Question

Perpetuity X has annual payments of 1,2,3,... at the end of each year. Perpetuity Y has annual payments of q, q, 2q, 2q, 3q, 3q, ... at the end of each year. The present value of X is equal to the present value of Y at an annual effective interest rate of 10%. Calculate q.

I'm new to perpetuities but basically understand how perpetuities work. I also have a formula for perpetuities that increase every year. I just can't figure out how to break down the "q, q, 2q, 2q, 3q, 3q, ..." portion that increases every other year so please be sure to elaborate how that portion of the problem is broken down.

Answer 1

You can consider this perpetuity as two perpetuities:
First perpetuity with cash flow as q,2q,3q,4q....in year 1,3,5,7 respectively
1,3,5,7 can be considered as 1,2,3,4 with interest rate compounded biennially instead of annually=1.1^2-1=21%
Present value at t=1:=(q/21%+q/(21%*21%))*1.21
Present value at t=0:=(q/21%+q/(21%*21%))*1.1

Second perpetuity with cash flow as q,2q,3q,4q....in year 2,4,6,8 respectively
2,4,6,8 can be considered as 1,2,3,4 with interest rate compounded biennially instead of annually=1.1^2-1=21%
Present value at t=2:=(q/21%+q/(21%*21%))*1.21
Present value at t=0:=(q/21%+q/(21%*21%))

Sum=57.61904762*q