Question

Twins Jane and Hal each inherited $150,000 exactly ten years ago. Jane invested the entire amount in a brokerage account to fund her retirement. Her account has been earning 8% per year since she invested it, and she expects it to earn 5% per year for the next 20 years. Hal spent all of his inheritance and has not saved anything for retirement. Assume there are no taxes.

a. How much is Jane expected to have in her account at retirement (20 years from now)?

b. Due to sibling rivalry, Hal wants to have at least $100,000 more saved at retirement (20 years from now) than Jane is expected to have at that time. He plans to make an equal deposit each year in an account earning the same annual interest rate as Jane’s, i.e., 5%, with the first deposit occurring one year from today and the last occurring 20 years from today. How much must Hal deposit each year in order to achieve his goal?

Answer 1

a]

Future value = present value * (1 + rate)^t

where t = number of years

Value of account at retirement = $150,000 * (1 + 8%)¹⁰ * (1 + 5%)²⁰

Value of account at retirement = $859,240.61

b]

Future value of annuity = P * [(1 + r)ⁿ - 1] / r,

where P = periodic payment. We need to calculate this.

r = periodic rate of interest. This is 5%

n = number of periods. This is 20

$959,240.61 = P * [(1 + 5%)²⁰ - 1] / 5%

P =  $959,240.61 * 5% / [(1 + 5%)²⁰ - 1]

P =  $29,009.92

Hal must save $29,009.92 each year