FM for Actuaries

Rates of Return 109
where y 4 is the IRR on 4-month interval. The nominal rate of return on 4-monthly
compounding is 3y 4 . The effective annual rate of return is y =(1+y 1 ) 12 − 1=
(1 + y 4 ) 3 − 1.
The above equations of value have to be solved numerically for y 1 or y 4 . Using
Excel to solve for y 1 , we enter 100 into Cell A1, −20 into Cells A5 and A9, and
−80 into Cell A25. The rest of the cells in column A, from A2 up to A24 are
filled with zeros. Using IRR with A1:A25 as the cash stream, we obtain 1.0406%
as the IRR per month, namely, y 1 . The effective annualized rate of return is then
(1.010406) 12 −1=13.23%. The nominal annualized rate is 12×0.0104 = 12.49%
for monthly compounding. On the other hand, solving for y 4 with Excel, we obtain
the answer 4.2277%. Hence, the annualized effective rate is (1.042277) 3 − 1=
13.23%, which is equal to the effective rate computed using y 1 .
When cash flows occur irregularly, we can define y as the annualized rate and
express all time of occurrence of cash flows in years. Suppose there are n +1cash
flows occurring at time 0, t 1 , ···, t n , with cash amounts C 0 , C 1 , ···, C n . Equation
(4.2) is rewritten as
n∑ C j
C 0 = −
(1 + y) t . (4.3)
j
j=1
Note that C j occurs at time t j and is discounted by (1 + y) t j
. Equation (4.3)
requires numerical methods for the solution of y, which can be computed using
Excel Solver. Alternatively, we may also use the Excel function XIRR to solve for
y. Unlike IRR, XIRR allows the cash flows to occur at irregular time intervals.
The specification of XIRR is as follows:
Excel function: XIRR (values, dates, guess)
values = C j , an array of values containing the cash flows in (4.3)
dates = t j , an array of values containing the timing of the cash flows in (4.3)
guess = guess starting value, set to 0.1 if omitted
Output = IRR, value of annualized y satisfying (4.3)
To enter the values of the timing of the cash flows for XIRR, two methods may
be used. First, we may use the DATE function in Excel, with input values of the
calendar year, month and day of the time of cash flows. Second, we may enter
the numerical values of the number of days from the first payment. Example 4.4