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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 approach

• weighted 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 Returns• Time-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 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

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THE USE OF MARKET INDICES

• INDICES– are used to indicate performance but depend

upon• the securities used to calculate them

• the 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 VOLATILITY 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

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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 LINE

arp

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 LINE

arp

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 value

• recall the linear regression equation

y = + x + e• alpha 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 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

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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

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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