Question

The present value of an annual perpetuity immediate of 150 is equal to the present value of an annual perpetuity immediate that pays 100 at the end of the first 20 years and 200 at the end of year 21 and each year thereafter. Calculate i.

Answer 1

Suppose i is the interest rate. Then;
Present value of perpetuity is:
= 150/(i)

Present value of an annual perpetuity immediate that pays 100 at the end of the first 20 years and 200 at the end of year 21 and each year thereafter is calculated as:
=100/i*(1-1/(1+i)^20) + (200/(1+i)^20)/i
=(100/i)*(1-1/(1+i)^20) + (200/(1+i)^20)/i
Equating both the sides, we get;
150/(i)=(100/i)*(1-1/(1+i)^20) + (200/(1+i)^20)/i
=>150=(100)*(1-1/(1+i)^20) + (200/(1+i)^20)
=>3=2*(1-1/(1+i)^20) + (4/(1+i)^20)
=>3=2-2/(1+i)^20 + 4/(1+i)^20
=>1=4/(1+i)^20 - 2/(1+i)^20
=>1=2/(1+i)^20
=>(1+i)^20=2
=>1+i=(2)^(1/20)
=>i=1.035264924 -1
=>i=0.035264924 or 3.5265%

Answer: Hence, the value of i is 3.5265%