Portfolio Management

Performance Evaluation

One period returns

• Gross return

• Net return

0

1Payoff1

P

Pr

00

01

0

1 gain CapitalPayoffPayoff1

Payoff

PP

PP

P

Pr

Average Returns

• Arithmetic Mean

• Geometric Mean

n

t

tAM

n

rr

1

n

tt

nGM rr

1

)1(1

1)1(/1

1

nn

ttGM rr

Example

Period Price Dividend

0 501 53 22 54 2

%66.553

25354

%1050

25053

2

1

r

r

Example

• Arithmetic Mean

• Geometric Mean

%83.72

66.510

AMr

0781.110566.110.1 2/1 GMr

0566.0110.0.112

GMr

Arithmetic vs Geometric

• Past Performance - generally the geometric mean is preferable to arithmetic

• Predicting Future Returns- generally the arithmetic average is preferable to geometric

Example

• A stock price doubles or halves

• Same probability

• We observePeriod Price

0 10

1 20

2 10

Example

• (True) Average mean

• (Observed) Geometric mean

%252

50100

AMr

%015.02 2/1 GMr

ExamplePeriod Price Return

0 10 0

1 20 100%

2 10 -50%

Time weighted return = arithmetic average return

(100-50) = 25%

Example

Period PriceNumber of shares bought

0 10 100

1 20 100

2 10 -200

Dollar weighted return = Internal Rate of Return

21

2000

1

20001000

rr

% 8.26r

Measuring Returns

Dollar-weighted returns

• Internal rate of return considering the cash flow from or to investment

• Returns are weighted by the amount invested in each stock

Time-weighted returns

• Not weighted by investment amount

• Equal weighting

Adjusting for risk

• Mean returns are not enough and one must also adjust for risk

• Find the appropriate comparison universe

• Mean-variance risk adjustments

The Sharpe Ratio

• Sharpe’s measure: expected excess return per unit of risk (measured as total volatility)

• Apropriate scenario: Evaluate a portfolio which represents the entire investor’s initial wealth

• Slope of the CAL

P

fPP

rrES

The Sharpe Ratio and M2

• Equates the volatility of the managed portfolio with the market by creating a hypothetical portfolio made up of T-bills and the managed portfolio

• If the risk is lower than the market, leverage is used and

• The hypothetical portfolio is compared to the market

The Sharpe Ratio and M2

• Find the value

where a is the value for which

• Then:

M

PMPMP r

rarrarr

22222*

PfPraErarE 1*

MPrErEM *

2

The Sharpe Ratio and M2E(r)

P*

rf

M

P

σMσP

M2

Jensen’s alpha & AP

• Jensen’s measure: the expected return of the portfolio above its CAPM counterpart

• The appraisal ratio: alpha divided by the portfolio’s nonsystematic risk

fMPfPP rrErrE

PP

P eAR

Jensen’s alpha & AP

• The AP is used in situations where the portfolio to be evaluated will be mixed with the market

• Why? For the optimal mix, the complete portfolio’s sharpe ratio is

• It measures improvement in the Sharpe ratio

2

22

P

PMC eSS

Treynor’s measure

• Treynor’s measure: excess expected return per unit of systematic risk (measured as beta)

• Appropriate when the portfolio is part of a large investment portfolio

• The slope of the T-line

P

fPP

rrET

Treynor’s measure

E(r)

E(rM)

rf

SML

= 1.0

Slope(SML)=TM=E(rM)- rf

Q

P

TQ

TP

P

P

PMPP TTT

2

Some Issues

• Assumptions underlying measures limit their usefulness– Constant distributions– Preferences

• When the portfolio is being actively managed, basic stability requirements are not met– An example: market timing

Market Timing

• Adjusting portfolio for up and down movements in the market– Low Market Return - low ßeta

– High Market Return - high ßeta

• Regression:

pfMPfMPPfP errcrrbirr 2

An Example of Market Timing

******

**

**

**

**

**

**

****

****

******

******

****

****

rp - rf

rm - rf

Steadily Increasing the Beta

Market Timing

• A simple alternative:– Beta is large if the market does well– Beta is small otherwise

• Regression

ppfMPfMPPfP eDrrcrrbirr

Market timing

the Beta takes only two values

rp - rf

**

****

**

******

**

**

**

**

******

****

******

********

rm - rf

ppfMPfMPPfP eDrrcrrbirr

Performance Attribution• Decomposing overall performance into

components

• Components are related to specific elements of performance

• Example components– Broad Allocation– Industry– Security Choice

Performance Attribution

• Set up a ‘Benchmark’ or ‘Bogey’ portfolio– Use indexes for each component: depends on

the asset class– Use target weight structure: neutral, depend

on preferences of the client

• BKM give the example:– 10% cash, 15% bonds and 75% equity for

risk-tolerant client. – 45% cash, 20% bonds and 35% equity for

risk-averse.

A question

• The bond-to-equity ratio is

– 15/75 = 0.2 (low risk aversion)

– 20/35 = 0.57 (high risk aversion)

• If cash is riskless, does it make sense according to standard assumptions?

Asset allocation puzzle

• Canner, Mankiew and Weil:

”Popular financial advisors appear not to follow the mutual-fund separation theorem. When these advisors are asked to allocate portfolios among stocks, bonds, and cash,

they recommend more complicated strategies than indicated by the theorem”

• And so do BKM!!!!

Performance Attribution

• Calculate the return on the ‘Bogey’ and on the managed portfolio

• Explain the difference in return based on component weights or selection

• Summarize the performance differences into appropriate categories

Performance Attribution

PiBi

n

iPi

n

iBiBiPiBiBi

n

iPiPi

n

iBiBi

n

iPiPiBP

n

iPiPiP

n

iBiBiB

wrrrwwrwrw

rwrwrr

rwrrwr

)()(

(managed) (bogey)

111

11

11

Asset Allocation Security Selection

Performance Attribution

Contribution for asset allocation (wpi - wBi) rBi

+ Contribution for security selection wpi (rpi - rBi)

= Total Contribution from asset class wpirpi -wBirBi

Style analysis

• Regress the returns under evaluation on a sufficiently representative set of asset classes

• This allows identification of the capital allocation decision

• The proportion not explained: security selection

Style analysis

• Magellan Fund– Growth stocks 47%– Medium cap 31%– Small stocks 18%– European stocks 4%

Some Complications

• Two major problems– Need many observations even when portfolio

mean and variance are constant– Active management leads to shifts in

parameters making measurement more difficult

• To measure well– You need a lot of short intervals– For each period you need to specify the

makeup of the portfolio