Shaking a ladder Monte Carlo in everyday life

In Flaw of Averages, Sam Savage uses the illustration of shaking a ladder to explain why someone would want to use Monte Carlo simulation. Before climbing a ladder, most people shake the ladder a little to make sure it’s sturdy.

When you position a ladder and climb it immediately, you’re saying that you’re satisfied that the ladder’s position is safe. You’re assuming it will stay in the position you placed it. But when you shake the ladder, you’re testing how it will behave at a variety of nearby positions. If the ladder remains sturdy, you have more confidence that accidental motions while you’re on the ladder are not likely to cause an accident. When you stick a single number into a model, you’re acting like someone climbing a ladder immediately. When you stick in a series of random inputs, it’s like you’re shaking the ladder.

The latest INFORMS podcast features an interview with Sam Savage (audio). He mentions the ladder analogy and adds an observation that I don’t recall seeing in his book. One criticism of Monte Carlo methods is that the validity of your results depends on the validity of your input distributions. It’s better to have realistic input distributions, but it’s better to perform a simulation with an incorrect distribution than to not try random input at all. The distribution of random forces likely to result from working while standing on a ladder is different from the distribution of forces from shaking the ladder with your hands. But that doesn’t mean it isn’t a good idea to shake the ladder anyway.

Mortgages, banks, and Jensen’s inequality

Sam Savage’s new book Flaw of Averages has a brilliantly simple explanation of why volatility in the housing market caused such problems for banks recently. When housing prices drop, more people default on their mortgages and obviously that hurts banks. But banks are also in trouble when the housing market is volatile, even if on average house prices are good.

Suppose there’s no change in the housing market on average. Prices go up in some areas and down in other areas. As long as the ups and downs average out, there should be no change in bank profits, right? Wrong!

When the housing market goes up a little bit, mortgage defaults drop a little bit and bank profits go up a little bit. But when the market goes down a little bit, defaults go up more than just a little bit, and bank profits go down more than a little bit. There is much more down-side potential than up-side potential. Say 95% of homeowners pay their mortgages. Then a good housing market can only improve repayments by 5%. But a bad housing market could decrease repayments by much more.

In mathematical terminology, the bank profits are a concave function of house prices. Jensen’s inequality says that if f() is a concave function (say bank profits) and X is a random variable (say, house prices) then the average of f(X) is less than f(average of X). Average profit is less than the profit from the average.