Problem:

Suppose you invest $1,000 in a financial asset earning an annual interest rate of 6%.

1. How much interest will you earn after one year? How much money will be available to you (principal and interest) at the end of one year?
2. If the money is allowed to compound annually at 6%, how much money will be available at the end of 2 years? 5 years?
3. Suppose you have the opportunity to purchase a risk-free asset that will return $1,000 one year from now. If the risk-free interest rate is currently 5%, what is the present value of this asset?
4. Alternatively, suppose the risk-free asset will return $1,000 two years from now. Still assuming a risk-free interest rate of 5%, what is the present value of this asset?
5. You are considering purchasing a risk-free asset that will return $1,000 at the end of one year, another $1,000 at the end of the second year, and another $1,000 at the end of the third year. What is present value of this asset if the risk-free interest rate is 5%?
6. Why does the present value of an asset tell you the greatest amount you should pay for the asset?

Answer:

1. At a rate of 6%, an investment of $1,000 will earn .06 x $1,000 = $60 in interest. Added onto the original $1,000, a total of $1,060 will be available. This total can be computed alternately using the formula $1,000 x (1 + .06) = $1,060.
2. The $1,060 available at the end of the first year will earn .06 x $1,060 = $63.60 in additional interest over the second year. The total available at the end of the second year is then $1,060 +$63.60 = $1,123.60. Alternatively, $1,123.60 = $1,000 x (1.06)², or $1,000 x 1.06 x 1.06. At the end of 5 years, the total available will be $1,000 x (1.06)⁵ = $1,000 x 1.06 x 1.06 x 1.06 x 1.06 x 1.06 = $1,338.23.
3. At 5%, the present value of $1,000 one year from now is $1,000 / (1.05) = $952.38.
4. The present value is $1,000 / (1.05)² = $907.03.
5. As determined in parts c and d, the present value of $1,000 received at the end of one year is $952.38, while the present value of $1,000 received at the end of two years is $907.03. Similarly, the present value of $1,000 received at the end of three years is $1,000 / (1.05)³ = $863.84. The present value of a risk-free asset returning $1,000 at the end of each of the first three years is the sum: $952.38 + $907.03 + $863.84 = $2,723.25.
6. Paying any amount greater than the present value would imply receiving a lower rate of return than could be obtained by investing in the risk-free asset.