How to calculate Present Value- PV and Future Value-FV of annuity?

 (This post elucidates on what is annuity, what is ordinary annuity, how to calculate present and future value of annuity along with formulas for calculation of annuity values)

An annuity can be defined as a sequence of equal payments made at equal intervals of time. Equated Monthly Installments (EMIs) of loan repayments, monthly remittance of Recurring Deposit installments, Payments of Life Insurance Premium etc. are examples of annuities.

Expressions used in annuity: The time period between successive payments is called payment period or payment interval such as weekly, quarterly, half yearly, annually etc. The period between beginnings of the annuity payment to last annuity payment is known as term or duration of the annuity. The equal amount of each annuity payment is known as the periodic payment of the annuity. The person who receives the payment is called annuitant. The sum of all payments made in a year is called annual rent.

Ordinary annuity:  The series of periodic payments where payments are made at the end of each payment period. In ordinary annuity (or immediate annuity), amount of annuities certain, the no. of payments are fixed. For example, you borrow some loan from a bank which is repayable in 30 EMIs and as per the sanction terms; the first installment of loan repayment will be started one month after the date of borrowings. Here, payment is at the end of each payment interval. Thus, it is an example of ordinary annuity where annuities certain, the no. of installments are fixed and payments are made at the end of each payment period.

Annuity Due: When the equal amount of each annuity payment made at the beginning of each period is known as ‘Annuity Due’. Here, payment is at the beginning of each payment interval. Payments of life insurance premium, payments of recurring deposit installments in the bank are examples of annuity due. Here, annuities certain, the no. of payments are fixed.

Future value of annuity:

The total amount (Principal plus accrued compound interest) due at the end of the term of the annuity is called as ‘Future Value of annuity’. In the other words, future value of annuity is the sum total of each installment kept on compound interest till the end of the term. To calculate the future value of an annuity (to find what the value at a future date would be for a series of periodic payments) following formula is used.

FV=PMT [(1+r) ^n-1) ÷ r] where PMT=Periodic Payment, r=rate of interest per period, n=number of periods.

Present value of annuity:

The current value of a sequence of equal periodic payments made over some time is called the ‘Present Value of annuity’. In other words, the present value of an annuity determines the series of future periodic payments at a given time. Thus, the present value of the annuity worked out based on the concept of the time value of money, in that one hundred rupee of the present day is worth more than those same Rupees at a future date.

Note:  (1) To find the annuity value due, where payments are made at the end of each payment period we have to apply the following formula;

P VA= PMT x [(1 – (1 / (1 + r) ^ n) ÷r]

(2)To find the value of an annuity due where payments are made at the beginning of the payment period, multiply the above formula by a factor of (1 + r);

P VA= PMT x [(1 – (1 / (1 + r) ^ n) ÷r] x (1 + r)

[PVA=Present Value of Annuity, PMT=Periodic Payment, r=rate of interest per period, n=number of periods.]

Example (1):

A company offers to its employee, who is retiring under a voluntary retirement scheme, an annuity that pays Rs.250000 at the end of every year for the next 20 years with a 6 percent discount rate or a Rs.2900000/- (Rupees twenty nine lakh) lump-sum payment. The employee needs to determine the more rational option. You need to advise that employee which is the better option.

PV= PMT[1-(1/1+r)^n]/r

Present Value of the annuity= 250000[1-(1/1.06)^20]/0.06

PVA=250000[0.688195273]/0.06=2867480.30. The lump sum offer is 2900000/-

Given this information, the annuity is worth Rs.32520 less on a time-adjusted basis, so the employee should choose the lump-sum payment over the annuity

Example  2: A company offers to its employee, who is retiring under a voluntary retirement scheme, an annuity that pays Rs.250000 at the beginning of every year for the next 20 years with a 6 percent discount rate or an Rs.2900000/- (Rupees twenty-nine lakh) lump-sum payment. The employee needs to determine the more rational option. You need to advise that employee which is the better option.

Calculation: With an annuity due, the payments are made at the beginning of the period in question, and then the formula for present value will be.

PV A= PMT x ((1 – (1 / (1 + r) ^ n)) / r) x (1 + r)

Present Value of the annuity= 250000[1-(1/1.06)^20)/0.06] x 1.06

PV =250000[0.688195273]/0.06]x(1.06)= 250000[0.688195273]/0.06]x1.06= 3039529/-  The lump sum offer is only  Rs.2900000/-. Given this information, the annuity is worth Rs.139529 more on a time-adjusted basis, so the employee should choose the annuity payment over the lump sum payment.