Question

At an annual interest rate of i, i ≥ 0, the present value of a perpetuity paying 10 at the end of each three-year period, with the first payment at the end of year 3, is 34.21. At the same annual effective rate of i, the present value of a perpetuity paying 1 at the end of each four-month period, with the first payment at the end of four months, is X.

Calculate X.

Answer 1

Part-1.

Year	0	3	6	9	12-----------
payment	0	10	10	10	10-------

Let j be the periodic interest rate for 3 year interval of time

i is the effective interest per year

hence ,

1+j = (1+i)^3

=> j = (1+i)^3-1---------------------------(1)

here PVP = present value of the perpetuity

A = Perpetuity amout per period

j= periodic interest rate

hence,

  

Putting the value of j from equation(1), we will get,

=>i = 0.0892372379

Part-2

Month	0	4	8	12	16-----------
payment	0	1	1	1	1-------

Let y be the periodic rate (i.e. rate for 4 month)

and i is the annual effective rate

hence

(1+y)^3 = 1+i

Now potting the value of i from part 1, we will get

(1+y)^3 = 1+0.0892372379

=>1+y =1.0289023522

=>y = 0.0289023522

Hence the value of X = 34.60