Problem 8 - Applied Calculus

Question

(a) Find the present and future value of an income stream of \(\$ 6000\) per year for a period of 10 years if the interest rate, compounded continuously, is \(5 \%\) (b) How much of the future value is from the income stream? How much is from interest?

Step-by-Step Solution

Answer

PV = $46,977.46, FV = $76,815.73; Income Stream: $60,000; Interest: $16,815.73.

Step 1: Define the Present Value Formula

The present value (PV) of a continuous income stream is given by the formula:\[ PV = \int_{0}^{T} P(t) e^{-rt} \, dt \]where \(P(t)\) is the rate of income per year, \(r\) is the interest rate, and \(T\) is the time period in years. Here, \(P(t) = 6000\), \(r = 0.05\), and \(T = 10\).

Step 2: Calculate Present Value

Substitute the values into the formula:\[ PV = \int_{0}^{10} 6000 \, e^{-0.05t} \, dt \]Evaluate this integral:\[ PV = 6000 \left[ -\frac{1}{0.05} e^{-0.05t} \right]_{0}^{10} \]\[ PV = 6000 \left( -20(e^{-0.5}) + 20(1) \right) \]\[ PV = 6000 (20 - 20e^{-0.5}) \]Calculate the numerical value to find \(PV\).

Step 3: Define the Future Value Formula

The future value (FV) of a continuous income stream is calculated by accumulating the present value at the continuous compound rate:\[ FV = PV \cdot e^{rT} \]You can find \(PV\) from the previous step and then use this formula to evaluate \(FV\).

Step 4: Calculate Future Value

After finding the present value, use \(FV = PV \cdot e^{0.05 \times 10}\) to calculate the future value. Compute this numerical value.

Step 5: Determine Contribution from Income Stream

The total contribution from the income stream itself is simply the yearly income multiplied by the number of years. Hence, it is:\[ 6000 \times 10 = 60000 \]

Step 6: Determine Contribution from Interest

The contribution from interest is the difference between the future value and the total income stream contribution:\[ \text{Interest Contribution} = FV - 60000 \]

Key Concepts

Present Value Calculation

When planning to evaluate a continuous income stream, calculating the present value is crucial. It helps you determine how much that future income is worth today. Think of it like a reverse compound interest calculation. Instead of accepting future money at face value, present value considers how much you need today to grow into that future sum at a certain interest rate.

For continuous income streams, we use an integral calculus formula to calculate the present value:

• Formula: \[ PV = \int_{0}^{T} P(t) e^{-rt} \, dt \]
• Here, \(P(t)\) is the annual income rate, \(r\) is the interest rate, and \(T\) is the time period in years.

This approach accounts for the flow of income continuously, applying the exponential discounting factor \(e^{-rt}\) to each small segment of income over the timeframe.

Future Value Calculation

Future value calculations help to understand what your investment, made today, will grow into over time. In our exercise, after determining the present value, it's useful to know how this value will compound over time to become the future value. This step uses the concept of continuous compounding interest.

Using the formula:

• \[ FV = PV \cdot e^{rT} \]

We take the present value previously calculated and apply the compound interest formula. Here:\
\(PV\) is the present value, \(e^{rT}\) is the continuous growth factor for the period \(T\) years at interest rate \(r\). With these values, you can determine exactly what today's investment will amount to in the future, illustrating the power of compound growth.

Compound Interest

Compound interest is the magic ingredient that allows present investments to grow into larger amounts in the future. This growth is exponential, meaning it accelerates over time as interest earns not only on your initial principal but also on the accumulated interest from previous periods.

Continuously compounded interest, as used in this problem, takes this to another level. Instead of calculating interest at regular intervals, continuous compounding assumes every moment is interest earning:

• Formula: \( A = P e^{rt} \)

This is a powerful concept because it maximizes the potential growth of an investment, and is highly relevant in financial mathematics and investment strategies. Understanding this will give you insight into why small differences in interest rate and time period can significantly affect the outcome.

Integral Calculus

Integral calculus comes into play when dealing with continuous functions. Unlike standard formulas that deal with fixed amounts at intervals, integrals help calculate accumulations over a continuous range. It's like drawing the complete picture, taking into account all moments in time when calculating values like present and future value of continuous income streams.

The formula for present value, as shown earlier, incorporates integral calculus because it involves summing up the value of income flow at infinitesimally small segments over the entire time period:

• \[ PV = \int_{0}^{T} P(t) e^{-rt} \, dt \]

This approach is precise and accurate, allowing for detailed planning and assessment of continuous cash flows.