Question

Karim deposits $100 every two years for 40 years into an account that earns an effective annual interest rate i. The accumulated value after 20 years is X. The accumulated value after 40 years is Y.

a) i = 2%. Find X. Find Y.

b) i is unknown, but Y = 4X. Find i. Find X.

Answer 1

Future value after t payments of amount deposited every two years=x/(1+i)^2*(1-(1/(1+i)^2)^t)/(1-(1/(1+i)^2))*(1+i)^(2*t)

a) Answer

X=100/(1+i)^2*(1-(1/(1+i)^2)^10)/(1-(1/(1+i)^2))*(1+i)^20

=>X=100/(1+2%)^2*(1-(1/(1+2%)^2)^10)/(1-(1/(1+2%)^2))*(1+2%)^20

X=1202.840089

Y=100/(1+i)^2*(1-(1/(1+i)^2)^20)/(1-(1/(1+i)^2))*(1+i)^40

=>Y=100/(1+2%)^2*(1-(1/(1+2%)^2)^20)/(1-(1/(1+2%)^2))*(1+2%)^40

Y=2990.197187

b)Answee

Y=100/(1+i)^2*(1-(1/(1+i)^2)^20)/(1-(1/(1+i)^2))*(1+i)^40

X=100/(1+i)^2*(1-(1/(1+i)^2)^10)/(1-(1/(1+i)^2))*(1+i)^20

Y=4X

=>100/(1+i)^2*(1-(1/(1+i)^2)^20)/(1-(1/(1+i)^2))*(1+i)^40=4*(100/(1+i)^2*(1-(1/(1+i)^2)^10)/(1-(1/(1+i)^2))*(1+i)^20)

=>i=0.056467

X=100/(1+0.056467)^2*(1-(1/(1+0.056467)^2)^10)/(1-(1/(1+0.056467)^2))*(1+0.056467)^20

X=1722.31