1 Chapter 19 Performance Evaluation Portfolio Construction, Management, & Protection, 5e, Robert A....

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Chapter 19

Performance Evaluation

Portfolio Construction, Management, & Protection, 5e, Robert A. StrongCopyright ©2009 by South-Western, a division of Thomson Business & Economics. All rights reserved.

2

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

3

Importance ofMeasuring 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

2( ) ( ),E U f E R

4

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

5

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

6

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)

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.

8


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

9


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

10

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

11

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

12


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

13


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?

14


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

15

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

16

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

17

Traditional Performance Measures

Sharpe Measure Treynor Measures Jensen Measure Performance Measurement in Practice

18

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

19

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

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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?

21


Example (cont’d)

Solution: First, compute the average excess return for Stock XYZ:

1.86% 5.09% 1.99% 1.72%

40.88%

R

22


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

23

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

25

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

26

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

27

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

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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

30

Industry Issues (cont’d) Net selectivity measures the portion of the

return from selectivity in excess of that provided by the “diversification” component

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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

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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

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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

34

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

35

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

36

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 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 to or withdrawals

38

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

40

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

41

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

42

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

43

Table19-4 Mutual Fund Account Value

Date Description $ Amount Price SharesTotal

Shares Value

January 1 balance forward

$7.00 1,080.011 $7,560.08

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

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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

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Table19-5 Daily Valuation Worksheet

Date Sub Period MVB Cash FlowEnding Value MVE MVE/MVB

January 1 $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%

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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 percent over the seven-month period (see next slide)

48

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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Intuition

1 0

0

The approximation formula can be rearranged and written as:

Net Cash Flow Net Cash Flow(1 ) (1 )

2 2

The initial value is compounded forward at the interest rate R

because P is a "time-zero" value.

P R P R

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

1

end and thus does not

need any compounding since its "spot on the timeline" is the same

as the final portfolio value P .

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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

51

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)

52

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

53

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

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55


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

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

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Table19-9 Modified BAI Method Worksheet (Using Interest Rate of 42.1%)

WeekUnoptioned

Portfolio

Long Stock Plus Option

Premiums

Less

Short Optionsa

Equals

Optioned Portfolio

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 13,164 208,707aOption 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.

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 11.3 percent per year to the overall performance of the fund

59

Table19-10 IRAR WorksheetUnoptioned 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

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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

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 the 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:

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

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

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.

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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

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