In: Finance
Tom decides to get an early start on retirement saving and, beginning at age 22, he invests $4,000 per year in a Roth IRA for 10 years in a row. At that point, he stops contributing to the account but leaves the money invested until age 65 (a period of 33 years). Harry doesn’t start investing until he’s 32 but from then on invests $5,000 in a Roth IRA each year for 33 years until retirement at age 65. If both men earn 9.875 percent per year on their investments, compounded annually, what are their final account values? Who has accumulated more money? How much more?
Step 1- Calculation of value of Tom savings when he turns 32 years old | |||||||
We can use the future value of annuity formula to calculate this value. | |||||||
Future value of annuity = P x {[(1+r)^n -1]/r} | |||||||
Future value of annuity = value of Tom savings when he turns 32 years old = ? | |||||||
P = annual savings = $4000 | |||||||
r = interest rate on investment = 9.875% per year | |||||||
n = number of years from age 22 years to 32 years = 10 | |||||||
Future value of annuity = 4000 x {[(1+0.09875)^10 -1]/0.09875} | |||||||
Future value of annuity = 4000 x 15.84221 | |||||||
Future value of annuity = 63,368.85 | |||||||
Value of Tom savings when he turns 32 years old = $63,368.85 | |||||||
Step 2- Calculation of value of Tom savings when he turns 65 years old | |||||||
We can use future value of lumsum formula to calculate this value. | |||||||
Future value of lumsum = P x (1+r)^n | |||||||
Future value of lumsum = value of Tom savings when he turns 65 years old = ? | |||||||
P = value calculated in step 1 = $63,368.85 | |||||||
r = interest rate on investment = 9.875% per year | |||||||
n = number of years from age 32 years to 65 years = 33 | |||||||
Future value of lumsum = 63368.85 x (1+0.09875)^33 | |||||||
Future value of lumsum = 63368.85 x 22.36986 | |||||||
Future value of lumsum = 1417552.34 | |||||||
Value of Tom savings when he turns 65 years old = $14,17,552.34 | |||||||
Step 3- Calculation of value of Harry's savings when he turns 65 years old | |||||||
We can use the future value of annuity formula to calculate this value. | |||||||
Future value of annuity = P x {[(1+r)^n -1]/r} | |||||||
Future value of annuity = value of Harry's savings when he turns 65 years old = ? | |||||||
P = annual savings = $5000 | |||||||
r = interest rate on investment = 9.875% per year | |||||||
n = number of years from age 32 years to 65 years = 33 | |||||||
Future value of annuity = 5000 x {[(1+0.09875)^33 -1]/0.09875} | |||||||
Future value of annuity = 5000 x 216.4037 | |||||||
Future value of annuity = 1082018.33 | |||||||
Value of Harry's savings when he turns 65 years old = $10,82,018.33 | |||||||
Answer | |||||||
Final account value | |||||||
Tom | $1,417,552.34 | ||||||
Harry | $1,082,018.33 | ||||||
Tom has accumulated more money by | $335,534.01 | ||||||