Performance Evaluation

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Performance Evaluation

• Timothy R. Mayes, Ph.D.
• FIN 4600

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Performance and the Market Line
E(Ri)
Undervalued
ML
M
E(RM)
RF
Overvalued
RiskM
Riski
Note Risk is either b or s
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Performance and the Market Line (cont.)
E(Ri)
B
ML
A
M
E(RM)
C
E
RFR
D
RiskM
Riski
Note Risk is either b or s
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The Treynor Measure

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

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The Sharpe Measure

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

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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)

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Sharpe Treynor Examples
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Jensens Alpha
a gt 0
a 0

• Jensens 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

a lt 0
Risk Premium
0
Market Risk Premium
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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.

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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.

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Famas Decomposition

• Fama decomposed excess return into two main
  components
• Risk
• Managers risk
• Investors risk
• Selectivity
• Diversification
• Net selectivity
• Excess return is defined as that portion of the
  return in excess of the risk-free rate

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Famas Decomposition (cont.)
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Famas Decomposition Risk

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

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Famas Decomposition Investors 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
  investors risk and managers risk.
• Investors risk is the risk premium that would
  have been earned if the portfolio beta was
  exactly equal to the target beta

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Famas Decomposition Managers 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 managers took

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Famas 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.

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Famas 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

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Famas Decomposition Net Selectivity

• Selectivity is made up of two components
• Net Selectivity
• Diversification
• Diversification is included because part of the
  managers 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

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Additive Attribution

• Famas 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.

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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

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Additive Attribution (cont.)

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

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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 sectors 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.

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Additive Attribution An Example