Question

1.6 Derive an analytical relationship between simple payback period and internal rate of return (IRR) over a 15-year assessment period for a project with a single fixed capital payment (K) at the beginning of year 1 and equal constant-dollar annual net benefits over this period (B). Hint: the simple payback period will be K/B. Use the equation for IRR. Then solve this equation iteratively using Excel for payback periods between 1 and 15, and plot the corresponding graph of IRR vs Payback Period

Answer 1

Initial Fixed Capital Payment = $ K, Annual Net Benefit = $ B, Project Horizon = 15 years

Let Internal Rate of Return be denoted by IRR % and simple payback period be denoted by PP

The IRR is the discount rate at which the NPV of the project would be zero or the initial fixed capital payment be equal to Present Value (PV) of all annual net benefits summed together.

Therefore, K = B x (1/IRR) x [1-{1/(1+IRR)^(15)}]

Payback Period = PP = K/B

Hence, (K/B) = (1/IRR) x [1-{1/(1+IRR)^(15)}]

PP = (1/IRR) x [1-{1/(1+IRR)^(15)}]