Question

4-122 Will receives $4000 from his aunt when he turns 21 and he immediately invests the money in a savings account. The account earns 12% annual interest, with continuous compounding. He gets his first job after 5 years.
a) Determine the accumulated savings in this account at the end of 5 years.
b) Will wants to retire from work in 20 years. If he deposits $100 into his account every month for the first 10 years, and $200 every month for the next 10, how much will he have after 20 years? Assume he continues to earn 12% annual interest with continuous compounding.

Answer 1

12% = 0.12

Computation of Effective interest rate per year:
= e^0.12 - 1
= 1.1274968 - 1
= 0.1274968

Computation of Nominal interest rate per month:
= 0.12 / 12
= 0.01

Computation of Effective interest rate per month:
= e^0.01 - 1
= 1.010050 - 1
= 0.010050

(a) Determining the accumulated savings in this account at the end of 5 years:

F = P *(1+i)^t

Future worth of account after 5 years:
= 4000 * (1 + 0.1274968)^5
= 4000 * (1.1274968)^5
= 4000 * 1.8221188
= $7288.47

(b) How much will he have after 20 years:

T = 10 * 12 = 120 months for first investment and then second too.

Future Worth (FW) of the account is computed as under:

= 100 * (F/A,1.01005%,120) * (F/P,1.01005%,120) + 200 * (F/A,1.01005%,120) + 7288.475 * (F/P,12.74968%,20)
= 100 * ((1 + 0.01005)^120-1) / 0.010050 * (1 + 0.01005)^120 + 200 * ((1 + 0.01005)^120-1) / 0.01005 + 7288.475 * (1 + 0.1274968)^20
= 100 * ((1.01005)^120-1) / 0.010050 * (1.01005)^120 + 200 * ((1.01005)^120-1) / 0.01005 + 7288.475 * (1.1274968)^20
= 100 * 230.850847 * 3.320051 + 200 * 230.850847 + 7288.475 * 11.023166
= $203155.90