Question

Consider the following scenario to determine the future value of the payments. Annual payments of 100 are made for 5 years at an effective rate of r, followed by annual payments of 200 for the next 3 years at the same rate. Set up three different equation of values for this situation.

Answer 1

Solution:

We are making the payment of 100 for 5 years at the rate r and then 200 for next 3 years at the rate r as well.

Future value for first investment = 100*(1+r)^8

Future value for Second investment = 100*(1+r)^7

Future value for third investment = 100*(1+r)^6

Future value for fourth investment = 100*(1+r)^5

Future value for fifth investment = 100*(1+r)^4

Future value for Sixth investment = 200*(1+r)^ 3

Future value for Seventh investment = 200*(1+r)^ 2

Future value for Eighth investment = 200*(1+r)^ 1

We can create the equation for 100 payment as Future value = 100*(1+r)^8 + 100*(1+r)^7 +100*(1+r)^6+100*(1+r)^5+100*(1+r)^4 = 100 * (1+r)^4[ 1+(1+r) +(1+r)^2+(1+r)^3 +(1+r)^4]

= 100 * (1+r)^4 * [ (1+r)^4-1]/(1+r-1) [Sum of GP = a * (r^n-1)/(r-1)]

= 100 * (1+r)^4 * { (1+r)^4-1}/r

Similarly for 200 payment

Future value =  200*(1+r)^ 3 + 200*(1+r)^ 2 + 200*(1+r)^1

=  200*(1+r) {(1+r)^2 -1}/r