Question

The future worth in year 10 of an arithmetic gradient cash flow series for years 1 through 10 is $725,000. If the gradient increase each year, G, is $2500, determine the cash flow in year 1 at an interest rate of 9% per year.

Answer 1

Let us assume the amount is A.

First convert the future value into present value.

PW = 725,000(P/F, 9%, 10)

> 725,000 PW = 1 10 = $306, 247.8349

The cash flow in year 1 is A, in year 2 is A+1750, in year 3 it is A + 3500 likewise till tenth year. The present value of this gradient series can be written as

PW - A A+ 1750 1 002 + ... A + 15, 750 + - 1.0910 1.097

PW = A(P/A, 9%, 10) + 1750(P/G,9%, 10)

> PW = 6.41765 A+ 1750 x 24.3727

=> PW = 6.41765 A+42.652.3539

Now the two present value must be equal

S PW = 6.41765 A+ 42,652.3539 = 306, 247.8349

6.41765 A = 306, 247.8349 – 42,652.3539

6.41765 A = 263,595.48

=> A = 263,595.48 6.41765

A = $ 41,073.47

Cash flow in year 1 = $ 41,073.

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