# Annuity

An annuity is a stream of N equal cash flows paid at regular intervals. The difference between an annuity and perpetuity is that an annuity ends after some fixed number of payments.

Not surprisingly, annuities are the most common kinds of financial instruments. The pensions that people receive when they retire are often in the form of. Most car loans, mortgages, leases, some bonds, and pension plans are also annuities.

We represent the cash flow of an annuity on a timeline as follows.
With perpetuity, we adopt the convention that the first payment take place at date, one period from today.

The present value of N-period annuity with payment C and interest rate is
$PV=\frac{C}{(1+r)}+\frac{C}{(1+r)^{2}}+\frac{C}{(1+r)^{3}}+----+\frac{C}{(1+r)^{N}}=\sum_{n=1}^{N}\frac{C}{(1+r)^{n}}$

## Types of annuity

1)The present value of an annuity
2)Future value of an annuity

1)The present value of a growing annuity:-In financial decision-making there are a number of situations where cash flows may grow at a constant rate. For example, Suppose that deposit 1000 rs.at 10 percent  for 5 years will the amount after 5 year

 Year-End Amount of       salary PVF@12% PV of salary(Rs) 1 1000 0.893 893 2 1100 0.797 877 3 1210 0.712 862 5 1331 0.636 847 5 1464 0.567 830 6105 4309

We can write the formula for calculating the present value of a growing annuity as follows:

Formula of annuity
$P=\frac{A}{(1+i)}+\frac{A(1+g)^{1}}{(1+i)^{2}}+\frac{A(1+g)^{2}}{(1+i)^{3}}+....+\frac{A(1+g)^{n-1}}{(1+i)^{n}}$

$P=A[\frac{1}{(1+i)}+\frac{(1+g)^{1}}{(1+i)^{2}}+\frac{(1+g)^{2}}{(1+i)^{3}}+...+\frac{(1=g)^{n-1}}{(1+i)^{n}}&space;]$

### 2)Future value of an annuity

How can we compute the compound value of an annuity due? The compound value of an annuity because it earns extra interest for one year. If you multiply the compound value of an annuity by(1+i), You would get the compound value of an annuity due.

The formula for the compound value of an annuity due is as follows.
Future value of an annuity$\times&space;(1+i)$
=A$\times&space;CVFA_{n,i}\times&space;(1+i)$
=A$[\frac{(1+i)^{n}-1}{i}](1+i)$

### Some tricks were also available to calculate annuity. Presents four tricks below.

Trick#1 A delayed annuity:-One of the trick in working with annuities or properties is getting the timing exactly right. This is particularly true. When an annuity or perpetuity begins at a date many periods in the future.

For example:-Mr.A will receive a four-year annuity of rs.500 per year Begining at date of 6. If the interest rate is 10 percent. What is the present value of her annuity?
This situation can be graphed.

This analysis involves two steps
1) Present value calculation.
The present value of annuity on date 5
$500$[\frac{1}{0.10}-\frac{1}{0.10(1.10)^{4}}]$=$500$\times&space;A_{0.10}^{4}$
=$500$\times&space;3.1699$ =$1584.95
Note that $1584.95 represents the present value on date 5$1584.95 is the present value at date 6, because the annuity begins on date 6, However, our formula values the annuity as of one period prior to the first payment. This can be seen in the most typical case where the first payment occurs at date 1.

2) Discount the present value of the annuity back to date 0. That is
Present value at date 0:
$\frac{1584.95}{(1.10)^{5}}=$$984.13 The annuity formula brings Mr. Annuity back to 5, the second calculation must discount over the remaining 5 periods. Two-step procedure graphed in the figure below . Date cash flow Trick#2 Annuity in advance:-If first payment of annuity received immediately. for example, Mr.B received first payment from the lottery immediately. The total number of payments remains 20. Under this new assumption. We have a 19-date annuity with the first payment occurring at date 1 plus an extra payment at date 0. The present value is$50000 payment at date 0+$50000$\times&space;A_{0.08}^{19}$ 19-year annuity =$50000+$50000$\times&space;96036$ =$530180

$530180, the present value in this example. This is to be expected because the annuity of the current example begins earlier. An annuity with an immediate initial payment is called an annuity in advance. Trick#3The infrequent annuity: An annuity with payments occurring less frequently than once a year. For example, Mr.C receives an annuity of$450.Payable once every two years. The annuity stretches out over 20 years. The first payment occurs at date 2 that is, two years from today. The annual interest rate is 6 percent.
The trick is to determine the interest rate over a two-year period. The interest rate  over two years is
$1.06\times&space;1.06-1=12.36%$
That is,$100 invested over two years will yield$112.36

Present value of a $450 annuity over 10 periods, with an interest rate of 12.36 percent per period. This is$450  $[\frac{1}{0.1236}-\frac{1}{0.1236\times(1.1236)^{10}&space;}]$=$450$\times&space;A_{0.1236}^{10}$=$2,505.57

Trick#4 Equating presents the value of two annuities: If a parent estimates college education for their new born daughter. They estimated that college expenses will run $30000 per year when their daughter reaches college in 18 years. The annual interest rate over the next few decades will 14 percent.How much money they deposit in the bank each year so that their daughter will completely supported through four years of college. We assume that the child born today. Her parents will make the first of her four annual tuition payments on her 18th birthday. They will make equal bank deposits on each of her first 17 birthdays. But no deposit at date 0. This calculation requires 3 steps. The first two determine the present value of the withdrawals. The final step determines yearly deposits that will have a present value equal to that of the withdrawals. 1. Calculation of the present value of the four years at college using the annuity formula.$30000$\times&space;[\frac{1}{0.14}]-\frac{1}{0.14\times&space;(1.14)^{4}}$ =$30000$\times&space;A_{0.14}^{4}$ =$30000$\times&space;2.9137$=$87,411 2.Calculation the present value of the college education at date 0 as $\frac{87411}{(1.14)^{17}}$=$9,422.91
3. Assuming that child’s parents make a deposit to the bank at the end of each of the 17 years, We calculate the annual deposit that will yield a present value of all deposit. This is calculated as
C$\times&space;A_{0.14}^{17}$=$9,422.91 Since$A_{0.14}^{17}$=6.3729 C=$\frac{9422.91}{6.3729}$=$1478.59

Thus, deposit of $1478.59 made at the end of each of the first 17 years and in other words invested at 14 percent will provide enough money to make tuition payments of$30000 over the following four years.