Interest-rate problems (which may require a calculator): a....

Interest-rate problems (which may require a calculator): a. You invest $\$ 2000$ at an interest rate of 13.5 percent per year. What is your total balance after 6 months? b. Interest is said to be "compounded" when you earn interest on whatever interest has already been paid; most interest rates quoted today are compounded. If you invest $\$ 10,000$ for 3 years at a compound annual interest rate of 10 percent, what is the total value of the investment at the end of each year? c. Consider the following data: The consumer price index in 1977 was $60.6,$ and in 1981 it was $90.9 .$ Interest rates on government securities in 1978 through 1981 (in percent per year) were 7.2,10.0 11.5, and $14.0 .$ Calculate the average nominal and real interest rates for the 4 -year period $1978-1981 .$ d. Treasury bills (T-bills) are usually sold on a discounted basis; that is, a 90-day T-bill for $\$ 10,000$ would sell today at a price such that collecting $\$ 10,000$ at maturity would produce the market interest rate. If the market interest rate is 6.6 percent per year, what would be the price on a 90 -day $\$ 10,000$ T-bill?

Interest-rate problems (which may require a calculator):
a. You invest $\$ 2000$ at an interest rate of 13.5 percent per year. What is your total balance after
6 months?
b. Interest is said to be "compounded" when you earn interest on whatever interest has already been paid; most interest rates quoted today are compounded. If you invest $\$ 10,000$ for 3 years at a compound annual interest rate of 10 percent, what is the total value of the investment at the end of each year?
c. Consider the following data: The consumer price index in 1977 was $60.6,$ and in 1981 it was $90.9 .$ Interest rates on government securities in 1978 through 1981 (in percent per year) were 7.2,10.0 11.5, and $14.0 .$ Calculate the average nominal and real interest rates for the 4 -year period $1978-1981 .$
d. Treasury bills (T-bills) are usually sold on a discounted basis; that is, a 90-day T-bill for $\$ 10,000$ would sell today at a price such that collecting $\$ 10,000$ at maturity would produce the market interest rate. If the market interest rate is 6.6 percent per year, what would be the price on a 90 -day $\$ 10,000$ T-bill?

Key Concepts

Simple Interest

Simple interest is the method of calculating interest where the interest charge is based solely on the original principal, without any compounding. This concept is essential when determining the amount earned or owed over a period by multiplying the principal amount, the interest rate, and the time period involved.

Compound Interest

Compound interest involves earning interest on both the initial principal and the accumulated interest from previous periods. This key concept is fundamental in finance as it demonstrates how investments or debts can grow at an accelerated rate over time due to interest being earned on interest.

Time Period Conversion in Interest Calculations

Time period conversion is critical in interest calculations because interest rates are typically quoted on an annual basis, yet investments or loans may be for shorter or non-standard periods. Converting the annual rate to a rate appropriate for the given time period ensures accurate interest computation.

Nominal versus Real Interest Rates

The distinction between nominal and real interest rates is crucial in understanding the true return on an investment. The nominal rate is the stated interest rate, while the real rate adjusts for inflation, revealing the effective increase in purchasing power over time. This concept is key when analyzing economic growth and investment performance in an inflationary environment.

Discounted Securities Pricing

Discounted securities pricing involves determining the present value of a financial instrument that is sold at a discount to its face value, with the return being the difference at maturity. This concept is particularly relevant for instruments like treasury bills, where the pricing reflects a discounted purchase to yield the market interest rate when held to maturity.

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