The present value of an annuity is calculated by discounting each payment and adding them together, taking into account the different times when payments are made. Formulas are used to calculate payments on fully amortizing loans, such as residential mortgages. Actuarial data is used to estimate the number of payments for uncertain annuities.
The present value of an annuity, or a finite stream of payments of equal size, is calculated by determining the discounted value of each payment and adding them together. This value takes into account the different times when payments are made: a payment made in the future is worth less than the same amount today due to factors such as uncertainty and opportunity cost. To calculate it, divide the payment amount by 1 plus the discount rate for the first period; This is the present value of the first period. For the second period, divide the payment amount by 1 plus the discount rate for the first period multiplied by 1 plus the discount rate for the second period; repeat for each subsequent period.
Calculating the present value of an annuity yields the formula: PV = C / (1 + r1) + C / ((1 + r1) (1 + r2)) + C / ((1 + r1) (1 + r2) ) ( 1 + r3)) +… + C / ((1 + r1) (1 + r2)… (1 + rT-1) (1 + rT)). In the formula, C is the amount of the annuity payment, also called the coupon. The discount rate for each period is represented by rt, and T is the number of periods.
If the discount rate is constant for the entire time that the annuity makes payments, you can use the formula PV = C / r * (1-1 / (1 + r)T). This formula is derived from the stepwise method of calculating the present value of an annuity. If the discount rate is always r, then the present value of the first payment is C / (1 + r). The present value of the second payment is C / (1 + r) ^ 2, and so on. Therefore, the present value of an annuity is represented by: PV = C / (1 + r) + C / (1 + r) 2 +… + C / (1 + r) T-1 + C / (1 + r ) T.
An annuity can be thought of as a truncated perpetuity. This means that it would be an infinite series if the payments never stopped. Since the annuity payments are finite, you must calculate the sum of a finite series. To do this, calculate the sum of the infinite series as if the payments go on forever, then subtract the sum of the infinite series representing the payments that will never be made. The present value of the series of payments after the annuity stops is calculated with the formula: PV = C / (1 + r) T + 1 + C / (1 + r) T + 2 +…
The sum of an infinite geometric series in which the terms are described by A(1/b)k, where k varies from zero to infinity, is represented by A/(1-(1/b)). For an annuity with a constant discount rate, A is C / (1 + r) and b is (1 + r). The sum is C / r. For the series of payments that will never be made, A is C / (1 + r) T + 1 and b is (1 + r). The sum is C / (r * (1 + r)T). The difference gives the present value of an annuity that is finite: C / r * (1-1 / (1 + r)T).
Formulas for the present value of an annuity are used to calculate the payments on fully amortizing loans, or loans in which a finite number of equal-sized payments return interest and principal. An example of a fully amortizing loan is a residential mortgage. Since payments are often made on a monthly basis while fees are annualized, you need to adjust the numbers when doing the math. Use the number of payments for T and divide r by the number of payments per year. If the number of payments is uncertain, such as in a lifetime annuity, then actuarial data is used to estimate the number of payments that will be made, and that number is used to calculate present value.
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