Problem 11
Question
Find the accumulated future value of each continuous income stream at rate \(\mathrm{R}(t),\) for the given time \(\mathrm{T}\) and interest rate \(k\) compounded continuously. $$ R(t)=\$ 400,000, \quad T=20 \mathrm{yr}, \quad k=4 \% $$
Step-by-Step Solution
Verified Answer
The accumulated future value is \$12,255,400.
1Step 1: Understanding the Problem
We are asked to find the accumulated future value of a continuous income stream with a constant rate of \( R(t) = \$400,000 \) over \( T = 20 \) years, with an interest rate of \( k = 4\% \) compounded continuously.
2Step 2: Formula for Future Value of Continuous Income Stream
Use the formula to find the future value of a continuous income stream: \[ FV = \int_0^T R(t) e^{k(T-t)} \, dt \]. Here, since \( R(t) \) is constant, it simplifies to \( R(t) = 400,000 \), and \( k = 0.04 \).
3Step 3: Setting Up the Integral
The integral becomes \[ FV = \int_0^{20} 400,000 \, e^{0.04(20-t)} \, dt \]. This integral represents the sum of all continuous payments made over time, each compounded continuously at \( 4\% \).
4Step 4: Simplifying the Exponential
Rewrite the exponential term as \( e^{0.8 - 0.04t} \), so the integral becomes \[ FV = 400,000 \int_0^{20} e^{0.8 - 0.04t} \, dt \].
5Step 5: Integrating the Exponential Function
The integral of \( e^{0.8 - 0.04t} \) is \( -25 e^{0.8 - 0.04t} \) (consider the negative reciprocal of \( -0.04 \)). Thus, \[ FV = 400,000 \left[ -25 e^{0.8 - 0.04t} \right]_0^{20} \].
6Step 6: Calculating the Definite Integral
Substitute the limits of integration: \[ FV = 400,000 \left( -25 e^{0.8 - 0.04 \times 20} + 25 e^{0.8} \right) \].
7Step 7: Evaluating the Expressions
Calculate \( e^{0.8 - 0.8} = e^0 = 1 \) and \( e^{0.8} \approx 2.22554 \). Substitute these into the expression to get: \[ FV = 400,000 \left( -25 \times 1 + 25 \times 2.22554 \right) \].
8Step 8: Final Computation
Compute \( -25 + 25 \times 2.22554 \approx -25 + 55.6385 = 30.6385 \). Thus, \( FV = 400,000 \times 30.6385 \approx 12,255,400 \).
9Step 9: Conclusion
The accumulated future value of the continuous income stream is \$12,255,400 at the end of 20 years with a \( 4\% \) continuously compounded interest rate.
Key Concepts
Calculus Applications in FinanceContinuous CompoundingIntegral Calculus
Calculus Applications in Finance
Calculus is an essential tool in financial analysis, particularly when dealing with complex financial scenarios like continuous income streams. In finance, calculus helps in understanding and solving problems related to rate changes, growth, and decay over time.
One common application in finance is calculating the future value of cash flows that occur continuously over a period. This is often seen when businesses or individuals want to know the value of a steady income stream in the future, taking into account interest rates.
The problem is typically addressed by using integral calculus, which allows us to sum all infinitesimally small income slices over a period. This finds its roots in the fundamental theorem of calculus, linking differentiation with integration. By applying calculus, we can manage the continuous changes and growth accurately, proving its significant role in financial decision-making.
One common application in finance is calculating the future value of cash flows that occur continuously over a period. This is often seen when businesses or individuals want to know the value of a steady income stream in the future, taking into account interest rates.
The problem is typically addressed by using integral calculus, which allows us to sum all infinitesimally small income slices over a period. This finds its roots in the fundamental theorem of calculus, linking differentiation with integration. By applying calculus, we can manage the continuous changes and growth accurately, proving its significant role in financial decision-making.
Continuous Compounding
Continuous compounding is a concept where interest is calculated and added to the initial amount continuously, rather than at discrete intervals (such as monthly or annually). This means that the interest is constantly being calculated and applied.
The formula used for continuous compounding is derived from the exponential function, where the future value is given by the expression:
- \[ FV = P imes e^{kt} \] - where \( P \) is the principal amount, \( k \) is the annual interest rate, and \( t \) is the time in years.
Continuous compounding allows for the most accurate calculation of growth over time, as it accounts for the added interest at each possible tiny fraction of time. In finance, this is especially useful when determining the future value of investments and comparing different investment opportunities that may compound interest at varying intervals.
The formula used for continuous compounding is derived from the exponential function, where the future value is given by the expression:
- \[ FV = P imes e^{kt} \] - where \( P \) is the principal amount, \( k \) is the annual interest rate, and \( t \) is the time in years.
Continuous compounding allows for the most accurate calculation of growth over time, as it accounts for the added interest at each possible tiny fraction of time. In finance, this is especially useful when determining the future value of investments and comparing different investment opportunities that may compound interest at varying intervals.
Integral Calculus
Integral calculus plays a crucial role in determining the accumulation of quantities over time, which is particularly applicable to continuous financial flows. By using integrals, you can calculate the total sum of an infinite number of instantaneous values over a certain period, capturing the entire picture of financial growth.
In the context of our exercise, we use the integral:
- \[ FV = \int_0^T R(t) e^{k(T-t)} \, dt \] - where \( R(t) \) is the continuous rate, \( T \) is the total time in years, and \( k \) represents the interest rate. The key here is that the integral function accounts for the entire timeframe, incorporating continuous changes in the income stream and interest rate application.
The result of this integral gives us a precise figure for what the future accumulated value will be, offering more rigorous insights compared to simple interest calculations.
In the context of our exercise, we use the integral:
- \[ FV = \int_0^T R(t) e^{k(T-t)} \, dt \] - where \( R(t) \) is the continuous rate, \( T \) is the total time in years, and \( k \) represents the interest rate. The key here is that the integral function accounts for the entire timeframe, incorporating continuous changes in the income stream and interest rate application.
The result of this integral gives us a precise figure for what the future accumulated value will be, offering more rigorous insights compared to simple interest calculations.
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