Question

1. At an annual effective interest rate of i, i > 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 3, is $40.

   At the same annual effective rate of i, the present value of a perpetuity paying $20 at the end of each 3-month period, with the first payment at 3 months, is X.

   Calculate X.

Answer 1

We are given
PV=40
10/j=40
10/((1+i)^3-1)=40
10=40*((1+i)^3-1)
50=40*(1+i)^3
=>(1+i)^3=(50/40)
=>(1+i)=(50/40)^(1/3)

Next:
Let k be the periodic rate
So
(1+k)^4=(1+i)
(1+k)=(1+i)^(1/4)
k=(1+i)^(1/4)-1

Then X
=20*1/k
=20*1/((1+i)^(1/4)-1)
=20*1/(((50/40)^(1/3))^(1/4)-1)
=20*53.2786

=1065.5718