FM for Actuaries

118 CHAPTER 4
Indeed, while the arithmetic mean of the bond fund exceeds the geometric mean by
only 8 basis points (i.e., 0.08%), the arithmetic mean of the stock fund exceeds the
geometric mean by 67 basis points.
As the arithmetic mean is always larger than the geometric mean, there is reason
for fund managers to report their performance using the arithmetic mean. However,
if the purpose is to measure the return of the fund over the holding period of
m years, the geometric mean rate of return is the appropriate measure. It provides
the annualized average rate of return over m years assuming a buy-and-hold portfolio.
On the other hand, the arithmetic mean rate of return describes the average
performance of the fund for a single year taken randomly from the sample period.
If there are more data about the history of the fund, alternative measures of the
performance of the fund can be used. The methodology of the time-weighted rate
of return in Section 4.2 can be extended to beyond 1 period (year). We first measure
the returns over subperiods between cash injections and withdrawals. Suppose
there are n subperiods, with returns denoted by R 1 , ··· ,R n , over a period of m
years. Then we can measure the m-year return by compounding the returns over
each subperiod to form the TWRR using the formula
⎡
⎤
n∏
R T = ⎣ (1 + R j ) ⎦
j=1
1
m
− 1. (4.11)
We shall use the notation R T for the TWRR of both 1- and multiple-year returns.
Equation (4.11) is a generalization of (4.5) for m ≥ 1, with a slight change of
notation for the number of subperiods in the m-year period. The computation of
the TWRR over m years is also similar to the calculation of the geometric mean
rate of return R G in equation (4.10). However, while the returns R j in equation
(4.10) are over equal intervals of 1 year, the subinterval for each return observation
in equation (4.11) is determined by fund injections and withdrawals.
We can also compute the return over a m-year period using the IRR. To do this,
we extend the notations for cash flows in Section 4.2 to the m-year period. Suppose
cash flows of amount C j occur at time t j for j =1, ··· ,n,where0 <t 1 < ··· <
t n <m. Let the fund value at time 0 and time m be B 0 and B 1 , respectively. We
treat −B 1 as the last transaction, i.e., fund withdrawal of amount B 1 . The rate of
return of the fund is calculated as the IRR which equates the discounted values of
B 0 , C 1 , ··· ,C n , and −B 1 to zero. Unlike the TWRR defined in (4.11), this rate is
sensitive to the amount of cash flows C j . Hence, it is referred to as the DWRR over
the m-year period. Like the DWRR for the 1-year return given in (4.7), we shall