Performance Evaluation of Portfolios

1Dr. Vinita Kalra

Performance Evaluation of Portfolios

Risk-Adjusted Performance Measures Jensen Index Treynor Index Sharpe Index CAPM Measures When You Can Lend But Cannot

Borrow at the Risk-Free Rate CAPM Measures When the Market Index is

Inefficient Performance Measures Based on the APT

Risk Adjusted Performance Measures

How do you discriminate between higher returns due to skillful management, and higher returns due simply to higher risk? In other words, how can we rank order risk-adjusted performance?

Three Widely Used Risk Adjusted Performance Measures Based on the Capital

Asset Pricing Model Assumptions: (1) The CML and the SML are

applicable to the pricing of securities. (2) Borrowing and lending takes place at the risk-free rate. (3) Construction of the CML and the SML is a function of publicly available information.

Given the above assumptions, investors may attempt to employ private information to identify undervalued and overvalued securities. One source of legal private information is the output of unique techniques of analysis of publicly available data.

Jensen Index(Sometimes Called Jensen’s Alpha)

Jensen’s Index is the vertical distance from the SML.

Evaluation of Expected Returns:

Evaluation of Past ReturnsSML on the Return

Return Expected Expected

)β]r)[E(r(r )E(r J jFMFjj

]β)rr(r[rJ jFMFj

Jensen Index (Continued)

The Jensen Index is sensitive only to depth and not to breadth: Depth: Magnitude of excess returns. Breadth: Magnitude of residual variance (e.g.,

Is the portfolio well diversified?) Note: Since beta is the risk measure:

Only systematic risk, and not residual variance is relevant.

The Jensen Index has been used for individual securities as well as portfolios. (No one expects individual securities to be well diversified).

The Jensen Index:A Graphical Illustration

02468

1012141618

0 1 2 3

Expected Return

SML

Beta Coefficient

+Jj

-JjE(rM)

rF

Treynor Index

Treynor’s Index is the slope of a straight line going through the risk-free rate of return. The Treynor Index may also be defined as the risk premium earned per unit of risk taken, where beta is the risk measure.

Evaluation of Expected Returns:

Evaluation of Past Returns:

Treynor Index (Continued)

Similarities With the Jensen Index: Since the beta coefficient is the risk

measure, the Treynor Index (like the Jensen Index) is insensitive to breadth (i.e., it ignores residual variance).

Furthermore, with beta as the risk measure, the Treynor Index is applicable for individual securities as well as for portfolios.

The Treynor Index:A Graphical Illustration

-2.5

2.5

7.5

12.5

17.5

22.5

0 0.5 1 1.5 2 2.5

Expected Return

Beta Coefficient

A

B

C

SML

TA > TB > TC

An Advantage of the Treynor Index Over the Jensen Index

The Treynor Index is advantageous over the Jensen Index in that it takes the opportunity to lever excess returns into account when ranking alternatives.

Example on the Following Graph: An investor could borrow at the risk-free rate,

and invest the proceeds in security (A) in order to obtain portfolio (C). Note that portfolio (C) dominates security (B):

E(rC) > E(rB) Yet C = B

Treynor Index Versus Jensen Index: TA > TB However JA = JB

Treynor Index Versus the Jensen Index

0

4

8

12

16

20

0 0.5 1 1.5 2 2.5

Expected Return

Beta Coefficient

A

C

B

SML

Sharpe Index

Sharpe’s Index is the slope of a straight line going through the risk-free rate of return. The Sharpe Index may also be defined as the risk premium earned per unit of risk taken, when the standard deviation of return is the risk measure.

Evaluation of Expected Returns:

Evaluation of Past Returns:

Sharpe Index (Continued)

The Sharpe Index is sensitive to both: Depth: Magnitude of excess returns Breadth: Diversification (residual

variance)

Note: Since the standard deviation of returns is the risk measure, the Sharpe Index is only appropriate for portfolios and not for individual securities.

The Sharpe IndexA Graphical Illustration

0

4

8

12

16

0 10 20 30

Expected Return

Standard Deviation of Returns

CMLA M

B

SA > SM > SB

CAPM Measures When You Can Lend But Cannot Borrow at the Risk-Free Rate

0

0.25

0 0.480

0.25

0 0.5 1 1.5

E(r) E(r)

(r)

E(rM) E(rM)

E(rZ) E(rZ)

SML

LM

X

rF

CAPM Measures When You Can Lend But Cannot Borrow at the Risk-Free Rate

(Continued) On the preceding graph, the CML is: (rF to L to M to

X) Recalling the Sharpe Index:

Note that Sp,(Point L) > Sp,(Point M) > Sp,(Point X) Yet, points L, M, and X, all lie on the CML.

Therefore, the Sharpe Index is upward biased towards low risk portfolios, and downward biased towards high risk portfolios. Furthermore, there is no easy way to correct the Sharpe Index for this problem.

CAPM Measures When You Can Lend But Cannot Borrow at the Risk-Free Rate

(Continued) Whereas the Sharpe Index is

difficult to correct for this situation, the Jensen Index and the Treynor Index may be modified as follows:

Revised Jensen Index:

Revised Treynor Index:)β)]E(r)[E(r)(E(r )E(r J jZMZjj

j

Zjj β

)E(r)E(rT

CAPM Measures When the Market Index is Inefficient

Note that if the market index used is inefficient, securities and portfolios plot above and below the Security Market Line. Therefore, we cannot tell if a portfolio’s position relative to the SML is due to performance, or simply due to the inefficiency of the market index.

In other words, a security’s or portfolio’s position relative to the SML is sensitive to the inefficient proxy chosen to represent the true market portfolio.

CAPM Measures When the Market Index is Inefficient (Continued)

0

0.25

0 0.480

0.25

0 0.5 1 1.5

E(r)

E(rM)

E(rz)

M

(r)

E(rM)

E(rz)

E(r)

M’

Performance Measure Based on the APT

Two Factor Model Example:

Note: The measure above is similar to the CAPM Jensen Index. It reflects only depth, and not breadth of performance.

Problem: What is the appropriate factor structure?

Equation APTthe on Based Return Expected

)β)]E(r)[E(Iβ)]E(r)[E(I +)(E(r)E(rA p2,Z2p1,Z1Zpp

Final Note on Performance Measurement Using Asset Pricing Models

In the case of CAPM, you can never know whether portfolio performance is due to management skill or to the fact that you have an inaccurate index of the true market portfolio. In the case of APT, given the freedom to select factors without restriction, you can literally make the performance of a portfolio anything you want it to be.