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 \(100 t^{1 / 2}\). One hundred cases of wine are available for sale, and the interest rate is 10 percent. a. 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? b. Suppose that at the time of purchase, someone offers you \(\$ 130\) per case immediately. Should you take the offer? c. How would your answers change if the interest rate were only 5 percent?

Short answer

You should buy 100 cases, sell them immediately. You will receive $10,000 from their sale. If offered $130 per case, you should accept the offer and sell all the cases for a total of $13,000. If the interest rate decreases to 5 percent, this does not change the decision to sell immediately or reconsider the $130 offer.

Step-by-step solution

Case Value Growth Function Over Time

The value of a case of wine held for \(t\) years is given as \(100 t^{1 / 2}\). This implies that the value of the case increases the longer it is held, but at a decreasing rate.

Calculate Maximized case value

To maximize profit, derive the case value function with respect to time and equate it to 0 to find when the value would be maximized. Solve \(50 t^{-1 / 2} = 0\) for \(t\). This results in the wine value not having a maximum value, implying holding onto the wine indefinitely would maximize its value in an environment without interest rates.

Consider Effects of Interest Rates

However, there are interest rates, which means that money now is worth more than the same amount of money in the future. Thus, find the present value of the case of wine at time \(t\), which is \(PV = \frac{{100 t^{1 / 2}}}{{(1+0.1)^t}}\). Holding a case longer decreases its present value, so you should sell it as soon as possible, at t = 0, for each case bought.

Calculate Total Profit

You buy 100 cases at $100 each for a total cost of $10,000. You can sell them immediately at their purchase price, thus making no profit or loss.

Evaluate Selling Price Offer

If an offer of $130 per case is made immediately after purchase, the selling price would be $13,000, making a profit of $3000. That is a better option, thus you should accept the offer.

Evaluate Effect of Lowered Interest Rate

With a lowered interest rate, calculate the present value function as \(PV = \frac{{100 t^{1 / 2}}}{{(1+0.05)^t}}\). As interest rate decreases, present value of future cash inflows increases. This would make waiting to sell the wine more potentially profitable. However, it still doesn't have a maximum point, so in a purely theoretical sense, holding indefinitely would maximize present value. In practical scenarios assuming cases cannot be held indefinitely, selling immediately might still be best.

Key concepts

Interest Rates

Interest rates are crucial when making investment decisions. They represent the cost of borrowing money or the reward for saving money over time. In our fine wine scenario, a 10% interest rate means money today is worth more than the same amount in the future. This is because 10% interest would grow your money considerably over time if invested elsewhere.

• Higher rates increase the difference in value between immediate and future amounts.
• Investors prefer immediate returns unless future gains are sufficiently large.

If someone offers you $130 per case now and the interest rate is at 10%, taking the immediate offer might be preferred. However, if the rate drops to 5%, waiting becomes more appealing as the future value of money is closer to its present value.

Present Value

Present value (PV) measures how much a future sum of money is worth today. It's calculated by discounting the future value using the interest rate. Present value helps investors decide whether receiving money now or later is more beneficial. In the wine investment, the PV formula used is:
\[ PV = \frac{100 t^{1/2}}{(1+0.1)^t} \]

• A higher interest rate reduces present value because future earnings are worth less.
• Lower rates mean future values lose less value when brought back to today's terms.

By understanding present value, you find that selling the wine now at $100 per case is better because the value decreases the longer you wait.

Profit Maximization

Maximizing profit involves finding the best time to buy and sell to ensure the highest returns. In our exercise, buying 100 cases at $100 each and selling them immediately offers no profit without the $130 offer. However, maximizing profit means considering other factors:

• Immediate $130 offer results in a $3000 profit.
• Unchanged wine value means no added benefit from delaying the sale.

If interest rates drop, future selling becomes a bit more advantageous, but selling immediately is still often most profitable in practical terms. Always aim to compare both the present and potential future values when deciding to maximize profits.