Jake inherits a perpetuity that will pay him $10, 000 at the end of the first year increasing by $10, 000 per year until a payment of $150, 000 is made at the end of the fifteenth year. Payments remain level after the fifteenth year at $150, 000 per year. Determine the present value of this perpetuity, assuming a 7.5% annual interest rate

The present value of this perpetuity is $120,389,126.79.

Explanation:

This can be calculated using the following steps:

Step 1: Calculation of present value of $10,000 increasing by $10,000 per year

This can be calculated using the formula for calculating the present value (PV) of a growing annuity is used as follows:

PVga = (P / (r - g)) * (1 - {(1 + g) / (1 + r))^n) (1)

Where;

PVga = Present value of the growing annuity = ?

P = First year payment= $10,000

r = annual interest rate = 7.5%, or 0.075

g = growth rate of annual payment = $10,000 / $10,000 = 1.00

n = number years applicable to the increase of $10,000 per year = 15

Substituting the values into equation (1), we have:

PVga = ($10,000 / (0.075 – 1.00)) * (1 – ((1 + 1.00) / (1 + 0.075))^15)

PVga = $119,713,194.75

Step 2: Calculation of the present value of annual payments of $150,000 per year after the fifteenth year

To determine this, we need to first calculate the present value of the perpetuity in 15 years’ time using the formula for calculating the present value of a perpetuity as follows:

PVp15 = C / r ……………….. (2)

Where;

PVp15 = Present value of perpetuity in 15 years’ time = ?

C = annual payment = $150,000

r = annual interest rate = 7.5%, or 0.075

Substituting the values into equation (2), we have:

PVp15 = $150,000 / 0.075

PVp15 = $2,000,000

We can now calculate the today’s present value of PVp15 using the simple present value as follows:

TPV = PVp15 / (1 + r)^n ……………………….. (3)

Where;

TPV = Today's present value of PVp15 = ?

PVp15 = $2,000,000

r = annual interest rate = 7.5%, or 0.075

n = number of years = 15

Substituting the values into equation (3), we have:

TPV = $2,000,000 / (1 + 0.075)^15

TPV = $675,932.04

Step 3: Calculation of the present value of this perpetuity

The present value of this perpetuity can be calculated adding the present value of the growing annuity (PVga) obtained in Step 1 above and the present value of the today's present value of perpetuity in 15 years’ time (TPV) as follows:

Present value of this perpetuity = PVga + TPV

Present value of this perpetuity = $119,713,194.75 + $675,932.04

Present value of this perpetuity = $120,389,126.79

Therefore, the present value of this perpetuity is $120,389,126.79.

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