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# Interest Compounding, Frequency, Future Value and Present Value

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1. Compounding frequency and future value

You plan to invest \$2,000 in an individual retirement account (IRA) today at a nominal rate of 8 percent, which is expected to apply to all future years.

a. How much will you have in the account after 10 years if the interest is compounded:
1. Annually
2. Semi-Annually
3. Daily (assume a 360-day year)
4. Continuously

b. What is the effective annual rate, EAR, for each compounding period in a?

c. How much greater will your account balance be at the end of ten years if interest is compounded continuously rather than annually?

d. How does the compounding frequency affect the future value and effective annual rate for a given deposit? Explain in terms of your finding in a through c.

2. Present value and discount rates

You just won a lottery that promises to pay you \$1,000,000 exactly 10 years from today. Because the \$1,000,000 payment is guaranteed by the state in which you live, opportunities exist to sell the claim today for an immediate lump sum cash payment.

a. What is the least you would sell your claim for if you could earn the following rates of return on similar-risk investments during the 10-year period?
1. 6 percent
2. 9 percent
3. 12 percent

b. Rework (a) under the assumption that the \$1,000,000 payment will be received in 15 rather than 10 years.

c. Based on your findings in (a) and (b), discuss the effect of both the size of the rate of return and the time until receipt of payment on the present value of a future sum.

You plan to retire in exactly 20 years. Your goal is to create a fund that will allow you to receive \$20,000 per year for 30 years between retirement and death (a physic told you would die after 30 years). You know that you will be able to earn 11 percent per year during the 30-year retirement period.

a. How large a fund will you need when you retire in 20 years to provide the 30-year, \$20,000 retirement annuity?

b. How much would you need today as a lump sum to provide the amount calculated in (a) if you earn only 9 percent per year during the 20 years preceding retirement?

c. What effect would an increase in the rate could earn both during and prior to retirement have on the values found in (a) and (b)?