# Annuity (finance Theory) - Annuity-immediate

Annuity-immediate

An annuity is a series of payments made at fixed intervals of time. If the number of payments is known in advance, the annuity is an annuity-certain. If the payments are made at the end of the time periods, so that interest is accumulated before the payment, the annuity is called an annuity-immediate, or ordinary annuity. Mortgage payments are annuity-immediate, interest is earned before being paid.

 ↓ ↓ ... ↓ payments ——— ——— ——— ——— — 0 1 2 ... n periods

The present value of an annuity is the value of a stream of payments, discounted by the interest rate to account for the payments are being made at various moments in the future. The present value is given in actuarial notation by:

where is the number of terms and is the per period interest rate. Present value is linear in the amount of payments, therefore the present value for payments, or rent is:

$PV(i,n,R) = R times a_{overline{n}|i}$

In practice, often loans are stated per annum while interest is compounded and payments are made monthly. In this case, the interest is stated as a nominal interest rate, and .

The future value of an annuity is the accumulated amount, including payments and interest, of a stream of payments made to an interest-bearing account. For an annuity-immediate, it is the value immediately after the n-th payment. The future value is given by:

$s_{overline{n}|i} = frac{(1+i)^n-1}{i}$

where is the number of terms and is the per period interest rate. Future value is linear in the amount of payments, therefore the future value for payments, or rent is:

$FV(i,n,R) = R times s_{overline{n}|i}$

Example: The present value of a 5 year annuity with annual interest rate 12% and monthly payments of \$100 is:

$PV(0.12/12,5times 12,100) = 100 times a_{overline{60}|0.01} = 4,495.50$

The rent is understood as either the amount paid at the end of each period in return for an amount PV borrowed at time zero, the principal of the loan, or the amount paid out by an interest-bearing account at the end of each period when the amount PV is invested at time zero, and the account becomes zero with the n-th withdrawal.

Future and present values are related as:

$s_{overline{n}|i} = (1+i)^n times a_{overline{n}|i}$

and

$frac{1}{a_{overline{n}|i}} - frac{1}{s_{overline{n}|i}} = i$