There is a specific satisfaction in watching a net present value come out positive. You have entered your best estimate for each assumption — demand, price, cost, the discount rate — pressed calculate, and the model has returned a single, confident, black number. The project clears the hurdle. The decision, it seems, has made itself. I have felt that satisfaction many times, and I have learned to distrust it, because a positive NPV is an answer to a question I did not quite ask.
The cleanest way to see the problem is an old joke about a statistician who drowned while fording a river that was, on average, three feet deep. The river was a foot deep at the banks and well over his head in the middle; the average was true and useless, and the gap between the two killed him. Sam Savage, a Stanford professor who turned this into a small crusade, calls it the flaw of averages, and it sits quietly inside almost every single-point NPV.
The average of the inputs is not the input of the average
The flaw comes in two strengths. The weak version is the more familiar one: a single number throws away the range of things that might happen, so a forecast of “1,000 units” tells you nothing about the months of 400 or 1,600. The strong version is sharper and less intuitive, and it has a name. A Danish mathematician, Johan Jensen, proved in 1906 that whenever the relationship between inputs and output is curved rather than straight, the output of the average input is not the same as the average of the outputs. Business schools teach this as Jensen’s inequality and then watch their graduates go straight back to plugging average assumptions into models, surprised every time by results that come in below projection and behind schedule.
A small example makes it concrete. Suppose your plan assumes monthly demand of 1,000 and shows a tidy positive NPV. Real demand, though, is closer to 1,400 in good months and 600 in poor ones, which does indeed average your assumed 1,000. In the good months your capacity caps you at, say, 1,100, so you cannot bank the extra 300 of demand; in the poor months you serve 600 and still pay the rent and staff sized for 1,100. Average those two real months together and your profit lands well below what the average-demand model promised. The model did not lie about the input. It lied about what that input produces, because the path from demand to profit bends — capped on the upside, exposed on the downside — and a bent relationship is exactly where averaging the inputs misleads you. A single NPV computed from central assumptions is, more often than not, a biased estimate dressed as a precise one.
Even the correct expected value is not a decision
Suppose you fix that — you run the model across the full range of demand and compute the true expected NPV rather than the NPV at average demand. You now have a better number, and you still do not have a decision.
Expected value is an average over many parallel worlds. It tells you what you would earn if you could run this venture thousands of times and bank the mean. A founder, an investor committing a fund, a consultant advising a single client, does not get thousands of attempts; they get one. A bet with a handsome positive expected value — a 60% chance of a fortune, a 40% chance of insolvency — has a mean that looks excellent and a reality that ends four times in ten with the company gone. The expected value cannot see the difference between a loss you can absorb and a loss that removes you from the game, because averaging treats ruin as just another number to fold into the mean.
A real decision therefore needs three things the NPV alone withholds: the shape of the distribution, not just its centre; the size and likelihood of the downside specifically; and an honest reckoning of whether you can survive that downside and still be standing to enjoy the upside. The drunk in Savage’s other example is instructive — staggering down the centre of a motorway, his average position is the safe white line, while his average fate is roadkill. The mean is alive; the man is dead.
And it says nothing about when
There is a final blind spot worth naming, because it is the one that empties bank accounts fastest. NPV collapses an entire timeline into a present-value number and so says nothing about the order in which the cash arrives. A project with a healthy positive NPV can still run out of money in month fourteen, because the costs land early and the returns land late, and a business that is solvent on paper over five years can be insolvent in practice in year two. The mean outcome over the life of the project is silent about the path taken to get there, and the path is where you actually have to keep paying wages.
From a number to a decision
None of this is an argument against NPV, which remains one of the better summaries of a project’s economics. It is an argument about its job. A positive NPV is an input to a decision rather than the decision itself, and it earns that role only once it is surrounded by the things it cannot tell you on its own: how the figure behaves when the assumptions move, how heavy the left tail is, and whether the cash holds together month by month.
Producing that surrounding context is, deliberately, most of what FinModeler does after it generates the base case. The deterministic engine gives you a model whose arithmetic is fixed and inspectable; the scenarios, sensitivity analysis and Monte Carlo simulation then run on top of it to show the NPV as a distribution rather than a point, with the downside and the drivers carrying it made explicit. Savage’s own prescription for the flaw of averages was to stop representing an uncertain quantity with one number and start representing it with many, which is precisely what simulation does — a theme worth the separate article it already has. The result is not a different NPV. It is an NPV you can actually decide with.
So by all means enjoy the moment the number turns positive. Then treat it as the beginning of the analysis rather than the end of it, and check how deep the river is in the middle before you wade in.
Test the downside before you fund the upside — model your idea and see the full distribution on FinModeler.
FAQs
What is the flaw of averages?
The systematic error that creeps in when an uncertain quantity is replaced by a single average value. In its strong form — Jensen’s inequality — the output of the average input differs from the average of the outputs whenever the relationship between them is non-linear, which in financial models it almost always is.
If NPV is flawed, what should I use instead?
Keep the NPV, but stop treating it as the answer. Run the model across the range of each key assumption to see the distribution of outcomes, pay particular attention to the downside and to the timing of cash, and weigh the result against your capacity to absorb a bad year. Scenario analysis and Monte Carlo simulation are the standard tools for this.
Does a positive expected NPV mean I should proceed?
Not on its own. Expected value is an average over many repetitions, and most real commitments happen once. A positive mean that carries a meaningful chance of ruin can still be a poor decision for anyone who cannot survive the bad outcome.
Sources
- The flaw of averages, the drowned statistician and the strong/weak distinction: Sam L. Savage, The Flaw of Averages: Why We Underestimate Risk in the Face of Uncertainty (and his 2002 Harvard Business Review article of the same name).
- The formal underpinning: Jensen’s inequality (Johan Jensen, 1906), as taught in management science and applied to non-linear loss relationships.