SEATTLE UNIVERSITY
Prof. Ken Shah
A REVIEW OF COMPOUNDING
Suppose you purchase a security at price P0 and hold it for an interval t and sell it at price Pt. In
that period, you receive total cash disbursement (e.g. from dividends or interest payments) of Dt.
The Holding Period Return (HPR) is a measure of how well you did as a result of buying and
holding the security over the entire time period t:
Pt  P0  Dt
(1)
P0
Note that the holding period 0 to t can be a month, a quarter, a year or several years. However, it
is customary to report returns on an annual basis. Hence it is often necessary to convert HPR
over a particular interval to an annual rate. Let us consider the two general situations: i)
converting smaller interval returns into a larger interval return ii) converting larger interval return
into average return over smaller intervals.
HPR =
1. Converting smaller interval returns into a return over a larger interval
Examples of this are:
 From quarterly returns get annual return
 From monthly returns get annual return
 From daily returns get annual return
 From annual returns get multi-year return
For example, let R1, R2, R3, R4 represent periodic HPRs. Then the total return over intervals 1 to
4 is:


Return over intervals 1 to 4 = R1to 4 = 1  R1 1  R2 1  R3 1  R4   1



(2)
For instance, if R1-R4 represent four quarterly returns in 1993, then the above formula
gives annual return over the entire year 1993
If we R1-R4 are four annual returns in 1991, 1992, 1993 and 1994 then the above
formula then we would give the total return over four years 1991 to 1994
As another example, if you had monthly returns R1-R12 in 1994, then you would
multiply twelve (1+Ri) and subtract 1 from the product to get annual return in 1994
2. Converting a larger interval HPR into average return over smaller interval
Example of this are:
 From a 5 year HPR get average annual return over the 5 years
In general, the formula to calculate the average return when there are n periods in a holding
period interval is
2

Average return = R  1  HPR
1/ n
 1
(3)
Suppose in the HPR formula (1) above t = 5, i.e., we have measured the HPR over a 5-year
interval. In this case, Dt will represent the total cash disbursement in the entire 5 year interval.
For example, the stock is purchased on Jan 1 1990 at P0 and sold on Dec. 31 1994 at Pt and Dt is
the total dollar amount of dividends paid over this 5 year interval. We compute the HPR over the
5-year interval using the HPR equation (1). Now we want the average annual return over the 5
year interval. Then

Average Annual Return in Years 1990-1994 = R  1  HPR90  94 
1/ 5
 1
Sometimes, one may wish to get average quarterly returns from HPR measured over a longer
interval. Say we compute HPR in (1) over two years, 1991 and 1992. Now there are eight
quarters in two years. Then the average quarterly return over the 1991 and 1992 is

Average Quarterly Return in Years 1991 and 1992 = R  1  HPR91 92 
1/ 8
 1
It is important to realize that the average return calculated by equation (3) takes into effect the
effect of compounding. It is quite different from the simple average of returns
(R1,+R2+R3+R4+R5)/5. Let's consider the following example:
PERIOD
1990
1991
1992
1993
1994
HPR
-0.31
0.96
0.19
0.41
-0.55
SIMPLE AVG
0.14
PRODUCT OF (1+R)
CMPD AVG ANNUALLY
CMPD AVG QTRLY
CMPD AVG MNTHLY
1+Ri
0.69
1.96
1.19
1.41
0.45
VALUE OF
$100 INVESTMENT
$69.00
$135.24
$160.94
$226.92
$102.11
= 14%
1.0211
= 2.11% HPR OVER
5 YEARS
0.004192 = 0.4% ANNUALLY
0.001046 = 0.1% QUARTERLY
0.000349 = 0.03% MONTHLY
In the above example, there is wide year-to-year fluctuation in return ranging from -55% in 1994
to +96% in 1991. The simple average of the five annual returns is calculated as
[-.31+.96+.19+.41+(-.55)]/5 = .14 or 14%. This would seem like a decent return by historical
standards. However, you will notice that the value of $100 investment earning the five annual
returns at the end of 1994 is only $102.11. Hence, the total return over 1990-1994 is actually
only 2.11%:
[(1+(-.31))(1+.96)(1+.19)(1+.41)(1+(-.55))] - 1 = .0211 or 2.11%
3
The average returns as calculated by equation (3) is:

Average Annual Return in Years 1990-1994 = R  1.0211
1/ 5
  1 = .004 or .4%.
Clearly this is quite different from 14% simple average. The simple average is actually
misleading in measuring how well or poorly one did as a result of an investment in the past. The
correct way to measure ex-post average returns is to take into account the effect of compounding,
which is precisely what equation (3) does. It basically says that if one's investment earned
compounded annual return of 0.4% every year for 5 years, it is the same as earning 2.11% over
the entire 5 year interval. This measure is appropriate for measuring past performance because it
measures the number of extra dollars you actually had at the end of the period from an
investment of $1 at the beginning of the period. The answer in this case is, when all is said and
done, investing $1 in 1990 left you with $1.0211 at the end of 1994. Clearly, this was not a great
investment.
Similarly,

  1 = .001 or .1%
Average Monthly Return in Years 1990-1994 = R  1.0211   1 = .0003 or .03%
Average Quarterly Return in Years 1990-1994 = R  1.0211
1/ 20
1/ 60
The simple average of 14% can have a useful purpose. If we can assume that periodic returns in
the past (R1, …, RN) are drawn randomly from a distribution of returns, then it is an unbiased
forecast of future return. This is an ex-ante forecast of what one could expect the return will be
in the next period. But this is quite different from asking the question: how well or poorly we did
in the past?
SUMMARY:
1. The total HPR over t intervals is
 n

HPR1ton  (1  R1 )(1  R2 )(1  R3 )...(1  Rn )  1   (1  Ri )  1
 i 1

2. Average return over a smaller interval when there are t such intervals in the HPR measurement
period is

R  1  HPR
1/ n
 1
A Final Note:
Note that we have so far assumed that Pt and P0 are known. That is, we are measuring how well
an investment did as a result of buying and selling decisions at some point in the past. We can
also compute a forward-looking measure of return. That is, we know the price today, P0, and we
have an estimate or an expectation of the price at which we are likely to sell at time t. We also
have an estimate of the total cash disbursement over the interval 0 to t. Then we can calculate
4
the Expected HPR as
Expected HPR =
E Pt   P0  E Dt 
P0
where E(•) denote expectations.