Chapter 19 Performance Evaluation

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Chapter 19

Performance Evaluation

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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

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Outline Introduction Importance of measuring portfolio risk Traditional performance measures Performance evaluation with cash deposits

and withdrawals Performance evaluation when options are

used

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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

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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

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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

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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

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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

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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

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A Lesson from A Few Mutual Funds (cont’d)

Year44 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%

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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

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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

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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

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Why the Arithmetic Mean Is Often Misleading (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

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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

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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?

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Why the Arithmetic Mean Is Often Misleading (cont’d)

Example

Solution: The geometric mean is computed as follows:

1/

1

1

(0.60)(1.50)(1.00) 1

0.10 10%

nn

ii

GM R

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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

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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 $250,00012% 44% 12.10%

$40,250,000 $40,250,000

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Traditional Performance Measures

Sharpe and Treynor measures Jensen measure Performance measurement in practice

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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

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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

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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?

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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

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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

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Jensen Measure The Jensen measure stems directly from the

CAPM:

it ft i mt ftR R R R

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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

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Jensen Measure (cont’d) The Jensen measure is generally out of

favor because of statistical and theoretical problems

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Performance Measurement in Practice

Academic issues Industry issues

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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

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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

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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

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Performance Evaluation With Cash Deposits & Withdrawals

Introduction Daily valuation method Modified Bank Administration Institute

(BAI) Method An example An approximate method

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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 time-weighted rate of return:• Daily valuation method• Modified BAI method

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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

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Daily Valuation Method (cont’d)

The daily valuation method solves for R:

daily1

1

where

n

ii

i

i

R S

MVES

MVB

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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

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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

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Modified 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

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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

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An Example (cont’d) The daily valuation method returns a time-

weighted return of 40.6% over the seven-months period• See next slide

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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)

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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

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An Approximate Method (cont’d)

Using the approximate method in Table 19-6:

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

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Performance Evaluation When Options Are Used

Introduction Incremental risk-adjusted return from

options Residual option spread Final comments on performance evaluation

with options

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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

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Introduction (cont’d) Consider two alternative methods when

options are included in a portfolio:• Incremental risk-adjusted return (IRAR)

• Residual option spread (ROS)

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Incremental Risk-Adjusted Return from Options

Definition An IRAR example IRAR caveats

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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

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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

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Definition (cont’d)

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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

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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

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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

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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

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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

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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

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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

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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

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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

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Final Comments IRAR and ROS both focus on whether an

optioned portfolio outperforms an unoptioned portfolio• Can overlook subjective considerations such as

portfolio insurance