Problem 47 - Calculus Single Variable

Question

An income stream \(f(t)\) is given (in dollars per year with \(t=0\) corresponding to the present). The income will commence \(T_{1}\) years in the future and continue in perpetuity. Calculate the present value of the income stream assuming that the discount rate is \(5 \%\). $$ f(t)=1000+50 t ; T_{1}=0 $$

Step-by-Step Solution

Answer

The present value of the income stream is $60,000.

Step 1: Understanding the Problem

We are given a continuous income stream function \(f(t) = 1000 + 50t\) (dollars per year) that starts immediately (since \(T_1 = 0\)) and continues indefinitely. Our task is to calculate the present value of this income stream with a discount rate of 5%.

Step 2: Identify the Formula for Present Value

The present value of a continuous income stream that begins at \(T_1\) and continues indefinitely, or in perpetuity, is given by the formula: \[ PV = \int_{T_1}^{ ext{infinity}} e^{-rt}f(t) \, dt \] where \(r\) is the discount rate, \(T_1 = 0\), and \(f(t)\) is the income stream.

Step 3: Substitute Values and Set Up the Integral

Substituting \(f(t) = 1000 + 50t\), \(r = 0.05\), and \(T_1 = 0\), the present value integral becomes: \[ PV = \int_{0}^{ ext{infinity}} e^{-0.05t}(1000 + 50t) \, dt \]

Step 4: Integrate the Function

To solve \(\int_{0}^{ ext{infinity}} e^{-0.05t}(1000 + 50t) \, dt\), we can break it into two parts: 1. \(\int_{0}^{ ext{infinity}} 1000e^{-0.05t} \, dt\)2. \(\int_{0}^{ ext{infinity}} 50te^{-0.05t} \, dt\)

Step 5: Calculate the First Integral

For part 1: \(\int_{0}^{ ext{infinity}} 1000e^{-0.05t} \, dt\), Using the fact that the integral of \(k e^{-at}\) from 0 to infinity is \(\frac{k}{a}\) for \(a > 0\), so it becomes:\[ \frac{1000}{0.05} = 20000 \]

Step 6: Calculate the Second Integral

For part 2: \(\int_{0}^{ ext{infinity}} 50te^{-0.05t} \, dt\), we use integration by parts:Let \(u = t\) and \(dv = 50e^{-0.05t}dt\).Then \(du = dt\) and \(v = -1000e^{-0.05t}\).Thus, \(\int u\, dv = uv - \int v\, du \), giving:\[ = -50t(20e^{-0.05t}) \bigg|_0^{ ext{infinity}} - \int_0^{ ext{infinity}} (-20)e^{-0.05t} dt \]Evaluate to get:\[ = 0 + 20(1000)e^{-0.05t} \bigg|_0^{ ext{infinity}} = 0 + 40000 = 40000 \]

Step 7: Sum the Values for Total Present Value

Add the two results together: \(PV = 20000 + 40000 = 60000\).

Key Concepts

continuous income stream

A continuous income stream refers to an ongoing flow of income that is received over time without interruption. In the context of this exercise, it refers to the income function \( f(t) = 1000 + 50t \), where \( t \) is time in years. It represents a scenario where the income starts immediately from \( t = 0 \) and continues indefinitely. This is different from income that is received at discrete intervals, as it allows for smooth and constant accrual of value over time.

To evaluate such continuous income streams in financial analysis, it's essential to calculate the present value. The present value allows us to understand what a future stream of income is worth in today's dollars, considering the time value of money, which is affected by factors like the discount rate.

discount rate

The discount rate is a crucial component in calculating the present value of future cash flows. In simple terms, it's the interest rate used to determine the present worth of future income or costs. The rate reflects the risk and opportunity cost of capital over time, and in this exercise, it is given as 5% (\( r = 0.05 \)).

The discount rate has an important role because it adjusts future cash flows to reflect their present value. A higher discount rate would decrease the present value of future income, showing that money received in the future is worth less today, considering inflation and investment opportunities. Conversely, a lower discount rate increases the present valuation, indicating a smaller opportunity cost or risk associated with future income.

• Discount rate reflects investment risk and opportunity costs.
• Higher rates reduce present values; lower rates increase present value.

integration by parts

Integration by parts is a calculus technique that simplifies the integration of products of functions. It is based on the product rule for differentiation and creates a new integration formula for two functions \( u \) and \( v \):\[\int u \, dv = uv - \int v \, du\]

In this exercise, integration by parts is used to solve the integral \( \int_{0}^{\infty} 50te^{-0.05t} \, dt \). Here, we chose \( u = t \) and \( dv = 50e^{-0.05t}dt \). This allows us to find \( du = dt \) and \( v = -1000e^{-0.05t} \), leading to a simplified integration solution.

Integration by parts helps us manage complex integrals by breaking them down into easier parts. When tackling continuous income streams, it can be particularly useful for handling terms involving exponential functions.

• Simplifies integral calculations for product functions.
• Involves selecting appropriate \( u \) and \( dv \) to make integration easier.

perpetuity

A perpetuity refers to a series of cash flows that continue indefinitely. In terms of an income stream, it means the payments go on forever without an end. This is an important concept in financial mathematics, as it allows for simplified valuation models where the income does not have a predetermined end date.

In the context of the problem, the income stream \( f(t) = 1000 + 50t \) is considered a perpetuity because it starts at \( t = 0 \) and extends infinitely into the future. Calculating the present value of a perpetuity involves integrating from the start time to infinity, incorporating the discount rate to account for the time value of money.

• Perpetuities provide endless cash flows.
• Present value of perpetuity calculated using continuous compounding.