Problem 36 - Calculus and its applications

Question

Dale was a furnace maintenance employee who earned a salary of \(\$ 70,000\) per year before he was injured on the job. Through a settlement with his employer's insurance company, he is granted a continuous income stream of \(\$ 40,000\) per year for 25 yr. Dale invests the money at \(5 \%,\) compounded continuously. a) Find the accumulated future value of the continuous income stream. Round your answer to the nearest \(\$ 10\) b) Thinking that he might not live for \(25 \mathrm{yr},\) Dale negotiates a flat sum payment from the insurance company, which is the accumulated present value of the continuous stream plus \(\$ 100,000 .\) What is that amount? Round your answer to the nearest \(\$ 10 .\)

Step-by-Step Solution

Answer

a) $ $1992240 b) $ $670790

Step 1: Understand the Future Value of a Continuous Income Stream

The future value (FV) of a continuous income stream where \(R\) is the rate of the stream, \(r\) is the interest rate, and \(T\) is the time period, is given by the formula:\[ FV = R \left(\frac{e^{rT} - 1}{r}\right) \]In Dale's case, \(R = 40000\), \(r = 0.05\), and \(T = 25\). Plug these values into the formula.

Step 2: Calculate Future Value

Using the formula from Step 1, substitute the values:\[ FV = 40000 \left(\frac{e^{0.05 \times 25} - 1}{0.05}\right) \]Calculate this expression to find the future value.

Step 3: Perform Calculations for Future Value

Calculate the exponential component first:\[ e^{0.05 \times 25} \approx e^{1.25} \approx 3.4903 \]Thus, the future value calculation becomes:\[ FV = 40000 \left(\frac{3.4903 - 1}{0.05}\right) \approx 40000 \times 49.806 \approx 1992240 \]Round this to the nearest $$10, which gives $$1992240.

Step 4: Understand Present Value of Continuous Income Stream

The present value (PV) of a continuous income stream is given by:\[ PV = R \left(\frac{1 - e^{-rT}}{r}\right) \]Using \(R = 40000\), \(r = 0.05\), and \(T = 25\), substitute these into the formula to find the present value.

Step 5: Calculate Present Value

Substitute the values and calculate the expression:\[ PV = 40000 \left(\frac{1 - e^{-0.05 \times 25}}{0.05}\right) \]Calculate this expression to find the present value.

Step 6: Perform Calculations for Present Value

Calculate the exponential component first:\[ e^{-0.05 \times 25} \approx e^{-1.25} \approx 0.2865 \]Thus, the present value calculation becomes:\[ PV = 40000 \left(\frac{1 - 0.2865}{0.05}\right) \approx 40000 \times 14.27 \approx 570800 \]Round this to the nearest $$10, which gives $$570790.

Step 7: Calculate the Negotiated Flat Sum Payment

Dale negotiates a payment which is the accumulated present value plus an extra $100,000:\[ \text{Flat Sum} = 570790 + 100000 = 670790 \]

Step 8: Finalize Answer

The future value of Dale's continuous income stream is $ $1992240 rounded to the nearest $ $10, and the negotiated flat sum payment is $ $670790.

Key Concepts

Future Value of Continuous Income Stream

When you're dealing with a continuous income stream, like the one Dale has with \\(40,000 per year for 25 years, you want to know how much this stream will be worth in the future. This is called the future value, and it's all about calculating how much money you'll accumulate over time with the power of compound interest.

To find this, we use a specific formula: \[FV = R \left(\frac{e^{rT} - 1}{r}\right)\]Where:

• \(FV\) is the future value.
• \(R\) is the rate of income per year, which is \\)40,000 for Dale.
• \(r\) is the interest rate, which is 5\% or 0.05.
• \(T\) is the time in years, which is 25 for Dale's stream.

The expression \(e^{rT}\) involves \(e\), which is a mathematical constant approximately equal to 2.71828. We multiply our interest rate by the time period and take the power of \(e\), then subtract 1 and divide by the interest rate. This equation calculates how the income stream grows over time by taking into account continuous compounding, which means interest is being added at every possible instant.

By plugging Dale's numbers into the formula, we find his income stream will grow significantly over 25 years, illustrating the power of continuous compounding in transforming a steady income stream into substantial savings.

Present Value Calculation

When you need your money's worth in today’s terms, you calculate the present value (PV) of a future cash flow. This concept helps you understand the value of a future income stream if it were worth something today. This is particularly useful for decisions like Dale's, where he's considering taking a lump sum now instead of regular payments over time.

The formula for calculating the present value of a continuous income stream is:\[PV = R \left(\frac{1 - e^{-rT}}{r}\right)\]Here, the components stay quite similar:

• \(PV\) represents the present value, the worth of the future income stream right now.
• The \(1 - e^{-rT}\) factor adjusts the future values back to present-day terms, countering the growth represented by \(e\) in the future value formula.
• Just like before, \(R\) is the income rate, and \(r\) is the interest rate.
• \(T\) is the total time in years.

By using Dale's specific figures in this formula, we translate his future income over 25 years into a single present value sum. This allows him to negotiate a flat sum payment right now, which can be more relevant if he's unsure about the duration over which he'll need funds.

Ultimately, this calculation enables decisions between one large payment or many small payments over time, making it a crucial tool in financial planning.

Continuous Compounding

Continuous compounding involves calculating interest so frequently that it's essentially earned constantly. This contrasts with traditional compounding, which occurs at regular intervals like annually or monthly. By assuming the compounding happens continuously, the formula uses the mathematical constant \(e\).

The growth of an investment with continuous compounding over time is represented by the formula \(A = Pe^{rt}\), where:

• \(A\) is the amount of money accumulated after n years, including interest.
• \(P\) is the principal amount (initial investment).
• \(r\) is the annual interest rate (as a decimal).
• \(t\) is the time the money is invested for in years.

What's key to understanding is that \(e\) operates as a growth multiplier, which makes your money grow faster than it would under simpler compounding methods. The idea is that the frequency of compounding can greatly affect the final amount, especially with larger sums and longer time periods.

In Dale’s context, using continuous compounding allows his income stream to grow as aggressively as possible, maximizing the future and present values of his settlement, thanks to the profoundly powerful effect of compound interest working all the time. Thus, understanding continuous compounding helps in seeing how small regular payments can lead to significant accumulated wealth over time.