Question

Given a continuous income stream at the constant rate of \(\$ 1,000\) per year: (a) What will be the present value \(\Pi\) if the income stream lasts for 2 years and the continuous discount rate is 0.05 per year? (b) What will be the present value \(\Pi\) if the income stream terminates after exactily 3 years and the discount rate is \(0.04 ?\)

Short answer

(a) $1,904, (b) $2,827.50.

Step-by-step solution

Define the Problem

We need to find the present value of a continuous income stream of $1,000 per year over a specified time period with given continuous discount rates. The formula to use is: \[\Pi = \int_0^T R(t) \cdot e^{-rt} \, dt\] where \(R(t)\) is the income rate, \(r\) is the continuous discount rate, and \(T\) is the time duration.

Understand the Given Values for Part (a)

For part (a), the income \(R(t)\) is $1,000 per year, the duration \(T\) is 2 years, and the continuous discount rate \(r\) is 0.05 per year.

Apply the Formula for Part (a)

The present value formula becomes: \[\Pi = \int_0^2 1000 \cdot e^{-0.05t} \, dt\]. We need to evaluate this integral.

Solve the Integral for Part (a)

The integral can be solved as: \[\Pi = 1000 \left[-\frac{1}{0.05} e^{-0.05t} \right]_0^2 = 1000 \left[-20 e^{-0.05t} \right]_0^2 \]. Calculating this gives: \[= 1000 \left(-20 e^{-0.1} + 20 \right) \].

Calculate Values for Part (a)

Calculate the exponents as follows: \(e^{-0.1} \approx 0.9048\). Thus, \(-20 \times 0.9048 + 20 = 1.904\) and \(1000 \times 1.904 = 1904\). The present value \(\Pi\) is $1,904.

Understand the Given Values for Part (b)

For part (b), the income \(R(t)\) remains $1,000 per year, the duration \(T\) is 3 years, and the continuous discount rate \(r\) is 0.04 per year.

Apply the Formula for Part (b)

The present value formula becomes: \[\Pi = \int_0^3 1000 \cdot e^{-0.04t} \, dt\]. We must evaluate this integral.

Solve the Integral for Part (b)

The integral can be solved as: \[\Pi = 1000 \left[-\frac{1}{0.04}e^{-0.04t} \right]_0^3 = 1000 \left[-25 e^{-0.04t} \right]_0^3 \]. Calculating gives \[= 1000 \left(-25 e^{-0.12} + 25 \right)\].

Calculate Values for Part (b)

Calculate the exponents as follows: \(e^{-0.12} \approx 0.8869\). Thus, \(-25 \times 0.8869 + 25 = 2.8275\) and \(1000 \times 2.8275 = 2,827.50\). The present value \(\Pi\) is $2,827.50.

Key concepts

Continuous Income Stream

A continuous income stream is a financial setup where money flows continuously into an account or investment at a constant rate. Imagine water flowing steadily into a container. That's precisely how a continuous income stream works, except with money.

• The term "continuous" implies that the income enters seamlessly over time, without interruptions.
• When calculating financial metrics like present value, this approach is more sophisticated and realistic than one-time or periodic payments.
• In real life, examples might include ongoing rental income, perpetual annuities, or royalties from intellectual property.

The key benefit of understanding continuous income streams is its ability to represent long-term cash flows accurately, allowing for better financial planning and decision-making.

Continuous Discount Rate

The continuous discount rate is a concept in finance that applies to continuously compounded interest. Think of it as the rate at which the value of future cash flows is reduced to reflect their present worth. Mathematically, it makes use of the constant base of Euler's number, approximately 2.718, noted as "e".

• This rate is continuous because it applies at every moment, rather than at discrete intervals like month-end or year-end.
• The formula involving continuous discounting is often written with the exponential function, such as \( e^{-rt} \), where \( r \) is the continuous discount rate and \( t \) is time.
• This approach might feel abstract, but it’s more aligned with how real-world financial systems operate, offering precision in calculations.

Using a continuous discount rate provides a refined estimation of current value in comparison to traditional, simpler discount methods.

Definite Integration

Definite integration is a core mathematical concept used here to calculate the net present value of a cash flow over a certain time period. It involves taking the integral of a function over a specified range, providing the total value of what lies "under the curve".

• In the context of finance, the integration process adjusts our income stream for the effects of continuous discounting.
• By integrating across time, we accumulate and sum up the tiny slices of discounted income.
• For example, in the given problem, the present value formula uses definite integration from 0 to 2 years or 3 years, depending on the case.

Through definite integration, one can precise quantify continuous cash flows and their present values, even when dealing with continuous rates.

Present Value Formula

The present value formula is an essential tool in finance for determining how much future cash flows are worth today. For continuous cash flows, this formula is adapted to include integration to account for the unbroken nature of the income.

• In its continuous form, the present value formula is written as \( \Pi = \int_0^T R(t) \cdot e^{-rt} \, dt \).
• Here, \( R(t) \) is the rate of income at time \( t \), while \( e^{-rt} \) adjusts each dollar to reflect its current worth, using the continuous discount rate \( r \).
• This formula provides a holistic value of future income, allowing individuals and businesses to make informed investment or spending decisions.

Understanding the present value formula and its application in scenarios involving continuous cash flows is vital for anyone assessing long-term financial projects.