An annuity is a loan, mortgage, or similar contract to make fixed payments of money at regular intervals over a certain period of time in exchange for a sum of money.  Interest is paid on the money at a fixed rate and is compounded each interval.


Derive 's financial functions are based on the equation

                            n               
       n    p·(1+i·t)·((1+i)  - 1)          
v·(1+i)  + ———————————————————————— + f = 0 
                       i                    

that applies to an annuity with a present value v in which a fixed payment p is paid for n regular periods at a periodic interest rate of i with a future value of f.  Positive numbers represent money you receive; negative numbers represent money you pay out.


The financial functions have an optional fifth argument t, the time of payment.  Make this argument 0 if the payments are made at the end of each period, 1 if the payments are made at the beginning of each period, or a proportionate fraction between 0 and 1 if the payments are made within each period.  The default value of t is 0, indicating that payments are made at the end of each period.


Because of catastrophic cancellation in the above formula, it is advisable to use 10 or more significant digits of precision when using these functions (see the Precision fieldPrecision_field of the Options > Mode Settings > Simplification command19_L5FP).


PVAL(i, n, p, f, t) is the present value of the contract.  The future value argument f defaults to 0.


FVAL(i, n, p, v, t) is the future value of the contract.  The present value argument v defaults to 0.


PMT(i, n, v, f, t) is the periodic payment of the contract.  The present value argument v and the future value argument f default to 0.


NPER(i, p, v, f, t) is the number of payment periods.  The present value argument v and the future value argument f default to 0.


RATE(n, p, v, f, t) is the periodic interest rate.  Note that RATE returns the interest rate per period, not necessarily the annual interest rate.  The present value argument v and the future value argument f default to 0.  If the interest rate is negative or greater than 1 (i.e. 100%), lower and upper bounds on the rate must be entered as the 6th and 7th argument of RATE, respectively.


For example, to compute the monthly payment on a 4 year car loan of $15,000 at a 10% annual interest rate, approximate

PMT(10%/12, 4·12, 15000)

using 10 significant digits of precision.  This gives a monthly payment of -380.4387515.  The amount is negative because it represents an outflow of money.


As another example, to compute the balance of a savings account in 5 years after making an initial deposit of $1,000, making monthly payments of $100 at the beginning of the month, and earning 5.5% annual interest compounded monthly, approximate

FVAL(5.5%/12, 5·12, -100, -1000, 1)

using 10 significant digits of precision.  This gives a future balance of 8235.356459.  The monthly payment and the initial deposit are negative because they represent an outflow of money.


Other Built-in Functions and ConstantsBuilt_in_Functions_and_Constants 

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