Question

Two payment streams have the same present value under effective annual interest rate of 8%:

1. 5 annual payments of 200, beginning in one year, followed by a monthly perpetuity of $X.

2. 20 payments of 900 every two years, beginning today.

Calculate X.

Answer 1

Given

Option 1

5 Annual payment P=200

Monthly perpetuity =X

Annual effective rate R=8%

Monthly rate i=(1+R)^(1/12)-1=(1+8%)^(1/12)-1 =0.643%

PV of this option PV1= P*(1-(1+R)^-5)/R + X/(i*(1+R)^5)

PV1=200*(1-(1+8%)^-5)/8% + X/(0.643%*(1+8%)^5)=798.54+105.78X

Option 2

Payment every two years P=900

N=20 payments

Annual effective rate R=8%

PV of this option PV2= P*(1+R)^2*(1-(1+R)^-40)/{(1+R)^2 -1}

PV2=900*(1+8%)^2*(1-(1+8%)^-40)/{(1+8%)^2-1} =$6018.26

Given

PV1=PV2

798.54+105.78X=6018.26

X=$49.35