Annuity

for more see shahjahan mina ( for valuation of share & time value of money)

Annuity means a stream or series of equal payments. For example, you have made an investment that will generate an interest income of $5,000 for you at the end of each year for five years. The income of $5,000 at the end of each year is an annuity.

Annuities may be in the form of regular deposits in one’s savings account, monthly insurance payments and monthly home mortgage payments. Payments can be weekly, monthly, yearly etc.
An annuity may be of the following types:


Annuity-due - Under it, payments are made at the beginning of each period. Some examples of annuity-due are deposits in savings, insurance payments, rent or lease payments etc.

Ordinary annuity (also annuity-immediate) - Payments are made by the end of each period.
An ordinary annuity is a series of equal payments, with all payments being made at the end of each successive period. An example of an ordinary annuity is a series of rent or lease payments. The present value calculation for an ordinary annuity is used to determine the total cost of an annuity if it were to be paid right now.

The value of money changes over time. The value of a dollar in hand today is more than the value of a dollar to be received a year from now because if you have a dollar in hand today you can invest it elsewhere and earn some interest on it.

The present value is computed either for a single payment or for a series of payments to be received in future. This article explains the computation of the present value of a single payment to be received at a single point in future.

The present value of a single payment in future can be computed by using present value formula.
Present value formula:

Where;
PV = Present value of the amount
PV = Future value of the amount (amount to be received in future).
i = Interest rate
n = Number of periods
The following examples explain the computation of the present value of a single payment.

Example 1:
A company is expecting to receive $5,000 four years from now. Compute present value of this sum if the current market interest rate is 10% and the interest is compounded annually.
Solution:
To find out the present value, the amount of $5,000 to be received in future would be discounted using the given interest rate of 10% the interest is compounded annually.
Solution:
To find out the present value, the amount of $5,000 to be received in future would be discounted using the given interest rate of 10%. We can do so using the present value formula given above
(1) Use of present value formula:

= $5,000/(1 + 10%)4
= $5,000/(1 + 0.1*)4
= $5,000/(1.1)4
= $5,000 / 1.4641
= $3,415
n = 4
i = 10%
*10/100 = 0.1
The amount of $5,000 to be received after four years has a present value of $3,415. It means if the amount of $3,415 is invested today @10% per year compounded annually, it will grow to $5,000 in 4 years.


payment to be received at a series of payments.
Capital investments usually involve in generating series of cash flows and managers need to take into account the present value of such series of cash flows (i.e., annuities) to know the true profitability of an investment project.
The formula to compute present value of an annuity is given below:
Formula:

Where;
A = Present value of an annuity
R = Amount of an annuity
i = Rate of interest
n = Number of periods
Example 1:
A company has made an investment in government bonds. The bonds will generate an interest income of $25,000 each year for 5 years. The interest rate is 10% compounded annually.
Required: Compute present value of the stream of interest income for 5 years.
Solution:

= $25,000 × [(1 + 0.1)5 – 1 / 0.1(1 + 0.1)5]
= $25,000 × [(1.1)5 – 1 / 0.1(1.1)5]
= $25,000 × [1.61051 – 1 / 0.1(1.61051)]
= $25,000 × [0.61051 / 0.161051]
= $25,000 × [3.791]
= $94,775
At 10% interest compounded annually, the present value of this annuity is $94,775