Last Day

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Last Day

• Started Portfolio Evaluation

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Today

• Portfolio Evaluation contd

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Portfolio Performance Evaluation Techniques

• As mentioned, one way to compare portfolios is to
  examine the return earned by alternative
  portfolios of the same risk
• The most examined type of funds are mutual funds
• Many authors have used the above technique to
  examine the performance of mutual funds under a
  number of scenarios

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Portfolio Performance Evaluation Techniques

• NOTE
• If the investor desired a risk different from
  that offered by the fund, she/he could modify
  risk by lending and/or borrowing.
• The relevant definition of performance may change
  if the problem is examined from the point of view
  of the fund manager
• This leads us directly to our second measure of
  performance
• Differential Return with Risk Measured by S.D.

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Portfolio Performance Evaluation Techniques

• Differential Return with Risk Measured by S.D.
• Recall mangers should be evaluated based on
  their ability to pick securities and combine them
  into a portfolio, given the risk level at which
  she/he is constrained to operate.
• To illustrate this evaluation technique, consider
  the following example.

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Portfolio Performance Evaluation Techniques

• Assume
• We are evaluating a manager of an all-equity
  portfolio
• Risk level is determined by the client
• The manager is the only equity manager, so total
  risk is important
• What should be evaluated in this case?
• The managers ability to adhere to clients wishes
  and pick securities accordingly for the portfolio

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Portfolio Performance Evaluation Techniques

• To evaluate the manager we could compare how well
  they do in relation to the naïve strategy
• This strategy relies on investing partly in the
  market portfolio and part in the riskless asset
• Sound familiar?
• It should, since weve dealt with this type of
  analysis already, but in a different context.

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Portfolio Performance Evaluation Techniques

• Consider the case of evaluating fund A, where
  risk is determined by the client
• To evaluate the managers performance, we compare
  the naïve strategys result to fund A.
• To examine how the manager has done, consider the
  following diagram

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Portfolio Performance Evaluation Techniques
Market Portfolio
A
M
Differential Return
A
RF
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Portfolio Performance Evaluation Techniques

• RECALL the slope of the ray connecting RF and M
  is
• Using the intercept, which we know is RF we can
  write the equation of the line as

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Portfolio Performance Evaluation Techniques

• How would we determine the return on portfolio A
  in our earlier diagram?
• It is determined by substituting the s.d. of A in
  the preceding formula and solving for the return
  of A
• The differential return (what were interested in
  calculating) is, of course, then just the
  difference between the return earned on A and A.

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Portfolio Performance Evaluation Techniques

• Example
• Given

We get
Differential Return
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Portfolio Performance Evaluation Techniques

• With this measure, funds would be ranked
  according to their differential return
• The best performing fund would obviously be the
  one with the highest differential return.
• NOTE
• It is important to realize that the relative
  ranking of funds will be affected by the choice
  of performance measure
• To see this consider the following example

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Portfolio Performance Evaluation Techniques
Which fund would be ranked first?
A
B
M
A
RF
B
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Portfolio Performance Evaluation Techniques

• What we have just done is evaluate the fund
  managers performance using the s.d. as the risk
  measure
• But, what if nondiversifiable risk is deemed to
  be the appropriate measure of risk?
• In that case, our analysis would change slightly,
  but luckily enough the two measures just
  discussed have counterparts to go along with this
  change is risk.

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Portfolio Performance Evaluation Techniques

• Excess Return to Nondiversifiable Risk
• This is similar to the first case we discussed
  (Sharpe measure), but is in expected return beta
  space this measure is often called the Treynor
  measure
• The slope of the line connecting the riskless
  asset and risky portfolio is now
• Once again the investor would prefer the most
  counterclockwise ray emanating form the riskless
  asset

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Portfolio Performance Evaluation Techniques
A
The Treynor Measure
B
C
RF
D
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Portfolio Performance Evaluation Techniques

• NOTE The equation of the line is the same as
  before, except with betas

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Portfolio Performance Evaluation Techniques

• The final measure examines differential return
  when beta is the risk measure
• Again, lets consider the line connecting the
  riskless rate and the market portfolio.
• A manager could obtain any point along this line
  by investing in the market portfolio and mixing
  this with the riskless asset to obtain the
  desired risk level.

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Portfolio Performance Evaluation Techniques

• Suppose managers choice is to actively manage
  the fund
• Then the managers performance is the difference
  in return earned by actively managing the fund
  and the return that would have been earned by
  passively managing the fund
• Passively managing here is referring to the rate
  that would have been earned if the manager
  invested in the market portfolio and riskless
  asset
• NOTE we are comparing the performance based on
  the same risk level

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Portfolio Performance Evaluation Techniques

• Given our comparison, the slope of the line
  connecting the riskless asset and the market
  portfolio is
• Given this, we can write the equation of the line
  as,

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Portfolio Performance Evaluation Techniques

• The differential return, in this case, is the
  actual return less the return on the portfolio of
  identical Beta
• NOTE the portfolio of identical Beta is lying on
  the line connecting the riskless asset and the
  market portfolio
• To calculate the return it is simply a matter of
  plugging the beta of the portfolio into the
  previous equation

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Portfolio Performance Evaluation Techniques

• Example
• Market return is 10
• Risk-free rate is 5
• Beta on the portfolio being evaluated is 0.8
• What is the expected return of a mixture of the
  market portfolio and the riskless asset to obtain
  a Beta of 0.8?

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Portfolio Performance Evaluation Techniques

• Example contd
• What is the differential return?
• It is just the difference between the return on
  the portfolio and the 9 we just calculated.
• NOTE
• This measure was first proposed by Jensen
• It is often referred to as the Jensen
  differential performance index

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Portfolio Performance Evaluation Techniques

• Jensen Measure
• Has special appeal because of its relationship to
  the CAPM.
• While weve presented the relationship between
  the return on a fund and the return on a
  portfolio constructed by mixing the riskless
  asset and the market portfolio to obtain the same
  risk, there is another way to view it

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Portfolio Performance Evaluation Techniques

• The other way to view the Jensen measure is to
  use the CAPM equation to calculate the
  differential return
• In this case the differential return is the
  difference in the return earned by the fund vs.
  the return the CAPM implies should be earned.
• To see this more clearly consider the example of
  the following slide

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Portfolio Performance Evaluation Techniques

• Example
• Assume a zero beta version of the CAPM
• This implies that the expected return on
  portfolio i is
• This formula is used to calculate the exp.return
  implied by the zero Beta CAPM for a portfolio
  with the same risk as the portfolio being
  evaluated
• The differential return is the return on the
  portfolio being evaluated, less the return
  implied by the zero Beta CAPM

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Portfolio Performance Evaluation Techniques

• Decomposition of overall evaluation
• When attempting to decompose performance, were
  trying to determine what factors led to the
  success/failure of a manager.
• That is, were interested in various aspects of
  performance that might affect overall
  performance.
• One of the most widely referenced decompositions
  if by Eugene Fama (this is the fellow who came up
  with EMH)
• His decomposition is illustrated in the following
  diagram

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Portfolio Performance Decomposition
Net Selectivity
Return from Selectivity
Diversification
M
Total Excess Return
Return from Managers Risk
Return from Investors Risk
RF
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Portfolio Performance Decomposition

• In Famas decomposition Diagram
• The line plotted represents all combinations of
  the riskless asset and portfolio M
• As discussed, one possible strategy is for the
  manager with a desired risk level to achieve this
  level by holding a portfolio composed of the
  riskless asset and the market (RFM, plots the
  return to all such points)
• The Jensen measure is A-A (that is the height
  above the line)

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Portfolio Performance Decomposition

• Contd
• NOTE Portfolios A and A have the same beta,
  hence they have the same nondiversifiable risk
• What they do not have is the same total risk!
• The risk of the naïve strategy comes about
  because of the fluctuations in the market
  portfolio
• thus the risk of portfolio A is completely
  nondiversifiable!

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Portfolio Performance Decomposition

• Contd
• Portfolio A, however, is not strictly a market
  portfolio
• Why not?
• Remember
• We are comparing A with a naïve portfolio, NOT
  with the same nondiversifiable risk, but rather
  with the same total risk