Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.
McGraw-Hill/Irwin
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Portfolio Performance Evaluation

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—Single period return
—Total return
—Definition of cash flow
—Multiple period return FV and PV
—Two common ways to measure average portfolio return:
—
1.Dollar-weighted returns
2.Time-weighted returns
3.
—Returns must be adjusted for risk.
Introduction

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—Dollar-weighted returns
—Internal rate of return considering the cash flow from or to investment
—Returns are weighted by the amount invested in each period:
—
Dollar- and Time-Weighted Returns

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Example of Multiperiod Returns


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Dollar-Weighted Return
Dollar-weighted Return (IRR):
-$50
-$53
$2
$4+$108

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—Time-weighted returns
—
—The geometric average is a time-weighted average.
—Each period’s return has equal weight.
—
Dollar- and Time-Weighted Returns

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Time-Weighted Return
rG = [ (1.1) (1.0566) ]1/2 – 1 = 7.81%

8
Averaging Returns
Arithmetic Mean:
Geometric Mean:
Example:
(.10 + .0566) / 2 = 7.83%
[ (1.1) (1.0566) ]1/2 - 1
= 7.808%
Example:
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—The arithmetic average provides unbiased estimates of the expected return of the stock. Use this
to forecast returns in the next period.
—
—The geometric average is less than the arithmetic average and this difference increases with the
volatility of returns.
—
—The geometric average is also called the time-weighted average (as opposed to the dollar weighted
average), because it puts equal weights on each return.
Geometric Average
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—The simplest and most popular way to adjust returns for risk is to compare the portfolio’s return
with the returns on a comparison universe.
—The comparison universe is a benchmark composed of a group of funds or portfolios with similar
risk characteristics, such as growth stock funds or high-yield bond funds.
—
—
—
Adjusting Returns for Risk

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Figure 24.1 Universe Comparison
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—1)  Sharpe Index
Risk Adjusted Performance: Sharpe
rp  = Average return on the portfolio
rf  = Average risk free rate
p
= Standard deviation of portfolio
return


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—2)  Treynor Measure
Risk Adjusted Performance: Treynor
rp = Average return on the portfolio

rf = Average risk free rate
ßp = Weighted average beta for portfolio

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Risk Adjusted Performance: Jensen
3)  Jensen’s Measure
p
= Alpha for the portfolio
rp  =  Average return on the portfolio
ßp =  Weighted average Beta
rf  =  Average risk free rate
rm =  Average return on market index portfolio


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Information Ratio
Information Ratio = ap / s(ep)
The information ratio divides the alpha of the portfolio by the nonsystematic risk.
Nonsystematic risk could, in theory, be eliminated by diversification.

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M2 Measure
—Developed by Modigliani and Modigliani
—Create an adjusted portfolio (P*)that has the same standard deviation as the market index.
—Because the market index and P* have the same standard deviation, their returns are comparable:

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M2 Measure: Example
Managed Portfolio:  return = 35% standard deviation = 42%
Market Portfolio:  return = 28% standard deviation = 30%
T-bill return = 6%
P* Portfolio:
30/42 = .714 in  P  and  (1-.714) or .286 in  T-bills
The return on P* is (.714) (.35) + (.286) (.06) = 26.7%
Since this return is less than the market, the managed portfolio underperformed.

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Figure 24.2 M2 of Portfolio P
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—It depends on investment assumptions
1)If the portfolio represents the entire risky investment , then use the Sharpe measure.
—
—2) If the portfolio is one of many combined into a larger investment fund, use the Jensen or the
Treynor measure. The Treynor measure is appealing because it weighs excess returns against
systematic risk.
—
Which Measure is Appropriate?

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Table 24.1 Portfolio Performance
t24.1.bmp
Is Q better than P?

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Figure 24.3 Treynor’s Measure
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Table 24.3 Performance Statistics
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Interpretation of Table 24.3
—If P or Q represents the entire investment, Q is better because of its higher Sharpe measure and
better M2.
—If P and Q are competing for a role as one of a number of subportfolios, Q also dominates because
its Treynor measure is higher.
—If we seek an active portfolio to mix with an index portfolio, P is better due to its higher
information ratio.
—
—

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Performance Measurement for
Hedge Funds

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Performance Measurement with Changing Portfolio Composition
—We need a very long observation period to measure performance with any precision, even if the
return distribution is stable with a constant mean and variance.
—What if the mean and variance are not constant? We need to keep track of portfolio changes.
—

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Figure 24.4 Portfolio Returns
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Style Analysis
—Introduced by William Sharpe
—Regress fund returns on indexes representing a range of asset classes.
—The regression coefficient on each index measures the fund’s implicit allocation to that “style.”
—R –square measures return variability due to style or asset allocation.
— The remainder is due either to security selection or to market timing.
—
—
—
—

Table 24.5 Style Analysis for Fidelity’s Magellan Fund
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—
—
Monthly returns on Magellan Fund over five year period.
Regression coefficient only positive for 3.
They explain 97.5% of Magellan’s returns.
2.5 percent attributed to security selection within asset classes.
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Figure 24.7 Fidelity Magellan Fund Cumulative Return Difference
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