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Chapter 5: Problem 6

What's the future value of a \(7 \%,\) 5-year ordinary annuity that pays \(\$ 300\) each year? If this was an annuity due, what would its future value be?

Short Answer

Expert verified
Ordinary annuity: $1725.22; Annuity due: $1845.19.

Step by step solution

01

Understanding the Problem

We are tasked with calculating the future value of a 5-year ordinary annuity that pays $300 annually at an interest rate of 7%. Additionally, we need to calculate the future value if it were an annuity due.
02

Determine the Formula for Future Value of an Ordinary Annuity

The future value of an ordinary annuity is calculated using the formula: \[ FV_{ordinary} = P \, \frac{(1 + r)^n - 1}{r} \]where \( P \) is the payment per period ($300), \( r \) is the annual interest rate (7% or 0.07), and \( n \) is the number of periods (5 years).
03

Calculate Future Value of the Ordinary Annuity

Plug in the known values into the formula: \[ FV_{ordinary} = 300 \, \frac{(1 + 0.07)^5 - 1}{0.07} \]Calculate the expression:\[ FV_{ordinary} = 300 \, \frac{(1.402552) - 1}{0.07} \]\[ FV_{ordinary} = 300 \, \frac{0.402552}{0.07} \]\[ FV_{ordinary} = 300 \, \times 5.750743 \approx 1725.22 \]
04

Determine the Formula for Future Value of an Annuity Due

The future value of an annuity due is found using the formula:\[ FV_{due} = P \, \frac{(1 + r)^n - 1}{r} \times (1 + r) \]We multiply the future value of the ordinary annuity by \( (1 + r) \) to adjust for the additional compounding period.
05

Calculate Future Value of the Annuity Due

Using the calculated ordinary annuity future value:\[ FV_{due} = 1725.22 \, \times (1 + 0.07) \]\[ FV_{due} = 1725.22 \, \times 1.07 \approx 1845.19 \]
06

Conclusion

The future value of the 5-year ordinary annuity is approximately $1725.22. If it is an annuity due, its future value is approximately $1845.19.

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Key Concepts

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

Ordinary Annuity
An ordinary annuity is a series of equal payments made at regular intervals, such as monthly or annually. However, in an ordinary annuity, each payment is made at the end of the period. For example, rent payments or monthly savings where you contribute at the end of each month are ordinary annuities.

The future value of an ordinary annuity is the worth of these payments at a future date, accounting for interest earned over time. This is crucial when planning for future financial goals as it helps determine how much you need to save regularly.
  • Formula: \[ FV_{ordinary} = P \frac{(1 + r)^n - 1}{r} \]
  • Variables include the payment amount (\(P\)), interest rate (\(r\)), and the number of periods (\(n\)).
  • It assumes interest is applied at the end of a period.
Annuity Due
An annuity due is similar to an ordinary annuity, with the key difference being that payments are made at the beginning of each period. This might include insurance premiums or lease agreements where payment is required upfront.

This structure results in a higher future value due to an additional period of compounding interest on each payment. Understanding the impact of timing on interest can substantially affect financial planning.
  • Formula: \[ FV_{due} = P \frac{(1 + r)^n - 1}{r} \times (1 + r) \]
  • Multiplying by \((1 + r)\) adjusts for the extra compounding period.
  • Essential for scenarios where earlier investment return is critical.
Financial Mathematics
In finance, mathematical models are pivotal in forecasting and valuing investments. Understanding these calculations is vital for anyone making financial decisions, whether consumers or professionals.

Financial mathematics equips you with tools for evaluating the performance of different financial products and decisions.
  • It helps in evaluating loan payments, savings plans, and investment growth.
  • Concepts such as present value and future value are foundational, indicating how much future earnings are worth today or will be worth in the future.
  • Integral for understanding the time value of money, which states money today is worth more than the same amount in the future due to its potential earning capacity.
Mastering these concepts allows for informed decisions and strategic financial planning, ensuring future security and optimized investments.

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Most popular questions from this chapter

EVALUATING LUMP SUMS AND ANNUITIES Crissie just won the lottery, and she must choose between three award options. She can elect to receive a lump sum today of \(\$ 61\) million, to receive 10 end-of-year payments of \(\$ 9.5\) million, or to receive 30 end-of-year payments of \(\$ 5.5\) million. a. If she thinks she can earn \(7 \%\) annually, which should she choose? b. If she expects to earn \(8 \%\) annually, which is the best choice? c. If she expects to earn \(9 \%\) annually, which option would you recommend? d. Explain how interest rates influence the optimal choice.

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 \(9 \%\) in the future. a. If she follows your advice, how much money will she have at \(65 ?\) b. How much will she have at \(70 ?\) c. She expects to live for 20 years if she retires at 65 and for 15 years if she retires at 70 . If her investments continue to earn the same rate, how much will she be able to withdraw at the end of each year after retirement at each retirement age?

If you deposit money today in an account that pays \(6.5 \%\) annual interest, how long will it take to double your money?

Shalit Corporation's 2008 sales were \(\$ 12\) million. Its 2003 sales were \$6 million. a. At what rate have sales been growing? b. Suppose someone made this statement: "Sales doubled in 5 years. This represents a growth of \(100 \%\) in 5 years; so dividing \(100 \%\) by \(5,\) we find the growth rate to be \(20 \%\) per year." Is that statement correct?

Erika and Kitty, who are twins, just received \(\$ 30,000\) each for their 25 th birthday. They both have aspirations to become millionaires. Each plans to make a \(\$ 5,000\) annual contribution to her "early retirement fund" on her birthday, beginning a year from today. Erika opened an account with the Safety First Bond Fund, a m煤tual fund that invests in high-quality bonds whose investors have earned \(6 \%\) per year in the past. Kitty invested in the New Issue Bio-Tech Fund, which invests in small, newly issued bio-tech stocks and whose investors have earned an average of \(20 \%\) per year in the fund's relatively short history. a. If the two women's funds earn the same returns in the future as in the past, how old will each be when she becomes a millionaire? b. How large would Erika's annual contributions have to be for her to become a millionaire at the same age as Kitty, assuming their expected returns are realized? c. Is it rational or irrational for Erika to invest in the bond fund rather than in stocks?
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