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Chapter 2: Problem 19

Future value of an annuity Your client is 40 years old, and she wants to begin saving for retirement, with the first payment to come one year from now. She can save \(\$ 5,000\) per year, and you advise her to invest it in the stock market, which you expect to provide an average return of 9 percent in the future. a. If she follows your advice, how much money would she have at \(65 ?\) b. How much would she have at \(70 ?\) c. If she expects to live for 20 years in retirement if she retires at 65 and for 15 years at \(70,\) and her investments contintue to earn the same rate, how much could she withdraw at the end of each year after retirement at each retirement age?

Short Answer

Expert verified
At age 65, her fund is nearly $438,870, enabling $47,000 annual withdrawals for 20 years. At age 70, her fund reaches about $756,561, allowing $97,597 annually for 15 years.

Step by step solution

01

Define the problem

We need to find the future value of an annuity when a client invests $5,000 annually with an expected return of 9% for two scenarios: when the client starts withdrawing at age 65 and at age 70. We will also calculate the periodic withdrawals she can make if she retires at these ages.
02

Calculate Future Value at Age 65

Since the client starts saving at age 40, she will make 25 payments by age 65. The future value of an ordinary annuity is given by the formula: \[ FV = P \frac{(1 + r)^n - 1}{r} \]where \(P\) is the annual payment, \(r\) is the interest rate, and \(n\) is the number of payments.Plugging in the values: \(P = 5000\), \(r = 0.09\), and \(n = 25\), we calculate the future value.
03

Calculate Future Value at Age 70

If she continues investing until age 70, she will have made 30 payments. Using the same future value formula: \[ FV = 5000 \frac{(1 + 0.09)^{30} - 1}{0.09} \]Calculate this to find her savings at age 70.
04

Retirement Withdrawals from Age 65

When the client retires at age 65, we need to determine the constant annual withdrawal she can make over 20 years (until age 85) assuming a 9% earning rate. We use the annuity formula:\[ W = \frac{FV}{\frac{(1 - (1 + r)^{-t})}{r}} \]With her future value from age 65, interest rate \(r = 0.09\), and \(t = 20\) years, calculate the annual withdrawal.
05

Retirement Withdrawals from Age 70

For retirement at age 70, the withdrawal period is 15 years (until age 85). Reuse the annuity formula:\[ W = \frac{FV}{\frac{(1 - (1 + r)^{-t})}{r}} \]Using her future value from age 70, \(r = 0.09\), and \(t = 15\), find the annual withdrawal she can sustain.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Retirement Planning
Retirement planning is a crucial step for ensuring a comfortable future once you stop working. It's all about setting aside money now so you have enough to support yourself when you retire. The best time to start planning for retirement is as early as possible. When starting retirement planning, consider factors such as your current age, desired retirement age, and how much you want to retire with. These will help determine how much you need to save each year. Starting early enables compound interest to work in your favor鈥攖his means your savings will grow faster as time goes on. Retirement planning also allows you to set realistic financial goals based on your expected future expenses, lifestyle choices, and life expectancy.
Ordinary Annuity Formula
The ordinary annuity formula is essential when calculating the future value of regular payments or investments made over time. An ordinary annuity means that each payment is made at the end of a period, which is typically a year. The formula for the future value of an ordinary annuity is: \[ FV = P \frac{(1 + r)^n - 1}{r} \] where:
  • \( FV \) is the future value of the annuity,
  • \( P \) is the annual payment,
  • \( r \) is the interest rate per period,
  • \( n \) is the total number of payments.
Using this formula, you can determine how much your regular contributions will accumulate over time with compound interest. It鈥檚 particularly useful in scenarios like retirement savings, where you make periodic payments towards a retirement fund.
Investment Strategy
An investment strategy outlines how you plan to build wealth over time. It's important to choose a strategy that aligns with your financial goals, risk tolerance, and timeline. For a long-term goal like retirement, investing in the stock market can be beneficial due to its historical returns. Diversifying your portfolio by including stocks, bonds, and other investment vehicles helps manage risk while you aim for higher returns. It's also critical to regularly review and adjust your investment strategy to ensure it stays tuned to changes in your life or market conditions. Staying informed about economic trends and potential investments will help you make sound decisions and ensure maximum growth of your retirement fund.
Annual Withdrawal Calculation
Once you retire, you'll need to calculate how much you can safely withdraw from your savings each year. This is known as the annual withdrawal rate. Determining this involves using the annuity formula, which helps find the amount you can withdraw so that it lasts for your entire retirement period. The formula is: \[ W = \frac{FV}{\frac{(1 - (1 + r)^{-t})}{r}} \] where:
  • \( W \) is the annual withdrawal amount,
  • \( FV \) is the future value of your investments at the start of retirement,
  • \( r \) is the annual return rate during retirement,
  • \( t \) is the number of years you'll be withdrawing for.
By understanding and applying this calculation, you ensure that your funds are distributed evenly across your retirement years, allowing you to maintain your lifestyle without depleting your savings too early.

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Most popular questions from this chapter

Future value of an annuity Find the future values of these ordinary anmuities. Compounding occurs once a year. a. \(\quad \$ 400\) per year for 10 years at 10 percent. b. \(\quad \$ 200\) per year for 5 years at 5 percent. c. \(\quad \$ 400\) per year for 5 years at 0 percent. d. Rework parts a \(b\), and c assuming that they are ammities due.

Present and future values for different periods Find the following values, using the equations and then a financial calculator. Compounding/discounting occurs annually. a. An initial \(\$ 500\) compounded for 1 year at 6 percent. b. \(\quad\) An initial \(\$ 500\) compounded for 2 years at 6 percent. c. The present value of \(\$ 500\) due in 1 year at a discount rate of 6 percent. d. The present value of \(\$ 500\) due in 2 years at a discount rate of 6 percent.

Time for a lump sum to double If you deposit money today in an account that pays 6.5 percent annual interest, how long will it take to double your money?

Effective rate of interest Find the interest rates earned on each of the following: a. You borrore \(\$ 700\) and promise to pay back \(\$ 749\) at the end of 1 year. b. You lend \(\$ 700\) and the borrower promises to pay you \(\$ 749\) at the end of 1 year. c. You borrow \(\$ 85,000\) and promise to pay back \(\$ 201,229\) at the end of 10 years. d. You borrow \(\$ 9,000\) and promise to make payments of \(\$ 2,684.80\) at the end of each year for 5 years.

Amortization schedule with a balloon payment You want to buy a house that costs \(\$ 100,000 .\) You have \(\$ 10,000\) for a down payment, but your credit is such that mortgage companies will not lend you the required \(\$ 90,000\). However, the realtor persuades the seller to take a \(\$ 90,000\) mortgage (called a seller take-back mortgage) at a rate of 7 percent, provided the loan is paid off in full in 3 years. You expect to inherit \(\$ 100,000\) in 3 years, but right now all you have is \(\$ 10,000,\) and you can only afford to make payments of no more than \(\$ 7,500\) per year given your salary. (The loan would really call for monthly payments, but assume end-of-year annual payments to simplify things.) a. If the loan were amortized over 3 years, how large would each annual payment be? Could you afford those payments? b. If the loan were amortized over 30 years, what would each payment be, and could you afford those payments? c. \(\quad\) To satisfy the seller, the 30 -year mortgage loan would be written as a "balloon note," which means that at the end of the 3 rd year you would have to make the regular payment plus the remaining balance on the loan. What would the loan balance be at the end of Year \(3,\) and what would the balloon payment be?
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