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CHAPTER
TWENTY-FOUR
PORTFOLIO
PERFORMANCE
EVALUATION
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
MEASURES OF RETURN

TWO MEASURES OF RETURN
• Dollar-Weighted Returns
uses discounted cash flow approach
weighted because the period with the greater
number of shares has a greater influence on
the overall average
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
MEASURES OF RETURN

TWO MEASURES OF RETURN
• Comparison of Time-Weighted to DollarWeighted Returns
Time-weighted useful in pension fund
management where manager cannot control
the deposits or withdrawals to the fund
MAKING RELEVANT
COMPARISONS

PERFORMANCE
• should be evaluated on the basis of a
relative and not an absolute basis
this is done by use of a benchmark portfolio
• BENCHMARK PORTFOLIO
should be relevant and feasible
reflects objectives of the fund
reflects return as well as risk
THE USE OF MARKET
INDICES

INDICES
• are used to indicate performance but
depend upon
the securities used to calculate them
the calculation weighting measures
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
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
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
ARITHMETIC V.
GEOMETRIC AVERAGES

GEOMETRIC MEAN FRAMEWORK
GM = (P HPR)1/N - 1
where P = the summation of the
product of
HPR= the holding period returns
n= the number of periods
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
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
RISK-ADJUSTED MEASURES
OF PERFORMANCE

THE REWARD TO VOLATILITLY RATIO
(TREYNOR MEASURE)
• There are two components of risk
risk associated with market fluctuations
risk associated with the stock
• Characteristic Line (ex post security line)
defines the relationship between historical
portfolio returns and the market portfolio
TREYNOR MEASURE

TREYNOR MEASURE
• Formula
RVOL p 
where
arp  ar f
bp
arp = the average portfolio return
arf = the average risk free rate
bp = the slope of the characteristic
line during the time period
TREYNOR MEASURE
THE CHARACTERISTIC LINE
arp
SML
bp
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
TREYNOR MEASURE
THE CHARACTERISTIC LINE
arp
SML
bp
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 sp
THE SHARPE RATIO

SHARPE RATIO
• formula:
SR p 
where
ar p  ar f
s
p
SR = the Sharpe ratio
sp = the total risk
THE SHARPE RATIO

SHARPE RATIO
• indicates the risk premium per unit of total
risk
• uses the Capital Market Line in its analysis
THE SHARPE RATIO
arp
CML
sp
THE JENSEN MEASURE OF
PORTFOLIO PERFORMANCE

BASED ON THE CAPM EQUATION
E ( ri )  RFR  b [ E ( rm )  RFR ]
• 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
THE JENSEN MEASURE OF
PORTFOLIO PERFORMANCE

THE JENSEN MEASURE
• known as the portfolio’s alpha value
recall the linear regression equation
y = a + bx + e
alpha is the intercept
THE JENSEN MEASURE OF
PORTFOLIO PERFORMANCE

DERIVATION OF ALPHA
• Let the expectations formula in terms of
realized rates of return be written
R jt  RFR t  b j Rmt  RFR t   u jt
• subtracting RFR from both sides
R jt  RFR t  b j Rmt  RFR t   u jt
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 bj times a market
risk premium plus a random error term
THE JENSEN MEASURE OF
PORTFOLIO PERFORMANCE

DERIVATION OF ALPHA
• to measure superior portfolio performance,
you must allow for an intercept a
• a superior manager has a significant and
positive alpha because of constant positive
random errors
COMPARING MEASURES
OF PERFORMANCE

TREYNOR V. SHARPE
• SR measures uses s as a measure of risk
while Treynor uses b
• SR evaluates the manager on the basis of
both rate of return performance as well as
diversification
COMPARING MEASURES
OF PERFORMANCE
• for a completely diversified portfolio
SR and Treynor give identical rankings because
total risk is really systematic variance
any difference in ranking comes directly from a
difference in diversification
CRITICISM OF RISK-ADJUSTED
PERFORMANCE MEASURES

Use of a market surrogate
Roll: 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
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
END OF CHAPTER 24