FM for Actuaries

106 CHAPTER 4
Learning Objectives
• Internal rate of return (yield rate)
• 1-period rate of return of a fund: time-weighted rate of return and dollarweighted
(money-weighted) rate of return
• Rate of return over longer periods: geometric mean rate of return and arithmetic
mean rate of return
• Portfolio return
• Return of a short-selling strategy
• Crediting interest: investment-year method and portfolio method
• Capital budgeting and project appraisal
4.1 Internal Rate of Return
Consider a project with initial investment C 0 to generate a stream of future cash
flows. For simplicity, we assume the cash flows occur at regular intervals, say,
annually. The project lasts for n years and the future cash flows are denoted by
C 1 , ··· ,C n . We adopt the convention that cash inflows to the project (investments)
are positive and cash outflows from the project (withdrawals) are negative. We
define the internal rate of return (IRR) (also called the yield rate) as the rate of
interest such that the sum of the present values of the cash flows is equated to zero.
Denoting the internal rate of return by y,wehave
n∑
j=0
C j
=0, (4.1)
(1 + y)
j
where j is the time at which the cash flow C j occurs. This equation can also be
written as
n∑ C j
C 0 = −
(1 + y) j . (4.2)
j=1
The right-hand side of the above equation is the present value of future outflows
(withdrawals) minus the present value of future inflows (investments). Thus, the
net present value of all future withdrawals (injections are negative withdrawals)
evaluated at the IRR is equal to the initial investment C 0 .