# Annuity

Annuity is a sequence of periodic payments (or receipts) regularly over a specified period of time. When we pay a fixed amount of money periodically over a specified time period, we create an annuity. For example, payment for life insurance premium, rent of house, etc.In case of an annuity, a series of payments (or receipts) must have the following features:

- Amount paid (or received) must be constant over the period of annuity
- Time interval between two consecutive payments (or receipts) must be the same

There are two types of annuity

**Annuity Regular:**In annuity regular first payment/receipt takes place at the end of first period.**Annuity Due or Annuity Immediate:**In this case, the first receipt or payment is made at the beginning of the annuity.

# Future value

Future value is the cash value of an investment in the future.

Suppose, you invest â‚¹10,000 in a bank at 10% p.a. After 1 year, you will get back â‚¹11,000. Hence, we can say that the future value of â‚¹10,000 after 1 year at 10% p.a. rate of interest is â‚¹11,000.

Future value of a single cash flow can be computed using the relation:

*FV* = *CF* (1 + *r*)^{n}

where *FV* â†’ Future value

* CF* â†’ single cash flow

* r* â†’ rate of interest in decimal (*R*/100)

* n* â†’ time period of the annuity

*FV*=

*CF*(1 +

*r*)

*.*

^{n}*CF*= Cash flow = â‚¹5000

*r*= rate of interest = 0.08

*n*= time period = 2

Substituting the values,

*FV*= â‚¹5000 (1 + 0.08)

^{2}

= â‚¹5000 Ã— 1.1664 = â‚¹5832

# Future value of an annuity regular

Consider an annuity in which a certain sum *P* is deposited at the end of every conversion period for *n* conversion periods at a rate of interest *r* (rate of interest in decimals). Let *A* be the future value of the annuity. Then,

^{7 }= 1.9487.

*P*= â‚¹1,000

*n*= 7,

*r*= 10% = 0.10

*A*= â‚¹9487

# Future value of an annuity due

We know that in annuity due, the first payment is made at the beginning of the annuity period.Future value of annuity due can be calculated by first considering it as an annuity regular and using the method shown above and then multiplying the value obtained with (1 + *r*).

So, future value of an annuity due = future value of an annuity regular Ã— (1 + *r*)

[Given: (1 + 0.06)

^{20}= 3.207]

Future value of annuity due = 183916.67 Ã— (1 + 0.06) = â‚¹194951.67

# Present value

Present value of a sum of money to be paid or received at some point of time in future is its value today or at present. It means that this amount can be invested today (at a suitable interest rate) to get the desired amount in future.The present value *PV* of an amount *A* due at the end of *n* interest periods at a rate of interest *r* (in decimals) can be obtained by the following relation

*A*= â‚¹10,

*r*= 10% = 0.1,

*n*= 2

# Present value of an annuity regular

Let*PV*be the present value of an annuity

*A*. Then,

*PV*is the sum of the present values of payments.

*PV*can be calculated using the relation

One important application of this formula is to calculate periodic installments to repay a loan amount. Loan amount would be the present value, if the rate of interest and time are known, we can calculate what is the periodic payment required.

(1.12

^{20}= 9.6463)

*PV*= â‚¹10, 00,000

*n*= 20,

*r*= 12 %, p.a. = 0.12

# Present value of an annuity due

To find the present value of an annuity due, first consider it as an annuity regular and find the present value for a time period 1 less than the given time period. To this value, add the initial payment.*PV*= 79242.2 + 25000 = â‚¹104242.5

# Sinking Fund

It is the fund credited for a specified purpose by way of a series of periodic payments over a period of time at a specified rate of interest.The size of the sinking fund is computed using the relation

where *A* is the amount to be saved (future value)

* P* is the periodic payment

* **n* is the payment period

It is exactly like calculating future value of an annuity.

*A*= 500000,

*n*= 5,

*r*= 0.1