Question

It is given that the effective rate of interest for the n-th year period is in = 0.01 + e^(−n)

(a) Find a(t) for t being an integer.

(b) If the principal is $100, find the total amount of interest earned in year 3, 4 and 5.

Answer 1

a) A = P * (1+R1) (1+R2) (1+R3)……………….(1+Rt)

A: Amount

P: Principal

R1, R2, R3 etc: effective interest rate for 1st, 2nd, 3rd year and so on

n: number of years

R1 (interest rate for first year) = 0.01 + e^-1

R2 (interest rate for 2nd year) = 0.01 + e^-2

R3 (interest rate for third year) = 0.01 + e^-3

Rt (interest rate for t year) = 0.01 + e^-t

Amount at the end of t years is given as

A (t) = P * (1+0.01+e^-1) (1+0.01+e^-2) (1+0.01+e^-3)………………..(1+0.01+e^-t)

A (t) = P * (1.01+e^-1) (1.01+e^-2) (1.01+e^-3)………………..(1.01+e^-t)

b)

A (2) = 100 * (1.01+e^-1) (1.01+e^-2) = 157.8134 approx

A (3) = 100 * (1.01+e^-1) (1.01+e^-2) (1.01+e^-3) = 167.2486 approx

A (4) = 100 * (1.01+e^-1) (1.01+e^-2) (1.01+e^-3) (1.01+e^-4) = 171.9843 approx

A (5) = 100 * (1.01+e^-1) (1.01+e^-2) (1.01+e^-3) (1.01+e^-4) (1.01+e^-5) = 174.8630 approx

Interest earned in 3rd year = A(3) - (A2) = $ 9.4352 approx

Interest earned in 4th year = A(4) - (A3) = $ 4.7358 approx

Interest earned in 5th year = A(5) - (A4) = $ 2.8787 approx