Find the present value of a continuous income stream F(t) = 20 + 5t, where t is in years and F is in thousands of dollars per year, for 5 years, if money can earn 2.5% annual interest, compounded continuously- Present value = thousand dollars.
Find the present value of a continuous income stream F(t) = 20 + 5t, where t is in years and F is in thousands of dollars per year, for 5 years, if money can earn 2.5% annual interest, compounded continuously- Present value = thousand dollars.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Present Value of Continuous Income Stream:**
Given a continuous income stream \( F(t) = 20 + 5t \), where \( t \) is in years and \( F \) is in thousands of dollars per year. The timeline is for 5 years, and money can earn 2.5% annual interest, compounded continuously.
**Objective:** Find the present value \( PV \) of this continuous income stream.
The present value \( PV \) can be found using the formula:
\[
PV = \int_{0}^{T} F(t) e^{-rt} \, dt
\]
where:
- \( T \) is the total time (5 years),
- \( F(t) = 20 + 5t \),
- \( r \) is the continuous compounding interest rate (2.5% or 0.025).
**Step-by-Step Calculation:**
1. **Substitute the values into the present value formula:**
\[
PV = \int_{0}^{5} (20 + 5t) e^{-0.025t} \, dt
\]
2. **Solve the integral (requires calculus):**
\[
PV = \int_{0}^{5} 20 e^{-0.025t} \, dt + \int_{0}^{5} 5t e^{-0.025t} \, dt
\]
These integrals can be evaluated using integration by parts or a definite integral calculator.
3. **Compute the result:**
The final answer will provide the present value of the continuous income stream.
**Result:**
\[
\text{Present value} = \_\_\_\_ \text{ thousand dollars}.
\]
(The blank is for calculation or instructional purposes, to be filled in with the computed value.)
This mathematical exercise demonstrates how to find the present value of a continuous income stream, which is essential in financial mathematics for assessing investment opportunities and understanding the time value of money.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d56d87b-1ee4-40b0-9831-f06655864a4b%2F27d1ffba-3008-4fe9-8173-d011be9524a1%2F60q0sdj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Present Value of Continuous Income Stream:**
Given a continuous income stream \( F(t) = 20 + 5t \), where \( t \) is in years and \( F \) is in thousands of dollars per year. The timeline is for 5 years, and money can earn 2.5% annual interest, compounded continuously.
**Objective:** Find the present value \( PV \) of this continuous income stream.
The present value \( PV \) can be found using the formula:
\[
PV = \int_{0}^{T} F(t) e^{-rt} \, dt
\]
where:
- \( T \) is the total time (5 years),
- \( F(t) = 20 + 5t \),
- \( r \) is the continuous compounding interest rate (2.5% or 0.025).
**Step-by-Step Calculation:**
1. **Substitute the values into the present value formula:**
\[
PV = \int_{0}^{5} (20 + 5t) e^{-0.025t} \, dt
\]
2. **Solve the integral (requires calculus):**
\[
PV = \int_{0}^{5} 20 e^{-0.025t} \, dt + \int_{0}^{5} 5t e^{-0.025t} \, dt
\]
These integrals can be evaluated using integration by parts or a definite integral calculator.
3. **Compute the result:**
The final answer will provide the present value of the continuous income stream.
**Result:**
\[
\text{Present value} = \_\_\_\_ \text{ thousand dollars}.
\]
(The blank is for calculation or instructional purposes, to be filled in with the computed value.)
This mathematical exercise demonstrates how to find the present value of a continuous income stream, which is essential in financial mathematics for assessing investment opportunities and understanding the time value of money.
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