Chapter 18
Portfolio
Performance
Evaluation
McGraw-Hill/Irwin
Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.
18.1 Risk-Adjusted Returns
18-2
Types of management revisited
• Passive management
1. Capital allocation between cash and the risky portfolio
2. Asset allocation within the risky portfolio.
– How passive the management actually is varies
• “Set it and forget it”
• Change allocations in 1 or 2 according to perceptions
of risk to keep current with portfolio goals.
• Active management
– Forecasting future rates of return on either/both asset
classes and individual securities
18-3
Performance evaluation
• Purpose:
– Are the returns worth the risk and the fees?
• Measures
– Average return by itself is an insufficient measure. Why?
• Average return may not = expected return
• Risk return relationship
• Major component of returns is the market
performance
– Need a measure of abnormal performance
18-4
Abnormal Performance
How is abnormal performance measured?
Comparisons to peer groups
Rank fund performance within a given category
Benchmark portfolio such as an index
Really need measures that consider risk:
Calculate reward to risk measures such as the alpha,
Sharpe Measure (or variations)
18-5
Factors That Lead to Abnormal
Performance
• Successful across asset allocations
• Superior allocation within each asset class
– Sectors or industries
• Overweight better performing sectors, underweight
poorer performers
• Individual security selection
• Pick the right stocks, those with performance
better than expected
18-6
Risk Adjusted Performance
• To measure abnormal performance, must measure
normal performance:
•The single index model can be used:
RPt  βPRMt  αP  ept
From this the expected return [E(RP)] can be
found:
E(RP )  βPE(RMt )  αP
18-7
Risk Adjusted Performance:
Jensen
3) Jensen’s Alpha Measure
αP  RP - βP RM
αP  Portfolio alpha
RP  Average excessreturn on the portfolio
βP  Portfolio beta
RM  Averageexcessreturn on the market
18-8
Risk Adjusted Performance:
Jensen (cont.)
3) Jensen’s Alpha Measure
αP  RP - βP RM
• Measure of abnormal return
• Must establish statistical significance via regression
• Possible conceptual problem:
• Greater abnormal return may be due to greater risk
• Unless you can hedge out the risk with short sales one
should rank portfolio performance by the adjusted
alpha which is p / p
18-9
Measuring Risk Adjusted
Performance: Sharpe Ratios
• 1) Historical Sharpe Ratios
rp  r f
Sharpe Ratios 
σp
r p  Av eragereturn on the portf olio
r f  Av eragerisk f reerate
Use:
When choosing
among competing
portfolios that will not
be mixed.
σ p  Portf oliostandarddev iation
In practice:
Used when one
manager handles the
(entire) portfolio.
18-10
Risk Adjusted Performance:
Treynor
• 2) Historical Treynor Ratios
Treynor Ratios 
rp  r f
p
r p  Av eragereturn on the portf olio
r f  Av eragerisk f reerate
p  Portf oliobeta
Use:
Evaluate a portfolio
when portfolio is a
piece of a larger
portfolio that has
different managers.
18-11
M2 Measure
rp  r f
Sharpe Ratios 
σp
• Variation of the Sharpe Ratio that is easier to interpret.
– Concept of the Sharpe is easy to interpret but the
Sharpe number is not.
– Developed by Modigliani and Modigliani; hence M2
– Use M2 to compare performance of a managed
portfolio (MP) with a market index.
• The M2 measure creates a hypothetical complete
portfolio that is composed of T-bills and the managed
portfolio that has the same standard deviation as the
market index.
18-12
2
M
Measure: Example
Managed Portfolio
Market
T-bill
Return
32%
18%
6%
Stan. Dev
40%
25%
0%
Purpose: Create a complete portfolio w/ same risk as the market
Market = WMPMP + (1 – WMP)T-bill
25% = WMP40% + (1 – WMP)0;
WMP = 25% / 40% = 62.5%
&
WT-bills = 37.5%
E(rPComplete) = (0.625)(0.32) + (0.375)(0.06) = 22.25%
Since this return is more than the market the managed portfolio
outperformed the market on a risk adjusted basis by 4.25%.
M2 = 22.25% - 18% = 4.25%
18-13
Information Ratio
When evaluating a portfolio to be mixed with a
position in the passive benchmark portfolio we
must draw on insights of the Treynor-Black
Model (See Chapter 6)
Choose the portfolio, which when combined with
the passive benchmark, generates an efficient
frontier with the best return per unit of risk as
measured by the standard deviation.
This is found by combining the Sharpe ratio of the
benchmark M with the information ratio of
2
portfolio P:




α
2
S Optimal  sqrtS M
 P  
σ  

 p 

18-14
Summary of measures and usage
Performance Definition
Measure
Sharpe
Treynor
Information
ratio
Rp / 
Application
When choosing
among portfolios
competing as the
optimal risky portfolio
Rp / 
When ranking many
portfolios that will be
mixed to form the
optimal risky portfolio
p / e
When evaluating a
portfolio to be mixed
with a position in the
passive benchmark
portfolio
18-15
More on Alpha
Alpha and the Sharpe measure
P
SP  SM  SM (  1) 
P
 = Correlation between RP.RM
• Positive alpha does not guarantee a higher Sharpe than
the benchmark because SM(-1) < 0.
• Necessary but not a sufficient condition for net
performance improvement
– The alpha must be large enough to offset increase in
residual risk from moving away from the diversified
optimum.
18-16
More on Alpha
Alpha and the Treynor measure
P
TP  TM 
P
• Positive alpha does not guarantee a
higher Treynor ranking because you have
to know the Beta as well.
18-17
More on Alpha
Alpha and the Information Ratio:
αP
Information Ratio 
σ eP
• Positive alpha does not guarantee a
higher square of the information ratio
because higher alpha may come with
higher residual risk.
18-18
Alpha Capture & Transport
• If an analyst finds an undervalued security and
invests in it, market moves may wipe out any
gains.
– Can hedge out market risk via shorting stock index or
stock index futures to establish a market neutral
position.
– Process is called alpha capture or alpha transport.
– When short positions and leverage are allowed a
significant non-zero alpha is a sufficient condition for
an improvement in the Sharpe and information ratio.
18-19
Performance measures for P, Q, M
& cash
18-20
Evaluation with a multi-index
model
• Evidence indicates we should use a multiindex model such as the Fama-French
model to establish the expected return:
RPt  βPRMt  βSMBrSMBt  βHML rHMLt  αP  ept
RPt  βPRMt  βSMB r SMBt  βHML rHMLt  αP
• This allows an estimation of alpha:
18-21
T2 (Treynor Square) Measure
• Used to convert the Treynor Measure into
percentage return basis
• Makes it easier to interpret and compare
• Equates the beta of the managed portfolio with
the market’s beta of 1 by creating a hypothetical
portfolio made up of T-bills and the managed
portfolio
• If the beta is lower than one, leverage is used
and the hypothetical portfolio is compared to the
market
18-22
T2 Example
Port. P.
Market
Risk Prem. (r-rf)
13%
10%
Beta
0.80
1.0
Alpha
Treynor Measure
5%
16.25
0%
10
Weight to match Market w = M/P = 1.0 / 0.8
Adjusted Return RP* = w (RP) = 16.25%
T2P = RP* - RM = 16.25% - 10% = 6.25%
18-23
18.2 Style Analysis
18-24
Style Analysis
• Complex method of performance
evaluation introduced by William Sharpe
• Recent studies of mutual fund
performance show that > 90% of variation
in returns can be explained by the funds’
allocations to bills, bonds and stocks.
18-25
Style Analysis
• Explaining percentage returns by
allocation to style
• Style Analysis has become popular with
the industry
18-26
Sharpe’s Style Portfolios for
the Magellan Fund
97.5% of the fund’s return
• ______
variability is explained by
asset allocation among
these 7 factors, & actually
only three
_____ variables explain
the fund’s investments.
2.5% is due
• The remaining _____
to security selection.
18-27
18.3 Morningstar’s
Risk-Adjusted Ratings
18-28
Morning Star’s Risk Adjusted
Rating
•
•
•
Companies are put into peer groups based on Morningstar style
definitions
Risk adjusted fund performance is ranked and then Stars are assigned
according to the following table (5 stars is the highest rating)
Percentile
Stars
0-10
1
10-32.5
2
32.5-67.5
3
67.5-90
4
90-100
5
Star ratings are highly correlated to Sharpe measure
rankings
18-29
Figure 18.6 Relationship of Morningstar’s
Category RARs and Excess Return
Sharpe Ratios
18-30
18.4 Risk Adjustment With
Changing Portfolio
Composition
18-31
Problems with performance
measures
• Measures assume a fund maintains a constant
level of risk
– Particularly problematic for funds that engage in
active asset allocation
Sharpe M = 0.4
First 4 qtrs Sharpe = 0.5
2nd 4 qtrs Sharpe = 0.5
Overall Sharpe = 0.37
What happened?
18-32
Problems with performance
measures
• In a large universe of funds, some funds
will have abnormal performance in every
period just by chance.
– Requires statistical work
• Volatility is quite high and creates large errors in
estimation
18-33
18.5 Performance
Attribution Procedures
18-34
Performance Attribution
• Decomposing overall performance into
components
• Performance is determined by specific
portfolio choices,
• Major performance determinants:
– Broad asset allocation among types of
securities,
– Industry weighting in equity portfolio,
– Security choice,
– Timing of purchases and sales.
18-35
Process of Attributing
Performance to Components
• Set up a benchmark or ‘bogey’ portfolio
• Use indexes for each component in the
bogey
• Use target weight structure in the bogey
18-36
Process of Attributing
Performance to Components
• Calculate the return on the ‘bogey’ and on
the managed portfolio
• Explain the difference in return based on
component weights or selection
• Summarize the performance differences
into appropriate categories
18-37
Performance Attribution
Example
• Managed portfolio with monthly return of
5.34%.
Component
Weight
Monthly
Return
Equity
70%
7.28%
Bonds
7%
1.89%
Cash
23%
0.4857%
Total
5.34%
• The portfolio was comprised of
18-38
Performance of the Managed Portfolio
Bogey Performance
and Extra Return
Bogey
Performance
Benchmark Index Return
and
Extra Return
Component
Weight
during Month
Benchmark
Index
Return
Bonds (Shearson-Lehman Index)
0.3
1.45%
Component
Weight
during
Month
Equity (S&P500 Index)
0.6
5.81%
BondsCash
(Shearson-Lehman
Index)
0.3
1.45%
(Money Market)
0.1
0.48%
Equity (S&P500 Index)
0.6
5.81%
Return
 (0.30)(1.45%)  (0.60)(5.8
1%)  (0.10)(0.4
 3.97%
Cash Bogey
(Money
Market)
0.1
0.48%8%)
Return of the Managed Portfolio
5.34%
- Return of the Bogey Portfolio
Return of the Managed Portfolio
Extra Return of the Managed
- Return
of the Bogey Portfolio
Portfolio
Extra Return of the Managed
Portfolio
5.34%
3.97%
3.97% 1.37%
1.37%
• The “Bogey” is a benchmark portfolio comprised of three
component indices with the given weights.
• The Bogey return represents the return on an unmanaged
portfolio
• The Bogey weights are representative of a standard portfolio for
the typical risk tolerance of the given type of client or for the
typical fund in this category
18-39
Performance of the Managed Portfolio
• The “Bogey” is a benchmark portfolio comprised of three
indices with the given weights.
• The Bogey return represents the return on an
unmanaged portfolio.
• The Bogey weights are representative of a standard
portfolio for the typical risk tolerance of the given type of
client or for the typical fund in this category.
18-40
Performance Attribution Example
A. Contribution of Asset Allocation to Performance
Market
Actual Weight
Benchmark Weight
Bonds
0.07
0.30
Stock
0.70
0.60
Cash
0.23
0.10
Excess Weight
-0.23
Index Return
Minus Bogey
x
1.45-3.97 = -2.52%
=
x
0.10
5.81-3.97 = 1.84%
=
0.13
0.48-3.97 = -3.49%
x
=
Contribution of asset allocation:
Contribution
0.5796%
0.1840%
-0.4537%
0.3099%
Contribution to performance from broad asset allocation decision
Note this uses the index return, not the actual managed portfolio return
“How did the difference in weights in the managed portfolio affect portfolio
performance?”
Superior performance is generated by overweighting investments in
classes that perform better than the bogey.
18-41
Performance Attribution
Contribution to performance from broad asset allocation decision
Note that Column 4 is the INDEX return, not the actual managed
portfolio return.
“How did the difference in weights in the managed portfolio affect
portfolio performance?”
Superior performance is generated by overweighting investments in
classes that perform better than the bogey.
18-42
Performance Attribution
B. Contribution of Selection to Total Performance
Market
Portfolio
Return
Index
Return
Extra
Return
Portfolio
Weight
Contribution to
Performance
x
x
0.07
0.0308%
0.70
1.0290%
0.000057 x
0.23
0.000013%
Bonds
0.0189
0.0145
0.0044
Stock
0.0728
0.0581
0.0147
Cash
0.004857
0.0048
Contribution of Selection
1.0611%
• Material superior performance in both Bond and Stock
Sectors
• Table B delineates contribution to performance of both
sector and security selection
18-43
Performance Attribution
18-44
Performance Attribution
18-45
Sector Selection within the Equity Market
C. Contribution of Equity Sector Allocation to Total Performance
Weights
Sector
Portfolio
S&P 500
Excess
Weight
Sector
Return
Contribution of
Sector Allocation
Basic Materials
0.0196
0.083
-0.0634 x
6.90%
-0.437%
Business Services
0.0784
0.041
0.0374 x
7.00%
0.262%
Capital Goods
0.0187
0.078
-0.0593 x
4.10%
-0.243%
Consumer Cyclicals
0.0847
0.125
-0.0403 x
8.80%
-0.355%
Consumer Noncyclicals
0.4037
0.204
0.1997 x 10.00%
1.997%
Credit Sensitive
0.2401
0.218
0.0221 x
5.00%
0.111%
Energy
0.1353
0.142
-0.0067 x
2.60%
-0.017%
Technology
0.0195
0.109
-0.0895 x
0.30%
-0.027%
Contribution of Sector Allocation
1.290%
18-46
Sector Selection within the
Equity Market
18-47
Portfolio Attribution: Summary
D. Portfolio Attribution Summary
0.3099%
1. Asset Allocation
2. Selection
a. Equity Extra Return
i. Sector Allocation
1.290%
ii. Security Selection
0.180%
b. Bonds Extra Return
Weights
1.470%
x
70.00%
=
1.029%
0.4400%
x
7.00%
=
0.031%
See Table B
Table B
c. Cash Extra Return
x
23.00%
= 0.0013%
Total Extra Return on the portfolio
1.3710%
0.0057%
The Security Selection component is inferred as follows:
_________________________________
The
total equity extra return = 1.47%
1.29% of the total so Security
Sector allocation accounted for ______
- 1.29% or 0.18%
Selection must have resulted in 1.47%
_____________________.
18-48
Portfolio Attribution: Summary
18-49
18.6 Market Timing
18-50
Market Timing
• Adjust the asset allocation for movements
in the market
– Shift between stocks and money market
instruments or bonds
– With perfect ability to forecast behaves like an
option
– Little evidence of market timing ability
18-51
Figure 18.8 Rate of Return of a
Perfect Market Timer
18-52
Value of Market Timing
Invest $1 on December 1, 1926
Strategy
Money Markets
Stocks
Perfect Timing*
Value in 2008
Geom Avg.
Return
$20
3.71%
$1,626
9.44%
$36,699,302,473
34.54%
*Perfect Timing: Every month 100% of the funds are
placed in either stocks or cash based on which would
have the higher return.
18-53
With Imperfect Ability to
Forecast
• Takes a long time horizon to judge the
ability
• Judge proportions of correct calls
– Bull markets and bear market calls
18-54
Market Timing & Performance
Measurement
• Timer adjusts portfolio for up and down
movements in the market
– Low Market Return - low ßeta
– High Market Return - high ßeta
18-55
Characteristic Lines
18-56
Selected Problems
18-57
Problem 1
Portfolio A
Portfolio B
Market index
Risk-free asset
E(r)
11%
14%
12%
6%

10%
31%
20%
0%

0.8
1.5
1.0
0.0
11% – [6% + 0.8(12% – 6%)] = 0.2%
14% – [6% + 1.5(12% – 6%)] = –1.0%
• The alphas for the two portfolios are:
• a. A =
•
B =
• Might add A if you believe the alpha is significant, might short
B if you believe its alpha is significant, should check Treynor.
18-58
Problem 1
Portfolio A
Portfolio B
Market index
Risk-free asset
b.
•
•
•
11%  6%
 0.5
10%
E(r)
11%
14%
12%
6%

10%
31%
20%
0%

0.8
1.5
1.0
0.0
14%  6%
 0.26
31%
If you hold only one of the two portfolios, then the Sharpe measure is the appropriate criterion:
SA =
SB =
Therefore, using the Sharpe criterion, Portfolio A is preferred.
18-59
Problem 1
Portfolio A
Portfolio B
Market index
Risk-free asset
• M2 Measure for A:
WAA + (1 – WA)T-bill
20% = WA10% + (1 – WA)0 ;
• Market =
-1
20% / 10% = 2
•
(2)(0.11)&+WT-bills
(-1)(0.06)
WA =
= = 16%
• E(rPComplete) =
E(r)
11%
14%
12%
6%

10%
31%
20%
0%

0.8
1.5
1.0
0.0
16% - 12% = + 4%
• M2 measure for A =
18-60
Problem 1
• M2 Measure for B:
Portfolio A
Portfolio B
Market index
Risk-free asset
E(r)
11%
14%
12%
6%

10%
31%
20%
0%

0.8
1.5
1.0
0.0
WBB + (1 – WB)T-bill
• Market = 20% = W 31% + (1 – W )0
B
B
;
•
WB = 20% / 31% = 0.645
& WT-bills = 0.355
• E(rPComplete) = (0.645)(0.14) + (0.355)(0.06) = 11.16%
• M2 measure for B =
11.16% - 12% = -0.84%
18-61
Problem 2
a. Actual: (0.70)(2.0%) + (0.20)(1.0%) + (0.10)(0.5%) = 1.65%
• Bogey: (0.60)(2.5%) + (0.30)(1.2%) + (0.10)(0.5%) = 1.91%
•
Relative performance = 1.65% – 1.91% = -0.26%
(Underperformance)
18-62
Problem 2
•
Security Selection:
b.
Market
Portfolio
Performance
Index
Performance
Excess
Performance
Manager’s
Portfolio Weight
Contribution
Equity
2.0%
2.5%
-0.5%
0.70
-0.35%
Bonds
1.0%
1.2%
-0.2%
0.20
-0.04%
Cash
0.5%
0.5%
0.0%
0.10
0.00%
Contribution of security selection:
-0.39%
18-63
Problem 2
•
Asset Allocation
c.
Market
Actual Weight
Benchmark Weight
Excess Weight
Index Return
Minus Bogey
Contribution
Equity
0.70
0.60
0.10
0.59%
0.059%
Bonds
0.20
0.30
-0.10
-0.71%
0.071%
Cash
0.10
0.10
0.00
-1.41%
0.000%
Contribution of asset allocation:
0.130%
Summary
Security selection
-0.39%
Asset allocation
0.13%
Excess performance
-0.26%
18-64