chapter twenty-four portfolio performance evaluation
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CHAPTER TWENTY-FOUR
PORTFOLIO PERFORMANCE
EVALUATION
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MEASURES OF RETURN
MEASURES OF RETURN•complicated by addition or withdrawal
of money by the investor
•percentage change is not reliable when the base amount may be changing
•timing of additions or withdrawals is important to measurement
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MEASURES OF RETURN
TWO MEASURES OF RETURN•Dollar-Weighted Returns
uses discounted cash flow approachweighted because the period with the
greater number of shares has a greater influence on the overall average
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MEASURES OF RETURN
TWO MEASURES OF RETURN•Time-Weighted Returns
used when cash flows occur between beginning and ending of investment horizon
ignores number of shares held in each period
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MEASURES OF RETURN
TWO MEASURES OF RETURN•Comparison of Time-Weighted to
Dollar-Weighted ReturnsTime-weighted useful in pension fund
management where manager cannot control the deposits or withdrawals to the fund
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MAKING RELEVANT COMPARISONS PERFORMANCE
•should be evaluated on the basis of a relative and not an absolute basisthis is done by use of a benchmark
portfolio
•BENCHMARK PORTFOLIOshould be relevant and feasiblereflects objectives of the fundreflects return as well as risk
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THE USE OF MARKET INDICES INDICES
•are used to indicate performance but depend uponthe securities used to calculate themthe calculation weighting measures
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THE USE OF MARKET INDICES INDICES
•Three Calculation Weighting Methods:price weighting
– sum prices and divided by a constant to determine average price
– EXAMPLE: THE DOW JONES INDICES
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THE USE OF MARKET INDICES INDICES
•Three Calculation Weighting Methods:value weighting (capitalization method)
– price times number of shares outstanding is summed
– divide by beginning value of index– EXAMPLE:
• S&P500• WILSHIRE 5000• RUSSELL 1000
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THE USE OF MARKET INDICES INDICES
•Three Calculation Weighting Methods:equal weighting
– multiply the level of the index on the previous day by the arithmetic mean of the daily price relatives
– EXAMPLE:• VALUE LINE COMPOSITE
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ARITHMETIC V. GEOMETRIC AVERAGES GEOMETRIC MEAN FRAMEWORK
GM = ( HPR)1/N - 1where = the summation of the
product of HPR= the holding period returns n= the number of periods
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ARITHMETIC V. GEOMETRIC AVERAGES GEOMETRIC MEAN FRAMEWORK
•measures past performance well
•represents exactly the constant rate of return needed to earn in each year to match some historical performance
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ARITHMETIC V. GEOMETRIC AVERAGES ARITHMETIC MEAN FRAMEWORK
•provides a good indication of the expected rate of return for an investment during a future individual year
•it is biased upward if you attempt to measure an asset’s long-run performance
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RISK-ADJUSTED MEASURES OF PERFORMANCE THE REWARD TO VOLATILITLY
RATIO (TREYNOR MEASURE)•There are two components of risk
risk associated with market fluctuationsrisk associated with the stock
•Characteristic Line (ex post security line)defines the relationship between historical
portfolio returns and the market portfolio
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TREYNOR MEASURE
TREYNOR MEASURE•Formula
where arp = the average portfolio return
arf = the average risk free rate
p= the slope of the characteristic
line during the time period
p
fpp
ararRVOL
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TREYNOR MEASURE
THE CHARACTERISTIC LINEarp
p
SML
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TREYNOR MEASURE
CHARACTERISTIC LINE•slope of CL
measures the relative volatility of portfolio returns in relation to returns for the aggregate market, i.e. the portfolio’s beta
the higher the slope, the more sensitive is the portfolio to the market
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TREYNOR MEASURE
THE CHARACTERISTIC LINEarp
p
SML
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THE SHARPE RATIO
THE REWARD TO VARIABILITY (SHARPE RATIO)•measure of risk-adjusted performance
that uses a benchmark based on the ex-post security market line
•total risk is measured by p
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THE SHARPE RATIO
SHARPE RATIO•formula:
where SR = the Sharpe ratio
p = the total risk
p
fpp
ararSR
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THE SHARPE RATIO
SHARPE RATIO•indicates the risk premium per unit of
total risk
•uses the Capital Market Line in its analysis
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THE SHARPE RATIO
arp
p
CML
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THE JENSEN MEASURE OF PORTFOLIO PERFORMANCE BASED ON THE CAPM EQUATION
•measures the average return on the portfolio over and above that predicted by the CAPM
•given the portfolio’s beta and the average market return
])([)( RFRrERFRrE mi
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THE JENSEN MEASURE OF PORTFOLIO PERFORMANCE THE JENSEN MEASURE
•known as the portfolio’s alpha valuerecall the linear regression equation
y = + x + ealpha is the intercept
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THE JENSEN MEASURE OF PORTFOLIO PERFORMANCE DERIVATION OF ALPHA
•Let the expectations formula in terms of realized rates of return be written
•subtracting RFR from both sides
jttmtjtjt uRFRRRFRR
jttmtjtjt uRFRRRFRR
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THE JENSEN MEASURE OF PORTFOLIO PERFORMANCE DERIVATION OF ALPHA
•in this form an intercept value for the regression is not expected if all assets are in equilibrium
•in words, the risk premium earned on the jth portfolio is equal to j times a market risk premium plus a random error term
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THE JENSEN MEASURE OF PORTFOLIO PERFORMANCE DERIVATION OF ALPHA
•to measure superior portfolio performance, you must allow for an intercept
•a superior manager has a significant and positive alpha because of constant positive random errors
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COMPARING MEASURES OF PERFORMANCE TREYNOR V. SHARPE
•SR measures uses as a measure of risk while Treynor uses
•SR evaluates the manager on the basis of both rate of return performance as well as diversification
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COMPARING MEASURES OF PERFORMANCE
•for a completely diversified portfolioSR and Treynor give identical rankings
because total risk is really systematic variance
any difference in ranking comes directly from a difference in diversification
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CRITICISM OF RISK-ADJUSTED PERFORMANCE MEASURES
Use of a market surrogateRoll: criticized any measure that
attempted to model the market portfolio with a surrogate such as the S&P500
– it is almost impossible to form a portfolio whose returns replicate those over time
– making slight changes in the surrogate may completely change performance rankings
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CRITICISM OF RISK-ADJUSTED PERFORMANCE MEASURES measuring the risk free rate
using T-bills gives too low of a return making it easier for a portfolio to show superior performance
borrowing a T-bill rate is unrealistically low and produces too high a rate of return making it more difficult to show superior performance
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END OF CHAPTER 24