Measuring Portfolio Performance With Asset Pricing Models (Chapter 11)

1
Measuring Portfolio PerformanceWith Asset
Pricing Models(Chapter 11)

• 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

2
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?

3
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.

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Jensen Index(Sometimes Called Jensens Alpha)

• Jensens Index is the vertical distance from the
  SML.
• Evaluation of Expected Returns
• Evaluation of Past Returns

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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).

6
The Jensen IndexA Graphical Illustration
Expected Return
SML
Jj
E(rM)
-Jj
rF
Beta Coefficient
7
Treynor Index

• Treynors 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

8
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.

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The Treynor IndexA Graphical Illustration
Expected Return
B
SML
A
C
TA gt TB gt TC
Beta Coefficient
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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) gt E(rB) Yet ?C ?B
• Treynor Index Versus Jensen Index
• TA gt TB However JA JB

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Treynor Index Versus the Jensen Index
Expected Return
C
B
A
SML
Beta Coefficient
12
Sharpe Index

• Sharpes 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

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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.

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The Sharpe IndexA Graphical Illustration
Expected Return
CML
M
A
B
SA gt SM gt SB
Standard Deviation of Returns
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CAPM Measures When You Can Lend But Cannot Borrow
at the Risk-Free Rate
E(r)
E(r)
SML
X
E(rM)
E(rM)
M
L
E(rZ)
E(rZ)
rF
?
?(r)
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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) gt Sp,(Point M) gt
  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.

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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

18
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 portfolios position relative to
  the SML is due to performance, or simply due to
  the inefficiency of the market index.
• In other words, a securitys or portfolios
  position relative to the SML is sensitive to the
  inefficient proxy chosen to represent the true
  market portfolio.

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CAPM Measures When the Market Index is
Inefficient (Continued)
E(r)
E(r)
M
E(rM)
E(rM)
M
E(rz)
E(rz)
?
?(r)
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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?

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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.