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Chapter 17
Performance Evaluation
Portfolio Construction, Management, & Protection, 4e, Robert A. Strong
Copyright ©2006 by South-Western, a division of Thomson Business & Economics. All rights reserved.
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.
the
from
On Candide’s forcible
impressment into
Bulgarian army,
Voltaire’s
2
Outline






Introduction
Importance of Measuring Portfolio Risk
Traditional Performance Measures
Dollar-Weighted and Time-Weighted Rates of
Return
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 Report

The 1968 Bank Administration Institute’s
Measuring the Investment Performance
of Pension Funds concluded:
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
1)
7
A Lesson from History: The 1968 Bank
Administration Institute Report (cont’d)
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 greatly that
indiscriminate comparisons among funds
might reflect differences in these
circumstances rather than in the ability of
managers
3)
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%
Change in net asset value, January 1 through December 31.
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: A Review

The arithmetic mean may give misleading
information
 e.g.,
a 50 percent decline in one period
followed by a 50 percent increase in the next
period does not produce an average return of
zero
13
Why the Arithmetic Mean
Is Often Misleading: A Review (cont’d)

The proper measure of average
investment return over time is the
geometric mean: 1/ n
 n

GM   Ri   1
 i 1 
where Ri  the return relative in period i
14
Why the Arithmetic Mean
Is Often Misleading: A Review (cont’d)

The geometric means in the preceding
example are:
 44
Wall Street: 7.9 percent
 Mutual Shares: 22.7 percent

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: A Review (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: A Review (cont’d)
Example
Solution: The geometric mean is computed as follows:

GM  

n

i 1

Ri 

1/ n
1
 (0.60 )(1.50 )(1.00 )1/ 3  1
 0.0345  3.45 %
17
Why Dollars Are More
Important Than Percentages

Assume two funds:
 Fund A has
$40 million in investments and
earned 12 percent last period
 Fund
B has $250,000 in investments and
earned 44 percent last period
18
Why Dollars Are More Important
Than Percentages (cont’d)

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 percent, –5.09 percent, –1.99 percent,
and 1.72 percent. The standard deviation of XYZ stock
returns is 3.07 percent. 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
 Arbitrage
pricing model, 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 return 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
Industry Issues (cont’d)

Diversification is the difference between
the return corresponding to the beta
implied by the total risk of the portfolio and
the return corresponding to its actual beta
 Diversifiable
risk decreases as portfolio size
increases, so if the portfolio is well diversified
the “diversification return” should be near zero
34
Industry Issues (cont’d)

Net selectivity measures the portion of the
return from selectivity in excess of that
provided by the “diversification”
component
35
Dollar-Weighted and
Time-Weighted Rates of Return

The dollar-weighted rate of return is
analogous to the internal rate of return in
corporate finance
 It
is the rate of return that makes the present value of
a series of cash flows equal to the cost of the
C3
C1
C2
investment:
cost 


(1  R)
(1  R) 2
(1  R)3
36
Dollar-Weighted and
Time-Weighted Rates of Return (cont’d)

The time-weighted rate of return measures the
compound growth rate of an investment
 It
eliminates the effect of cash inflows and outflows by
computing a return for each period and linking them
(like the geometric mean return):
time - weighted return  (1  R1 )(1  R2 )(1  R3 )(1  R4 )  1
37
Dollar-Weighted and
Time-Weighted Rates of Return (cont’d)

The time-weighted rate of return and the dollarweighted rate of return will be equal if there are
no inflows or outflows from the portfolio
38
Performance Evaluation with
Cash Deposits and Withdrawals
Introduction
 Daily Valuation Method
 Modified Bank Administration Institute
(BAI) Method
 An Example
 An Approximate Method

39
Introduction

The owner of a fund often takes 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
40
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 to or
withdrawals
41
Daily Valuation
Method (cont’d)

The daily valuation method solves for R:
n
Rdaily   Si  1
i 1
MVEi
where S 
MVBi
42
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
43
Modified Bank Administration
Institute (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
44
Modified Bank Administration
Institute (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
45
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
46
47
An Example (cont’d)

The daily valuation method returns a timeweighted return of 40.6 percent over the
seven-month period
 See
next slide
48
49
An Example (cont’d)

The BAI method requires use of a
computer

The BAI method returns a time-weighted
return of 42.1 percent over the sevenmonth period (see next slide)
50
51
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
52
An Approximate
Method (cont’d)

Using the approximate method in Table
17-6:
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%
53
Performance Evaluation
When Options Are Used
Introduction
 Incremental Risk-Adjusted Return from
Options
 Residual Option Spread
 Final Comments on Performance
Evaluation with Options

54
Introduction

Inclusion of options in a portfolio usually
results in a non-normal return distribution

Beta and standard deviation lose their
theoretical value if the return distribution is
nonsymmetrical
55
Introduction (cont’d)

Consider two alternative methods when
options are included in a portfolio:
 Incremental
 Residual
risk-adjusted return (IRAR)
option spread (ROS)
56
Incremental Risk-Adjusted
Return from Options
Definition
 An IRAR Example
 IRAR Caveats

57
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
58
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
59
Definition (cont’d)
60
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
61
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
62
63
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 percent per year to the
overall performance of the fund
64
65
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
66
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 the stock
A positive ROS does not necessarily mean that
the incremental return is appropriate given the
risk
67
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
68
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
69
70
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
71
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

72
Final Comments

IRAR and ROS both focus on whether an
optioned portfolio outperforms an
unoptioned portfolio
 Can
overlook subjective considerations such
as portfolio insurance
73