2.3 Present Value of Annuities

Introduction

The present value of any annuity is equal to the sum of the present values of all the annuity payments when they are moved to the beginning of the first payment interval. For example, assume you will receive [latex]\$1,000[/latex] annual payments at the end of every payment interval for the next three years from an investment earning [latex]10\%[/latex] compounded annually. How much money needs to be in the annuity at the start to make this happen? To find out how much money you need in the investment to sustain these payments, you would apply the fundamental concept of the time value of money to move each payment amount to the start of the first payment interval (the focal date) and sum the values to arrive at the present value.

Although you could use this technique to solve all present value of an annuity situations, the computations become increasingly cumbersome as the number of payments increases. In the above example, what if the person instead received quarterly payments of [latex]\$250[/latex] for three years? That is [latex]12[/latex] payments over three years, resulting in [latex]12[/latex] separate present value calculations. Or if they received monthly payments, the [latex]36[/latex] payments over three years would result in [latex]36[/latex] separate present value calculations! Clearly, solving this would be tedious and time consuming—not to mention prone to error.

[latex]\boxed{2.4}[/latex] Present Value of Ordinary Annuities

[latex]{\color{red}{PV}}\;\text{is the Present Value of the Annuity:}[/latex] The present value is the sum of the present value of all of the annuity payments at the start of the first payment interval.

[latex]{\color{blue}{PMT}}\;\text{is the Annuity Payment:}[/latex] The amount of money that is invested or paid after each payment interval.

[latex]{\color{green}{i_2}}\;\text{is the Periodic Interest Rate:}[/latex] for the interest rate whose compounding frequency matches the payment frequency}. If the annuity is a simple annuity, the compounding frequency and the payment frequency are equal and the given interest rate can be used to calculate the periodic interest rate. If the annuity is a general annuity, the compounding frequency and the payment frequency are different and an interest conversion must be done first to find the equivalent interest rate with the compounding frequency equal to the payment frequency.

[latex]{\color{purple}{n}}\;\text{is the Total Number of Payments made during the Annuity:}[/latex] [latex]n=P/Y \times t[/latex] where [latex]P/Y[/latex] is the payment frequency and [latex]t[/latex] is the time in years.

The amount of interest ([latex]I[/latex]) earned by the annuity is

[latex]\displaystyle{I=n \times PMT-PV}[/latex]

Example 2.3.1

Rodriguez is planning on having an annual gross income of [latex]\$50,000[/latex] at the end of every year when he retires at age [latex]65[/latex]. He is planning for the account to be emptied by age [latex]78[/latex], which is the average life expectancy for a Canadian man. If the account earns [latex]5.1\%[/latex] compounded annually, how much money needs to be in the account when he retires? How much interest will he receive from the fund?

Solution

The timeline for Rodriguez’s account appears below.

Step 1:The given information is

The payments are at the end of the payment intervals, and both the compounding frequency and the payment frequency are the same (both annually). This is an ordinary simple annuity. Because this is a simple annuity, an interest rate conversion is not required.

[latex]\begin{eqnarray*} PMT & = & \$50,000 \\ P/Y & = & 1 \\ I/Y & = & 5.1\% \\ C/Y & = & 1 \\ t & = & 13 \mbox{ years} \end{eqnarray*}[/latex]

Step 2: Calculate the periodic interest rate.

[latex]\begin{eqnarray*} i_2 & = & \frac{I/Y}{C/Y} \\ & = & \frac{5.1\%}{1}\\ & = & 5.1\%\end{eqnarray*}[/latex]

Step 3: Calculate the number of payments.

[latex]\begin{eqnarray*} n & = & P/Y \times t \\ & = & 1 \times 13 \\ & = & 13 \end{eqnarray*}[/latex]

Step 4: Calculate the present value.

[latex]\begin{eqnarray*} PV & = & PMT \times \left[1-\frac{(1+i_2)^{-n}}{i_2}\right] \\ & = & 50,000 \times \left[ \frac{1-(1+0.051)^{-13}}{0.051}\right] \\ & = & \$466,863.69 \end{eqnarray*}[/latex]

Step 5: Calculate the interest earned.

[latex]\begin{eqnarray*} I & = & n \times PMT-PV \\ & = & 13 \times 50,000-466,863.69 \\ & = & \$183,136.31 \end{eqnarray*}[/latex]

Step 6: Write as a statement.

Rodriguez will need to have [latex]\$466,863.69[/latex] in his account when he turns [latex]65[/latex] if he wants to receive [latex]13[/latex] years of [latex]\$50,000[/latex] payments. He will receive [latex]\$183,136.31[/latex] in interest from the fund.

[latex]\boxed{2.5}[/latex] Present Value of Annuities Due

[latex]{\color{red}{PV}}\;\text{is the Present Value of the Annuity:}[/latex] The present value is the sum of the present value of all of the annuity payments at the start of the first payment interval.

[latex]{\color{blue}{PMT}}\;\text{is the Annuity Payment:}[/latex] The amount of money that is invested or paid after each payment interval.

[latex]{\color{green}{i_2}}\;\text{is the Periodic Interest Rate:}[/latex] for the interest rate whose compounding frequency matches the payment frequency. If the annuity is a simple annuity, the compounding frequency and the payment frequency are equal and the given interest rate can be used to calculate the periodic interest rate. If the annuity is a general annuity, the compounding frequency and the payment frequency are different and an interest conversion must be done first to find the equivalent interest rate with the compounding frequency equal to the payment frequency.

[latex]{\color{purple}{n}}\;\text{is the Total Number of Payments made during the Annuity:}[/latex] [latex]n=P/Y \times t[/latex] where [latex]P/Y[/latex] is the payment frequency and [latex]t[/latex] is the time in years.

The amount of interest ([latex]I[/latex]) earned by the annuity is

[latex]\displaystyle{I=n \times PMT-PV}[/latex]

Using a Financial Calculator

The above examples were fairly straightforward annuity questions. But even these simple examples, which did not require interest conversions, are cumbersome, and time-consuming, to solve using the formulas. A financial calculator can quickly solve annuity problems, with the added bonus of not requiring an interest conversion in situations where the payment frequency and compounding frequency are not equal.

_{Ordinary Annuity Calculations (PV, PMT, FV) by Joshua Emmanuel [4:31] (transcript available).}

_{Annuity Due Calculations Using BAII Plus-Part 2 by Joshua Emmanuel [2:05] (transcript available).}

Example 2.3.3

Rodriguez is planning on having an annual gross income of [latex]\$50,000[/latex] at the end of every year when he retires at age [latex]65[/latex]. He is planning for the account to be emptied by age [latex]78[/latex], which is the average life expectancy for a Canadian man. If the account earns [latex]5.1\%[/latex] compounded annually, how much money needs to be in the account when he retires?

Solution

The timeline for the Rodriguez’s account appears below.

PMT Setting	END
[latex]N[/latex]	[latex]1 \times 13=13[/latex]
[latex]PV[/latex]	?
[latex]FV[/latex]	[latex]0[/latex]
[latex]PMT[/latex]	[latex]50,000[/latex]
[latex]I/Y[/latex]	[latex]5.1[/latex]
[latex]P/Y[/latex]	[latex]1[/latex]
[latex]C/Y[/latex]	[latex]1[/latex]

[latex]PV=\$466,863.69[/latex]

Rodriguez will need to have [latex]\$466,863.69[/latex] in his account when he turns [latex]65[/latex] if he wants to receive [latex]13[/latex] years of [latex]\$50,000[/latex] payments.

When you purchase an expensive or large-ticket item such as a car or house, you often need to take out a loan to make the purchase. After all, most people do not have large sums of cash available to pay for these items outright. With these expensive items, you often make an up-front partial payment on the purchase price called a down payment, and then take out a loan for the outstanding purchase price.

For example, suppose you want to purchase a [latex]\$100,000[/latex] luxury sports car. You pay [latex]\$10,000[/latex] as a down payment and take out a loan for the balance. The [latex]\$10,000[/latex] down payment is cash you are paying up-front on the purchase price. You would then need to take out a loan for the remaining [latex]\$90,000[/latex] of the purchase price. The [latex]\$90,000[/latex] is a loan, so you would be charged interest for borrowing the money and then you would need to make regularly scheduled payments to pay back the loan. Because the [latex]\$10,000[/latex] down payment is paid up-front, there is no interest charged on this [latex]\$10,000[/latex]. By making the [latex]\$10,000[/latex] down payment, you are reducing the amount of money you need to borrow, and consequently reducing the amount of interest you need to pay.

In general

[latex]\displaystyle{\mbox{Purchase Price}=\mbox{Down Payment}+\mbox{Loan Amount}}[/latex]

Example 2.3.5

Rodriguez is planning on having an annual gross income of [latex]\$50,000[/latex] at the beginning of every year when he retires at age [latex]65[/latex]. He wants to leave a [latex]\$100,000[/latex] inheritance for his children, assuming he dies at age [latex]78[/latex]. If the account earns [latex]5.1\%[/latex] compounded semi-annually, how much money needs to be in the account when he retires?

Solution

The timeline for the Rodriguez’s account appears below.

PMT Setting	BGN
N	[latex]1 \times 13=13[/latex]
PV	?
FV	[latex]100,000[/latex]
PMT	[latex]50,000[/latex]
I/Y	[latex]5.1[/latex]
P/Y	[latex]1[/latex]
C/Y	[latex]2[/latex]

[latex]PV=\$541,027.07[/latex]

Rodriguez will need to have [latex]\$541,027.07[/latex] in his account when he turns [latex]65[/latex] if he wants to receive [latex]13[/latex] years of [latex]\$50,000[/latex] payments and leave a [latex]\$100,000[/latex] inheritance for his children.

Example 2.3.6

Rodriguez is planning on having an annual gross income of [latex]\$50,000[/latex] at the beginning of every year when he retires at age [latex]65[/latex] until age [latex]71[/latex]. Realizing he needs to make some type of adjustment to his annual gross income to account for the rising cost of living, starting at age [latex]71[/latex] he wants his annual payments to increase to [latex]\$60,000[/latex] until age [latex]78[/latex]. He wants to leave a [latex]\$100,000[/latex] inheritance for his children, assuming he dies at age [latex]78[/latex]. If the account earns [latex]5.1\%[/latex] compounded semi-annually, how much needs to be in the account when he retires?

Solution

The timeline for the Rodriguez’s account appears below.

Step 1: Find the present value at the start of time segment [latex]1[/latex].

PMT Setting	BGN
[latex]N[/latex]	[latex]1 \times 7=7[/latex]
[latex]PV[/latex]	?
[latex]FV[/latex]	[latex]100,000[/latex]
[latex]PMT[/latex]	[latex]60,000[/latex]
[latex]I/Y[/latex]	[latex]5.1[/latex]
[latex]P/Y[/latex]	[latex]1[/latex]
[latex]C/Y[/latex]	[latex]2[/latex]

[latex]PV_1=\$433,232.035...[/latex]

Step 2: Calculate the present value at the start of time segment [latex]2[/latex].

The present value from time segment [latex]1[/latex] becomes the future value for time segment [latex]2[/latex]: [latex]PV_1=\$433,232.035...=FV_2[/latex].

PMT Setting	BGN
[latex]N[/latex]	[latex]1 \times 6=6[/latex]
[latex]PV[/latex]	?
[latex]FV[/latex]	[latex]433,232.035...[/latex]
[latex]PMT[/latex]	[latex]50,000[/latex]
[latex]I/Y[/latex]	[latex]5.1[/latex]
[latex]P/Y[/latex]	[latex]1[/latex]
[latex]C/Y[/latex]	[latex]2[/latex]

[latex]PV_2=\$585,742.42[/latex]

Step 3: Write as a statement.

Rodriguez needs to have [latex]\$585,742.42[/latex] invested in his retirement fund at age [latex]65[/latex].

Section 2.3 Exercises

Attribution

“11.3: Present Value of Annuities” from Business Math: A Step-by-Step Handbook Abridged by Sanja Krajisnik; Carol Leppinen; and Jelena Loncar-Vines is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

“11.3: Present Value of Annuities” from Business Math: A Step-by-Step Handbook (2021B) by J. Olivier and Lyryx Learning Inc. through a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License unless otherwise noted.