Probability and Statistics for Finance

Chapter 15. Conditional Probability and Bayes' Rule

In Chapter 8 we explained that one interpretation of the probability of an event is that it is the relative frequency of that event when an experiment is repeated or observed a very large number of times. Here are three examples: (1) a mortgage company observed that over the past 10 years, 8% of borrowers are delinquent in making their monthly mortgage payments; (2) the risk manager of a bank observed that over the past 12 years, 5% of corporate loans defaulted; and (3) an asset management firm has observed that over the past eight years, in 20% of the months the managers of its stock portfolios underperformed a client's benchmark by more than 50 basis points. For these examples, suppose that the following information is available: (1) during recessionary periods, the mortgage service company observes that 15% of borrowers are delinquent in making their monthly mortgage payment, (2) the risk manager of the bank observes that during recessionary periods, 11% of corporate loans defaulted, and (3) during periods of a declining stock market, the asset management firm observes that in 30% of the months its stock portfolio managers under-performed a client's benchmark by more than 50 basis points.

These three hypothetical examples suggest that taking into account the available additional knowledge (economic recession in our first two examples and a declining stock market in the third example) could result in revised (and more accurate) ...