Problem 37 - Calculus and its applications

Question

Lucky Larry wins \(\$ 1,000,000\) in a state lottery. The standard way in which a state pays such lottery winnings is at a constant rate of \(\$ 50,000\) per year for 20 yr. a) If Lucky invests each payment from the state at \(4.4 \%,\) compounded continuously, what is the accumulated future value of the income stream? Round your answer to the nearest \(\$ 10 .\) b) What is the accumulated present value of the income stream at \(4.4 \%,\) compounded continuously? This amount represents what the state has to invest at the start of its lottery payments, assuming the \(4.4 \%\) interest rate holds. c) The risk for Lucky is that he doesn't know how long he will live or what the future interest rate will be; it might drop or rise, or it could vary considerably over \(20 \mathrm{yr}\). This is the risk he assumes in accepting payments of \(\$ 50,000\) a year over 20 yr. Lucky has taken a course in business calculus so he is aware of the formulas for accumulated future value and present value. He calculates the accumulated present value of the income stream for interest rates of \(3 \%, 4 \%,\) and \(5 \% .\) What values does he obtain? d) Lucky thinks "a bird in the hand (present value) is worth two in the bush (future value)" and decides to negotiate with the state for immediate payment of his lottery winnings. He asks the state for \(\$ 750,000\). They offer \(\$ 600,000 .\) Discuss the pros and cons of each amount. Lucky finally accepts \(\$ 675,000 .\) Is this a good decision? Why or why not?

Step-by-Step Solution

Answer

Lucky's decision to accept $675,000 is reasonable as it aligns with the present value given interest variations. It's a balanced choice considering uncertainty and the state's offer.

Step 1: Calculating the Accumulated Future Value

The future value of a continuous income stream is calculated using the formula \[ FV = P \cdot \left( e^{rt} - 1 \right) \] where \( P \) is the payment amount per year, \( r \) is the interest rate, and \( t \) is the number of years. With \( P = 50000 \), \( r = 0.044 \), and \( t = 20 \), calculate \[ FV = 50000 \cdot \left( e^{0.044 \cdot 20} - 1 \right) \] to find the future value for each year and sum for 20 years. The total accumulated future value is calculated to be approximately \( \$1,600,000 \).

Step 2: Calculating the Accumulated Present Value

To calculate the present value of the income streams, use the formula for continuous compounding: \[ PV = P \cdot \frac{1 - e^{-rt}}{r} \] where \( P = 50000 \), \( r = 0.044 \), and \( t = 20 \). Calculate \[ PV = 50000 \cdot \frac{1 - e^{-0.044 \cdot 20}}{0.044} \] which will give the total present value needed now. The accumulated present value is calculated to be approximately \( \$700,000 \).

Step 3: Present Value for Different Interest Rates

Using the formula \[ PV = P \cdot \frac{1 - e^{-rt}}{r} \], repeat the calculation for \( r = 0.03, 0.04, 0.05 \) with \( P = 50000 \) and \( t = 20 \). For \( r = 0.03 \), \[ PV \approx 742,034 \]. For \( r = 0.04 \), \[ PV \approx 707,212 \]. For \( r = 0.05 \), \[ PV \approx 673,449 \]. These calculations illustrate how a varying interest rate impacts the present value.

Step 4: Analyzing Immediate Payment Options

Lucky is considering immediate payment options of \( \\(750,000 \) and \( \\)600,000 \) against the calculated present value of \( \\(700,000 \). **Pros of \\)750,000:** Immediate access to a larger sum compared to the computed present value. **Pros of \\(600,000:** Immediate guarantee, although less favorable compared to the computed present value. **Lucky's Acceptance of \\)675,000:** This decision is in the middle of the state's offer and his demand, a compromise that is above the computed present value of \( \\(673,449 \) at the maximum interest scenario but below his desired \\)750,000.

Key Concepts

Accumulated Future Value

When thinking about accumulated future value, consider it as the amount of money you'll end up with in the future after consistently saving or investing over time. In Lucky's situation, the future value of his winnings is determined by how much he invests, how often he invests, and for how long.
The formula for calculating the accumulated future value of a continuously compounded income stream is: \[ FV = P \cdot \left( e^{rt} - 1 \right) \] where:

• \( P \) is the periodic payment amount
• \( r \) is the interest rate
• \( t \) is the time period in years

In Lucky's case, he invests \( P = 50,000 \) dollars every year for 20 years at a rate of 4.4%. To find out how much this will grow in 20 years, you substitute these values into the formula.
This allows Lucky to see that his \( 50,000 \) yearly payments could turn into about \( 1,600,000 \) dollars over 20 years due to continuous compounding. This means that over time, his money grows significantly because interest is continually added to the principal, rather than only added at the end of each period such as yearly or monthly.

Present Value

The present value is the concept of determining how much a future amount of money is worth today. This might seem the opposite of calculating future value because it requires thinking about today's worth instead of tomorrow's gains. If someone offered you \( 1,000 \) dollars today or \( 1,000 \) dollars in a year, you'd probably take the money now. Because, in a year, it won't have the same value due to inflation and missed investment opportunities.
The formula for calculating the present value of continuous payments is: \[ PV = P \cdot \frac{1 - e^{-rt}}{r} \] Here,

• \( P \) is the payment amount
• \( r \) is the interest rate
• \( t \) is the time in years

Using this formula with Lucky's scenario, by plugging in a payment \( P \) of \( 50,000 \) dollars for 20 years at an interest rate of 4.4%, he finds the current value to be about \( 700,000 \). This indicates what the state should invest at the beginning to provide the \( 50,000 \) yearly outflows, assuming the same interest rate.

Continuous Compounding

Continuous compounding means that your investment is growing continuously, without interruption. Instead of seeing your money grow periodically, like quarterly or annually, continuous compounding lets your investments grow at every possible moment.
With other types of compounding, interest is calculated at specific intervals (e.g., annually, quarterly). But with continuous compounding, the interest is added at an infinitesimally small intervals, which is mathematically represented by the constant \( e \), approximately 2.718.
This concept is crucial to understanding Lucky's decision-making, as it affects both accumulated future and present value. The formula used extensively in these calculations, \( e^{rt} \), represents this continuous growth and can be crucial in maximizing financial outcomes. With continuous compounding, Lucky's investments grow not just annually but every second, minute, and hour, magnifying the growth rate.

Interest Rates

Interest rates are a fundamental part of both finance and economics, affecting the cost of borrowing and the return on investment. They denote the percentage at which your investments or borrowings grow over time.
In Lucky's scenario, the interest rate is 4.4% per annum, compounded continuously. However, it’s essential to consider that interest rates can fluctuate due to economic conditions, impacting the value of future income streams. Different rates can provide different present values, as illustrated by Lucky's calculations at 3%, 4%, and 5%.

• The higher the interest rate, the less the present value, assuming the investment grows faster to reach the same future value.
• Conversely, lower rates boost the present value as future growth potential shrinks.

For retirees, investors, or Lucky, understanding how these rates influence investment decisions is crucial in optimizing financial strategies.

Business Calculus

Business calculus applies mathematical principles to solve financial and economic problems. Part of its utility is in understanding how to operate with functions and derivatives within business-related scenarios.
Lucky’s application of calculus is through formulas like those determining future and present values using continuous compounding, fundamentally rooted in exponential functions. These allow him to make educated decisions regarding his winnings.
Because uncertainty in life is a constant challenge, tools from business calculus help evaluate different financial scenarios, accounting for variables like changing interest rates. Mastering these concepts equips one with the ability to logically analyze and project financial outcomes, a skill Lucky uses to weigh his options for accepting or negotiating his prize. They form the toolkit required for efficient business analysis and subsequent decision-making.