Annuity Formula

Mr.John gets an annuity of $500 a week for 5 years. What is an annuity? And, how does he get such an income? Well, because he planned these future periodic payments by buying an annuity for himself!

What is annuity

Money is often an interesting topic to discuss. Let's learn about the annuity and its formula.

Lesson Plan

What Are Annuities?

An annuity is a fixed amount of income that is given annually or at regular intervals.

Who gives the fixed money to whom?

Well, one can buy an annuity. An annuity is an agreement with an insurance company in which you make a lump sum payment (one-time big payment) or series of payments and, in return, receive a regular fixed income, beginning either immediately or after some predefined time in the future.

Example:

John bought an annuity plan for $10,000 for 5 years by paying a lump sum of $10,000 to the insurance company. Under this plan, he will get an annuity of $200 monthly for 5 years. 

What is annuity

So, John paid them one large amount, now they will give him back small payments at predecided regular intervals.
It cost him $10,000, and in return, he gets $200 a month for 5 years.

But that would be a good deal only if John gets more money than he had actually put in, right?

Let us calculate that. He gets \($200 \times 12 \times 5 = $12,000\)

So yes, it is indeed a good deal for John. He gets $2,000 extra on his initial investment of $10,000

How did the company manage to give John more money than he had invested?

The company invested the received $20,000 and earned interest on it, or did other financially profitable deals to make more money.


Understanding the Time Value of Money

Congratulations!!! You have won a cash prize of $10,000!

what is time value of money

But you are given two options:

1. Receive the $10,000 now

Or

2. Receive $10,000 later after five years

What will you choose?

Okay, the above offer is imaginary, but let's play along.

Most people would like to receive $10,000 now. Of course, five years is a very long time to wait and patience is a rare trait. If all things are equal, why not have the money now rather than later?

time value of money illustration

Keeping the virtue of patience aside, having $10,000 today has more value than having the money later.

Why? Because you get a chance to increase the future value of your money by investing and gaining interest over a period of time.

For the other option, the payment received in five years (future value) would be the same because you never had the money in your hand to get to grow it. Having money now is clearly a smarter choice.


Time Value of an Annuity

We need to understand two types of the time value of an amount(annuity).

1. Future value of an annuity

2. Present value of an annuity

Why is it important for us to know the time value of the money in the future and in the present?

It is important to know the time value of money to compare the worth of the investments with different periods.

present value of annuity

To know the future and present values of an annuity we need to calculate the interest on the money in consideration for a specific time.

Example: What will be the future value of $100 after a year with a yearly interest rate of 10%?

$100 now could earn $100 x 10% = $10 in a year.

So, $100 invested today will bring $110 next year (at 10% interest).

interest

We can interpret the above image in two ways:

1. The future value of today's $100, after a year is $110 (at 10% interest.)

Or

2. The present value of $110 next year is $100 today (at 10% interest.)

So, at 10% interest:

  • To find the future value: multiply by 1.10
  • To find the present value: divide by 1.10

What Is the Formula for Calculating the Present Value of Annuity?

Before we learn the formula for calculating the present value of an annuity let's imagine that you bought a plan to receive an annuity of $500 yearly for 3 years. The interest rate is 10% per annum.

The interest rate of 10% means 1.10 in decimal.

Also, note that $500 paid yearly for 3 years would make a total of $1500

Let us understand the present value concept with each yearly payment involved.

Future payments

to be received 

 

Steps for calculating the

present value

Present value (PV)
First

Your first payment of $500 is next year. What is its worth today? 

To go from next year to now, 

divide the annuity amount by 1.10:

\(→ 500 \div 1.10 \)= \(454.55\)

$454.55 now
Second

Your second payment is 2 years from now.

Bring it back one year,

then bring it back by another year:

\(→ 500 \div 1.10 \div 1.10 \)= \(413.22\)

$413.22 now
Third

Your third payment is 3 years from now.

Similarly:

\(→ 500  \div 1.10 \div 1.10 \div 1.10\) = \(375.66\)

$375.66 now
     

Total PV:

 $1243.43

3 annual payments of $500 at 10% interest is worth $1243.43 now. This means the annuity plan should cost you equal to or lesser than $1243.43.

This can be interpreted in both ways:

1. $1243.43 today has the potential of making $1500 after 3 years at a 10% interest rate.

2. The present value of the above annuity is $1243.43

To sum up, it is a profitable deal as the money invested is growing steadily.

The present value of a yearly annuity of $500 for 3 years is simple to calculate. This would not be this easier if we wish to calculate it for 8 years. This would mean 8 separate calculations.

Thankfully the following formula makes it easier. If we don't use the formula we will have to do 8 separate calculations. 

present value annuity

Example:

We can now easily calculate the present value of 8 annual payments in the future using the above formula. We will need an exponent calculator to solve it easily.

Here's one:

The interest rate is 10%, so \(r = 0.10\)

The number of annuity payments is 8, so \(n =8\)

Each payment is $500, so \(\text P = $500\)

\[\begin{align*} \text{PV} &= \text P \times \dfrac{ 1−(1+r)^{-n} }{ r} \\ &= $500 \times \dfrac {1 − 1.10^{-8}}{0.10} \\  &= $500 \times \dfrac {1 − 0.46650}{0.10} \\ &= $500 \times 5.335...\\ &=  $2,667.5 \end{align*} \]

The present value of 8 annual payments of $500 at 10% yearly interest is $2,667.5 now. 

This means the annuity plan should cost you equal to or lesser than $2,667.5

The future value of 8 annual payments of $500 at 10% yearly interest is \($500 \times 8 = $4000\)

To conclude, the present value of an annuity is the amount of money that would be required as a first payment to produce the desired future payments.


What Is the Formula for Calculating Annuity Payment?

What if you know the amount you want to invest and want to work out annuity payments (i.e., the sum of the annuity)?

Let us understand the sum of the annuity formula with the help of an example.

Example: Lara has a budget of $20,000 to buy an annuity plan and she wants to plan a monthly income for herself for 10 years. Assuming a monthly interest rate of 0.5%, how much annuity would she get each month?

Future value of annuity

The monthly interest rate is 0.5%, so \(r\) will be \(0.005\)

10 years x 12 months = 120 monthly payments, so \(n = 120\), and \(PV = $20,000\)

The same formula will be in use

\[\text {PV} = \text P \times \dfrac {1 − {1+r }^n}{r}\]

\(\text P\) is the value of each annuity payment which she will receive monthly. Readjusting the above formula to get \(\text P\) on the LHS we get:

Sum of annuity formula

\[ \begin{align*} \text P &= \text {PV} \times \dfrac{ r}{1 − (1+r)^{−n}} \\ &= $20,000 \times \dfrac{0.005}{1 − (1.005)^{−120}} \\ &= $20,000 \times \dfrac{ 0.005}{1 − 0.54963} \\ &= $20,000 × 0.011... \\ &=  $220\end{align*} \]

Lara would get an annuity of $220 monthly for 10 years.

10 years of $220 a month will be \($220 \times 10 \times 12 = $26,400\)

The extra money Lara makes on top of her investment \(= $26,400 - $20,000 = $6,400\)


What Is the Future Value of Annuities?

The steps to calculate the future value of an annuity is similar to finding the present value of an annuity.

The following picture depicts the relation between the present value and future value:

Present and future value

The future value of the annuity is the total value of the payments at the end of a specific period of time. We can simply find the future value of an annuity using the following formula:

future value of annuity formula

Example:

Say you are getting $100 at the end of each year for 5 years at an interest rate of 5%. What will be the future value of this annuity at the end of 5 years?

Here \(r\) will be 0.05

5 years  = 5 yearly payments, so n = 5, and P = $100

\(\text{FV} = \text P \times \dfrac{(1+r)^{n} −1 }{ r} \)

\[ \begin{align*} \text{FV} &= \text P \times \dfrac{(1+r)^{n} −1 }{ r} \\  &= $100 \times \dfrac{(1+0.05)^{5} −1 }{ 0.05} \\ &= 100 \times 55.256 \\  FV &= $552.56 \end{align*}\]
Hence, the future value of annuity after the end of 5 years is $552.56


Present Value of Annuity Calculator

This present value annuity calculator computes the present value of a series of equal payments to be received in the future.

Use this calculator to find out what a future cash stream is worth in today's dollars. Be it from an annuity, business, real estate, or other assets, this calculator will do the desired job for you.

 

 
Challenge your math skills
Challenging Question
  1. Susan has savings of $50,000 and she wishes to receive an annuity quarterly for 10 years. At the monthly interest rate is 0.15%, calculate the profit percentage that she will enjoy on her investment? 

What Is the Formula for Calculating Annuity Interest?

The rate of interest of an annuity can be calculated using the following formula when both the present value and the future value are known to us: 

annuity

Example:

Julia figured out that the present value of the annuity is $536 while its future value is $875.6 at the end of each year for 4 years. What will be the interest rate of this annuity?

4 years  = 4 yearly payments, so \(n = 4\), and \(\text{PV} = $536\), and \(\text{FV} = $875.6\)

Using the formula:

\( r = {\left(\dfrac {FV}{PV} \right)}^{\frac 1n}- 1 \)

\[ \begin{align*} r &= {\left(\dfrac {FV}{PV} \right)}^{\frac 1n}- 1 \\  &= {\left(\dfrac {875.6}{536} \right)}^{\frac 14}- 1\\ &= 1.63^{\frac 14} -1 \\   &= 1.129-1 \\  & \approx  0.13 \end{align*}\]

\(r \approx 0.13\)

Hence, the yearly interest rate of the annuity is approximately 13%

 
important notes to remember
Important Notes
  1. An annuity is an agreement with an insurance company for receiving regular fixed income.
  2. The present value of an annuity is the amount of money that would be required as a first payment to produce the desired future payments. 
    \(\text{PV} = \text P \times \dfrac{ 1−(1+r)^{-n} }{ r} \)
  3. The future value of the annuity is the total value of the payments at the end of a specific period of time. \(\text{FV} = \text P \times \dfrac{(1+r)^{n} −1 }{ r} \)

Solved Examples

Example 1

 

 

Mr. Kennedy is planning to receive an annuity of $800 each year for 5 years. If the annual rate is 10%. How will he calculate the present value of the annuity?

Calculate present value of annuity

Solution

We can find out the present value of the annuity by using this formula:

\[\text{PV} = \text P \times \frac{1-(1+r)^{-n}}{r}\]

Here, \(\text P = 800\), \(n = 5\) and \(r = 0.1\)

Therefore,

\[\begin{align*} \text{PV} &= \text P \times \frac{1-(1+r)^{-n}}{r}\\ &= 800 \times \frac{1-(1.1)^{-5}}{0.1} \\ &\approx 3032.629 \end{align*}\]

\(\therefore\) The present value of the annuity is \(\approx $3032.629\)
Example 2

 

 

Aleena is planning her finances. She has $7,500 saved and she wishes to receive an annuity quarterly continuously for 12 years. The current interest rate is 1.5%. How much money will she receive each year?

Annuity formula

Solution

To calculate the yearly payment we will use the formula:

\[\text P=\text{PV} \times \frac{r}{1-(1+r)^{-n}}\]

Here, \(\text{PV} = 7,500, n = 4 \times 12 = 48\) and \(r = 0.015\)

Using the formula:

\[\begin{align*}\text P&=\text{PV} \times \frac{r}{1-(1+r)^{-n}}\\
&=7,500 \times \frac{0.015}{1-(1.015)^{-48}}\\
&=7,500 \times \frac{0.015}{1-0.48936 \ldots}\\
&=220.312 \ldots
\end{align*}\]

So, Aleena will receive \($220.31\) each quarter. 
Note that \($220.31 \times 4 \times 12 = $10,574.88\), which is substantially more than the invested money of \($7,500.\)
This is because interest has been added.

\(\therefore\) Aleena will receive \($220.31\) in each quarter for 12 years.
Example 3

 

 

Mr. Donald has a saved amount of  $20,000 and he wishes to receive an annuity monthly for 10 years.
At the monthly interest rate is 0.1% how much extra money will he enjoy on the invested money?

Annuity

Solution

Use the formula \[P=P V \times \frac{r}{1-(1+r)^{-n}}\]
In this case \(\text{PV} = 20,000\), \(n=10 \times 12=120\) and \(r=0.001\)
Therefore, \[\begin {align*}P&=20,000 \times \dfrac{0.001}{1-(1.001)^{-120}}\\ &=20,000 \times \dfrac{0.001}{1-0.88697 \cdots}\\ &=176.949 \cdots \end {align*}\]
So, each month he will receive \(\approx $176.95\)

The total amount he will receive over the 10-year period \(= 10 \times 12 \times 176.95= $21,234\)

The extra money made: \($21,234 - $20,000 = $ 1,234\)
Therefore, Gary will receive \($ 1,234\) extra.

\(\therefore\) Gary will receive \($ 1,234\) extra money on top of his invested money.


Interactive Questions on Annuity Formula

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 

 


Let's Summarize

The mini-lesson targeted the fascinating concept of the annuity formula. The math journey around the annuity formula starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


Frequently Asked Questions

1. What is the formula for future value annuity with examples?

The future value of the annuity is the total value of the payments at the end of a specific period of time. We can simply find the future value of an annuity using the following formula:

Example:
Say you are getting $100 at the end of each year for 5 years at an interest rate of 5%. What will be the future value of this annuity at the end of 5 years?

Here \(r\) will be 0.05

5 years  = 5 yearly payments, so \(n\) = 5, and \(\text P = $100\)

\[\begin{align*} \text{FV} = \text P \times \dfrac{(1+r)^{n} −1 }{ r} \\ \text{FV} &= \text P \times \dfrac{(1+r)^{n} −1 }{ r} \\  &= 100 \times \dfrac{(1+0.05)^{5} −1 }{ 0.05} \\ &= 100 \times 5.5256 \\  FV &= $552.56 \end{align*}\]
Hence, the future value of annuity after the end of 5 years is $552.56

2. What is the formula for loans?

The loan formula is the same formula that is used to calculate payments on an ordinary annuity. A loan almost exactly is an annuity, in that it consists of a series of future periodic payments. 

present value annuity

3. What is the formula of payout annuity?

\(\text P_{0}=\frac{d\left(1-\left(1+\frac{r}{k}\right)^{-N k}\right)}{\left(\frac{r}{k}\right)}\)

P0 = starting amount (principal.)
d = Amount withdrawn each year, each month, etc.
r = Annual interest rate (in decimals.)
k = number of compounding periods per annum.
N = number of years planned for withdrawals.

  
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