Chapter 5: The Rate of Return
For annuity and bonds, we considered the interest rate at which the amount invested is equal
to the amounts received. For example, for an annuity the investment at t = 0 was the present
value of the regular periodic payments based on a given interest rate. Similarly, for bonds,
the initial investment or purchase price was the present value of the coupon and redemption
amounts based on the corresponding yield rate. The interest rate in these examples is
referred to as the internal rate of return (IRR).
Thus, finding the IRR means solving for the interest rate in the corresponding equation of
value. That is, with the total values of the payments received and the payments made out are
set equal when adjusted to the same point in time using v = ( 1+i )-1 (possibly for t = 0),
solving for i gives the IRR.
This equation of value can be constructed for any sequence of cash flows. Let A0, A1, …, An
represent the payments received at times t0, t1, …, tn, corresponding to the first n interest
periods. Similarly, let B0, B1, …, Bn be the payments made out at the same points in time,
then at the end of the kth period (t = tk) there is a net amount received of Ck = Ak – Bk.
Note Ck may be negative. Also, Ak and/or Bk may be zero for any given period.
Example
Suppose that for investments of $500 initially and $500 at the end of the first year, I will
receive payments of $300 at the end of each six months during the first year and $400 at the
end of each six months during the second year. What is my rate of return for this
investment?
0
1
2
3
4
Ak
0 300 300 400 400
Bk 500
0 500
0
0
Ck -500 300 -200 400 400
net amounts: -500
300
-200
400
400
Thus, the equation of value
500  500v 2  300v  300v 2  400v3  400v 4
may be written in terms of the net amounts as
500  300v  200v 2  400v3  400v 4  0 .
Using my graphing calculator, I solved for the zero of this quartic polynomial and found
1
 0.81213556 , and so j  0.2313215 . But the same answer
v  0.81213556 . Thus,
1 j
may be found by entering these net cash flows into your business calculator and using the
IRR button. The result is IRR = 23.13215. This is the rate of return percentage for the 6month period.
As demonstrated in the above example, the equation of value based on t = 0 may be written
in the form
n
C v
k 0
k
k
0.
But, as we’ve seen before, an equation of value may be constructed by adjusting all the
amounts to any time ti to form an equivalent equation
n
C v
k 0
k
k i
 0.
In many situations, the IRR may be used to compare between two investment opportunities.
But we note that solving such polynomial equations may result in multiple solutions. That
is, there may be more than one interest rate that satisfies the equation. Example 5.2 on page
270 illustrates such potential pitfalls.
Note sections 5.1.3 and 5.1.4 are excluded from the recommended syllabus.
Dollar-Weighted Rate of Return
The dollar-weighted and time-weighted rates of return are introduced in section 5.2. These
are methods for measuring an investment performance on an annual basis.
The dollar-weighted rate of return is similar to the IRR but is based on simple interest, not
compound interest. That is, sum the initial amount in the fund at the beginning of the year
along with each net amount received during the year, all accumulated to the end of the year
using simple interest. Setting this sum equal to the fund balance at the end of the year forms
the equation of value. Solving this equation for the interest rate yields the dollar-weighted
rate of return. Note this value does not differ much from the IRR for this shorter time
period.
For example, suppose an initial investment of $1000 is made at the beginning of the year.
At the end of the first two quarters $300 is deposited and at the end of the second two
quarters $500 is deposited. Payments of $400 are received from the fund every six months
and at the end of the year the fund has a value of $2100. Find the dollar-weighted rate of
return and compare this with the IRR for this same fund.
month:
Ak
Bk
net, Ck
0
6
9
12
0
0 400
0 2500
1000 300 300 500 500
-1000 -300 100 -500 2000
The resulting equation of value is given by
3
1000(1  i)  300 1  129 i   100 1  126 i   500 1  123 i   2000  0
Solving this equation algebraically, we find
i
1000  300  100  500  2000
300

 0.23077
1000  300( 129 )  100( 126 )  500( 123 ) 1300
The dollar-weighted rate of return for this entire year is i = 0.23077. Computing the IRR
gives j = 0.05378 for the 3-month periods, or i = (1.05378)4-1 = 0.2331.
Note that in the calculation above i 
300
where 300 equals the total interest earned
1300
during the year and 1300 is the average amount on deposit during the year.
Time-Weighted Rate of Return
Like the dollar-weighted rate, the time-weighted rate of return is another method for
measuring an investment performance on an annual basis. This approach uses the rate of
return for each successive portion of a year to determine the overall annual yield.
For example if an account earns a 6-month rate of 4%, earns a 3-month rate of 3%, and
earns 2% during the remaining 3 months of the year, then (1.04)(1.03)(1.02) = 1.0926 and so
the annual time-weighted rate of return is 9.26%. It’s not relevant that the periods were not
of equal length, only that we knew the rate j for the length of period indicated. Typically,
we will break the year into fractions defined by the points at which transactions occur.
Example 5.4 (pg. 281)
initial balance transaction new balance
first of Jan.
1,000,000
end of Feb.
200,000
1,240,000
end of Aug.
200,000
1,600,000
end of Oct.
-500,000
1,080,000
end of Dec.
-200,000
900,000
For the first period (1/1 – 3/1), the total interest earned is given by
j = current balance – previous balance – payment received + payment paid out
= 1240000 – 1000000 – 200000 + 0 = 40000
40000
 0.04, or 4% .
Thus the percent increase over this time period is
1000000
Similarly, the percent increase for the following 3 periods are found to equal 0.1290,
- 0.0125, and 0.0185, respectively. Therefore, the time-weighted rate of return for this year
is computed to be (1 + 0.04)(1 + 0.1290)(1 – 0.0125)(1 + 0.0185) – 1 = 0.1809, or 18.09%.
For comparison, the dollar-weighted rate for this fund, found by solving the equation
1,000,000(1  i)  200,000(1  56 i)  200,000(1  62 i)  500,000(1  16 i)  1,100,000  0 ,
equals 0.1739. The time-weighted rate “eliminates the impact of money flows in and out of
the fund” and is often used to compare the relative performance of investment fund
managers.