Performance Evaluation

Timothy R. Mayes, Ph.D.

FIN 4600

Performance and the Market Line

Riski

E(Ri)

M

RF

RiskM

E(RM)

MLUndervalued

Overvalued

Note: Risk is either or

Performance and the Market Line (cont.)

Riski

E(Ri)

M

RFR

RiskM

E(RM)

ML

A

B

C

D

E

Note: Risk is either or

The Treynor Measure

The Treynor measure calculates the risk premium per unit of risk (i)

Note that this is simply the slope of the line between the RFR and the risk-return plot for the security

Also, recall that a greater slope indicates a better risk-return tradeoff

Therefore, higher Ti generally indicates better performance

The Sharpe Measure

The Sharpe measure is exactly the same as the Treynor measure, except that the risk measure is the standard deviation:

Sharpe vs Treynor

The Sharpe and Treynor measures are similar, but different: S uses the standard deviation, T uses beta S is more appropriate for well diversified portfolios,

T for individual assets For perfectly diversified portfolios, S and T will give

the same ranking, but different numbers (the ranking, not the number itself, is what is most important)

Sharpe & Treynor ExamplesPortfolio Return RFR Beta Std. Dev. Trenor Sharpe

X 15% 5% 2.50 20% 0.0400 0.5000Y 8% 5% 0.50 14% 0.0600 0.2143Z 6% 5% 0.35 9% 0.0286 0.1111

Market 10% 5% 1.00 11% 0.0500 0.4545

Risk vs Return

0%

5%

10%

15%

0.00 0.50 1.00 1.50 2.00 2.50Beta

Ret

urn M

X

Y

Z

Risk vs Return

0%

5%

10%

15%

0% 5% 10% 15% 20%Std. Dev.

Ret

urn

M

X

YZ

Jensen’s Alpha

Jensen’s alpha is a measure of the excess return on a portfolio over time

A portfolio with a consistently positive excess return (adjusted for risk) will have a positive alpha

A portfolio with a consistently negative excess return (adjusted for risk) will have a negative alpha

Ris

k Pre

miu

mMarket Risk Premium

0

> 0

= 0

< 0

Modigliani & Modigliani (M2)

M2 is a new technique (Fall 1997) that is closely related to the Sharpe Ratio.

The idea is to lever or de-lever a portfolio (i.e., shift it up or down the capital market line) so that its standard deviation is identical to that of the market portfolio.

The M2 of a portfolio is the return that this adjusted portfolio earned. This return can then be compared directly to the market return for the period.

Calculating M2

The formula for M2 is:

As an example, the M2 for our example portfolios is calculated below:

Recall that the market return was 0.10, so only X outperformed. This is the same result as with the Sharpe Ratio.

ffii

M2 RRRM

062.005.005.006.009.011.0M

074.005.005.008.014.011.0M

105.005.005.015.020.011.0M

2Z

2Y

2X

Fama’s Decomposition

Fama decomposed excess return into two main components: Risk

Manager’s risk Investor’s risk

Selectivity Diversification Net selectivity

Excess return is defined as that portion of the return in excess of the risk-free rate

Fama’s Decomposition (cont.)

M anager 's R isk Investo r 's R isk

R isk P rem ium D ue to R isk

D iversification N et S electiv ity

R isk P rem ium Due to S electiv ity

T otal R isk P rem ium

Fama’s Decomposition: Risk

This is the portion of the excess return that is explained by the portfolio beta and the market risk premium:

Fama’s Decomposition: Investor’s Risk

If an investor specifies a particular target level of risk (i.e., beta) then we can further decompose the risk premium due to risk into investor’s risk and manager’s risk.

Investors risk is the risk premium that would have been earned if the portfolio beta was exactly equal to the target beta:

fMTskInvestorRi RRRP

Fama’s Decomposition: Manager’s Risk

If the manager actually takes a different level of risk than the target level (i.e., the actual beta was different than the target beta) then part of the risk premium was due to the extra risk that the manager’s took:

fMTikManagerRis RRRP

Fama’s Decomposition: Selectivity

This is the portion of the excess return that is not explained by the portfolio beta and the market risk premium:

Since it cannot be explained by risk, it must be due to superior security selection.

Fama’s Decomposition: Diversification

This is the difference between the return that should have been earned according to the CML and the return that should have been earned according to the SML

If the portfolio is perfectly diversified, this will be equal to 0

Fama’s Decomposition: Net Selectivity

Selectivity is made up of two components: Net Selectivity Diversification

Diversification is included because part of the manager’s skill involves knowing how much to diversify

We can determine how much of the risk premium comes from ability to select stocks (net selectivity) by subtracting diversification from selectivity

Additive Attribution

Fama’s decomposition of the excess return was the first attempt at an attribution model. However, it has never really caught on.

Other attribution systems have been proposed, but currently the most widely used is the additive attribution model of Brinson, Hood, and Beebower (FAJ, 1986)

Brinson, et al showed that the portfolio return in excess of the benchmark return could be broken into three components: Allocation describes the portion of the excess return that is due to

sector weighting different from the benchmark Selection describes the portion of the excess return that is due to

choosing securities that outperform in the benchmark portfolio Interaction is a combined effect of allocation and selection.

Additive Attribution (cont.)

The Brinson model is a single period model, based on the idea that the total excess return is equal to the sum of the allocation, selection, and interaction effects.

Note that Rt is the portfolio return, Rt bar is the benchmark return, and At, St, and It are the allocation, selection, and interaction effects respectively:

ttttt ISARR

Additive Attribution (cont.)

The equations for each of the components of excess return are:

N

1it,it,it,it,it

N

1it,it,it,it

N

1itt,it,it,it

RRwwI

RRwS

RRwwA

Additive Attribution (cont.)

So, looking at the formulas it should be obvious that: Allocation measures the relative weightings of each sector in

the portfolio and how well the sectors performed in the benchmark versus the overall benchmark return. A positive allocation effect means that the manager, on balance, over-weighted sectors that out-performed in the index and under-weighted the under-performing sectors.

Selection measures the sector’s different returns versus their weightings in the benchmark. A positive selection effect means that the manager selected securities that outperformed, on balance, within the sectors.

Interaction measures a combination of the different weightings and different returns and is difficult to explain. For this reason, many software programs allocate the interaction term into both allocation and selection.

Additive Attribution: An ExampleSector Portfolio Benchmark

Weight Return Weight Return

Equities 70.00% 7.00% 60.00% 8.00%Bonds 20.00% 7.50% 40.00% 6.00%Cash 10.00% 6.00% 0.00% 5.00%Total 100.00% 7.00% 100.00% 7.20%

Sector Allocation Selection Interaction Total

Equities 0.08% -0.60% -0.10% -0.62%Bonds 0.24% 0.60% -0.30% 0.54%Cash -0.22% 0.00% 0.10% -0.12%Total 0.10% 0.00% -0.30% -0.20%