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CHAPTER TWENTY
PERFORMANCE MEASUREMENT
AND PRESENTATION
© 2001 South-Western College Publishing
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
2
Outline

Performance Presentation Standards
Composite Results
 International Portfolios
 Leverage and Derivatives

3
Risk, Return, and Utility

Proper performance evaluation should
recognize both the return and the
riskiness of the investment.

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).
4
Arithmetic vs. Geometric Averages

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?

The proper measure of average return over
time with investments is the geometric mean
1
return :
n
n
xGM



  Ri 
 i 1 
the return relative
where R i = in period i
To get around the problem of negative returns,
returns are transformed into return
relatives by adding 1.0 to them.
5
Traditional Performance Measures

The Sharpe measure relates return to total
risk. It can be used effectively with a portfolio
where unsystematic risk has been diversified
away.
Ri  Rf
Sharpe measure 
i
where R i = arithmetic mean return of security i
Rf = risk free rate
 i = standard deviation of returns on security i
6
Traditional Performance Measures

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.
Treynor measure 
Ri  R f
i
7
Traditional Performance Measures

The Jensen measure stems directly from the
implications of the capital asset pricing
model as estimated by the market model.
Ri  Rf   i   i Rmarket  Rf
or


 i  Ri  Rf    i R market  Rf 
According to finance theory, i should be
zero. So, a positive alpha that is statistically
different from zero indicates an aboveaverage performance, and vice versa.
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mean return
Traditional Performance Measures

standard deviation
The Sharpe performance
measure can be interpreted as
the slope of a line relating the
security’s return with its risk.

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.
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mean return
Traditional Performance Measures

beta

It is also possible to plot the
returns of securities against
their levels of systematic risk,
or beta.
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.
10
Performance Measurement

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.
11
AIMR-Required Calculations
 Total return, including realized and unrealized
gains and losses plus income, is required.
Total return 

Valueending  Valuebeginning  Income
Valuebeginning
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.
12
AIMR-Required Calculations

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.
13
Daily Valuation Method
n
Rdaily   Si  1
where Si 
MVEi
i 1
MVBi
MVEi  market value of the portfolio at the end of period i
before any cash flows in period i but including
accrued income the period
MVBi  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
14
Modified BAI Method
n
MVE   Fi 1 R
Wi
i 1
where Fi  the sum of the cash flows during the period (with
opposite signs for inflows and outflows)
MVE market value at the end of the period, including
accrued income
F0  market value at the start of the period
Wi 
CD Di
CD
CD  total number of days in the period
Di  number of days since the beginning of the period in
which cash flow Fi occurred
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Recommended Calculations

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.
16
Performance Presentation Standards :
Composite Results

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.
17
Composite Results

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.
18
Performance Presentation Standards


International portfolios : AIMR
standards require disclosure on
whether returns are net or gross
of withholding taxes on dividends,
interest, or capital gains.
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.
19
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
20
Review

Performance Presentation Standards
Composite Results
 International Portfolios
 Leverage and Derivatives

21