Problem 4 - Applied Calculus

Question

Find the present and future values of an income stream of \(\$ 12,000\) a year for 20 years. The interest rate is \(6 \%\) compounded continuously.

Step-by-Step Solution

Answer

Present value is $127,100.4, and future value is $464,004.

Step 1: Define the problem

We need to find both present and future values of a continuous income stream. The income is $12,000 per year for 20 years, and the interest rate is 6%, compounded continuously.

Step 2: Set up the formula for present value

The present value (PV) of a continuous income stream can be calculated using the formula: \[ PV = \int_{0}^{T} C \, e^{-rt} \, dt \]where \( C = 12000 \) is the annual income, \( r = 0.06 \) is the interest rate, and \( T = 20 \) is the number of years.

Step 3: Solve the integral for present value

Calculate the present value using the integral: \[ PV = \int_{0}^{20} 12000 \, e^{-0.06t} \, dt \]The solution to this integral is: \[ PV = 12000 \times \left[ -\frac{1}{0.06} e^{-0.06t} \right]_{0}^{20} \] Evaluate this expression further to find:\[ PV = 12000 \times \left[ -\frac{1}{0.06} e^{-1.2} + \frac{1}{0.06} \right] \] \[ PV = 12000 \times \left( 16.6667 \times (1 - e^{-1.2}) \right) \] \[ PV \approx 12000 \times 10.5917 \approx 127,100.4 \]

Step 4: Set up the formula for future value

The future value (FV) of a continuous income stream can be calculated using the formula: \[ FV = \int_{0}^{T} C \, e^{r(T-t)} \, dt \] Using the same values for \( C \), \( r \), and \( T \).

Step 5: Solve the integral for future value

Calculate the future value using the integral: \[ FV = \int_{0}^{20} 12000 \, e^{0.06(20-t)} \, dt \] The solution to this integral is: \[ FV = 12000 \times \left[ \frac{1}{0.06} e^{0.06(20-t)} \right]_{0}^{20} \] Evaluate this expression further to find:\[ FV = 12000 \times \left( \frac{1}{0.06} \times (e^{1.2} - 1) \right) \] \[ FV = 12000 \times 16.6667 \times (3.3201 - 1) \] \[ FV \approx 12000 \times 38.667 \approx 464,004 \]

Step 6: Interpret the results

The calculated present value of this income stream is approximately $127,100.4, and the future value is approximately $464,004 at the end of 20 years, with 6% interest compounded continuously.

Key Concepts

Present Value

The present value (PV) is essentially the value of a future amount of money today. It tells us how much a stream of income, received in the future, is worth right now in today's dollars. This concept is crucial when planning financial investments or future income streams.
For a continuous income stream, the formula to calculate the present value is:

• \( PV = \int_{0}^{T} C \, e^{-rt} \, dt \)

Where:

• \( C \) is the annual income (in this exercise, \(12,000),
• \( r \) is the annual interest rate (expressed as a decimal, so 6% becomes 0.06),
• \( T \) is the total time in years (20 years in this case).

By evaluating this integral, we find out the lump sum payment required today, which will be equivalent to receiving \)12,000 annually for 20 years.
This amount is contingent on an interest rate of 6%, compounded continuously.

Future Value

Future value (FV) refers to the value of an investment after it has grown over time, considering the continual compounding effect. It answers the question of what the worth of a regular cash flow will be in the future.
For continuous income streams, the future value is calculated using:

• \( FV = \int_{0}^{T} C \, e^{r(T-t)} \, dt \)

The same variables from the present value formula apply:

• \( C \) - the continuous yearly payment of $12,000,
• \( r \) - the interest rate (0.06 in our example),
• \( T \) - the number of years (20 years).

Solving this integral gives the amount of money you would have at the end of the 20-year period, if you let each cash flow grow at a rate of 6% compounded continuously.
This calculation is pivotal for assessing long-term financial goals or retirement plans.

Continuous Compounding

Continuous compounding refers to the process where interest is calculated and added to the account balance infinitely many times throughout a period.
This method assumes that as soon as you earn interest, it begins earning interest itself, leading to exponential growth over time.
Continuous compounding is more theoretical, yet it's an enlightening concept for understanding the potential growth of investments.

• The formula for continuous compounding is \( A = Pe^{rt} \).

This formula represents:

• \( A \) - the amount of money accumulated after time \( t \),
• \( P \) - the principal amount (initial amount of money),
• \( r \) - the rate of interest,
• \( t \) - the time period the money is invested or borrowed for.

In our example, continuous compounding ensures that every fraction of the $12,000 received per year immediately contributes to interest, leading to a larger future sum than traditional compounding.

Integral Calculus

Integral calculus plays a foundational role in determining the present and future values of continuous income streams.
It provides a framework for calculating accumulated quantities, which represents the heart of financial mathematics in these contexts. Calculus helps us understand how continuous income streams accumulate value over time through:

• Setting up integral expressions that model the financial scenario,
• Solving these integrals to find precise present or future values.

In this exercise, the integral \( \int_{0}^{T} C \, e^{-rt} \, dt \) and its future value counterpart \( \int_{0}^{T} C \, e^{r(T-t)} \, dt \) cover how continuous payments grow or increase in value through time.
Each breakdown in solving these integrals approximates the infinite sum of tiny, compounded interests gained each instant.
Integral calculus efficiently consolidates how money and interest interact continuously, expanding ways to comprehensively evaluate investment opportunities.