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Don founded Kamakura Corporation in April 1990 and currently serves as its chairman and chief executive officer where he focuses on enterprise wide risk management and modern credit risk technology. His primary financial consulting and research interests involve the practical application of leading edge financial theory to solve critical financial risk management problems. Don was elected to the 50 member RISK Magazine Hall of Fame in 2002 for his work at Kamakura. Read More

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Kamakura Blog

Jul 7

Written by: Donald van Deventer
7/7/2009 7:59 AM 

Chapters 10 and 11 of Advanced Financial Risk Management (van Deventer, Imai and Mesler, John Wiley & Sons, 2004) trace the development of yield curve models as they evolved from the 1973 Merton term structure model (see “The Theory of Rational Option Pricing,” reproduced as Chapter 8 of Continuous Time Finance, Basel Blackwell Inc., 1990, page 285, footnote 34). A sophisticated reader asks, “What are the implications of the credit crisis for the best practice modeling of the risk-free yield curve?”  Have things changed in light of recent events?  We think the answer is definitely yes.  We think a number of developments have made this change in thing very important:


  • Interest rates can indeed become negative for an extended period of time.  This is not consistent with the Cox-Ingersoll-Ross or Black-Derman-Toy frameworks. For an example, see section 5.3 of the Monthly Bulletin of the Hong Kong Monetary Authority:
  • Interest rates can be very important drivers of both default and credit spreads
  • Multiple factors are at work in driving defaults, and market participants have become increasingly reluctant to make strong simplifying assumptions about interest rate modeling.
  • Complex retail financial products are being offered with a series of embedded options that make a closed form valuation impossible under any of these term structure modeling frameworks
  • Central Banks from the Bank of Japan to the Federal Reserve have been forced to hold short term interest rates constant and near zero for extended periods of time in order to revive distressed economies.  The behavior of short term interest rates in particular has been significantly different from the assumptions embedded in the early term structure model literature.


The original academic work was pioneering in many senses.  It showed that the shape of the yield curve itself was heavily affected by assumptions about how short term interest rates moved.  For the term structure models with normally distributed single factor shocks to either short term interest rates (Vasicek, Hull and White) or parallel shifts (Merton, Ho and Lee) closed form solutions were derived for zero coupon bonds and European options on both zero coupon bonds and coupon bearing bonds. 


The fact that the current credit crisis has at its heart retail mortgages is a significant factor in how risk managers simulate interest rates.  Structures like the infamous “option ARMs” contain so many intertwined embedded options that even the most simple, stylized assumptions about interest rate movements do not allow a closed form valuation equation. Moreover, the long term nature of the mortgage (30 and even 40 years) and the dependence of payments on short term interest rate movements mean that the movement of the entire term structure of interest rates is important. Finally, the greater complexity in central bank management of the economies of Japan, the U.S. and the Euro zone have made the simple one factor term structure models discussed above increasingly obsolete because of their lack of realism. My colleague Prof. Robert Jarrow makes another important point. Omitting a “true” driver of the yield curve from one’s risk analysis can distort investment decisions, giving the appearance of arbitrage opportunities when in fact there are none.


What is the alternative?  Like credit portfolio management simulation where multiple macro factors drive risk, risk managers are choosing 3, 4, 5 and more random factors to drive the risk free yield curve forward.  These factors a decade ago would have been chosen by principal components analysis.  More recently, risk managers are choosing specific maturities or instruments as having random yields, because of the ability to hedge using these instruments.  For example, the overnight rate of interest, the 3 month rate of interest, and yields on 1 year, 5 year and 10 year instruments may be made “risk factors” in a modern monte carlo simulation.  This, of course, means that monte carlo simulation must also be used for valuation, but this would be necessary anyway given the complex options embedded in structures like “option ARMs.”


What about credit spreads?  As Jarrow, Li, Mesler and van Deventer showed (RISK Magazine, September 2007), the same multiple factor approach is needed to model credit spreads even in the presence of high quality estimates for default probabilities.  This is true because the spread represents much more than just the “expected loss” on the instrument.  The spread is the price that results from the intersection of supply and demand for credit, and that is a function of many factors.


Albert Einstein is said to have commented, “"Any intelligent fool can make things bigger, more complex, and more violent. It takes a touch of genius -- and a lot of courage -- to move in the opposite direction."  As attractive as this quote is, it was in fact the “genius” of the highly simplified copula method that led to nearly a trillion dollars of losses in the collateralized debt obligation market.  Wall Street, in fact, exploited those who chose simplification over accuracy.  The rich complexity of financial markets is now better understood, even though we may be no closer to predictions of the financial future than we are to accurate modeling of complex weather patterns.  To deny the complex nature of the problem is to make oneself a victim on Wall Street.


For that reason, an “N-factor” term structure model is essential bullet-proofing for accurate risk analysis. 


Comments and questions welcomed at


Donald R. van Deventer

Kamakura Corporation

Honolulu, July 7, 2009