Perpetuities - Annuities

Question

A perpetuity pays $2X one year from today. The payments increase by 5% per year thereafter. The effective annual interest rate on this perpetuity is 6%. The present value is $32,400. A second perpetuity pays $Y one year from now, and the annual payment increases by $X per year thereafter. The effective annual interest rate on this perpetuity is i and the present value $24000. A third perpetuity pays $Y per year, with the first payment one year from now. The effective annual interest rate on this perpetuity is i and the present value is $4000. To the nearest $10, what is $Y?

Answer 1

32400 = 2x/(.06-.05) = 2x/.01

So, 2x = .01*32400 = 324

x=162

24000 = y/(1+i) + (y+x)/(1+i)^2 + (y+2x)/(1+i)^3 + (y+3x)/(1+i)^4 + ...

= y/(1+i) + y/(1+i)^2 + y/(1+i)^3 + ... + x/(1+i)^2 + x/(1+i)^3 + x/(1+i)^4 + ... +

x/(1+i)^3 + x/(1+i)^4 + x/(1+i)^5 + ...

= y/i + x/i * (1/(1+i) + 1/(1+i)^2 + 1/(1+i)^3 + ...) = y/i + x/i * 1/i

= y/i + x/i^2

4000 = y/i

So, 24000 = y/i + x/i^2 = 4000 + x/i^2

or 20000 = x/i^2 = 162/i^2

20000*i^2 = 162

i^2 = 162/20000 = 0.0081

i = square root of 0.0081 = .09

4000 = y/i = y/.09

so, y = 0.09 * 4000 = 360