Question

Spring 2019 Chapter 2: Annuities MAT 3541 Part D Basic problems 1. A perpetuity has payments of w, w, 2w, 2w, 3w, 3w,.. with payments made at the end of each year. The present value using an annual effective interest rate of 10% of this perpetuity is equal to the present value of the geometrically increasing perpetuity with initial payment w and each subsequent payment increasing by a factor of 1+r. Calculate r. [Ans. 0.0826]

Answer 1

Let's denote the sum of the present values of these perpetuities as S. Also let's call the annual effective interest rate as i.

Annuities are: w, w, 2w, 2w, 3w, 3w,...and so on

Hence, S = PV of all annuities

$S = \frac{w}{1+i} + \frac{w}{(1+i)^{2}} + \frac{2w}{(1+i)^{3}}+\frac{2w}{(1+i)^{4}} + \frac{3w}{(1+i)^{5}}+ \frac{3w}{(1+i)^{6}} + .....$

Let's multiply both sides by 1/(1+i)². We will get the following equation:

$\frac{S}{(1+i)^{2}} = \frac{w}{(1+i)^3} + \frac{w}{(1+i)^{4}} + \frac{2w}{(1+i)^{5}}+\frac{2w}{(1+i)^{6}} + \frac{3w}{(1+i)^{7}}+ \frac{3w}{(1+i)^{8}} + .....$

Subtract this equation from the first one, we should get:

$S - \frac{S}{(1+i)^{2}} = \frac{w}{(1+i)}+ \frac{w}{(1+i)^2}+\frac{w}{(1+i)^3} + \frac{w}{(1+i)^{4}} + \frac{w}{(1+i)^{5}}+\frac{w}{(1+i)^{6}} + .....$

The RHS is an infinite geometric progression with w / (1 + i) as first time and 1 / (1 + i) as common ratio

Hence, RHS = First term / (1 - Common ratio) = [w / (1 + i)] / [1 - 1/(1+i)] = w / i

Hence, S [1 - 1/(1+i)²] = w / i

Hence, S x [1 - 1 / (1 + 10%)²] = w / 10%

Hence, S x 0.173553 = 10 x w

Hence, S = 57.62 x w ---------------------- Equation (A)   

Further, S = PV of a geometrically increasing perpetuity with first term w and common ratio of (1+r)

Hence, S = PV of perpetuities w, w(1+r), w(1+r)², w(1+r)³ and so on..... Thus,

$S = \frac{w}{(1+i)}+ \frac{w\times (1+r)}{(1+i)^2}+\frac{w\times (1+r)^2}{(1+i)^3} + \frac{w\times (1+r)^3}{(1+i)^{4}}+ .....$

The RHS is an infinite geometric progression with w / (1 + i) as first time and (1 + r) / (1 + i) as common ratio

Hence, RHS = First term / (1 - Common ratio) = [w / (1 + i)] / [1 - (1+r)/(1+i)] = w / (i - r)

Hence, S = w / (10% - r) ----------------Equation (B)

From equation (A) and (B):

S = 57.62 x w = w / (10% - r)

Hence, r = 10% - 1/ 57.62 = 0.082644628

hence, r = 0.0826 (rounded off to four places of decimal)