Question

You are planning to invest in fine wine. Each case costs \(100, and you know from experience that the value of a case of wine held for t years is 100t^1/2. One hundred cases of wine are available for sale, and the interest rate is 10 percent.

1. How many cases should you buy, how long should you wait to sell them, and how much money will you receive at the time of their sale?
2. Suppose that at the time of purchase, someone offers you \)130 per case immediately. Should you take the offer?
3. How would your answers change if the interest rate were only 5 percent?

Short answer

1. 100 cases of wine should be bought and sold after five years. After-sales, $223.61 will be earned.
2. No, selling today will result in a negative net present value.
3. No, selling today when interest rate change will result in a negative net present value.

Step-by-step solution

Explanation for part (a)

The number of years to hold the wine cases, it is assumed that there is continuous compounding. Buying a case today for $100 and selling it after t years will fetch $100t^0.5. Thus, the net present value of the investment is calculated below:

$$\begin{gathered} \text{r =}\frac{\text{10}}{\text{100}} \\ \text{= 0.1} \\ {\text{NPV = - 100 +e}}^{\text{- r}}\left({\text{100t}}^{\text{0.5}}\right) \\ {\text{= - 100 +e}}^{\text{- 0.1t}}\left({\text{100t}}^{\text{0.5}}\right) \end{gathered}$$

The number of the year (t) is calculated below:

$$\begin{gathered} \frac{\text{dNPV}}{\text{dt}}\text{=}\left({\text{e}}^{\text{- 0.1t}}\right)\left({\text{50t}}^{\text{- 0.5}}\right)\text{-}\left({\text{0.1e}}^{\text{- 0.1t}}\right)\left({\text{100t}}^{\text{0.5}}\right)\text{= 0} \\ \left({\text{e}}^{\text{- 0.1t}}\right)\left({\text{50t}}^{\text{- 0.5}}{\text{- 10t}}^{\text{0.5}}\right)\text{= 0} \\ {\text{50t}}^{\text{- 0.5}}{\text{- 10t}}^{\text{0.5}}\text{= 0} \\ \text{50 - 10t = 0} \\ \text{10t = 50} \\ \text{t = 5} \\ {\text{NPV = - 100 +e}}^{\left(\text{- 0.1}\right)\left(\text{5}\right)}\left(\text{100}\right)\left({\text{5}}^{\text{0.5}}\right) \\ \text{=-100+0.6065×100×2.2360} \\ =\text{- 100 + 135.73} \\ =\text{\$ 35.73} \end{gathered}$$

Thus, buying 100 cases and holding for five years will be profitable as NPV is positive. The value received after a case sale will be $223.61 (=100*5^0.5).

Explanation for part (b)

If the selling occurs immediately, the gain will be $30 (=$130 – 100). If held for five years, then the net present value will be $35.73. If the selling takes place immediately, the NPV will be -$5.73 (=$30 – $35.73). Thus, selling will result in a negative NPV. A negative NPV is not desirable by a rational investor; hence, holding the wine cases for five years will be chosen.

Explanation for part (c)

The rate of interest changes from 10% to 5%.

The NPV is calculated below:

$$\begin{gathered} \text{r =}\frac{\text{5}}{\text{100}} \\ \text{= 0.05} \\ \text{NPV = - 100 +}\left({\text{e}}^{\text{- 0.05t}}\right)\left({\text{100t}}^{\text{0.5}}\right) \end{gathered}$$

The number of years (t) is calculated below:

$$\begin{gathered} \frac{\text{dNPV}}{\text{dt}}\text{=}\left({\text{e}}^{\text{- 0.05t}}\right)\left({\text{50t}}^{\text{- 0.5}}\right)\text{-}\left({\text{0.05e}}^{\text{- 0.05t}}\right)\left({\text{100t}}^{\text{0.5}}\right)\text{= 0} \\ \left({\text{e}}^{\text{- 0.05t}}\right)\left({\text{50t}}^{\text{- 0.5}}{\text{- 5t}}^{\text{0.5}}\right)\text{= 0} \\ {\text{50t}}^{\text{- 0.5}}{\text{- 5t}}^{\text{0.5}}\text{= 0} \\ \text{50 - 5t = 0} \\ \text{5t = 50} \\ \text{t = 10} \\ {\text{NPV = - 100 +e}}^{\left(\text{- 0.05}\right)\left(\text{10}\right)}\left(\text{100}\right)\left({\text{10}}^{\text{0.5}}\right) \\ =-100+\left(0.6065\times 100\times 3.1623\right) \\ =-100+191.79\} \\ =\text{\$ 91.79} \end{gathered}$$

Thus, if the wine is sold at $130 today,$91.79 could not be earned after ten years. Thus, the wine will be stored for ten years.