What Practitioners Need To Know
. . .
About Retum and Risk
14
Mark Kritzman
Windham Capital
Management
At first glance, return and risk
may seem to be straightforward
concepts. Yet closer inspection
reveals nuances that can have im
portant consequences for deter
mining the appropriate method
for evaluating financial results.
This column
reviews
various
measures of return and risk with
an emphasis on their suitability
for alternative uses.
Retum
Perhaps the most straightforward
rate of return is the
holding
period return.
It equals the in
come generated by an investment
plus the investment's change in
price during the period the in
vestment is held, all divided by
the beginning price. For example,
if we purchased a share of com
mon stock for $50.00, received a
$2.00 dividend, and sold the stock
for
S55.00, we would
have
achieved a holdingperiod return
equal to 14.00%. In general, we
can use Equation (1) to compute
holdingperiod returns.
Eq. 1
HPR = {I + E  B)/B
where
HPR = holdingperiod retum,
I = income,
E = ending price and
B = beginning price.
Holdingperiod returns are also
referred to as
periodic
retums.
DollarWeighted versus
TimeWeighted Rates of
Retum
Now let us consider rates of re
turn over multiple holding peri
ods.
Suppose that a mutual fund
generated the following annual
holdingperiod returns from 1988
through 1992:
1988
1989
1990
1991
1992
5.00%
15.20%
3.10%
30.75%
17.65%
Suppose further that we had in
vested $75,000 in this fund by
making contributions at the be
ginning of each year according to
the following schedule:
1988
1989
1990
1991
1992
S5,000
$10,000
$15,000
$20,000
$25,000
By the end of 1992, our invest
ment would have grown in value
to $103,804.56. By discounting
the ending value of our invest
ment and the interim cash flows
back to our initial contribution,
we can determine the invest
ment's
dollarweighted rate of
re
turn,
which is also referred to as
the
internal rate of return:
5000=

10000
15000
(1
H
r)
(1H
20000
25000
103805
7 "r
P,
(1 Hr)^
(1 +
DWR = 14.25%
We enter the interim contribu
tions as negative values, because
they are analogous to negative
dividend payments. Although we
cannot solve directly for the dol
larweighted rate of return, most
financial calculators and spread
sheet software have iterative algo
rithms that quickly converge to a
solution. In our example, the so
lution equals 14.25%.
The dollarweigh ted rate of re
tum measures the annual rate at
which our cumulative contribu
tions grow over the measurement
period. However, it is not a reli
able measure of the performance
of the mutual fund in which we
invested, because it depends on
the timing of the cash flows. Sup
pose,
for example, we reversed
the order of the contributions.
Given this sequence of contribu
tions,
our investment would have
grown
to a higher
value—
$103,893.76. The dollarweighted
rate of return, however, would
have been only 9.12%:
25000=

10000
20000
15000
(1 + r)
(1H r)^
5000
103894
DWR = 9.12%
In order to measure the underly
ing performance of the mutual
fund, we can calculate its
time
weighted rate of
retum.
This mea
sure does not depend on the
timing of cash flows.
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 Three '10
 YIPPIE
 Standard Deviation, riskless retum

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