Question

After graduating from UTD at age 25, John got his first job at Goldman Sachs with an annual salary of $60,000 a
year and a one-time signing bonus of $25,000. He bought a car using his signing bonus. Goldman Sachs offers a
401K retirement investment plan that will match employee’s contribution up to 10%. For example if John invests
1% in the 401K account, Goldman Sachs will put in another 1% into his account. John is expecting an annual
salary increase of 2.4% (APR on a monthly base. For simplicity, assume that the growth will start in the second
month). Suppose, the 401K investment plan will earn him an annual return of 8.4% (APR on a monthly base).
(Assume the beginning of age 25 is month 0 and salary is paid at the end of each month, i.e., beginning of age 65 is
the last period)
(a) What percentage of salary should John invest in his 401K account in order for him to have $2 million in the
account when he retires in 40 years?
(b) At the same contribution rate, if he retires in 35 years instead, how many percent less money will John have?
(c) Instead of buying a nice car, he brought a used car for $10,000, and saved the rest of signing bonus in a separate
investment account for retirement that pays 9.6% annual interest (APR on a monthly base). If John wants to
have $2 million when he retires in 35 years, what percentage of salary should John invest in his 401K account?After graduating from UTD at age 25, John got his first job at Goldman Sachs with an annual salary of $60,000 a
year and a one-time signing bonus of $25,000. He bought a car using his signing bonus. Goldman Sachs offers a
401K retirement investment plan that will match employee’s contribution up to 10%. For example if John invests
1% in the 401K account, Goldman Sachs will put in another 1% into his account. John is expecting an annual
salary increase of 2.4% (APR on a monthly base. For simplicity, assume that the growth will start in the second
month). Suppose, the 401K investment plan will earn him an annual return of 8.4% (APR on a monthly base).
(Assume the beginning of age 25 is month 0 and salary is paid at the end of each month, i.e., beginning of age 65 is
the last period)
(a) What percentage of salary should John invest in his 401K account in order for him to have $2 million in the
account when he retires in 40 years?
(b) At the same contribution rate, if he retires in 35 years instead, how many percent less money will John have?
(c) Instead of buying a nice car, he brought a used car for $10,000, and saved the rest of signing bonus in a separate
investment account for retirement that pays 9.6% annual interest (APR on a monthly base). If John wants to
have $2 million when he retires in 35 years, what percentage of salary should John invest in his 401K account?

Answer 1

a]

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

P = first payment

r = rate of return per period

g = growth rate per period

n = number of periods

P = first amount deposited into 401K account (including employer contribution)

r = rate of return per period = 8.4% / 12 = 0.007 (monthly return = annual return / 12)

n = number of periods = 40 * 12 = 480 (number of monthly deposits = number of years to retirement * 12)

g = 2.4% / 12 = 0.002 (monthly growth rate = annual growth rate / 12)

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

$2,000,000 = P * [(1 + 0.007)⁴⁸⁰ - (1 + 0.002)⁴⁸⁰] / (0.007 - 0.002)

$2,000,000 = P * 5169

P = $386.92

The first payment should be $386.92. This includes the employer contribution. John's contribution = $386.92 / 2 = $193.46

The contribution as a % of monthly salary = $193.46 / ($60,000 / 12) = 0.0387, or 3.87%

b]

If he retires in 35 years, the number of periods = 35 * 12 = 420

Future value = 386.92 * [(1 + 0.007)⁴²⁰ - (1 + 0.002)⁴²⁰] / (0.007 - 0.002)

Future value = $1,269,777

Decrease in future value of account = $2,000,000 - $1,269,777 = $730,223

% decrease = $730,223 / $2,000,000 = 0.3651, or 36.51%

c]

Remaining signing bonus = $25,000 - $10,000 = $15,000

Future value of $15,000 @ 9.6% annual interest for 35 years is calculated as below :

monthly interest rate = 9.6% / 12 = 0.008

number of periods = 35 * 12 = 420

Future value = $15,000 * (1 + 0.008)⁴²⁰ = $426,103

The monthly contributions should equal $2,000,000 - $426,103 = $1,573,897

$1,573,897 = P * [(1 + 0.007)⁴²⁰ - (1 + 0.002)⁴²⁰] / (0.007 - 0.002)

$1,573,897 = P * 3,281.73

P = $479.59

The first payment should be $479.59. This includes the employer contribution. John's contribution = $479.59 / 2 = $239.80

The contribution as a % of monthly salary = $239.80 / ($60,000 / 12) = 0.048, or 4.8%