Portfolio Performance
Evaluation
Chapter 24
McGraw-Hill/Irwin
Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.
Introduction
Complicated subject
Theoretically correct measures are difficult to
construct
Different statistics or measures are appropriate
for different types of investment decisions or
portfolios
Many industry and academic measures are
different
The nature of active management leads to
measurement problems
24-2
Dollar- and Time-Weighted Returns
Dollar-weighted returns
Internal rate of return considering the cash
flow from or to investment
Returns are weighted by the amount invested
in each stock
Time-weighted returns
Not weighted by investment amount
Equal weighting
24-3
Text Example of Multiperiod Returns
Period
Action
0
Purchase 1 share at $50
1
Purchase 1 share at $53
Stock pays a dividend of $2 per
share
2
Stock pays a dividend of $2 per
share
Stock is sold at $108 per share
24-4
Dollar-Weighted Return
Period
Cash Flow
0
-50 share purchase
1
+2 dividend -53 share purchase
2
+4 dividend + 108 shares sold
Internal Rate of Return:
 51
112
 50 

1
(1  r ) (1  r ) 2
r  7.117%
24-5
Time-Weighted Return
53  50  2
r1 
 10%
50
54  53  2
r2 
 5.66%
53
Simple Average Return:
(10% + 5.66%) / 2 = 7.83%
24-6
Averaging Returns
Arithmetic Mean:
Text Example Average:
n
rt
r
t 1 n
(.10 + .0566) / 2 = 7.81%
Geometric Mean:
Text Example Average:
1/ n


r   (1  rt )  1
 t 1

n
[ (1.1) (1.0566) ]1/2 - 1
= 7.83%
24-7
Geometric & Arithmetic Means Compared
Past Performance - generally the geometric
mean is preferable to arithmetic
Predicting Future Returns from historical
returns
Use a weighted average of arithmetic and
geometric averages of historical returns if the
forecast period is less than the estimation period
Use geometric if the forecast and estimation
period are equal
24-8
Abnormal Performance
What is abnormal?
Abnormal performance is measured:
Benchmark portfolio
Market adjusted
Market model / index model adjusted
Reward to risk measures such as the Sharpe Measure:
E (rp-rf) / p
Factors of abnormal performance
Market timing
Superior selection
Sectors or industries
Individual companies
24-9
Risk Adjusted Performance: Sharpe
1) Sharpe Index
rp - rf
p
rp = Average return on the portfolio
rf = Average risk free rate
=
Standard
deviation
of
portfolio
 p return
24-10
M2 Measure
Developed by Modigliani and Modigliani
Equates the volatility of the managed
portfolio with the market by creating a
hypothetical portfolio made up of T-bills
and the managed portfolio
If the risk is lower than the market,
leverage is used and the hypothetical
portfolio is compared to the market
24-11
M2 Measure: Example
Managed Portfolio: return = 35%
standard deviation = 42%
Market Portfolio: return = 28%
T-bill return = 6%
standard deviation = 30%
Hypothetical Portfolio:
30/42 = .714 in P (1-.714) or .286 in T-bills
(.714) (.35) + (.286) (.06) = 26.7%
Since this return is less than the market, the managed portfolio
underperformed
24-12
Risk Adjusted Performance: Treynor
2) Treynor Measure
rp - rf
ßp
rp = Average return on the portfolio
rf = Average risk free rate
ßp = Weighted average for portfolio
24-13
Risk Adjusted Performance: Jensen
3) Jensen’s Measure
 p= rp - [ rf + ßp ( rm - rf) ]
 p = Alpha for the portfolio
rp = Average return on the portfolio
ßp = Weighted average Beta
rf = Average risk free rate
rm = Avg. return on market index port.
24-14
Appraisal Ratio
Appraisal Ratio = p / (ep)
Appraisal Ratio divides the alpha of the portfolio
by the nonsystematic risk
Nonsystematic risk could, in theory, be
eliminated by diversification
24-15
Which Measure is Appropriate?
It depends on investment assumptions
1) If the portfolio represents the entire
investment for an individual, Sharpe
Index compared to the Sharpe Index for
the market.
2) If many alternatives are possible, use the
Jensen or the Treynor measure
The Treynor measure is more complete
because it adjusts for risk
24-16
Limitations
Assumptions underlying measures limit
their usefulness
When the portfolio is being actively
managed, basic stability requirements are
not met
Practitioners often use benchmark portfolio
comparisons to measure performance
24-17
Market Timing
Adjusting portfolio for up and down
movements in the market
Low Market Return - low ßeta
High Market Return - high ßeta
24-18
Example of Market Timing
rp - rf
* *
* *
*
*
* *
* **
* *
* * *
** *
*
*
* *
rm - rf
Steadily Increasing the Beta
24-19
Performance Attribution
Decomposing overall performance into
components
Components are related to specific
elements of performance
Example components
Broad Allocation
Industry
Security Choice
Up and Down Markets
24-20
Attributing Performance to Components
Set up a ‘Benchmark’ or ‘Bogey’ portfolio
Use indexes for each component
Use target weight structure
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
24-21
Formula for Attribution
n
n
i 1
i 1
rB   wBi rBi & rp   w pi rpi
n
n
i 1
i 1
rp  rB   w pi rpi   wBi rBi 
n
 (w
i 1
r  wBi rBi )
pi pi
Where B is the bogey portfolio and p is the managed portfolio
24-22
Contributions for Performance
Contribution for asset allocation
+ Contribution for security selection
(wpi - wBi) rBi
wpi (rpi - rBi)
= Total Contribution from asset class wpirpi -wBirBi
24-23
Complications to Measuring Performance
Two major problems
Need many observations even when portfolio
mean and variance are constant
Active management leads to shifts in
parameters making measurement more
difficult
To measure well
You need a lot of short intervals
For each period you need to specify the
makeup of the portfolio
24-24
Style Analysis
Based on regression analysis
Examines asset allocation for broad
groups of stocks
More precise than comparing to the
broad market
Sharpe’s analysis: 97.3% of returns
attributed to style
24-25