Question

An annuity can be defined as the difference between two perpetuities. calculate the present vale of an annuity that pays $2,500 per year for 10 years using a dicount rate of 6%. show how finding the difference between the two perpetuities produce the same answer as using the annuity formula.

Answer 1

Answer (a):

PV of ordinary annuity = PMT * (1 - 1 / (1+ Interest rate) ^{Number of Years} ) / Interest rate

Given:

PMT = $2500

Discount rate = 6%

Number of years = 10

Present value = 2500 * (1 - 1 / (1 + 6%) ¹⁰) / 6%

= $18,400.22

Present value = $18,400.22

Answer (b):

Above annuity can be also taken as difference between following 2 perpetuities:

1. Perpetuity giving annual returns of $2,500 at the year end from now with discount rate of 6%

2. Perpetuity giving annual returns of $2,500 starting from year end of year 11 onward with discount rate of 6%

PV of perpetuity 1 = 2500 / 6% = $41666.67

To get PV of perpetuity 2 first we need to calculate PV of this perpetuity at the end of year 10 which is = 2500 / 6% = $41666.67

PV of perpetuity 2 = 41666.67 / (1 + 6%) ¹⁰ = $23266.45

Difference between PV of perpetuity 1 and PV of perpetuity 2 = 41666.67 - 23266.45 = $18,400.22

This value is same as PV of annuity we calculated in part (a) above.

Hence:

Difference between the two perpetuities produce the same answer as using the annuity formula