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.

The Flaw of Averages: Why We Underestimate Risk in the Face of Uncertainty by Sam Savage

Related posts:

Convex optimization
Log-concave functions

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4 comments on “Mortgages, banks, and Jensen's inequality
  1. This is nice.

    You cannot safely ignore higher order statistics.

  2. John says:

    That is the essence of Savage’s book: you cannot ignore variation. Savage does a remarkable job of communicating this idea to a non-mathematical audience.

  3. Al Chou says:

    I have been trying for 10 years to get programmers around me to see beyond the arithmetic mean as the only statistic of interest, but no one wants to hear about standard deviation, average deviation, etc. Savage’s book is going on my wish list….

  4. nicolas says:

    Yup its hidden convexity.
    They charge for that though.

    But how do you estimate how much ?
    They cant. no one can.
    So commercial pressure make it go to almost 0 and we end up with messy situation.

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