharpe ranked it #4. Otherwise, the rankings are similar.
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Solutions for Chapter 18: Questions and Problems
2(a).
Portfolio MNO enjoyed the highest degree of diversification since it had the highest R 2
(94.8%). The statistical logic behind this conclusion comes from the CAPM which says
that all fully diversified portfolios should be priced along the security market line. R2 is a
measure of how well assets conform to the security market line, so R 2 is also a measure
of diversification.
2(b).
Note the mean returns are net of the risk-free rate. Doing the calculations we obtain:
Fund
Treynor
Sharpe
Jensen
ABC
0.975(4)
0.857(4)
0.192(4)
DEF
0.715(5)
0.619(5)
-0.053(5)
GHI
1.574(1)
1.179(1)
0.463(1)
JKL
1.262(2)
0.915(3)
0.355(2)
MNO
1.134(3)
1.000(2)
0.296(3)
2(c).
Fund
ABC
DEF
GHI
JKL
MNO
t(alpha)
1.7455(3)
-0.2789(5)
2.4368(1)
1.6136(4)
2.1143(2)
Only GHI and MNO have significantly positive alphas at a 95% level of confidence.
3(a).
(Information ratio) IRj = j/u where u = standard error of the regression
IRA = .058/.533 = 0.1088
IRB = .115/5.884 = 0.0195
IRC = .250/2.165 = 0.1155
3(b).
Annualized IR = (T)1/2(IR)
Annualized IRA = (52)1/2(0.1088) = 0.7846
Annualized IRB = (26)1/2(0.0195) = 0.0994
Annualized IRC = (12)1/2(0.1155) = 0.4001
3(c).
The higher the ratio, the better. Based upon the answers to part a, Manager C would be
rated the highest followed by Managers A and B, respectively. However, once the values
are annualized, the ranking change. Specifically, based upon the annualized IR, Manger
A is rated the highest, followed by C and B. (In both cases, Manager B is rated last).
Based upon the Grinold-Kahn standard for “good” performance (0.500 or greater), only
Manager A meets that test.
4(a).
Overall performance (Fund 1) = 26.40% - 6.20% = 20.20%
Overall performance (Fund 2) = 13.22% - 6.20% = 7.02%
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Solutions for Chapter 18: Questions and Problems
4(b).
E(Ri) = 6.20 + (15.71 – 6.20)
= 6.20 +  (9.51)
Total return (Fund 1) = 6.20 + (1.351)(9.51) = 6.20 + 12.85 = 19.05%
where 12.85% is the required return for risk
Total return (Fund 2) = 6.20 + (0.905)(9.51) = 6.20 + 8.61 = 14.81%
where 8.61% is the required return for risk
4(c)(i). Selectivity1 = 20.2% - 12.85% = 7.35%
Selectivity2 = 7.02% - 8.61% = -1.59%
4(c)(ii).Ratio of total risk1 = 1/m = 20.67/13.25 = 1.56
Ratio of total risk2 = 2/m = 14.20/13.25 = 1.07
R1 = 6.20 + 1.56 (9.51) = 6.20 + 14.8356 = 21.04%
R2 = 6.20 + 1.07 (9.51) = 6.20 + 10.1757 = 16.38%
Diversification1 = 21.04% – 19.05% = 1.99%
Diversification2 = 16.38% – 14.81% = 1.57%
4(c)(iii). Net Selectivity = Selectivity – Diversification
Net Selectivity1 = 7.35% - 1.99% = 5.36%
Net Selectivity2 = -1.59% - 1.57% = -3.16%
4(d).
Even accounting for the added cost of incomplete diversification, Fund 1’s performance
was above the market line (best performance), while Fund 2 fall below the line.
5.
a.
Year
1
2
3
4
5
6
7
8
9
10
Average
Mgr X
Return
-1.5
-1.5
-1.5
-1.0
0.0
4.5
6.5
8.5
13.5
17.5
4.5
Mgr Y Return
-6.5
-3.5
-1.5
3.5
4.5
6.5
7.5
8.5
12.5
13.5
4.5
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Solutions for Chapter 18: Questions and Problems
Std Dev
6.90
6.63
Semi-dev
0.65
4.20
Semi-deviation considers only the returns that are below the average.
b.
Sharpe ratio: (average return minus risk-free rate) / standard deviation
Mgr X:
0.435
Mgr Y:
0.452
Best performer
6.
6(a)(i). .6(-5) + .3(-3.5) + .1(0.3) = -4.02%
6(a)(ii). .5(-4) + .2(-2.5) + .3(0.3) = -2.41%
6(a)(iii). .3(-5) + .4(-3.5) + .3(0.3) = -2.81%
Manager A outperformed the benchmark fund by 161 basis points while Manager B beat
the benchmark fund by 121 basis points.
6(b)(i). [.5(-4 + 5) + .2(-2.5 + 3.5) + .3(.3 -.3)] = 0.70%
6(b)(ii). [(.3 - .6) (-5 + 4.02) + (.4 - .3) (-3.5 + 4.02) + (.3 -.1)(.3 + 4.02)] = 1.21%
Manager A added value through her selection skills (70 of 161 basis points) and her
allocation skills (71 of 161 basis points). Manager B added value totally through his
allocation skills (121 of 121 basis points).
7 (a). Dollar-Weighted Return
Manager L:
500,000 = -12,000/(1+r) - 7,500/(1+r)2- 13,500/(1+r)3 - 6,500/(1+r)4- 10,000/(1+r)5+
625,000/(1+r)5
Solving for r, the internal rate of return or DWRR is 2.75%
Manager M:
700,000 = 35,000/(1+r) + 35,000/(1+r)2+35,000/(1+r)3+35,000/(1+r)4+35,000/(1+r)5 +
625,000/(1+r)5
Solving for r, the internal rate of return or DWRR is 2.98%.
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Solutions for Chapter 18: Questions and Problems
7(b).
Time-weighted return
Manager L:
Periods
1
2
3
4
5
HPR
[(527,000 – 500,000) – 12,000]/500,000 = .03
[(530,000 – 527,000) – 7,500]/527,000 = -.0085
[(555,000 – 530,000) – 13,500]/530,000 = .0217
[(580,000 – 555,000) – 6,500]/555,000 = .0333
[(625,000 – 580,000) – 10,000]/580,000 = .0603
TWRR = [(1 + .03)(1 - .0085)(1 + .0217)(1 + .0333)(1 + .0603)]1/5 - 1
= (1.143) 1/5 – 1= 1.02712 – 1 = .02712 = 2.71%
Manager M:
Periods
1
2
3
4
5
HPR
[(692,000 – 700,000) + 35,000]/700,000 = .03857
[(663,000 – 692,000) + 35,000]/692,000 = .00867
[(621,000 – 663,000) + 35,000]/663,000 = -.01056
[(612,000 – 621,000) + 35,000]/621,000 = .04187
[(625,000 – 612,000) + 35,000]/612,000 = .0784
TWRR = [(1 + .03857)(1 + .00867)(1 - .01056)(1 + .04187)(1 + .0784)]1/5 - 1
= (1.1646) 1/5 – 1= 1.03094 – 1 = .03094 = 3.094%
EV – (1 – DW)(Contribution)
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