Stress Testing and Multiple-Factor Asset Pricing Models

Financial institutions regulators world-wide are introducing stress-testing of financial institutions’ value, net income, and capital ratios as a standard part of the regulatory process.  For many financial institutions, the breadth of the risk factors that are used in regulatory stress-testing has come as a surprise.  Stress tests are conducted with respect to risk factors, such as home prices, that were not used in the traditional risk management process at many institutions.  In addition, many institutions are unsure how to link risk factors to changes in the value of risky assets they hold.  These changes in value result from the impact of risk factor shifts on risk-free interest rates, the default probability of the issuer of the asset, prepayment probabilities on the risky asset, and the credit spread on the risk asset.  This blog provides a general framework for determining the response of asset values (and net income and capital ratios) to changes in key risk factors that takes advantage of the insights of Prof. Robert Jarrow (2013) in his recent paper “A Generalized Multiple-Factor Asset Pricing Model.”

A copy of the Jarrow paper is available at this link:

http://www.kamakuraco.com/LinkClick.aspx?fileticket=8BVYkVU4oNY%3D&tabid=36

Recent Trends in Regulatory Stress Testing

Regulatory stress tests have become a standard requirement for many wings of the financial services business in the wake of the 2006-2011 credit crisis.  The Federal Reserve Board in the United States recently announced the results of its Comprehensive Capital Analysis and Review stress testing program at this link:

http://www.federalreserve.gov/bankinforeg/ccar.htm

The Federal Reserve stress tests are based on 26 “risk factors” that are defined in detail at the link above:

• BBB-rated U.S. Corporate Bond Yield
• U.S. Inflation Rate
• Developing Asia Inflation Rate
• Euro Area Inflation Rate
• Japan Inflation Rate
• UK Inflation Rate
• Commercial Real Estate Price Index
• Dow Jones Total Stock Market Index
• Developing Asia Bilateral Dollar Exchange Rate
• Euro Area Bilateral Dollar Exchange Rate
• Japan Bilateral Dollar Exchange Rate
• Bilateral Dollar Exchange Rate
• U.S. House Price Index
• U.S. Mortgage Rate
• U.S. Nominal Disposable Income Growth
• U.S. Nominal GDP Growth
• U.S. Real Disposable Income Growth
• U.S. Real GDP Growth
• Developing Asia Real GDP Growth
• Euro Area Real GDP Growth
• Japan Real GDP Growth
• U.K. Real GDP Growth
• 10-Year U.S. Treasury Yield
• 3-Month U.S. Treasury Yield
• U.S. Unemployment Rate
• S&P Volatility Index (“VIX”)

Background on stress testing proposed by the U.K.’s Prudential Regulation Authority is given at this link:

http://www.bankofengland.co.uk/pra/Pages/supervision/activities/stresstesting.aspx

The Basel Committee on Banking Supervision has published basic principles for stress testing here:

http://www.bis.org/publ/bcbs155.pdf

An example of the European Banking Authority’s European Union-wide stress tests is given at this link:

http://www.eba.europa.eu/pdf/EBA_ST_2011_Summary_Report_v6.pdf

How should a sophisticated financial institution construct a highly accurate stress test that meets the requirements of both best practice risk management and key regulators?  We turn to that topic now.

Linking Risk Factors to Asset Values in a No Arbitrage Framework

In his April 24, 2013 paper, Prof. Robert Jarrow shows that risk factors and asset returns are tied together in a clear, logical framework based on the key assumption of the absence of arbitrage in financial markets.  In his paper, Prof. Jarrow assumes that a large number of risk factors affects the value of financial assets, although for any one type of security the number of risk factors driving returns on that security might be a small number.  The analysis is particularly clear when the risk factors are traded assets themselves.  This is a high quality assumption for almost all of the risk factors in the Fed’s CCAR stress testing program.  For example, while a commercial real estate index is not a traded asset per se, the buildings underlying the index are traded assets.

Prof. Jarrow explains that there is a set of primary assets or risk factors whose returns completely explain (except for data errors) the returns on all other securities.  From the perspective of a financial institution risk manager or regulator, this set of primary risk factors in its entirety can and should be the basis for stress testing.  Generalizing the arbitrage pricing theory of Ross (1976), Prof. Jarrow shows that in every scenario an asset’s return must be such that

Equation 1

where the relationship between the betas is such that

This expression can be rewritten

Equation 2

In every scenario, the return on asset i R_i is the sum of the risk-free rate r₀ and the sum of k terms that represent the influence of those k risk factors relevant to asset i. For each risk factor relevant to asset i, there is a “beta” coefficient that measures the return impact of the excess of the risk factor return r_j for risk factor j over the risk free rate r₀.  This result applies to every realized scenario, including the regulatory stress test scenarios, provided that the list of risk factors is complete.  This is a much more powerful result than a pricing model that concludes only that there is a similar relationship on an expected return basis:

We now turn to the implications of the Jarrow generalized multi-factor asset pricing model for the stress testing of financial institutions by risk managers and financial institutions regulators.

Using the Jarrow Generalized Multi-Factor Asset Pricing Model for Stress Testing

In the explanation above, we discussed only the returns on one asset, asset i. A financial institution is nothing but a portfolio of “securities” (broadly defined) which it holds either in long positions (Treasury bills, bonds, mortgage-backed securities, loans, etc.) or short positions (demand deposits, life insurance policies, savings deposits, bonds, convertible bonds, preferred stocks).  The number of risk factors affecting any given security will be finite (like the number of risk factors k above), but, in general, a different group of risk factors will affect each asset or liability.

We now outline how the Jarrow generalized multi-factor asset pricing model is used for stress testing in a step by step fashion:

Step 1: Determine the risk factors applicable to each transaction (asset or liability) and the value of the betas β_ij for each transaction i and risk factor j.

This can be done either by historical analysis, forward looking analysis, or both.  For example, it is well known that the value of interest rates, home prices, and the unemployment rate affect the value of both one mortgage or a portfolio of mortgages in the form of a mortgage-backed security.  One could use historical returns on mortgages and fit the betas for the three risk factors: interest rates, home prices, and unemployment.  For a transaction for which no history exists (say, the “option ARM” or option adjustable rate mortgage at the time it was first introduced), one would do a forward looking return analysis on the assumption of rational or partially rational default by a retail mortgage borrower (for example, when the mortgage principal is greater than the value of the house or when the borrower becomes unemployed).

Step 2: Collect the value of all of the relevant risk factors at both the current time 0 and in the stress test scenarios at each point in time for the stress tests.

For simplicity, let’s assume that the Federal Reserve Board was exactly right in its risk analysis and that all bank assets and liabilities have returns that are completely determined by the 26 risk factors given above, even though it is unlikely that all 26 apply to any one bank asset or liability.  Note that Jarrow (2013) assumes that all risk factors are traded.  This is largely true with respect to the Fed’s list of factors.  Moreover, the presence of exchange traded funds which mimic many of the factors makes the “traded” assumption a good one. The Fed requires three stress test scenarios 13 quarters forward.  The next step is to provide the values of the 26 risk factors at time zero and at the 13 forward points in time for all 3 scenarios, a total of 1+3(13)=40 values for each risk factor.  In the case of the Fed’s stress tests, this task is easy because all of these values are provided by the Federal Reserve itself.

Step 3: Convert the absolute values of the risk factors to returns

This step is simple arithmetic, converting the risk factor values to percentage changes.

Step 4: Substitute these risk factor values into either equation 1 or 2 for each asset and liability to get the returns on each asset or liability.

This can be done easily using an enterprise wide risk management system like Kamakura Risk Manager (“KRM”).

Step 5: Use the returns to calculate the new values of each asset or liability, the mark to market value of the institution, and related net income and capital ratios

The calculation of net income and other values that depend on generally accepted accounting principles will require an enterprise risk management system like KRM.  The net value of the financial institution (sometimes called the “market value of portfolio equity”) is an easier calculation that could even be done in a common spreadsheet if the number of assets and liabilities is manageable.

This completes the task of stress testing.  It is straightforward because of the very powerful implications of the Jarrow generalized multi-factor asset pricing model.

Conclusions

Stress-testing for risk management and regulatory purposes has been regarded as a great mystery to many members of financial institutions management.  Using the insights of a powerful asset pricing model like that proposed by Jarrow (2013), stress-testing becomes a clear and transparent process that is highly auditable even at the individual transaction level.  Kamakura Risk Manager and Kamakura Risk Information Services’ Credit Portfolio Manager provide a number of alternative approaches to stress testing in addition to the Jarrow approach.  For more information please contact us at Info@KamakuraCo.com.

Donald R. van Deventer
Kamakura Corporation
Honolulu, Hawaii
May 21, 2013

References

Heath, David, Robert Jarrow and Andrew Morton, “Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation,” Econometrica, 60 (1), 77-105.

Jarrow, Robert. “A Generalized Multiple-Factor Asset Pricing Model,” Kamakura Corporation and Cornell University memorandum, April 24, 2013.

Jarrow, Robert and A. Chatterjea, An Introduction to Derivative Securities, Financial Markets, and Risk Management, W.W. Norton & Co., New York, 2013.

Ross, S. “The Arbitrage Theory of Capital Asset Pricing,” Journal of Economic Theory, 1976, 13, 341-360.

van Deventer, Donald R., Kenji Imai and Mark Mesler, Advanced Financial Risk Management, 2nd Edition, John Wiley & Sons, Singapore, 2013.
Copyright © 2013 by Donald R. van Deventer. All Rights Reserved.