Simple annuities

From Wikicpa

An annuity is a sequence of periodic payments, usually equal and made at equal intervals of time. For example, familiar annuity payments are: rent & mortgage payments, insurance premiums or installment agreements. The period of time between consecutive payments is often called the payment interval and may be of any convenient length. (Originally the term 'annuity' referred to only payments made on an annual basis but is now usually used for payments with any set interval). The term of an annuity is the time from the beginning of the first payment to the end of the last payment.

Illustration: Say Mr Smith purchases a new vehicle for $2,000 down followed by 36 monthly installments of $500, the first due 1 month after the sale date. The monthly installments are an ordinary annuity whose payment term starts on the day of the sale and continues for 3 years. The payment interval is 1 months.

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Present value

The present value of an annuity is defined as the equivalent value of the set of payments due at the beginning of the term. When the payment interval and the interest conversion period are the same, the annuity is called a simple annuity.

Example: What is the present value of an annuity and the amount of an ordinary annuity consisting of 5 semiannual payments of $100 each assuming an interest rate factor of 4%

Solution: Let "A" represent the present value. and "B" represent the amount. The formula is calculated as follows:

$\mathrm{P} = 100(1.02)^{-1} + 100(1.02)^{-2} + 100(1.02)^{-3} + 100(1.02)^{-4}+ 100(1.02)^{-5} = $471.35\,$

Note: 2% is used (1.02) due to the payments being semiannual

$\mathrm{B} = 100+100(1.02) + 100(1.02)^{2} + 100(1.02)^{3} + 100(1.02)^{4} = $520.40\,$

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Present value formula (equivalant)

Source:Wikipedia.org
One hundred units 1 year from now at 5% interest rate is today worth:

${\rm Present\ value}=\frac{\rm future\ amount}{(1+{\rm interest\ rate})^{\rm term}}=\frac{100}{(1+.05)^1}=\ 95.23.$

So the present value of 100 units 1 year from now at 5% is 95.23 units.

The above is in regard to a single lump sum amount. There is a separate formula to calculate PV of annuities. For present value of annuities, use this formula:

$\mbox{PV annuity} = \frac{1-(1+r)^{-n}}{r}\cdot(\mbox{payment amount}).\,$