Problem 35 - Calculus and its applications

Question

A movie stuntman who receives an annual salary of \(\$ 180,000\) per year is injured and can no longer work. Through a settlement with an insurance company, he is granted a continuous income stream of \(\$ 120,000\) per year for \(20 \mathrm{yr}\). The stuntman invests the money at \(4 \%,\) 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 \(20 \mathrm{yr}\), the stuntman negotiates a flat sum payment from the insurance company, which is the accumulated present value of the continuous income stream. What is that amount? Round your answer to the nearest \(\$ 10\)

Step-by-Step Solution

Answer

a) Future value is approximately \( \$ 3,630,170 \). b) Present value is approximately \( \$ 1,660,450 \).

Step 1: Identify the Formula for Future Value of a Continuous Income Stream

To find the future value of a continuous income stream, use the formula \( FV = \int_{0}^{T} C e^{r(T-t)} \, dt \), where \( C \) is the income stream per year, \( r \) is the interest rate, and \( T \) is the number of years.

Step 2: Substitute the Given Values into the Future Value Formula

Given \( C = 120,000 \), \( r = 0.04 \), and \( T = 20 \), the formula becomes: \( FV = \int_{0}^{20} 120,000 e^{0.04(20-t)} \, dt \).

Step 3: Solve the Integral for Future Value

Evaluate \( \int_{0}^{20} 120,000 e^{0.04(20-t)} \, dt \). The indefinite integral \( \int C e^{rT} e^{-rt} \, dt = \frac{C}{r} e^{rT} (1 - e^{-rT}). \) Applying this to our problem: \( FV = \frac{120,000}{0.04} e^{0.8}(1 - e^{-0.8}) \).

Step 4: Calculate Future Value Using the Evaluated Integral

Calculate \( FV = \frac{120,000}{0.04} (1 - e^{-0.8}) \) and use \( e^{0.8} \approx 2.2255 \). After computations: \( FV \approx \$ 3,630,170 \).

Step 5: Identify the Formula for Present Value of a Continuous Income Stream

To find the present value of a continuous income stream, use the formula \( PV = \int_{0}^{T} C e^{-rt} \, dt \), where \( C \) is the income stream per year, \( r \) is the interest rate, and \( T \) is the number of years.

Step 6: Substitute the Given Values into the Present Value Formula

Given \( C = 120,000 \), \( r = 0.04 \), and \( T = 20 \), the formula becomes: \( PV = \int_{0}^{20} 120,000 e^{-0.04t} \, dt \).

Step 7: Solve the Integral for Present Value

Evaluate \( \int_{0}^{20} 120,000 e^{-0.04t} \, dt \). The indefinite integral is \( \frac{C}{r} (1 - e^{-rT}) \). Hence, \( PV = \frac{120,000}{0.04} (1 - e^{-0.8}) \).

Step 8: Calculate Present Value Using the Evaluated Integral

Calculate \( PV = \frac{120,000}{0.04} (1 - e^{-0.8}) \) using \( e^{-0.8} \approx 0.4493 \). After computations: \( PV \approx \$ 1,660,450 \).

Key Concepts

Future Value Calculation

Understanding how to calculate the future value of a continuous income stream is crucial in finance, particularly for investments. This calculation helps determine how much an investment will be worth in the future when compounded continuously.
Future value is significant because it projects the amount of money an investment will accumulate over time, factoring in consistent contributions and interest rates.
**Formula for Future Value**
The formula to find the future value (FV) of a continuous income stream is: \[FV = \int_{0}^{T} C e^{r(T-t)} \, dt\]Here,

• \(C\) is the continuous income per year.
• \(r\) is the annual interest rate (expressed as a decimal).
• \(T\) is the number of years.

By applying this formula to the initial problem where

• \(C = 120,000\)
• \(r = 0.04\)
• \(T = 20\)

the integral becomes: \[FV = \int_{0}^{20} 120,000 e^{0.04(20-t)} \, dt\] To solve this integral, we evaluate it using integration techniques to obtain the future value, which results in approximately \( \(3,630,170 \) after rounding to the nearest \)10.

Present Value Calculation

Present value (PV) gives the current worth of a future income stream, considering the ongoing discounting due to interest rates over time.
It is an essential concept for understanding the true worth of investments or settlements in present terms.
**Formula for Present Value**
The formula used to determine the present value of a continuous income stream is as follows: \[PV = \int_{0}^{T} C e^{-rt} \, dt\]For this formula, the elements are:

• \(C\) represents the constant income amount per year.
• \(r\) is the interest rate, expressed as a fraction of one.
• \(T\) denotes the time period in years.

In our specific case, using the same figures:

• \(C = 120,000\)
• \(r = 0.04\)
• \(T = 20\)

This results in the expression: \[PV = \int_{0}^{20} 120,000 e^{-0.04t} \, dt\] After solving this integral, the present value is approximately \( \(1,660,450 \) when rounded to the nearest \)10.

Continuous Income Stream

A continuous income stream is a financial concept representing a steady, ongoing flow of income over a period. Unlike discrete incomes, which are received at specific intervals, such as monthly or annually, continuous streams are considered to be unending small payments over the designated period.
This model is especially useful in financial mathematics for evaluating long-term investments or compensations like annuities or settlements.
**Key Aspects of Continuous Income Streams**

• Provides a smooth representation of income over time.
• Utilizes calculus for precise calculations, considering interest rate compounding effects.
• Ideal for long-term investment planning, creating predictable cash flows.

In our exercise, the stuntman's settlement is modeled as a continuous income stream, making it suitable for accurate financial projections and planning, given its uninterrupted nature over the agreed duration of 20 years.