FM for Actuaries

Rates of Return 111
Exhibit 4.3:
Solution of Example 4.4 using XIRR
In the above examples, C j , j =1, ··· ,n, are all negative so that there are
no subsequent outlays after the initial investment. Such projects are called simple
projects. For simple projects, (4.1) has a unique solution with y>−1, sothat
IRR is well defined. However, some projects may require further cash outlays after
time 0. In this circumstance, (4.1) generally has multiple roots, and IRR may not
be well defined. The example below illustrates this situation.
Example 4.5: A project requires an initial outlay of $8 million, generates returns
of $50 million 1 year later, and requires $50 million to terminate at the end of year
2. Solve for y in (4.1).
Solution: We are required to solve
8=50v − 50v 2 ,
which has v =0.8 and 0.2 as solutions. This implies y has multiple solutions of
25% and 400%.
The use of IRR as a measure of the rate of return of a project is based on some
implicit assumptions. First, as the same rate of interest y is used to discount all cash
flows irrespective of the time of occurrence, the term structure is assumed to be flat
and does not vary with the timing of the cash flows. Second, cash generated within
the duration of the project is assumed to earn the same rate of return y. Third,
both cash injections and withdrawals are discounted by the same rate of interest,
i.e., borrowing rate is assumed to be equal to lending rate. However, despite the
shortcomings of the IRR, it takes proper account of the time value of money in
evaluating cash flows and has a firm theoretical underpinning.