Problem 41 - Applied Calculus

Question

You win $$\$ 38,000$$ in the state lottery to be paid in two installments -$$\$ 19,000$$ now and $$\$ 19,000$$ one year from now. A friend offers you $$\$ 36,000$$ in return for your two lottery payments. Instead of accepting your friend's offer, you take out a one-year loan at an interest rate of \(8.25 \%\) per year, compounded annually. The loan will be paid back by a single payment of $$\$ 19,000$$ (your second lottery check) at the end of the year. Which is better, your friend's offer or the loan?

Step-by-Step Solution

Answer

Taking the loan is better, with a higher present value of \$36,555.56 compared to \$36,000 from your friend.

Step 1: Determine the Present Value of the Loan Repayment

To assess the attractiveness of not accepting your friend's offer, we need to calculate the present value of repaying the loan with the second lottery payment. The formula to calculate the present value (PV) is: \[ PV = \frac{FV}{(1 + r)^n} \]where FW is the future value, \(r\) is the interest rate, and \(n\) is the number of years. Thus, for the loan:\[ PV = \frac{19,000}{(1 + 0.0825)^1} \approx 17,555.56 \] This means the present value of the future repayment of \\(19,000 is approximately \\)17,555.56.

Step 2: Calculate the Total Value of Winning Distribution under the Loan

Now we calculate the total present value of the winnings if one decides to keep the lottery payments and take the loan. You will receive \\(19,000 now, plus the present value of the second installment:\[ \text{Total present value} = 19,000 + 17,555.56 = 36,555.56 \] Thus, the total present value when taking the loan is approximately \\)36,555.56.

Step 3: Compare with the Friend's Offer

Compare the total present value with your friend's offer of \\(36,000. Since the present value of keeping the lottery payments (\\)36,555.56) is higher than your friend’s offer (\$36,000), keeping the lottery payments and taking the loan is the better option.

Key Concepts

Future Value

The future value (FV) is an important concept in finance that refers to the amount of money an investment will grow over a period of time. In the context of our lottery problem, the future value is the second lottery payment of $19,000 that you will receive one year from now. Understanding FV helps you predict how much an investment made today will be worth in the future. It accounts for factors like the interest rate and the time period for the investment. This concept is vital when you are trying to compare different offers or investments over time.

Interest Rate

The interest rate plays a crucial role in determining both the future value and present value of money. In our scenario, you took a one-year loan with an annual interest rate of 8.25%. The interest rate indicates how much extra you will have to pay on top of the borrowed amount. It is a percentage of the borrowed sum. For instance, with compound interest, which is what we have here, the interest accumulates not just on the initial principal but also on the accumulated interest over previous periods. Thus, a higher interest rate usually means a higher cost of borrowing, impacting your financial decisions.

Lottery Payments

Lottery payments can be structured in various ways. In this exercise, you have won a lottery with split payments: $19,000 immediately and $19,000 after one year. The immediate payment is simple cash in hand, while the future payment is subject to current value considerations. The structured nature of your lottery winnings requires careful consideration of present and future values. The choice between accepting a lump sum or payments over time depends significantly on present value calculations and your financial strategies.

Loan Repayment

Loan repayment is vital to managing loans effectively. By the end of the year, you'll repay your loan using your second lottery installment. The key here is calculating the present value of this future repayment to determine if keeping the lottery payments while taking the loan is more beneficial than your friend's offer. This strategy allows you to evaluate whether the total value of your payments after accounting for the loan interest is worth more than the offer on the table. Proper calculation and comparison can help in making informed financial choices.