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Financial Algebra

Robert Gerver, Richard Sgroi

Chapter 9

Planning for Retirement

Educators


Problem 1

Explain how the quote can be interpreted.

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Problem 2

Ricky is 35 years old. He plans to retire when he is $63 .$ He has opened a retirement account that pays 3.2$\%$ interest compounded monthly. If he makes monthly deposits of $\$ 400$ , how much will he have in the account by the time he retires?

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Problem 3

Jay just graduated from college and he has decided to open a retirement account that pays 1.75$\%$ interest compounded monthly. If he has direct deposits of $\$ 100$ per month taken out of his paycheck, how much will he have in the account after 42 years?

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Problem 4

At the age of $30,$ Jasmine started a retirement account with $\$ 50,000$ which compounded interest semi-annually with an APR of 4$\%$ . She made no further deposits. After 25 years, she decided to withdraw 50$\%$ of what had accumulated in the account so that she could contribute towards her grandchild's college education. She had to pay a 10$\%$ penalty on the early withdrawal. What was her penalty?

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Problem 5

A taxpayer who pays 22$\%$ in taxes each year has these two accounts. Account $1 : \$ 10,000$ is placed in a tax-deferred account that pays 5$\%$ interest compounded annually for 25 years. Account $2 : \$ 10,000$ is placed in a taxable account that pays 5$\%$ interest compounded annually for 25 years.
a. How much is in Account 1 after the 25 -year period?
b. since the taxpayer pays 22$\%$ of all income in taxes, 22$\%$ of the interest he makes each year will go towards taxes. Therefore, his annual interest rate in actuality is 22$\%$ less than the 5$\%$ quoted rate. What is his real annual interest rate?
c. How much will he actually have made after the 25 -year period in Account 2 if taxes are taken into consideration?

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Problem 6

Laura has been contributing to a retirement account that pays 4$\%$ interest with pretax dollars. This account compounds interest monthly. She has put $\$ 500$ per month into the account. At the end of 10 years, she needed to pay some medical bills and had to withdraw 15$\%$ of the money that was in the account.
a. Rounded to the nearest dollar, how much did she withdraw?
b. Laura pays 23$\%$ of her income in taxes. What was her tax on the amount of the withdrawal (rounded to the nearest dollar)?
c. She had to pay a 10$\%$ early withdrawal penalty. How much was she required to pay, rounded to the nearest dollar?

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Problem 7

Fiona opened a retirement account that has an annual yield of 6$\% .$ She is planning on retiring in 20 years. How much must she deposit into that account each year so that she can have a total of $\$ 600,000$ by the time she retires?

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Problem 8

John is 60 years old. He plans to retire in two years. He now has $\$ 400,000$ in a savings account that yields 2.9$\%$ interest compounded continuously (see Lesson 3-7). He has calculated that his final working year's salary will be $\$ 88,000 .$ He has been told by his financial advisor that he should have $60-70 \%$ of his final year's annual income available for use each year when year's annual income available for use each year when he retires.
a. What is the range of income that his financial advisor thinks he must have per year once he retires?
b. Use the continuous compounding formula to determine how much he will have in his account at the ages of 61 and $62 .$
c. Assume that John is planning on using 65$\%$ of his current salary in each of his first 5 years of retirement. What should that annual amount be?
d. John has decided that he will need $\$ 20,000$ each year from his savings account to help him reach his desired annual income during retirement. Will John be able to make withdrawals of $\$ 20,000$ from his savings account for 20 years? Explain your reasoning.

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Problem 9

Bob can afford to deposit $\$ 400$ a month into a retirement account that compounds interest monthly with an APR of 3.996. His plan is to have $\$ 200,000$ saved so that he can then retire. Approximately how long will it take him to reach this goal?

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Problem 10

Jack contributed $\$ 400$ per month into his retirement account in pretax dollars during the last tax year. His taxable income for the year was $66,350$ . He files taxes as a single taxpayer.
a. What would his taxable income have been had he contributed to the account in after-tax dollars?
b. Use the tax table below to calculate his tax in both the pretax and after-tax contribution situations.
c. How much did Jack save in taxes during that year?

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Problem 11

Mark is an accountant who has been contributing to his retirement account for the last 15 years with pretax dollars. The account com- pounds interest semi-annually at a rate of 5\%. He contributes $X$ dollars after each 6 -month period, and this has not changed over the life of the account.
a. How much will he have in the account after 20 years of saving? Round numbers to the nearest hundredth.
b. After 20 years of contributions, he needed to withdraw 20$\%$ of the money in his account to pay for his children's education. Write an expression for the withdrawal amount.
c. Mark pays $T$ percent of his income in taxes. Write an algebraic expression for the combined total of his tax and the 10$\%$ early withdrawal penalty.

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Problem 12

Jhanvi is a 40 -year-old executive for a department store. She files taxes as head of household. She needed to withdraw $\$ 45,000$ from her tax-deferred retirement account to put a down payment on a new condominium. Jhanvis taxable income for that year was $\$ 110,550$ , excluding the $\$ 45,000$ early withdrawal from her retirement account.
a. Use the tax computation worksheet shown below to calculate Jhanvi's tax had she not made the early withdrawal.
b. Use the same worksheet to calculate her tax with an increase in her taxable income of $\$ 45,000 .$
c. How much more in taxes did she pay because of the early withdrawal?
d. If Jhanvi paid a 10$\%$ early withdrawal fee, what was her early withdrawal penalty?

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Problem 13

Nelson makes $\$ 120,000$ per year. His employer offers a 401 $\mathrm{k}$ plan in which they will match 40$\%$ of his contributions up to a maximum of 7$\%$ of his annual salary. His employer allows contributions up to a maximum of 15$\%$ of Nelson's salary per year. If Nelson contributes $\$ 200$ out of each biweekly paycheck, how much will his employer contribute to his 401 $\mathrm{k}$ )

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Problem 14

Mike makes $Y$ dollars per year. His company offers a matching retirement plan in which they agree to match $M$ percent of his contributions up to $P$ percent of his salary. Write an algebraic expression for the maximum value of the employer's matching contribution.

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