Term Structure Forecasting and Scenario Generation

A statistical model is developed to generate probability based scenarios for forecasting and risk management of long term risks. The model forecasts transformed daily forward interest rates over a 10 year horizon. The model is a reduced rank vector autoregression with time varying volatilities and correlations. A quasi-differencing version reduces the impact of autocorrelated measurement errors. Risk managers with a large book of fixed income assets are typically challenged to estimate the risk over both short and long run horizons. Familiar measures of market risk such as Value at Risk, VaR, are based on factor exposures to extreme events and will be time varying. This literature has been nicely surveyed by Gourieroux and Jasiak(2010) who also focus on longer term credit risks. Following this discussion, longer run risks involve estimates of what the term structure might look like in a year or in ten years. Most term structure models are not adequate to this task, so scenario analysis is used. However, designing scenarios is an art form which is particularly complicated in the fixed income asset class as the scenario should be internally consistent and arbitrage free and must have a probability assessment that motivates the response to risk. It furthermore will be useless if it does not stress the assets that are held in the firms portfolio. An alternative way to generate scenarios is to use probability based scenarios as described by Christensen, Lopez and Rudebusch(2014). The strategy is to construct a large number of equally likely scenarios which are drawn from the joint predictive distribution of the term structure. The value of the portfolio of assets can be calculated from each scenario over time and a Profit and Loss distribution computed at different horizons. Because each scenario has a known probability of occurring, the risks at different horizons in the future can be assessed. Their application is to the management of the FED’s enormous fixed income portfolio, but it could similarly be applied jointly to any financial firm’s assets and liabilities. This is essentially the proposal embodied in the Solvency II Directive, a new regulatory framework for the European insurance industry that adopts a more dynamic risk-based approach and 1 The authors appreciate financial and intellectual support from A.I.G in developing this model. Particular thanks go to Sid Dalal, David Li and Kevin Chen. We also would like to thank Glenn Rudebusch, Jim Hamilton, Michael Bauer and Cynthia Wu for useful conversations on this research. implements a non-zero failure regime. The directive mandates market valuation of assets and liabilities and a maximum 1 in 200 probability of failure over a long horizon. Both assets and liabilities incur risk and all risks may be correlated. This paper proposes an econometric approach to generating scenarios for the US treasury term structure. These scenarios can be interpreted as a predictive distribution for the term structure in the near or distant future. These predictions should be arbitrage free and should have sufficient range that the scenarios stress all realistic outcomes. 2. Literature Survey There is a vast literature on the term structure on non-defaultable securities. It is the main subject of countless books, courses and careers. The purpose of most of this literature is however different from the purpose of this paper. Most of the literature is concerned with estimating the fair market prices of securities that don’t have an observable price in terms of others that do. This is extremely important as fixed income securities and their derivatives trade only occasionally but must be evaluated based on other securities that do trade. Thus the goal is estimating the yield curve at a moment of time based on other prices recorded at that moment. Part of this analysis requires estimating the price at which a bond would trade and how this is decomposed into term and risk premiums. This analysis formally constrains the yield curve to be arbitrage free. Early models of this form treat the short rate as the state variable and by postulating a dynamic relation, derive the entire term structure. These include the Vasicek(1977), Cox, Ingersoll and Ross(1985), Ho and Lee(1986), and Hull and White(1990) among many others. Today short rate models are particularly problematic with the extended period of nearly zero short rates. A popular multi-factor specification is the affine structure which allows closed form expressions for these objects. See for example Duffie and Kan(1996) and a recent survey by Piazzesi(2010). Two features of these models are generally in conflict. To avoid arbitrage opportunities, nominal interest rates must be non-negative, at least if cash can be stored costlessly. However, to ensure that there are no arbitrage opportunities, it is sufficient to construct a risk neutral measure that prices all assets. Bjork(2009) shows that the affine family is the only tractable family that satisfies both conditions. Yet it is a very restricted specification which is claimed to be unable to model important features of the data such as the very low short interest rates we see today. See Duffee(2002) and Christensen and Rudebusch(2013) for example of these criticisms. To compute term premiums, it is necessary to compute the expectation of the future instantaneous short rate, r(s)