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8.6.6 Definitions

For your reference, here are the actual formulas used to compute Calc's financial functions.

Calc will not evaluate a financial function unless the rate or n argument is known. However, payment or amount can be a variable. Calc expands these functions according to the formulas below for symbolic arguments only when you use the a " (calc-expand-formula) command, or when taking derivatives or integrals or solving equations involving the functions.

These formulas are shown using the conventions of Big display mode (d B); for example, the formula for fv written linearly is ‘pmt * ((1 + rate)^n) - 1) / rate’.

                                             n
                                   (1 + rate)  - 1
     fv(rate, n, pmt) =      pmt * ---------------
                                        rate
     
                                              n
                                   ((1 + rate)  - 1) (1 + rate)
     fvb(rate, n, pmt) =     pmt * ----------------------------
                                              rate
     
                                             n
     fvl(rate, n, pmt) =     pmt * (1 + rate)
     
                                                 -n
                                   1 - (1 + rate)
     pv(rate, n, pmt) =      pmt * ----------------
                                         rate
     
                                                  -n
                                   (1 - (1 + rate)  ) (1 + rate)
     pvb(rate, n, pmt) =     pmt * -----------------------------
                                              rate
     
                                             -n
     pvl(rate, n, pmt) =     pmt * (1 + rate)
     
                                         -1               -2               -3
     npv(rate, [a, b, c]) =  a*(1 + rate)   + b*(1 + rate)   + c*(1 + rate)
     
                                             -1               -2
     npvb(rate, [a, b, c]) = a + b*(1 + rate)   + c*(1 + rate)
     
                                                  -n
                             (amt - x * (1 + rate)  ) * rate
     pmt(rate, n, amt, x) =  -------------------------------
                                                  -n
                                    1 - (1 + rate)
     
                                                  -n
                             (amt - x * (1 + rate)  ) * rate
     pmtb(rate, n, amt, x) = -------------------------------
                                             -n
                              (1 - (1 + rate)  ) (1 + rate)
     
                                        amt * rate
     nper(rate, pmt, amt) =  - log(1 - ------------, 1 + rate)
                                           pmt
     
                                         amt * rate
     nperb(rate, pmt, amt) = - log(1 - ---------------, 1 + rate)
                                       pmt * (1 + rate)
     
                                   amt
     nperl(rate, pmt, amt) = - log(---, 1 + rate)
                                   pmt
     
                                1/n
                             pmt
     ratel(n, pmt, amt) =    ------ - 1
                                1/n
                             amt
     
                             cost - salv
     sln(cost, salv, life) = -----------
                                life
     
                                  (cost - salv) * (life - per + 1)
     syd(cost, salv, life, per) = --------------------------------
                                       life * (life + 1) / 2
     
                                  book * 2
     ddb(cost, salv, life, per) = --------,  book = cost - depreciation so far
                                    life

In pmt and pmtb, ‘x=0’ if omitted.

These functions accept any numeric objects, including error forms, intervals, and even (though not very usefully) complex numbers. The above formulas specify exactly the behavior of these functions with all sorts of inputs.

Note that if the first argument to the log in nper is negative, nper leaves itself in symbolic form rather than returning a (financially meaningless) complex number.

rate(num, pmt, amt)’ solves the equation ‘pv(rate, num, pmt) = amt’ for ‘rate’ using H a R (calc-find-root), with the interval ‘[.01% .. 100%]’ for an initial guess. The rateb function is the same except that it uses pvb. Note that ratel can be solved directly; its formula is shown in the above list.

Similarly, ‘irr(pmts)’ solves the equation ‘npv(rate, pmts) = 0’ for ‘rate’.

If you give a fourth argument to nper or nperb, Calc will also use H a R to solve the equation using an initial guess interval of ‘[0 .. 100]’.

A fourth argument to fv simply sums the two components calculated from the above formulas for fv and fvl. The same is true of fvb, pv, and pvb.

The ddb function is computed iteratively; the “book” value starts out equal to cost, and decreases according to the above formula for the specified number of periods. If the book value would decrease below salvage, it only decreases to salvage and the depreciation is zero for all subsequent periods. The ddb function returns the amount the book value decreased in the specified period.