Question

1. Find the annual equivalent of an infinite cash flow which starts as $1 000 for the first 5 years , then increases as a geometric progression growing at a rate of 4% .use an 8% annual discount rate.

Answer 1

Ans. Cash flow in year 1 - 5, F = $1000

Growth rate ofcash flows, g = 4% or 0.04

Interest rate, r = 8% or 0.08

Present worth of the cash flows,

PW = F/(1+r) + F/(1+r)^2 + F/(1+r)^3 + F/(1+r)^4 + F/(1+r)^5 + F(1+g)/(1+r)^6 + F(1+g)^2/(1+r)^7+........

=> PW = 1000/(1+0.08) + 1000/(1+0.08)^2 + 1000/(1+0.08)^3 + 1000/(1+0.08)^4 + 1000/(1+0.08)^5 + 1000(1+0.04)/(1+0.08)^6 + 1000(1+0.04)^2 / (1+0.08)^7+.......

=> PW = 3992.71 + [1000*(1+0.04)/(1+0.08) + 1000/(1+0.04)^2 / (1+0.08)^2 +.........] * 1/(1+0.08)^5

[Using formula for perpetuity with first payment A(1+g)/(1+r),  growth rate g and discount rate r,

PW = A(1+g)/(r-g) ]
=> PW = 3992.71 + [1000*(1+0.04)/(0.08-0.04)] * 1/(1+0.08)^5

=> PW = $21687.8731

For annual equivalent cash flow, A, we will use the formula for present worth of a perpetuity with equivalent cash flows of A and discount rate, r,

PW = A/r

=> 21687.8731 = A/0.08

=> A = $1735.03

Thus, annual equivalent cash flow is $1735.03

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