Asset Management

Lecture 12

Outline of today’s lecture

Dollar- and Time-Weighted Returns Universe comparison Adjusting Returns for Risk

Sharpe measure Treynor measure Jensen measure Information ratio M2 measure

The choice of measure

Text Example of Multi-period Returns

0 1 2

Purchase 1 share at $50

Purchase 1 share at $53

Stock pays a dividend of $2 per share

Stock pays a dividend of $2 per share

Stock is sold at $54 per share

Text Example of Multi-period Returns

Dollar-Weighted Return Time-Weighted Return

%117.7

)1(

112

)1(

5150

21

r

rr%66.5

53

25354

%1050

25053

2

1

r

rInternal Rate of Return:

rG = [ (1.1) (1.0566) ]1/2 - 1 = 7.83%

•Internal rate of return considering the cash flow from or to investment; •Returns are weighted by the amount invested in each stock

Equal weighting

Universe comparisonComparison with other managers of

similar investment styleMay be misleading

Portfolio characteristics are not comparableSurvivorship bias

Universe comparison

95th percentile

5th percentile

The median

1) Sharpe Index

rp = Average return on the portfolio

rf = Average risk free rate

p= Standard deviation of portfolio

return

Risk Adjusted Performance: Sharpe

( )P f

P

r r

2) Treynor Measure

rp = Average return on the portfolio

rf = Average risk free rate

ßp = Weighted average for portfolio

Risk Adjusted Performance: Treynor

( )P f

P

r r

Risk Adjusted Performance: Jensen

3) Jensen’s Measure

p= Alpha for the portfolio

rp = Average return on the portfolio

ßp = Portfolio Beta

rf = Average risk free rate

rm = Average return on market index portfolio

( )P P f P M fr r r r

Risk Adjusted Performance: Information Ratio

Information Ratio

Information Ratio divides the alpha of the portfolio by the nonsystematic risk

p / (ep)

Nonsystematic risk could, in theory, be eliminated by diversification

Risk Adjusted Performance: M2

2*P MM r r

•rp* = return of a hypothetical portfolio made up of T-bills and the managed portfolio that has the same standard deviation as the market index portfolio

•rM = return of the market index portfolio

MMpMp SSrrM )(*2

Risk Adjusted Performance

( )P f

P

r r

( )P f

P

r r

( )P P f P M fr r r r

p / (ep)

2*P MM r r

Sharpe

Treynor

Jensen

Information ratio

M2

It depends on investment assumptions If the portfolio represents the entire

investment for an individual compared to the market (passive strategy) Sharpe Index or M2

If the portfolio is one of many portfolios combined in a large fund

There exist many alternative portfolios Jensen The Treynor measure: more complete because it

adjusts for risk

The choice of measure

Example: comparing two risky portfolios

%3%10*60.1%19

%2%10*90.0%11

Q

P

a

a Jensen’s measure:

Portfolio Q is preferred.

Example: comparing two risky portfolios

Nonsystematic risk will be diversified away in a well diversified fund.

Example: comparing two risky portfolios

Suppose that you form a portfolio with risk-free assets and portfolio P (or Q), then all possible portfolios lie along the TP line (or TQ line)

Treynor measure:

p

fpP

rrT

TP = 11% / 0.9 = 12.2%

TQ = 19% / 1.6 = 11.88%

An example of actual performance measurement

An example of actual performance measurement

Which portfolio to choose?

•If the portfolio stands for the entire investment fund

•If the portfolio is only a subportfolio of a larger fund

•If this is an active portfolio to be mixed with the index