**Annuity immediate- Present value and future value**

An annuity is a sequence of equal payment or a sequence of regular payment at regular intervals or in other words. it is a fixed sum paid at regular intervals under certain conditions.

The length of time during which the annuity is paid can either be until the death of the recipient or for a guaranteed minimum term of years, irrespective of whether the annuitant is alive or not.

The time between payments is called the payment interval, and the time which the money is to be paid is called the term of the annuity.

**Amount of an annuity**

Amount of an annuity is the total of all the installments left unpaid together with the compound interest of each payment for the period it remains unpaid.

The formulas to find the amount of annuity are given below-

**Present value**

Present value of an annuity is the sum of the present values of all payments (or installments) made at successive annuity periods.

The formulas to calculate present value ‘V’ of an annuity P are given below-

**The future value is calculated as-**

**Example: ****Sundar decides to deposit 20,000rs. at the end of each year in a bank which pays 10% p.a. compound interest.**

**If the installments are allowed to accumulate, what will be the total accumulation at the end of 9 years?**

**Solution:**

Suppose A Rs. be the total accumulation at the end of 9 years. Then we get-

Here P = 20,000 Rs., i = 10/100 = 0.1 and n = 9

Then

Hence the total required accumulation is 2,71,590rs.

**Example: Rajeev purchased a flat valued at 3,00,000rs. He paid 2,00,000rs. at the time of purchase and agreed to pay the balance with interest of 12% per annum compounded half yearly in 20 equal half yearly installments.**

**If the first installment is paid after six months from the date of purchase, find the amount of each installment.**

**[Given log 10.6 = 1.0253 and log 31.19 = 1.494]**

**Solution**

Here 2,00,000 has been paid at the time of purchase when cost of the flat was ` 3,00,000, we have to

Consider 20 equated half yearly annuity payments Rs. P when 12% is rate of annual interest compounded half

Yearly for present value of 1,00,000rs.

So that-

or

then

Hence the amount of each installment = 8,718.40 Suppose,

Taking log-

Hence

X = 0.3119