Download Performance Evaluation

Chapter 19
Performance Evaluation
Portfolio Construction, Management, & Protection, 5e, Robert A. Strong
Copyright ©2009 by South-Western, a division of Thomson Business & Economics. All rights reserved.
1
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
2
Importance of
Measuring Portfolio Risk
 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 
3
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
4
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
5
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
6
A Lesson from
a Few Mutual Funds (cont’d)
Year
44 Wall
Street
Mutual
Shares
Year
44 Wall
Street
Mutual
Shares
2010
184.1%
24.6%
2017
6.9
12.0
2011
46.5
63.1
2018
9.2
37.8
2012
16.5
13.2
2019
–58.7
14.3
2013
32.9
16.1
2020
–20.1
26.3
2014
71.4
39.3
2021
–16.3
16.9
2015
36.1
19.0
2022
–34.6
6.5
2016
-23.6
8.7
2023
19.3
30.7
Mean
19.3%
23.5%
Change in net asset value, January 1 through December 31.
7
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
Year
8
A Lesson from
a Few Mutual Funds (cont’d)
 44
Wall Street and Mutual Shares both had
good returns over the 2010 to 2023 period.
 Mutual
Shares clearly outperforms 44 Wall
Street in terms of dollar returns at the end of
2023.
9
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
10
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


GM   Ri   1
 i 1 
where Ri  the return relative in period i
n
11
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 2010 to 2023 period
12
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. [The average
rate of return is 3.3%]
What is the geometric mean over the three periods?
13
Why the Arithmetic Mean
Is Often Misleading: A Review (cont’d)
Example
Solution: The geometric mean is computed as follows:


GM  
Ri 
 i 1 
n

1/ n
1
 (0.60 )(1.50 )(1.00 )1/ 3  1
 0.0345  3.45 %
14
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
15
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%

 

16
Traditional
Performance Measures
 Sharpe
Measure
 Treynor Measures
 Jensen Measure
 Performance Measurement in Practice
17
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
18
Sharpe and
Treynor Measures (cont’d)
 The
Sharpe measure evaluates return
relative to total risk
• Appropriate for a well-diversified portfolio, but
not for individual securities
 The
Treynor measure evaluates the return
relative to beta, a measure of systematic risk
• It ignores any unsystematic risk
19
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?
20
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%
21
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
22
Jensen Measure
 The
Jensen measure stems directly from the
CAPM:
Rit  R ft     i  Rmt  R ft 
23
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
24
Academic Issues
Regarding Performance Measures
 The
use of Treynor and Jensen performance
measures relies on measuring the market
return and CAPM
• Difficult to identify and measure the return of
the market portfolio
 Evidence
continues to accumulate that may
ultimately displace the CAPM
• Arbitrage pricing model, multi-factor CAPMs,
inflation-adjusted CAPM
25
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
26
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
27
28
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
29
Industry Issues (cont’d)
 Net
selectivity measures the portion of the
return from selectivity in excess of that
provided by the “diversification”
component
30
Volume Weighted Average Price
 Portfolio
managers want to minimize the
impact of their trading on share price
• A buy order increases demand and price
• Portfolio managers frequently take the opposite
position in the pre-market trading
 Volume
weighted average price compares
the average market price to the average
price paid by (or received) by a manager
31
Trading Efficiency
Impacts on Performance (cont’d)



Implementation shortfall is the difference between the
value of a “on-paper” theoretical portfolio and portfolio
purchased
“On-paper” portfolio is based upon price of last reported
trade
Actual purchases are likely to lead to higher costs due to
• Commissions
• “Bid-Ask” Spread – the prior trade may have been at a bid, while
your trade may be at the ask price (reverse if selling)
• Market trend – are prices moving higher?
• Liquidity impact – your demand may exceed share available a
lower price
– Or, shares sold may exceed the number purchased at higher price
32
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:
C3
C1
C2
cost 


2
(1  R) (1  R) (1  R)3
33
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
34
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
35
Performance Evaluation with
Cash Deposits and Withdrawals

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
36
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
37
Daily Valuation
Method (cont’d)
 The
daily valuation method solves for R:
n
Rdaily   Si  1
i 1
MVEi
where S 
MVBi
38
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
39
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
40
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
41
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
42
Table19-4 Mutual Fund Account Value
Date
Description
$ Amount
Price
Shares
$7.00
Total
Shares
Value
1,080.011
$7,560.08
January 1
balance
forward
January 3
purchase
100
$7.00
14.286
1,094.297
$7,660.08
February 1
purchase
100
$7.91
12.642
1,106.939
$8,755.89
March 1
purchase
100
$7.84
12.755
1,119.694
$8,778.40
March 23
liquidation
5,000
$8.13
-615.006
504.688
$4,103.11
April 3
purchase
100
$8.34
11.900
516.678
$4,309.09
May 1
purchase
100
$9.00
11.111
527.789
$4,750.10
June 1
purchase
100
$9.74
10.267
538.056
$5,240.67
July 3
purchase
100
$9.24
10.823
548.879
$5,071.64
August 1
purchase
100
$9.84
10.163
559.042
$5,500.97
August 1 account value: $559.042 x $9.84 = $5,500.97
43
An Example (cont’d)
 The
daily valuation method returns a timeweighted return of 40.6 percent over the
seven-month period
• See next slide
44
Table19-5
Date
Sub Period
Daily Valuation Worksheet
MVB
Cash Flow
January 1
Ending
Value
MVE
MVE/MVB
$7,560.08
January 3
1
$7,560.08
100
$7,660.08
$7,560.08
1.00
February 1
2
$7,660.08
100
$8,755.89
$8,655.89
1.13
March 1
3
$8,755.89
100
$8,778.40
$8,678.40
0.991
March 23
4
$8,778.40
5,000
$4,103.11
$9,103.11
1.037
April 3
5
$4,103.11
100
$4,309.09
$4,209.09
1.026
May 1
6
$4,309.09
100
$4,750.10
$4,650.10
1.079
June 1
7
$4,750.10
100
$5,240.67
$5,140.67
1.082
July 3
8
$5,240.67
100
$5,071.64
$4,971.64
0.949
August 1
9
$5,071.64
100
$5,500.97
$5,400.97
1.065
Product of MVE/MVB values = 1.406;  R = 40.6%
45
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)
46
47
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
48
Intuition
The approximation formula can be rearranged and written as:
Net Cash Flow Net Cash Flow
P1  (1  R) P0  (1  R)

2
2
The initial value is compounded forward at the interest rate R
because P0 is a "time-zero" value. The rest of the cash flows
is split into two: half is modeled as being paid at the beginning
(and thus needs to be compounded forward at the rate R), while
the other half is modeled as being paid at the end and thus does not
need any compounding since its "spot on the timeline" is the same
as the final portfolio value P1.
49
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%
50
Performance Evaluation
When Options Are Used
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
 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
 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
52
Incremental Risk-Adjusted
Return from Options (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
53
Incremental Risk-Adjusted
Return from Options (cont’d)
54
Incremental Risk-Adjusted
Return from Options (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
55
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
56
Table19-9
Modified BAI Method Worksheet
(Using Interest Rate of 42.1%)
Unoptioned
Portfolio
Long Stock Plus
Option
Premiums
0
$200,000
$214,112
$14,112
$200,000
1
190,000
203,406
8,868
194,538
2
195,700
209,508
10,746
198,762
3
203,528
217,888
14,064
203,824
4
199,528
213,530
11,220
202,310
5
193,474
207,124
7,758
199,366
6
201,213
215,409
10,614
204,795
7
207,249
221,871
13,164
208,707
Week
Less
Short Optionsa
Equals
Optioned Portfolio
aOption
values are theoretical Black-Scholes values based on an initial OEX value of 300, a
striking price of 300, a risk-free interest rate of 8 percent, an initial life of 90 days, and a
volatility of 35 percent. Six OEX calls are written.
57
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
11.3 percent per year to the overall performance
of the fund
58
Table19-10 IRAR Worksheet
Unoptioned Portfolio
Relative Return
Optioned Portfolio
Relative Return
$200,000
0.9500
$200,000
$190,000
1.0300
$194,538
0.9727
$195,700
1.0400
$198,762
1.0217
$203,528
0.9800
$203,824
1.0255
$199,457
0.9800
$202,310
0.9926
$193,474
0.9700
$199,366
0.9854
$201,213
1.0400
$204,795
1.0272
$207,249
1.0300
$208,707
1.0191
Mean = 1.007
Mean = 1.0063
Standard Deviation = 0.0350
Standard Deviation = 0.0206
Risk-free rate = 8%/year = 0.15%/week
SHu = (0.0057 - 0.0015) ÷ 0.0350 = 0.1200
SHo = (0.0063 - 0.0015) ÷ 0.0206 = 0.2330
IRAR = (0.2330 – 0.1200) x 0.0206 = 0.0023 per week, or about 12% per year
59
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
60
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

61
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
62
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
63
Residual Options Spread Worksheet

Unoptioned Portfolio
• (0.95)(1.03)(1.04)(0.98)(0.97)(1.04)(1.03) = 1.03625

Optioned Portfolio
• (0.9727)(1.0217)(1.0255)(0.9926)(0.9854)(1.0272)(1.0191) = 1.04351

ROS = 1.04351 – 1.03625 = 0.00726

Given an initial investment of $200,000, the ROS translates
into a dollar differential of $200,000 x 0.00726 = $1,452.
64
Final Comments
 IRAR
and ROS both focus on whether an
optioned portfolio outperforms an
unoptioned portfolio
• Can overlook subjective considerations such as
portfolio insurance
65