Present Value
-------------

The `b P' (`calc-fin-pv') [`pv'] command computes the present value of
an investment.  Like `fv', it takes three arguments: `pv(RATE, N,
PAYMENT)'.  It computes the present value of a series of regular
payments.  Suppose you have the chance to make an investment that will
pay $2000 per year over the next four years; as you receive these
payments you can put them in the bank at 9% interest.  You want to
know whether it is better to make the investment, or to keep the money
in the bank where it earns 9% interest right from the start.  The
calculation `pv(9%, 4, 2000)' gives the result 6479.44.  If your
initial investment must be less than this, say, $6000, then the
investment is worthwhile.  But if you had to put up $7000, then it
would be better just to leave it in the bank.

Here is the interpretation of the result of `pv': You are trying to
compare the return from the investment you are considering, which is
`fv(9%, 4, 2000) = 9146.26', with the return from leaving the money in
the bank, which is `fvl(9%, 4, X)' where X is the amount of money you
would have to put up in advance.  The `pv' function finds the
break-even point, `x = 6479.44', at which `fvl(9%, 4, 6479.44)' is
also equal to 9146.26.  This is the largest amount you should be
willing to invest.

The `I b P' [`pvb'] command solves the same problem, but with payments
occurring at the beginning of each interval.  It has the same
relationship to `fvb' as `pv' has to `fv'.  For example `pvb(9%, 4,
2000) = 7062.59', a larger number than `pv' produced because we get to
start earning interest on the return from our investment sooner.

The `H b P' [`pvl'] command computes the present value of an
investment that will pay off in one lump sum at the end of the period.
For example, if we get our $8000 all at the end of the four years,
`pvl(9%, 4, 8000) = 5667.40'.  This is much less than `pv' reported,
because we don't earn any interest on the return from this investment.
Note that `pvl' and `fvl' are simple inverses: `fvl(9%, 4, 5667.40) =
8000'.

You can give an optional fourth lump-sum argument to `pv' and `pvb';
this is handled in exactly the same way as the fourth argument for
`fv' and `fvb'.

The `b N' (`calc-fin-npv') [`npv'] command computes the net present
value of a series of irregular investments.  The first argument is the
interest rate.  The second argument is a vector which represents the
expected return from the investment at the end of each interval.  For
example, if the rate represents a yearly interest rate, then the
vector elements are the return from the first year, second year, and
so on.

Thus, `npv(9%, [2000,2000,2000,2000]) = pv(9%, 4, 2000) = 6479.44'.
Obviously this function is more interesting when the payments are not
all the same!

The `npv' function can actually have two or more arguments.
Multiple arguments are interpreted in the same way as for the
vector statistical functions like `vsum'.
See Single-Variable Statistics.  Basically, if there are several
payment arguments, each either a vector or a plain number, all these
values are collected left-to-right into the complete list of payments.
A numeric prefix argument on the `b N' command says how many payment
values or vectors to take from the stack.

The `I b N' [`npvb'] command computes the net present value where
payments occur at the beginning of each interval rather than at the
end.