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The Luck of the Average

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Why you have to be very lucky to have average luck.

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MINYANVILLE ORIGINAL One of the most common risk errors is to do a computation assuming average values for uncertain inputs and treat the result as the average outcome.

For example, suppose we have a fair coin, that is one that has a 50% chance of flipping heads and a 50% chance of flipping tails, with each flip independent of prior flips. The probability of flipping one head is, of course, 50%. The probability of flipping nine heads in a row is 1/512 or 0.2%.

Suppose instead you hold a coin for which you have no idea of the probability of flipping heads. You think that probability is equally likely to be any number between 0% and 100%. The chance of flipping one head is still 50%. But the probability of flipping nine heads in a row is now 10%, not 0.2%. The reason is that if the coin has a high probability of flipping heads, say 95%, the chance of getting nine heads in a row is 63%, while if the coin has a low probability of flipping heads, say 5%, the chance of getting nine heads in a row is 0.0000000002%. Thus the high probability of heads coins--the 95%'s--add much more to the 0.2% probability than the low probability of heads coins -- the 5%'s -- takeaway.

For a practical example, consider a proposed government program that will tax 0.1% of gross domestic product (GDP) to fund some useful service. The real (that is, after inflation) tax revenues will increase with real GDP growth, the real program costs will also increase at some rate. Let's suppose we project average real growth rates for both real GDP and program real costs are 3% per year.

If we assume both growth rates are exactly 3% per year, the program will cost 0.1% of GDP. But suppose we instead assume there is some future uncertainty about the growth rates, that each month the rates can be 0.05% higher or lower than the previous month. So in the first month, the real GDP growth rate might be 2.95 / 12% or 3.05 / 12%, and the same for the real program costs. Some factors will make the growth rates positively correlated, for example expanding population will generally increase both GDP and program costs. Other factors will argue for negative correlation, for example bad economic times mean low GDP growth and increased need for government expenditures. We assume the changes in the two growth rates are independent, the positive and negative correlations offset.





The expected cost of this program is almost 0.2% of GDP, not 0.1%. Both average growth rate assumptions were correct, but the projected total cost was wildly incorrect. Like the coin, the reason is the asymmetry in costs. If GDP growth is slow and program costs rise quickly, the cost can easily be 1% of GDP or more. In the reverse circumstance, rapid GDP growth and slow growth in program costs, the program costs will likely be something like 0.03% or 0.04% of GDP. The high scenarios add more to the 0.1% projected cost than the low scenarios can subtract.

Or think about a project with a number of inter-related steps. Some will come in early and below budget, others will come in late and above budget. But the early steps won't reduce total project time much because we usually can't push up scheduling of later steps. We know, however, that the late steps will delay things, often causing cascading delays, so a week late in one step can mean months late to the final deliverable. Also, it's hard to save more than 10 or 20% in a step, but it's easy to go 100% or 200% over budget.

A UK study found that government programs cost an average of 44% over initial estimate. To improve decision-making, 44% was added to cost estimates. The result? Since people knew 44% would be added to their total, they used lowball estimates at each step, figuring that the final adjustment would take care of any slippage. There was no significant improvement in accuracy.

So far, we have covered only random errors. Another factor that makes outcomes worse than projected is incentives. The people commissioning a project and managers generally have incentives to bring it in early and under budget. But the planning is done from their point of view, so these incentives are incorporated in the schedule. Lots of other people gain from more spending (someone gets that extra money) and later deliveries (less work, less pressure). There may even be people against the project who can delay it, make it more expensive or even prevent completion altogether. Even people for the project can contribute to the problem, by making optimistic statements in order to secure agreement to undertake it.

People often do good-expected-bad case analyses to account for these effects, but these seldom capture the effect of genuine uncertainty. Within each good-expected-bad scenario, everything is certain. A better plan is to make good-expected-bad estimates for every factor, then take the average result of every combination of possible factors.

Bad outcomes are often blamed on risk-the claim is that managers took too much risk and failed. But often the bad outcome represents average luck, or even above-average. Good risk management has to start with a realistic idea of what the expected outcome is, in order to understand the uncertainty around that value.

Beware of any calculation that substitutes averages (or even good, expected, and bad values) for uncertain inputs. Your actual results are likely to be worse than the projections.
No positions in stocks mentioned.
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