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Annuity Dr. Vijay Kumar vijay.gahlawat@gmail.com

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In our day to day life we observe a lot of money transactions. In many money transactions payment is made in single transaction or in equal installments over a certain period of time. The amounts of these installations are determined in such a way that they compensate for their waiting time.

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In other cases, in order to meet future planned expenses, a regular saving may be done, i.e., at regular time intervals a certain amount may be kept aside, on which the person gains interest. In such cases the concept of annuity is used. A sequence of equal payments made/received at equal intervals of time is called annuity. The amount of regular payment of an annuity is called periodic payments. The time interval between two successive payments is called is called payment interval or period.

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Payment periods may be annual, half yearly, quarterly, monthly or any fixed duration of time. The time for which the payment of an annuity is made is called term of annuity, i.e., it is the time interval between the first payment and the last payment. The sum of all payments made and interest earned on them at the end of the term of annuities is called future value of an annuity, i.e., it is the total worth of all the payments at the end of the term of an annuity.

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The present and or capital value of an annuity is the sum of the present values of all the payments of the annuity at the beginning of the annuity, i.e. it is the amount of money that must be invested in the beginning of the annuity of purchase the payments due in future. (payments means yearly payments)

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Types of Annuity An annuity payable for a fixed number of years is called annuity certain. Installments of payment for a plot of land, bank security deposits, purchase of domestic durables are examples of annuities certain. Here the buyer/person knows the specified dates on which installments are to be made. Annuity Contingent:- An annuity payable at regular interval of time till the happening of a specific event or the date of which can not be foretold.

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Types of Annuity… For example, the premiums on a life insurance policy, or a fixed sum paid to an unmarried girl at regular intervals of time till her marriage takes place. Perpetual Annuity or Perpetuity:- An annuity payable forever. In it, beginning date is known but the terminal date is unknown, i.e., an annuity whose payments continues forever is called perpetuity. For example – the endowment funds of trust, where the interest earned is used for welfare activities only. The principal remains the same and activity continuous forever.

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1. All the above types of annuities are based on the number of their periods. 2. An annuity in which payments of installments are made at the end of each period is called ordinary annuity or annuity immediate, e.g. repayment of housing loan, car loan etc. 3. An annuity in which payments of installments are made in the beginning of each period is called annuity due. In annuity due every payment is an investment

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and earns interest. Next payment will earn interest for one period less and so on, the last payment will earn interest of one period, e.g. saving schemes, life insurance, life insurance payments etc. 4. An annuity which is payable after the lapse of a number of periods is called deferred annuity. In it, the term begins after certain time period termed as deferment period, e.g., pension plan of L.I.C. Many financial organizations give loan amount immediately

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and regular installments may start after specified time period.

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Formulae 1. Amount of Immediate Annuity or ordinary annuity. Let a be the ordinary annuity, i % = the rate of interest per period. The total annuity (A) for n period at i% rate of interest is:

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2. Present value of Immediate Annuity (or ordinary Annuity) (In the case of immediate annuity, payments are made periodically at the end of specified period) a = annual payment of an ordinary annuity n = number of years i% = rate of interest on one Rs. per year P = present value of the annuity.

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Formulae… 3. Amount of Annuity due at the End of n period a = annual payment of an ordinary annuity n = number of years i% = rate of interest on one Rs. per year

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Formulae… 4. Present value of Annuity due a = annual payment of an ordinary annuity n = number of years i% = rate of interest on one Rs. per year

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Formulae… 5. Perpetual Annuity It is an annuity whose payment continues forever. As such the amount of perpetuity is undefined as the amount increases without any limit as time passes on. We know that the present value P of immediate annuity is given by

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Formulae… Now as per the definition of perpetual annuity as n ∞, we get

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Amortization A loan is said to be amortized if it can be removed by a sequence of equal payments made over equal periods of time, which consists of the interest on the loan outstanding at the beginning of the payment period and part payments of the loan. With each payment the principal amount outstanding decreases and hence interest part on each payment decreases while the loan repayment of principal increases.

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Amortization… When a loan is amortized, the principal outstanding is the present value of remaining payments. Based on this, the following formulae are obtained that describe the amortization of an interest bearing loan of Rs. A at a rate of i per units per period, by n equal payments of Rs. a each when the payment is made at the end of each period.

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Amortization… 1.Amount of each periodic payment = 2.Principal outstanding of pth period = 3.Interest contained in the pth payment= 4.Principal contained in pth payment= 5.Total interest paid = (na-A)

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Sinking Fund It is the fund created by a company or person to meet predetermined debts or certain liabilities out of their profit at the end of every accounting year at compound rate of interest. This fund is also known as sinking fund. If a is the periodic deposits or payments, at the rate of iper units per year than after n years the sinking fund

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Sinking Fund… A is

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Derivation of Formulae… the end of first period. Therefore it earns interest for (n-1) periods, second installment earns interest for (n- 2) periods and so on. The last installment earns for (n- n) period, i.e. earns no interest. The amount of first annuity for (n-1) period at i % rate per period =

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