1
Chapter 17
Performance Evaluation
Portfolio Construction, Management, & Protection, 4e, Robert A. StrongCopyright ©2006 by South-Western, a division of Thomson Business & Economics. All rights reserved.
2
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.
On Candide’s forcible impressment into the Bulgarian army, from
Voltaire’s Candide
3
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
4
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
5
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
6
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
2( ) ( ),E U f E R
7
A Lesson from History: The 1968 Bank Administration Institute Report
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
8
A Lesson from History: The 1968 Bank Administration Institute Report (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 greatly that indiscriminate comparisons among funds might reflect differences in these circumstances rather than in the ability of managers
9
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
10
A Lesson from a Few Mutual Funds (cont’d)
23.5%
30.7
6.5
16.9
26.3
14.3
37.8
12.0
Mutual Shares
19.3%
19.3
–34.6
–16.3
–20.1
–58.7
9.2
6.9
44 Wall Street
Mean
1988
1987
1986
1985
1984
1983
1982
Year
8.7-23.61981
19.036.11980
39.371.41979
16.132.91978
13.216.51977
63.146.51976
24.6%184.1%1975
Mutual Shares
44 Wall StreetYear
Change in net asset value, January 1 through December 31.
11
A Lesson from a Few Mutual Funds (cont’d)
Mutual Fund Performance
$-$20,000.00$40,000.00$60,000.00$80,000.00
$100,000.00$120,000.00$140,000.00$160,000.00$180,000.00$200,000.00
Year
En
din
g V
alu
e (
$)
44 WallStreet
MutualShares
12
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
13
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
14
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/
1
1
where the return relative in period
nn
ii
i
GM R
R i
15
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
16
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?
17
Why the Arithmetic Mean Is Often Misleading: A Review (cont’d)
Example
Solution: The geometric mean is computed as follows:
%45.30345.0
1)00.1)(50.1)(60.0(
1
3/1
/1
1
nn
iiRGM
18
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
19
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 $250,00012% 44% 12.10%
$40,250,000 $40,250,000
20
Traditional Performance Measures
Sharpe and Treynor Measures Jensen Measure Performance Measurement in Practice
21
Sharpe and Treynor Measures The Sharpe and Treynor measures:
Sharpe measure
Treynor measure
where average return
risk-free rate
standard deviation of returns
beta
f
f
f
R R
R R
R
R
22
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
23
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?
24
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%
40.88%
R
25
Sharpe and Treynor Measures (cont’d)
Example (cont’d)
Solution (cont’d): Next, compute the Sharpe and Treynor measures:
0.88%Sharpe measure 0.29
3.07%
0.88%Treynor measure 0.73
1.20
f
f
R R
R R
26
Jensen Measure The Jensen measure stems directly from the
CAPM:
it ft i mt ftR R R R
27
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
28
Jensen Measure (cont’d) The Jensen measure is generally out of
favor because of statistical and theoretical problems
29
Performance Measurement in Practice
Academic Issues Industry Issues
30
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
31
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
32
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
33
34
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
35
Industry Issues (cont’d) Net selectivity measures the portion of the
return from selectivity in excess of that provided by the “diversification” component
36
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 investment:
33
221
)1()1()1(cost
R
C
R
C
R
C
37
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):
1)1)(1)(1)(1(return weighted-time 4321 RRRR
38
Dollar-Weighted and Time-Weighted Rates of Return (cont’d)
The time-weighted rate of return and the dollar-weighted rate of return will be equal if there are no inflows or outflows from the portfolio
39
Performance Evaluation with Cash Deposits and Withdrawals
Introduction Daily Valuation Method Modified Bank Administration Institute
(BAI) Method An Example An Approximate Method
40
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 time-weighted rate of return:• Daily valuation method• Modified BAI method
41
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
42
Daily Valuation Method (cont’d)
The daily valuation method solves for R:
daily1
1
where
n
ii
i
i
R S
MVES
MVB
43
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
44
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
45
Modified Bank Administration Institute (BAI) Method (cont’d) It solves for R:
1
0
(1 )
where the sum of the cash flows during the period
market value at the end of the period,
including accrued income
market value at the start of the period
to
i
nw
ii
ii
MVE F R
F
MVE
F
CD Dw
CDCD
tal number of days in the period
number of days since the beginning of the period
in which the cash flow occurrediD
46
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
47
48
An Example (cont’d) The daily valuation method returns a time-
weighted return of 40.6 percent over the seven-month period• See next slide
49
50
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 seven-month period (see next slide)
51
52
An Approximate Method Proposed by the American Association of
Individual Investors:
1
0
0.5(Net cash flow)1
0.5(Net cash flow)
where net cash flow is the sum of inflows and outflows
PR
P
53
An Approximate Method (cont’d)
Using the approximate method in Table 17-6:
1
0
0.5(Net cash flow)1
0.5(Net cash flow)
5,500.97 0.5( 4,200)1
7,550.08 0.5(-4,200)
0.395 39.5%
PR
P
54
Performance Evaluation When Options Are Used
Introduction Incremental Risk-Adjusted Return from
Options Residual Option Spread Final Comments on Performance
Evaluation with Options
55
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
56
Introduction (cont’d) Consider two alternative methods when
options are included in a portfolio:• Incremental risk-adjusted return (IRAR)
• Residual option spread (ROS)
57
Incremental Risk-Adjusted Return from Options
Definition An IRAR Example IRAR Caveats
58
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
59
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
60
Definition (cont’d)
61
Definition (cont’d) The IRAR calculation:
( )
where Sharpe measure of the optioned portfolio
Sharpe measure of the unoptioned portfolio
standard deviation of the optioned portfolio
o u o
o
u
o
IRAR SH SH
SH
SH
62
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
63
64
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
65
66
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
67
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
68
Residual Option Spread (cont’d)
The residual option spread (ROS) calculation:
1 1
1where /
value of portfolio in Period
n n
ot utt t
t t t
t
ROS G G
G V V
V t
69
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
70
71
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
72
The M2 Performance Measure (cont’d)
Calculate the risk-adjusted portfolio return as follows:
benchmarkrisk-adjusted portfolio actual portfolio
portfolio
benchmark
portfolio
1 f
R R
R
73
Final Comments IRAR and ROS both focus on whether an
optioned portfolio outperforms an unoptioned portfolio• Can overlook subjective considerations such as
portfolio insurance
Top Related