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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:

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

There are two types of annuity

  1. Annuity Regular: In annuity regular first payment/receipt takes place at the end of first period.
  2. 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 ₹10,000 in a bank at 10% p.a. After 1 year, you will get back ₹11,000. Hence, we can say that the future value of ₹10,000 after 1 year at 10% p.a. rate of interest is ₹11,000.
 

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

 

FV = CF (1 + r)n

 

where FV  Future value

         CF  single cash flow

         r  rate of interest in decimal (R/100)

        n  time period of the annuity
 

Example
Ram invests ₹5,000 in a two year investment that pays you 8% per annum. Calculate the future value of the investment.
Solution
We know the equation for calculating future value of a cash flow is FV = CF (1 + r)n.
CF = Cash flow = ₹5000
r = rate of interest = 0.08
n = time period = 2
Substituting the values, FV = ₹5000 (1 + 0.08)2
= ₹5000 × 1.1664 = ₹5832

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,

 

Description: 57295.png 
 

Example
Find the future value of an annuity of ₹1000 made annually for 7 years at an interest rate of 10% compounded annually. Given: (1.10)7 = 1.9487.
Solution
Here, annual payment P = ₹1,000
 
n = 7, r = 10% = 0.10
 
Future value of the annuity can be calculated using Description: 57342.png
 
 
Substituting the values in the formula, we get
 
Description: 57359.png 
Thus, A = ₹9487

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 × (1 + r)
 

Example
Mr. X invests ₹5,000 every year for the next 20 years, starting from today. If interest rate is 6 % p.a. compounded annually, calculate the future value of the annuity.
[Given: (1 + 0.06)20 = 3.207]
Solution
First let us consider the annuity to be an annuity regular and calculate the future value.
Description: 57366.png 
Future value of annuity due = 183916.67 × (1 + 0.06) = ₹194951.67

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

 

Description: 57377.png 

 

Example
What is the present value of ₹10 to be received after two years compounded annually at 10%?
Solution
Here, A = ₹10, r = 10% = 0.1, n = 2
 
Required present value Description: 57398.png
 
Description: 57410.png

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

 

Description: 57424.png 

 

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.
 

Example
Sunil borrows ₹10,00,000 to buy a house. If he pays equal installments for 20 years at 12% interest on outstanding balance, what will be the equal annual installment?
(1.1220 = 9.6463)
Solution
Given, present value PV = ₹10, 00,000
 
n = 20, r = 12 %, p.a. = 0.12
 
Description: 57486.png 

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.
 

Example
Suppose you start saving ₹25,000 in a bank every year for the next 5 years starting from today. The bank gives you 10% p.a. interest compounded annually. What is the present value of your savings?
Solution
First, let us compute the present value by considering the annuity to be an annuity regular.
 
Here, make sure that the time period you take is 1 less than the given period.
 
Description: 57527.png 
 
Now, you need to add the initial cash payment to this amount. Hence, the value of annuity due will be
 
PV = 79242.2 + 25000 = ₹104242.5

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

 

Description: 57509.png 

 

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.

 

Example
How much amount has to be invested every year to accumulate ₹5,00,000 at the end of 5 years if interest is compounded annually at 10%?
Solution
Here, final amount is given. We need to find the periodic instalments.
 
A = 500000, n = 5, r = 0.1
 
Description: 57533.png 





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