Question

You receive $4,000 from your aunt when you turn 21 and you immediately invest the money in a saving account. The account earns 12% annual rate, with continuous compounding. You get your first job after 5 years. a. Determine the accumulated saving in this account at the end of 5 years. b. You want to retire from work in 20 years. If you deposit $100 into your account every month for the first 10 years, and $200 every month for the next 10, how much will you have after 20 years? Assume you continue to earn 12% annual rate with continuous compounding?

You receive $4,000 from your aunt when you turn 21 and you immediately invest the money in a saving account. The account earns 12% annual rate, with continuous compounding. You get your first job after 5 years. a. Determine the accumulated saving in this account at the end of 5 years. b. You want to retire from work in 20 years. If you deposit $100 into your account every month for the first 10 years, and $200 every month for the next 10, how much will you have after 20 years? Assume you continue to earn 12% annual rate with continuous compounding?

Answer 1

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

nominal interest rate per month = 0.12 / 12 = 0.01

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

a. F = P *(1+i)^t

Future worth of account after 5 yrs = 4000 * (1+0.1274968)^5

= 4000 * (1.1274968)^5

= 4000 * 1.8221188

= 7288.475 ~ 7288.47

b.

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

Future worth of account = 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