You are thinking of building a new machine that will save your company $1,000 in the first year. The machine will then begin to wear out so that the savings decline at a rate of 2% per year forever. What is the present value of the savings if the interest rate is 5% per year? (Hint: this is a growing perpetuity.

The answer is $14,285.71

Explanation:

The question indicates that the savings decline at a rate of 2% per year indefinitely. The growing perpetuity(GP) formula is useful in this instance because it is used to calculate the present value of periodic cash flows which grow (or decline) at a constant rate infinitely. The following depicts the formula of a growing perpetuity:

Present value of a GP = Cash flows in period 1/ (discount rate - growth rate)

Note: This formula can only be used when the discount rate is greater than the growth rate. In this case, it is. To compute the present value of this perpetuity, the growth rate will be negative, to show that it is declining, and entered as such into the formula as shown below:

Cash flows in period 1: $1000

Discount rate: 5%

Growth rate: (-2)%

PV = $1000/(0.05 - (-0.02) )

     = $14, 285.71429

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