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#### 7.6.6 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.

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