annuity - Maple Help

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Finance

 annuity
 present value of an annuity

 Calling Sequence annuity(cash, rate, nperiods)

Parameters

 cash - cash flow rate - interest rate nperiods - number of periods

Description

 • The function annuity gives the present value at time=0 of an annuity of nperiods equal payments of the amount cash, starting at time=1.
 • Mortgages are examples of annuities.
 • Since annuity used to be part of the (now deprecated) finance package, for compatibility with older worksheets, this command can also be called using finance[annuity]. However, it is recommended that you use the superseding package name, Finance, instead: Finance[annuity].

Examples

 > $\mathrm{with}\left(\mathrm{Finance}\right):$

Present value of an annuity paying 100 units per year for 15 years starting next year. The interest rate is 10% per year.

 > $\mathrm{annuity}\left(100,0.10,15\right)$
 ${760.6079506}$ (1)

Monthly payments required for a mortgage of 10000 units, amortized over 25 years, not in advanced, with interest at 10% per year, compounded semiannually. There are 25*12 monthly payments.

 > $\mathrm{Npayments}≔25\cdot 12$
 ${\mathrm{Npayments}}{≔}{300}$ (2)

The semi-annual interest rate is 5% (10%/2). We need to find the monthly interest rate that when compounded give this 5% figure. This is calculated as follows. The range 0 .. 0.5, is to ensure we obtain the appropriate value.

 > $\mathrm{mrate}≔\frac{\mathrm{fsolve}\left(\mathrm{effectiverate}\left(r,6\right)=0.05,r,0..0.5\right)}{6}$
 ${\mathrm{mrate}}{≔}{0.008164846052}$ (3)

Verification: the future value of 1 units after 6 months should be 1.05

 > $\mathrm{futurevalue}\left(1.,\mathrm{mrate},6\right)$
 ${1.050000000}$ (4)

The value of an annuity of 1 units paid monthly for 25 years at the interest rate we calculated is then

 > $A≔\mathrm{annuity}\left(1,\mathrm{mrate},\mathrm{Npayments}\right)$
 ${A}{≔}{111.7958950}$ (5)

The required monthly payments are thus:

 > $P≔\frac{10000}{A}$
 ${P}{≔}{89.44872260}$ (6)

The total payments amount to:

 > $\mathrm{Tot}≔P\mathrm{Npayments}$
 ${\mathrm{Tot}}{≔}{26834.61678}$ (7)

From which the cost of the loan is readily obtained. The mortgage rules above are typical of mortgages obtained from banks in Canada. As with all legal matters, rules can change, so check first about their applicability.

The payments if the mortgage is paid in advanced (so the first payment is already deducted from the loan) are given by

 > $\mathrm{solve}\left(\mathrm{P1}+\mathrm{annuity}\left(\mathrm{P1},\mathrm{mrate},\mathrm{Npayments}-1\right)=10000\right)$
 ${88.72430239}$ (8)

Compatibility

 • The Finance[annuity] command was introduced in Maple 15.
 • For more information on Maple 15 changes, see Updates in Maple 15.

 See Also