Related Financial Functions
---------------------------

The functions in this section are basically inverses of the present
value functions with respect to the various arguments.

The `b M' (`calc-fin-pmt') [`pmt'] command computes the amount of
periodic payment necessary to amortize a loan.  Thus `pmt(RATE, N,
AMOUNT)' equals the value of PAYMENT such that `pv(RATE, N, PAYMENT) =
AMOUNT'.

The `I b M' [`pmtb'] command does the same computation but using `pvb'
instead of `pv'.  Like `pv' and `pvb', these functions can also take a
fourth argument which represents an initial lump-sum investment.

The `H b M' key just invokes the `fvl' function, which is the inverse
of `pvl'.  There is no explicit `pmtl' function.

The `b #' (`calc-fin-nper') [`nper'] command computes the number of
regular payments necessary to amortize a loan.  Thus `nper(RATE,
PAYMENT, AMOUNT)' equals the value of N such that `pv(RATE, N,
PAYMENT) = AMOUNT'.  If PAYMENT is too small ever to amortize a loan
for AMOUNT at interest rate RATE, the `nper' function is left in
symbolic form.

The `I b #' [`nperb'] command does the same computation but using
`pvb' instead of `pv'.  You can give a fourth lump-sum argument to
these functions, but the computation will be rather slow in the
four-argument case.

The `H b #' [`nperl'] command does the same computation using `pvl'.
By exchanging PAYMENT and AMOUNT you can also get the solution for
`fvl'.  For example, `nperl(8%, 2000, 1000) = 9.006', so if you place
$1000 in a bank account earning 8%, it will take nine years to grow to
$2000.

The `b T' (`calc-fin-rate') [`rate'] command computes the rate of
return on an investment.  This is also an inverse of `pv': `rate(N,
PAYMENT, AMOUNT)' computes the value of RATE such that `pv(RATE, N,
PAYMENT) = AMOUNT'.  The result is expressed as a formula like `6.3%'.

The `I b T' [`rateb'] and `H b T' [`ratel'] commands solve the
analogous equations with `pvb' or `pvl' in place of `pv'.  Also,
`rate' and `rateb' can accept an optional fourth argument just like
`pv' and `pvb'.  To redo the above example from a different
perspective, `ratel(9, 2000, 1000) = 8.00597%', which says you will
need an interest rate of 8% in order to double your account in nine
years.

The `b I' (`calc-fin-irr') [`irr'] command is the analogous function
to `rate' but for net present value.  Its argument is a vector of
payments.  Thus `irr(PAYMENTS)' computes the RATE such that `npv(RATE,
PAYMENTS) = 0'; this rate is known as the "internal rate of return".

The `I b I' [`irrb'] command computes the internal rate of return
assuming payments occur at the beginning of each period.