The present value of an annuity is calculated by discounting each payment and adding them together. The formula is PV = C / (1 + rt) + C / ((1 + r1) (1 + r2)) +…+ C / ((1 + r1) (1 + r2)… (1 + rT-1) (1 + rT)). For a constant discount rate, the formula is PV = C / r * (1-1 / (1 + r) T). The present value of a finite annuity is calculated by subtracting the sum of the infinite series that represents the payments that will never be made. These formulas are used to calculate payments for 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 in the present 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.
By calculating the present value of an annuity, the formula is obtained: 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 annuity payment amount, 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 as long as the annuity makes payments, then the formula PV = C / r * (1-1 / (1 + r) T) can be used. This formula is derived from the step method of calculating the present value of an annuity. If the discount rate is always r, 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 perpetual annuity. This means that it would be an infinite series if the payments never stopped. Since the annuity payments are finite, the sum of a finite series must be calculated. To do this, calculate the sum of the infinite series as if the payments continue forever, then subtract the sum of the infinite series that represents the payments that will never be made. The present value of the series of payments after the end of the annuity 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 ranges 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 a finite annuity: C / r * (1-1 / (1 + r) T).
Formulas for the present value of an annuity are used to calculate payments for fully amortizing loans, or loans in which a finite number of equally sized payments pay off the interest and principal. An example of a fully amortized loan is a residential mortgage. Because payments are often made monthly while rates are annualized, you’ll need to adjust the numbers as you make your calculations. Use the number of payments times T and divide r by the number of payments per year. If the number of payments is uncertain, such as in a life annuity, actuarial data is used to estimate the number of payments that will be made, and that number is used to calculate the present value.
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