# Future value of series of cash flows

Annuities are fixed amounts of money that are paid in each period. This can include

• if the cash flow occurs at the end of the period;

• annuities due,

• if the cash flow occurs at the start of the period; and

• perpetual annuities,

• if the cash flow occurs continuously.

If the cash flow A (= annuity) is paid at the end of the period, the future value after the N th payment of ordinary annuities, if compounded at r percent, equals

FVN = A·[((1 + r)N – 1)/r].

### Beispiel

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Suppose 400 Dollar is paid into an account on December 31, for three consecutive years: 2009, 2010 and 2011. Find the value of the account on December 31, 2011, if the money is compounded annually at 7 percent.

We calculate

FVN = A·[((1 + r)N – 1)/r]

= 400·[((1 + 0.07)3 – 1)/0.07]

= 400·3.2149

= \$1.285,96.

We can show that this is true by comparing the different payments. The first payment, on December 31, 2009, is worth 400·1.072 = 457.96 Dollar two years later. The second payment, on December 31, 2010, is worth 400·1.07 = 428 Dollar one year later. Therefore, the future value of the three payments equals 457.96 + 428 + 400 = 1,285.96, Dollar as calculated above.