Chapter 19
Performance Evaluation
1
And with that they clapped him into irons and hauled
him off to the barracks. There he was taught
“right turn,” “left turn,” and “quick march,”
“slope arms,” and “order arms,” how to aim and
how to fire, and was given thirty strokes of the
“cat.” Next day his performance on parade was a
little better, and he was given only twenty strokes.
The following day he received a mere ten and was
thought a prodigy by his comrades.
- From Candide by Voltaire
2
Outline
 Introduction
 Importance
of measuring portfolio risk
 Traditional performance measures
 Performance evaluation with cash deposits
and withdrawals
 Performance evaluation when options are
used
3
Introduction
 Performance
evaluation is a critical aspect
of portfolio management
 Proper
performance evaluation should
involve a recognition of both the return and
the riskiness of the investment
4
Importance of
Measuring Portfolio Risk
 Introduction
 A lesson
from history: the 1968 Bank
Administration Institute report
 A lesson from a few mutual funds
 Why the arithmetic mean is often
misleading: a review
 Why dollars are more important than
percentages
5
Introduction
 When
two investments’ returns are
compared, their relative risk must also be
considered
 People maximize expected utility:
• A positive function of expected return
• A negative function of the return variance
E (U )  f  E ( R),  2 
6
A Lesson from History

The 1968 Bank Administration Institute’s
Measuring the Investment Performance of
Pension Funds concluded:
1) Performance of a fund should be measured by
computing the actual rates of return on a
fund’s assets
2) These rates of return should be based on the
market value of the fund’s assets
7
A Lesson from History (cont’d)
3) Complete evaluation of the manager’s
performance must include examining a
measure of the degree of risk taken in the
fund
4) Circumstances under which fund managers
must operate vary so great that indiscriminate
comparisons among funds might reflect
differences in these circumstances rather than
in the ability of managers
8
A Lesson from
A Few Mutual Funds
 The
two key points with performance
evaluation:
• The arithmetic mean is not a useful statistic in
evaluating growth
• Dollars are more important than percentages
 Consider
the historical returns of two
mutual funds on the following slide
9
A Lesson from
A Few Mutual Funds (cont’d)
Year
44 Wall
Street
Mutual
Shares
Year
44 Wall
Street
Mutual
Shares
1975
184.1%
24.6%
1982
6.9
12.0
1976
46.5
63.1
1983
9.2
37.8
1977
16.5
13.2
1984
-58.7
14.3
1978
32.9
16.1
1985
-20.1
26.3
1979
71.4
39.3
1986
-16.3
16.9
1980
36.1
19.0
1987
-34.6
6.5
1981
-23.6
8.7
1988
19.3
30.7
Mean
19.3%
23.5%
10
A Lesson from
A Few Mutual Funds (cont’d)
Ending Value ($)
Mutual Fund Performance
$200,000.00
$180,000.00
$160,000.00
$140,000.00
$120,000.00
$100,000.00
$80,000.00
$60,000.00
$40,000.00
$20,000.00
$-
44 Wall
Street
Mutual
Shares
7
19
7
8
19
0
8
19
3
8
19
6
Year
11
A Lesson from
A Few Mutual Funds (cont’d)
 44
Wall Street and Mutual Shares both had
good returns over the 1975 to 1988 period
 Mutual
Shares clearly outperforms 44 Wall
Street in terms of dollar returns at the end of
1988
12
Why the Arithmetic Mean
Is Often Misleading
 The
arithmetic mean may give misleading
information
• E.g., a 50% decline in one period followed by a
50% increase in the next period does not return
0%, on average
13
Why the Arithmetic Mean
Is Often Misleading (cont’d)
 The
proper measure of average investment
return over time is the geometric mean:
1/ n


GM   Ri   1
 i 1 
where Ri  the return relative in period i
n
14
Why the Arithmetic Mean
Is Often Misleading (cont’d)
 The
geometric means in the preceding
example are:
• 44 Wall Street: 7.9%
• Mutual Shares: 22.7%
 The
geometric mean correctly identifies
Mutual Shares as the better investment over
the 1975 to 1988 period
15
Why the Arithmetic Mean
Is Often Misleading (cont’d)
Example
A stock returns –40% in the first period, +50% in the
second period, and 0% in the third period.
What is the geometric mean over the three periods?
16
Why the Arithmetic Mean
Is Often Misleading (cont’d)
Example
Solution: The geometric mean is computed as follows:
1/ n


GM   Ri   1
 i 1 
 (0.60)(1.50)(1.00)  1
 0.10  10%
n
17
Why Dollars Are More
Important than Percentages
 Assume
two funds:
• Fund A has $40 million in investments and
earned 12% last period
• Fund B has $250,000 in investments and earned
44% last period
18
Why Dollars Are More
Important than Percentages
 The
correct way to determine the return of
both funds combined is to weigh the funds’
returns by the dollar amounts:
 $40, 000, 000
  $250, 000

 $40, 250, 000 12%    $40, 250, 000  44%   12.10%

 

19
Traditional
Performance Measures
 Sharpe
and Treynor measures
 Jensen measure
 Performance measurement in practice
20
Sharpe and Treynor Measures
 The
Sharpe and Treynor measures:
Sharpe measure 
Treynor measure 
R  Rf

R  Rf

where R  average return
R f  risk-free rate
  standard deviation of returns
  beta
21
Sharpe and
Treynor Measures (cont’d)
 The
Treynor measure evaluates the return
relative to beta, a measure of systematic risk
• It ignores any unsystematic risk
 The
Sharpe measure evaluates return
relative to total risk
• Appropriate for a well-diversified portfolio, but
not for individual securities
22
Sharpe and
Treynor Measures (cont’d)
Example
Over the last four months, XYZ Stock had excess returns
of 1.86%, -5.09%, -1.99%, and 1.72%. The standard
deviation of XYZ stock returns is 3.07%. XYZ Stock has a
beta of 1.20.
What are the Sharpe and Treynor measures for XYZ
Stock?
23
Sharpe and
Treynor Measures (cont’d)
Example (cont’d)
Solution: First compute the average excess return for
Stock XYZ:
1.86%  5.09%  1.99%  1.72%
R
4
 0.88%
24
Sharpe and
Treynor Measures (cont’d)
Example (cont’d)
Solution (cont’d): Next, compute the Sharpe and Treynor
measures:
Sharpe measure 
Treynor measure 
R  Rf

R  Rf

0.88%

 0.29
3.07%
0.88%

 0.73
1.20
25
Jensen Measure
 The
Jensen measure stems directly from the
CAPM:
Rit  R ft     i  Rmt  R ft 
26
Jensen Measure (cont’d)
 The
constant term should be zero
• Securities with a beta of zero should have an
excess return of zero according to finance
theory
 According
to the Jensen measure, if a
portfolio manager is better-than-average,
the alpha of the portfolio will be positive
27
Jensen Measure (cont’d)
 The
Jensen measure is generally out of
favor because of statistical and theoretical
problems
28
Performance Measurement
in Practice
 Academic
issues
 Industry issues
29
Academic Issues
 The
use of traditional performance
measures relies on the CAPM
 Evidence
continues to accumulate that may
ultimately displace the CAPM
• APT, multi-factor CAPMs, inflation-adjusted
CAPM
30
Industry Issues
 “Portfolio
managers are hired and fired
largely on the basis of realized investment
returns with little regard to risk taken in
achieving the returns”
 Practical
performance measures typically
involve a comparison of the fund’s
performance with that of a benchmark
31
Industry Issues (cont’d)
 Fama’s
decomposition can be used to assess
why an investment performed better or
worse than expected:
• The return the investor chose to take
• The added return the manager chose to seek
• The return from the manager’s good selection
of securities
32
33
Performance Evaluation With
Cash Deposits & Withdrawals
 Introduction
 Daily
valuation method
 Modified Bank Administration Institute
(BAI) Method
 An example
 An approximate method
34
Introduction

The owner of a fund often taken periodic
distributions from the portfolio and may
occasionally add to it

The established way to calculate portfolio
performance in this situation is via a timeweighted rate of return:
• Daily valuation method
• Modified BAI method
35
Daily Valuation Method
 The
daily valuation method:
• Calculates the exact time-weighted rate of
return
• Is cumbersome because it requires determining
a value for the portfolio each time any cash
flow occurs
– Might be interest, dividends, or additions and
withdrawals
36
Daily Valuation
Method (cont’d)
 The
daily valuation method solves for R:
n
Rdaily   Si  1
i 1
MVEi
where S 
MVBi
37
Daily Valuation
Method (cont’d)

MVEi = market value of the portfolio at the end of
period i before any cash flows in period i but
including accrued income for the period

MVBi = market value of the portfolio at the
beginning of period i including any cash flows at
the end of the previous subperiod and including
accrued income
38
Modified BAI Method
 The
modified BAI method:
• Approximates the internal rate of return for the
investment over the period in question
• Can be complicated with a large portfolio that
might conceivably have a cash flow every day
39
Modified BAI Method (cont’d)
 It
solves for R:
n
MVE   Fi (1  R) wi
i 1
where F  the sum of the cash flows during the period
MVE  market value at the end of the period,
including accrued income
F0  market value at the start of the period
CD  Di
CD
CD  total number of days in the period
Di  number of days since the beginning of the period
wi 
in which the cash flow occurred
40
An Example
 An
investor has an account with a mutual
fund and “dollar cost averages” by putting
$100 per month into the fund
 The
following slide shows the activity and
results over a seven-month period
41
42
An Example (cont’d)
 The
daily valuation method returns a timeweighted return of 40.6% over the sevenmonths period
• See next slide
43
44
An Example (cont’d)
 The
BAI method requires use of a computer
 The
BAI method returns a time-weighted
return of 42.1% over the seven-months
period (see next slide)
45
46
An Approximate Method
 Proposed
by the American Association of
Individual Investors:
P1  0.5(Net cash flow)
R
1
P0  0.5(Net cash flow)
where net cash flow is the sum of inflows and outflows
47
An Approximate
Method (cont’d)
 Using
the approximate method in Table 19-
6:
P1  0.5(Net cash flow)
R
1
P0  0.5(Net cash flow)
5,500.97  0.5( 4, 200)

1
7,550.08  0.5(-4, 200)
 0.395  39.5%
48
Performance Evaluation
When Options Are Used
 Introduction
 Incremental
risk-adjusted return from
options
 Residual option spread
 Final comments on performance evaluation
with options
49
Introduction
 Inclusion
of options in a portfolio usually
results in a non-normal return distribution
 Beta
and standard deviation lose their
theoretical value of the return distribution is
nonsymmetrical
50
Introduction (cont’d)
 Consider
two alternative methods when
options are included in a portfolio:
• Incremental risk-adjusted return (IRAR)
• Residual option spread (ROS)
51
Incremental Risk-Adjusted
Return from Options
 Definition
 An
IRAR example
 IRAR caveats
52
Definition
 The
incremental risk-adjusted return
(IRAR) is a single performance measure
indicating the contribution of an options
program to overall portfolio performance
• A positive IRAR indicates above-average
performance
• A negative IRAR indicates the portfolio would
have performed better without options
53
Definition (cont’d)
 Use
the unoptioned portfolio as a
benchmark:
• Draw a line from the risk-free rate to its
realized risk/return combination
• Points above this benchmark line result from
superior performance
– The higher than expected return is the IRAR
54
Definition (cont’d)
55
Definition (cont’d)
 The
IRAR calculation:
IRAR  ( SH o  SH u ) o
where SH o  Sharpe measure of the optioned portfolio
SH u  Sharpe measure of the unoptioned portfolio
 o  standard deviation of the optioned portfolio
56
An IRAR Example
 A portfolio
manager routinely writes index
call options to take advantage of anticipated
market movements
 Assume:
• The portfolio has an initial value of $200,000
• The stock portfolio has a beta of 1.0
• The premiums received from option writing are
invested into more shares of stock
57
58
An IRAR Example (cont’d)
 The
IRAR calculation (next slide) shows
that:
• The optioned portfolio appreciated more than
the unoptioned portfolio
• The options program was successful at adding
about 12% per year to the overall performance
of the fund
59
60
IRAR Caveats
 IRAR
can be used inappropriately if there is
a floor on the return of the optioned
portfolio
• E.g., a portfolio manager might use puts to
protect against a large fall in stock price
 The
standard deviation of the optioned
portfolio is probably a poor measure of risk
in these cases
61
Residual Option Spread
The residual option spread (ROS) is an
alternative performance measure for portfolios
containing options
 A positive ROS indicates the use of options
resulted in more terminal wealth than only holding
stock
 A positive ROS does not necessarily mean that the
incremental return is appropriate given the risk

62
Residual Option
Spread (cont’d)
 The
residual option spread (ROS)
calculation:
n
n
t 1
t 1
ROS   Got   Gut
where Gt  Vt / Vt 1
Vt  value of portfolio in Period t
63
Residual Option
Spread (cont’d)
 The
worksheet to calculate the ROS for the
previous example is shown on the next slide
 The
ROS translates into a dollar differential
of $1,452
64
65
The M2
Performance Measure
 Developed
by Franco and Leah Modigliani
in 1997
 Seeks to express relative performance in
risk-adjusted basis points
• Ensures that the portfolio being evaluated and
the benchmark have the same standard
deviation
66
The M2 Performance
Measure (cont’d)
 Calculate
the risk-adjusted portfolio return
as follows:
Rrisk-adjusted portfolio
 benchmark

Ractual portfolio
 portfolio
  benchmark
 1 

 portfolio


 R f

67
Final Comments
 IRAR
and ROS both focus on whether an
optioned portfolio outperforms an
unoptioned portfolio
• Can overlook subjective considerations such as
portfolio insurance
68