1 Simulating and Validating a Multi-Factor Heath, Jarrow and Morton Model with Negative Interest Rates Robert A. Jarrow 1 and Donald R. van Deventer 2 First version: March 11, 2015 This version: March 17, 2015 1. Introduction Risk management in financial institutions, both for internal and regulatory uses, requires the computation of various risk measures that quantify the institution’s insolvency risk (the depletion of the institution’s equity). These risk measures include various modifications of value-at-risk (VaR) and running scenario analyses. For regulatory capital determination, the regulatory authorities provide the scenarios considered. This is certainly true for the European Union and the United States. When computing these risk measures, the entire balance sheet of the financial institution needs to be modeled. This modeling is necessary to determine the future value of the institution’s equity, which is the future value of the institution’s assets less liabilities. It is an understatement to say that modeling the institution’s assets and liabilities is a complex and daunting task. The task is sufficiently complex that simulation provides the only feasible methodology for the computation of the relevant risk measures. To simulate the institutions changing asset and liability values across time, one starts with the current values of these asset and liabilities, either marked-to-market or marked-to-model. This is the initial condition for the simulation. Then, a stochastic model for the evolution of the assets and liabilities is assumed. The importance of validating the assumed evolutions will be discussed below. These evolutions depend on parameters, which are estimated from historical data and/or calibrated to market prices. Given these estimated parameters, the evolution is simulated forward in time. The simulations generate paths of the asset and liability prices, from which the various risk measures can be computed. For the simulation to be “economically and statistically valid,” it needs to satisfy two conditions: (i) economic validity: the evolutions must be arbitrage-free, and (ii) statistical validity: the evolutions need to be consistent with the observed historical evolutions. If either of these conditions is violated, the simulation is misspecifed and it will provide invalid risk measures. These two conditions are essential for the model’s validity for the following reasons. For a simulation’s economic validity, the price evolutions must be arbitrage-free. This is because and arbitrage-free evolution is the weakest condition that one can impose 1 Samuel Curtis Johnson Graduate School of Management, Cornell University, Ithaca, New York 14853, and Kamakura Corporation, Honolulu, Hawaii 96815. E-mail: raj15@cornell.edu 2 Kamakura Corporation, Honolulu, Hawaii 96815.