Sections 7.6, 7.7
Consider the following one-year investment. An amount X is invested in
a fund at the beginning of the year. In six months, the fund is worth X/2
at which time the investor can decide to add to the fund, to withdraw
from the fund, or to do nothing. At the end of the year, the fund balance
is double the balance at six months.
Investor A initially invests X = $1000 and at six months withdraws half
the fund balance. Investor B initially invests X = $1000 and at six
months deposits an amount equal to the fund balance.
Note that two entities can be identified in a situation like this:
(1) the investor, deciding how much to deposit/withdraw at each stage,
(2) the manager, maintaining the fund.
We can choose to judge the fund’s performance based on the decisions of
the investor or based on the decisions of the manager.
The value of the fund to Investor A at the end of one year is $500.
What is the yield rate for investor A? 1000(1 + i) – 250(1 + i)1/2 = 500
From the quadratic formula, (1 + i)1/2 = 0.84307
The yield rate is i = –0.2892 or –28.92%
The value of the fund to Investor B at the end of one year is $2000.
What is the yield rate for investor B? 1000(1 + i) + 500(1 + i)1/2 = 2000
From the quadratic formula, (1 + i)1/2 = 1.18614
The yield rate is i = 0.4069 or 40.69%
The yield rate for Investor B is so much better than the yield rate for
investor A, because of the decisions made by each investor.
Now how can we evaluate the decisions made by the fund manager?
We find the yield rate for the first six months to be j1 = –0.5 or –50% .
We find the yield rate for the second six months to be j2 = 1.0 or 100% .
Letting i be the yield rate for the entire year, we combine the two half
year yield rates to obtain
1 + i = (1 + j1)(1 + j2) = (0.5)(2) = 1.
The yield rate for the year is then found to be i = 0.
This indicates that the manager did a poor job of maintaining the fund.
Yield rates computed by taking into account deposits and withdrawals
(which is what was previously done evaluate the decisions by Investors A
and B) are called dollar-weighted rates of interest.
Yield rates computed by considering only changes in interest rate over
time (which is what was previously done to evaluate the fund manager’s
performance) are called time-weighted rates of interest.
In general, a time-weighted rate of interest is defined by combining rates
of intervals for subintervals by points at which deposits or withdrawals
would occur.
Suppose m – 1 deposits or withdrawals are made during a year at times
t1 , t2 , …, tm–1 (so that t0 = 0 and tm = 1) which define m subintervals.
For k = 1, 2, …, m, we let jk be the yield rate over the kth subinterval;
that is,

 = balance at time t immediately before any
k
1 + jk = ————
k–1 + k–1
k
k
deposit/withdrawal
k = deposit/withdrawal at time tk
The overall yield rate i for the entire year is given by
1 + i = (1 + j1)(1 + j2)…(1 + jm) .
On January 1, an investment account is worth $500,000. On April 1, the
account value has increased to $530,000, and $120,000 of new principal
is deposited. On August 1, the account value has decreased to $575,000,
and $250,000 is withdrawn. On January 1 of the following year, the
account value is $400,000. Compute the yield rate using (a) the dollarweighted method and (b) the time-weighted method.
(a) The Dollar-Weighted Method:
total amount of interest earned
amount in fund at beginning
net principal contributed at time t

Instead of solving for i in the equation I = iA +
Ct  [(1 + i)1–t – 1]
t
(which can be difficult in general as we have seen), use the
approximate value for i based on the simple interest approximation)
total net principal contributed
given by
I
I = B – A – C and i  ————————
.
A + Ct  (1 – t)
amount in fund at end

t
I = B – A – C = 400,000 – 500,000 – (120,000 – 250,000) = 30,000
I
i  ———————— =
A + Ct  (1 – t)

t
30000
———————————————————— =
500000 + (120,000)(1 – 1/4) – (250,000)(1 – 7/12)
0.06175 or 6.175%
(b) The Time-Weighted Method:
1 + i = (1 + j1)(1 + j2)(1 + j3)
1 + i = (530000 / 500000) (575000 / 650000) (400000 / 325000)
1 + i = 1.1541
i = 0.1541 or 15.41%
Notice that the time-weighted method is not consistent with an
assumption of compound interest. Consider what would happen to the
value of i if say April 1 were changed to March 1 and August 1 were
changed to October 1.
Consider a fund consisting of the accounts for different entities
(individuals or companies). Each account is a pro rata share of the entire
fund (instead of one separate group among segregated assets). There are
two different approaches to allocating interest to the various accounts:
The portfolio method credits to each account an average rate of interest
based on the earnings of the entire fund. A drawback of this method is
that when interest rates tend to be high, potential new deposits could earn
a higher rate elsewhere because the portfolio rate is affected by
investments in the past yielding lower rates; this discourages new
deposits and even encourages withdrawals. However, the reverse is true
when potential new deposits earn a lower rate elsewhere because the
portfolio rate is affected by investments in the past yielding higher rates.
The investment year method is designed to recognize both the date of
investment and the current date; the rate for new deposits is often called
the new money rate. There is not just one simple investment year
method; decisions about how to credit past investments and when to
truncate the process must be made.
With the declining index system, the interest credited reflects the interest
rate on the remaining assets which are dwindling.
With the fixed index system, the interest credited reflects the interest rate
on the original investment modified by subsequent investment rates.
In practice, the investment year method is implemented by specifying a
two-dimensional table, such as Table 7.2 in the textbook. If y is the
calendar year of deposit, and m is the number of years for which the
investment year method is applicable, then the rate of interest credited
for the tth year of investment is denoted as ity for t = 1, 2, …, m.
For t > m, the portfolio method is applicable, and the portfolio rate of
interest credited for calendar year y is denoted as i y. Of course this can
all be modified to allow for quarterly or monthly credited rates.
In Table 7.2 on page 276 of the textbook, observe that
the column labeled i1y is a list of the new money rates for each of the
years from z through z +10,
the sequence of interest rates beginning from a given year of investment
runs horizontally through the row for that year and then down the last
column of rates,
the list of interest rates credited in a given year of investment runs
through the upward diagonal beginning in the row for that year.
An investment of $2000 is made at the beginning of calendar year z + 3
in an investment fund crediting interest according to the rates in Table
7.2 on page 276 of the textbook. Find the amount of interest credited in
calendar years z + 5 through z + 8 inclusive.
At the beginning of calendar year z + 5, the accumulated value of the
investment is 2000(1.09)2 = $2376.20.
At the end of calendar year z + 8, the accumulated value of the
investment is 2000(1.09)2(1.091)2(1.092)(1.0885)= $3361.89.
The amount of interest credited in calendar years z + 5 through z + 8
inclusive is $3361.89 – $2376.20 = $985.69.