Income Streams & Annuities

Income Streams

Definition and Notation

An income stream refers to income that is generated continuously and transferred into an account that earns interest at a fixed rate. Interest is assumed to be compounded continuously.

• \(R(t)\) = rate at which income is generated (in dollars per year)

• \(T\) = length of time (in years) of the income stream

• \(r\) = annual interest rate (compounded continuously)

Future Value

The future value of an income stream is the total of all of the money transferred plus all of the interest earned and is given by

\[A = e^{rT}\int_0^T R(t) e^{-rt} ~dt.\]

Present Value

The present value of an income stream is the principal investment, \(PV\), that yields the same accumulated value as the income stream when \(PV\) is invested for a period of \(T\) years at the same interest rate and is given by

\[PV = \int_0^T R(t) e^{-rt} ~dt.\]

Example 1

Future value of an income stream

Penn State Learning is expected to generate $3,000 a year for the next 2 years from the sales of its Calculus on Demand iOS app. Assuming the income is invested at an interest rate of 5%, what is the future value of this income stream?

Step 1:   Recall the formula for the future value of an income stream.

\[A = e^{rT} \int_0^T R(t) e^{-rt} ~dt\]

Step 2:   Plug in the given values.

\(R(t)=3000\), \(r=0.05\), and \(T=2\).

\[\begin{align*} e^{0.05(2)}\int_0^2 3000e^{-0.05t}~dt &= 3000 e^{0.1}\int_0^2 e^{-0.05t}~dt && \hbox{Constant multiple rule}\\ &= 3000 e^{0.1} \left(\frac{e^{-0.05t}}{-0.05}\right)\Biggr|^{2}_{0} && \hbox{Since $\displaystyle \int e^{ax} ~dx = \dfrac{e^{ax}}{a} + C$} \\ &= 3000 e^{0.1} \left(\frac{e^{-0.05(2)}}{-0.05} - \frac{e^{-0.05(0)}}{-0.05}\right) && \hbox{Plug in limits of integration}\\ &= 3000 e^{0.1} \left(\frac{e^{-0.1}}{-0.05} + \frac{1}{0.05}\right) && \hbox{Since $e^0 = 1$}\\ &= 60000 e^{0.1} \left(1- e^{-0.1}\right) && \hbox{Since $\frac{1}{0.05} = \frac{1}{5/100} = \frac{100}{5} = 20$}\\ &\approx \$6,310.26 \end{align*}\]

Example 2

Present value of an investment

An investment is expected to generate income at a rate of $300,000 per year for the next 6 years. Find the present value of this investment if the interest rate is 10% compounded continuously.

Step 1:   Recall the formula for the present value of an income stream.

\[PV = \int_0^T R(t) e^{-rt} ~dt\]

Step 2:   Plug in the given values.

\(R(t)=300000\), \(r=0.1\), and \(T=6\).

\[\begin{align*} \int_0^6 300000e^{-0.1t} ~dt &= 300000 \int_0^6 e^{-0.1t} ~dt && \hbox{Constant multiple rule}\\ &= 300000 \left(\frac{e^{-0.1t}}{-0.1}\right)\Biggr|^6_0 && \hbox{Since $\displaystyle \int e^{ax} ~dx = \dfrac{e^{ax}}{a} + C$}\\ &= 300000 \left(\frac{e^{-0.1(6)}}{-0.1}-\frac{e^{-0.1(0)}}{-0.1}\right) && \hbox{Plug in limits of integration}\\ &= 300000 \left(\frac{e^{-0.6}}{-0.1}+\frac{1}{0.1}\right) && \hbox{Since $e^0 = 1$}\\ &= 3000000\left(1- e^{-0.6}\right) && \hbox{Since $\frac{1}{0.1} = \frac{1}{1/10}=10$}\\ &\approx \$1,353,565.09 \end{align*}\]

Annuities

Definition and Notation

An annuity is a sequence of payments made at regular intervals.

• \(P=\) the size of each payment

• \(T=\) length of time (in years) that payments are made

• \(r\) = annual interest rate (compounded continuously)

• \(m=\) the number of payments per year

Annually	Semiannually	Quarterly	Monthly	Weekly	Daily
\(m=1\)	\(m=2\)	\(m=4\)	\(m=12\)	\(m=52\)	\(m=365\)

Amount of an Annuity

The amount of an annuity is the total of all of the payments plus all of the interest earned and is given by

\[A = \frac{mP}{r}\left(e^{rT} - 1\right)\]

Present Value of an Annuity

The present value of an annuity is the principal investment, \(PV\), that yields the same accumulated value as the annuity when \(PV\) is invested for a period of \(T\) years at the same interest rate and is given by

\[PV = \frac{mP}{r}\left(1 - e^{-rT}\right)\]

Observation

The above formulas for the amount and present value of an annuity are approximations and are based on treating an annuity as an income stream where \(R(t) = mP\).

Example 3

Amount of an annuity

A Math 110 student decides to make semiannual payments of $2,500 into a retirement account paying 2% interest per year compounded continuously. How much will the student have in their retirement account after 20 years?

Step 1:   Recall the formula for the amount of an annuity.

\[A = \frac{mP}{r}\left(e^{rT} - 1\right)\]

Step 2:   Plug in the given values.

\(m=2\), \(P=2500\), \(r=0.02\), and \(T=20\).

\[\begin{align*} \frac{(2)(2,500)}{0.02} \left(e^{(0.02)(20)}-1 \right) &= \frac{5000}{2/100}\left( e^{0.4}-1\right)\\ &= \frac{100}{2}\cdot 5000\left( e^{0.4}-1\right)\\ &= 50\cdot 5000\left( e^{0.4}-1\right)\\ &= 250000\left( e^{0.4}-1\right)\\ &\approx \$122,956.17 \end{align*}\]

Therefore, the student will have approximately $122,956.17 in their retirement account after 20 years.

Example 4

Present value of an annuity

Determine the present value of an annuity if payments of $100 are made monthly for the next 10 years and the account earns an interest rate of 10% per year compounded continuously.

Step 1:   Recall the formula for the present value of an annuity.

\[PV = \frac{mP}{r}\left(1 - e^{-rT}\right)\]

Step 2:   Plug in the given values.

\(m=12\), \(P=100\), \(r=0.1\), and \(T=10\).

\[\begin{align*} \frac{(12)(100)}{0.1} \left(1-e^{-(0.1)(10)} \right) &=\frac{1200}{1/10} \left(1-e^{-1} \right)\\ &=10\cdot 1200 \left(1-e^{-1} \right)\\ &=12000\left( 1-e^{-1}\right)\\ &\approx \$7,585.45 \end{align*}\]

Therefore, the annuity is worth a single lump sum of money worth approximately $7,585.45.