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#### 7.6.3 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.

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