CHAPTER TWENTY

PERFORMANCE MEASUREMENTAND PRESENTATION

© 2001 South-Western College Publishing

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Outline

Relating Risk and Return Risk, Return, and Utility Arithmetic vs. Geometric Averages Traditional Performance Measures The Capital Market Line The Security Market Line

Performance Measurement AIMR-Required Calculations Recommended Calculations

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Outline

Performance Presentation StandardsComposite ResultsInternational PortfoliosLeverage and Derivatives

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For most investors, the expected utility of an investment is a positive function of the expected return of the investment and a negative function of the variance of these returns : E(U) = f [ E(R), - 2 ]

Other relevant risk measures may include beta (for a stock portfolio) or duration (for a fixed income portfolio).

Risk, Return, and Utility Proper performance evaluation should

recognize both the return and the riskiness of the investment.

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To get around the problem of negative returns, returns are transformed into return relatives by adding 1.0 to them.

Suppose an initial investment of $100 falls by 50% in one period, and rises by 50% in the following period. What is the average return?

Arithmetic vs. Geometric Averages

The proper measure of average return overtime with investments is the geometric mean return : nn

iiGM Rx

1

1

where R i =the return relative in period i

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The Sharpe measure relates return to total risk. It can be used effectively with a portfolio where unsystematic risk has been diversified away.

Traditional Performance Measures

Sharpe measure

R i Rf

i

where = arithmetic mean return of security i= risk free rate= standard deviation of returns on security i

R iRfi

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The Treynor measure relates return to systematic risk, as measured by the security (or portfolio) beta. It is an appropriate measure for both single securities as well as for portfolios.

Traditional Performance Measures

Treynor measure

R i Rf

i

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According to finance theory, i should be zero. So, a positive alpha that is statistically different from zero indicates an above-average performance, and vice versa.

The Jensen measure stems directly from the implications of the capital asset pricing model as estimated by the market model.

Traditional Performance Measures

Ri Rf i i Rmarket Rf

i Ri Rf i Rmarket Rf or

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The line extending from the risk-free rate through the market portfolio on the efficient frontier is the capital market line.

Securities plotted above the capital market line show better-than-expected performance, and vice versa.

Traditional Performance Measuresm

ean

ret

urn

standard deviation

The Sharpe performance measure can be interpreted as the slope of a line relating the security’s return with its risk.

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The standard of comparison in this case is the security market line. This line extends from the risk-free rate through the point corresponding to the return associated with a beta of 1.0.

Traditional Performance Measuresm

ean

ret

urn

beta

It is also possible to plot thereturns of securities against their levels of systematic risk, or beta.

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Compliance with AIMR (Association for Investment Management and Research) Performance Presentation Standards is rapidly becoming a nonoptional practice in the money management business.

In order to be compliant with AIMR standards, certain information must be presented. Certain practices are recommended by AIMR too, though not required.

Performance Measurement

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Accrual accounting : Bonds accrue interest for each day that they are held, and such income is required to be calculated on an accrual basis. Earned dividends can be accrued too, though they are mostly accounted for on a cash basis.

Total return, including realized and unrealized gains and losses plus income, is required.

AIMR-Required Calculations

beginning

beginningending

Value

IncomeValueValue return Total

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Time-weighted rate of return : Returns should be measured with recognition of both the timing of the cash flows and their size. There are two ways of achieving this.

The daily valuation method calculates the exact time-weighted rate of return. Though cumbersome, it is the preferred method.

The modified BAI method approximates the internal rate of return for the investment over the period in question.

AIMR-Required Calculations

Daily Valuation Method

Rdaily Si 1

i1

n

Si

MVEiMVBi

where

market value of the portfolio at the end of period i before any cash flows in period i but including accrued income the period

market value of the portfolio at the beginning of period i , including any cash flows at the end of the previous subperiod and including accrued income

MVEi

MVBi

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Modified BAI Method

MVE Fi 1R Wi

i1

n

Fi where the sum of the cash flows during the period (with opposite signs for inflows and outflows)market value at the end of the period, including accrued incomemarket value at the start of the period

MVE

Wi

CD Di

CD

F0

total number of days in the periodnumber of days since the beginning of the period in which cash flow Fi occurred

CDiD

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Using the trade date : AIMR recommends that all calculations be done on the basis of the trade date rather than the settlement date.

Prior to fees : AIMR also recommends that investment results be presented before the deduction of management fees unless it would violate SEC advertising rules.

Before taxes : Performance should also generally be presented on a before-tax basis. If not, the tax rate used in the calculations must be disclosed.

Recommended Calculations

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Most investment management firms manage many accounts. A composite measure can be misleading if calculated inappropriately.

Include all portfolios : All portfolios under measurement must be included in at least one composite.

It is permissible to include a nonfee portfolio in a composite provided that such inclusion is disclosed.

Performance Presentation Standards :Composite Results

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Survivor bias : AIMR standards require that portfolios no longer under management must be included in a composite for the period in which they were in operation.

Treatment of convertibles : Convertible securities should be treated as equity instruments unless there is a clearly stated agreement to treat them differently.

At least a 10-year presentation of annual returns is required. A 20-year disclosure is preferred if the data is available.

Composite Results

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Leverage and derivatives : Their use and extent must be disclosed. The required information includes(1) a description of the use of the derivatives, (2) the amounts used, (3) the frequency of their use, and (4) a discussion of their characteristics.

Performance Presentation Standards

International portfolios : AIMR standards require disclosure onwhether returns are net or grossof withholding taxes on dividends,interest, or capital gains.

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Review

Relating Risk and Return Risk, Return, and Utility Arithmetic vs. Geometric Averages Traditional Performance Measures The Capital Market Line The Security Market Line

Performance Measurement AIMR-Required Calculations Recommended Calculations

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Review

Performance Presentation Standards Composite Results International Portfolios Leverage and Derivatives