**As Default Probabilities Rise, How Many U.S. Banks Will Fail?**

The chart above shows the upward trend in 10-year default probabilities for five parent companies of the "too big to fail" banking firms: Bank of America (NYSE:BAC) (NASDAQ:BLUE), Citigroup Inc. (NYSE:C) (red), JPMorgan Chase & Co. (NYSE:JPM) (yellow), Wells Fargo & Co. (NYSE:WFC) (green), and Morgan Stanley (NYSE:MS) (purple). This trend raises an important question for both investors in bank stocks and for the American taxpayer, who has repeatedly bailed out U.S. financial institutions insured by the Federal Deposit Insurance Corporation. How many U.S. banks will fail in the years ahead?

This is an easy question to answer using the Kamakura Risk Information Service U.S. Bank model, which provides default probabilities for all of the 5,909 banks insured by the FDIC. See the appendix for details on the KRIS U.S. Bank model. We take both a short-run one year view of default risk and an intermediate three-year view of default risk, all conditional on the current state of the economy. As of today, the Kamakura troubled company index indicates that worldwide corporate credit conditions are currently at the 82 nd percentile of the time period from 1990 to 2017. The 100th percentile indicates the all-time best credit conditions:

We answer the question "how many U.S. banks will default over the next one year and next three years" with both an average number of defaults for both time periods and a 10,000 scenario simulation that shows the range of possibilities with much more clarity than the average default number does. The average number of defaults is easy to calculate directly without simulation using an insight of Jarrow, Lando and Yu (2005): we simply add up the cumulative probability (expressed as a decimal, not a percent) of defaults for all banks. Here are the results

**Expected Number of U.S. Bank Defaults**

1 year ending March 9, 2018: 14.3 Banks

3 years ending March 9, 2020: 184.5 Banks

The first year number of bank defaults, on average, is very small at only 14.3 institutions because of the strong state of the economy. The longer run outlook is much worse, with 184.5 banks failing over the full three-year period.

We now simulate 10,000 scenarios of default/no default for each of the 5,909 banks insured by the FDIC. In scenario 1, we calculate how many defaults occur in 1 roll of the "default dice" for each bank. The dice for a bank with a 20% probability of default has one tail (default) and four heads (no default). We roll that dice for the first bank. If it is heads, we record zero defaults. If it's tails, we record one default. We repeat for the other 5,908 banks and get the total number of defaults in Scenario 1. That is the first result of our 10,000 simulations. We repeat 9,999 more times and get this distribution of defaults:

This video shows the compilation of the default distribution over the 10,000 scenarios:

The lowest number of bank failures in a one-year simulation was three banks, which occurred in three of the 10,000 scenarios. The highest number of bank failures to occur was 30 banks, which was recorded in two scenarios. The median number of defaults is 14, which occurred in 1,112 scenarios. The average converges to 14.3 as the number of simulations increases by the law of large numbers. The simulation takes advantage of Jarrow, Lando and Yu's proof that, conditional on the state of the economy at a point in time, the default/no default simulation for banks A and B are independent. Over time, their default probabilities are correlated as macro factors move the default probabilities, but at one point in time the Jarrow, Lando and Yu finding is the relevant one.

Over a three year horizon, the distribution of potential defaults is much wider:

This video shows the construction of the simulation for a 3-year horizon:

The lowest number of defaults simulated was 136, which occurred in only one of the 10,000 scenarios. The highest number of defaults was 241, which also occurred in only one scenario. The median number of defaults was 185 defaults, which happened in 343 of the 10,000 scenarios. As the number of scenarios gets larger, the average converges to 184.5.

**Using Credit Simulations for Bank Deposit Insurance and Bank Investments**

This simulation methodology can be done easily in a sophisticated enterprise-wide risk management system like Kamakura Risk Manager ("KRM") to assess both the safety and soundness of the FDIC insurance fund and to assess the default-adjusted performance of a bank equity and bond portfolio. At the individual investor level, investors with access to high quality default probabilities can easily replicate the simulation in this note with a statistical package or even common spreadsheet software.

**References**

Jarrow, Robert, David Lando, and Fan Yu, "Default Risk and Diversification: Theory and Applications," *Mathematical Finance*, January 2005, pp. 1-26

**Appendix**

The Kamakura U.S. Bank Model (abbreviated "KDP-BK1" for Kamakura Default Probability, Bank Model Version 1.0) was launched in 2014 after three years of development by Kamakura Risk Information Services. The model was developed using the insights of Prof. Robert A. Jarrow, Kamakura Managing Director of Research and former Senior Fellow at the Federal Deposit Insurance Corporation. In his role at the Federal Deposit Insurance Corporation, Prof. Jarrow co-authored the FDIC's 2003 Loss Distribution Model, which correctly forecast that the FDIC deposit insurance fund was significantly under-funded. The new model uses more than 2 million monthly observations of U.S. commercial banks and a time period that spans the full credit crisis experience for maximum accuracy. The KDP-BK1 model is even more accurate than the widely respected Kamakura public firm models. The Kamakura U.S. Bank Model is a modern reduced form model developed using the insights gleaned from Kamakura's public firm models, non-public firm models, and sovereign models.

**Disclosure:** I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.

I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.