Question

A perpetuity-immediate makes the following pattern of payments every 3 years. It pays 3 at t = 1, then 1 at t = 2, then 4 at t = 3. In a list the payments are 3,1,4,3,1,4,3,1,4... and so on. Find the present value of this perpetuity assuming 8% effective interest per year.

Answer 1

Given that, a perpetuity makes payment in following pattern.

It pays 3 at t = 1, then 1 at t = 2, then 4 at t = 3 and so on.

interest rate r = 8%

1st calculating value of three payment at year 3 using compounding formula FV = PV*(1+r)^t

So, $3 received at year 1 is valued 3*1.08^2 = $3.4992 at year 3

and $1 received at year 2 is valued at 1*1.08 = $1.08 at year 3

So, total value for year 1,2 and 3 can be converted to a single value at year 3 of (3.4992 + 1.08 + 4) = $8.5792

So, now this perpetuity pays $8.5792 every 3 years starting 3 year from now.

So 3 years interest rate = (1+r)^3-1 = 1.08^3 - 1 = 25.9712%

So, Present value of this annuity = Payment in 3 years/3-year rate = 8.5792/0.259712 = $33.0335

So, Present value of this annuity = $33.03