Question

At an annual effective interest rate of ?, ? > 0, the present value of a perpetuity paying 10 at the end of each 3-year period, with the first payment at the end of year 6, is 32. At the same annual effective rate of ?, the present value of a perpetuity-immediate paying 1 at the end of each 4-month period is X. Calculate X.

Answer 1

Given that

we have to calculate " X".

I have given the procedure to calculate X.

Annual Interest rate = i

Payment made at end of each 3 year = P = $10

Present value of the perpetuity = P/(1+i)³ + P/(1+i)⁶ + P/(1+i)⁹ + ...... = P/(1+i)³/ [1 - 1/(1+i)³] = P/[(1+i)³ - 1]

This is equal to $32

=> 10/[(1+i)³ - 1] = 32

=> (1+i)³ = (10/32) + 1 = 1.3125

=> i = 1.3125^1/3 - 1 = 0.0949 or 9.49%

For the second case, value paid after each 4 month = P₂ = $1

Interest Rate = i₂ = 0.0949/3

Present Value of perpetuity = P₂/(1+i₂) + P₂/(1+i₂)² + .... = P₂/i₂ = 1/(0.0949/3) = $31.61