Why not to look a gift horse in the mouth—don’t ask the value of a present.

Net present value, NPV, means the current value of a future sum you are expecting. To have and to hold, obviously.

The page on compound interest showed you that a sum held now will multiply by (1+i)ⁿ where i is the interest and n the number of periods for which it applies. At the end of this time, original sum A has become A* (1+i)ⁿ = R. Reversing this, R = A/(1+i)ⁿ. So amount R off in the future is worth A to you right now.

The point is, really, that you can’t add together piles of money that aren’t in your grubby paws right now; when some of them are future monies, you have to give it a ‘now’ value. If these invisible monies aren’t obvious right now, what about debts that are owed to you? These are future money. Similarly, you have to pay back loans, make good on promises and so on.

The interest rate that you choose should reflect what would have happened to an investment of similar risk; this is sometimes called the rate of return, or the discount rate. In general, cash flows in and out should be discounted back to their NPV, and equivalent risk items can be aggregated, so that net cash flow becomes the A amount in the formula above.

A thoroughly planned project might well come down to a decision about the net gain to the company; NPV>0 means yes, NPV means no and so on. NPV close to zero might well imply indifference (but I doubt that). My experience says there are too many variables (available for massage) to let a project be presented in different lights, plus there are intangible factors (happy customers, increased enthusiasm in the staff) so hard to put a value to (till it’s gone).

Wikipedia, wonderful site that it is, points to an integral form. Obvious, I thought.

NPV(i) = Σ Rt / (1+i^{)t } 0≤t≤N would change to integral of (1+i)^{-t} r(t) dt 0≤t

By redefining some terms, this can be made to look like many other functions, which is convenient because they, the other functions, model behaviour well. The target is

F(s) = ∫ e^{-st} f(t) dt where s is, obviously (I thought), s = ln (1+i)

FM [CIE] students will recognise that s could be complex, allowing for oscillation in the modelling.

Internal rate of return (and its other forms), adjusted present value and cost-benefit analysis are related terms. These are hard-headed decision measures that ignore the value of staff, flexibility of staff and management — what I might call the ‘soft’ costs.

Attached to this calculation is the idea of opportunity cost. That is the difference between this opportunity being considered and the fallback position, perhaps the second-best option. The opportunity cost of going to the pub includes the cost of missing the film on tv.

What is there about this that makes me so bored? I can’t get excited about what I deem as entirely obvious modelling. I think it is the attached jargon, which is what happens when people try to label all sorts of stuff and then get confused what the definitions are. By the time it gets sorted out, every term has a new, less imprecise, meaning — which meaning is still abused by those who don’t quite ‘get’ that the whole was never difficult, that it was only made difficult because of the muddle that the same terminology is supposed to eliminate. But doesn’t. Repeat cyclically until either clear/clean or grey beyond repair.

Yeah, I can work out all sorts of simple arithmetic on cash-flows. That’s what a computer is good at. If there are enough items and the money flow is big enough, this might be difficult. I guarantee that the apparently trivial decision of which rate to pick to reflect risk hides enough error to (in turn) guarantee that bad decisions are easily made over small differences. That’s called something fancy, like sensitivity analysis. Or to a Ferengi, another business opportunity.

I’d rather save ‘sensitivity analysis’ to what happens when I run my hand over an interested lady’s skin.....

DJS 20130209