Measuring Risk in Fixed Income Portfolios using Yield Curve Models

Abstract

We propose a novel approach to measure risk in fixed income portfolios in terms of value-at-risk (VaR). We obtain closed-form expressions for the vector of expected bond returns and for its covariance matrix based on a general class of dynamic factor models, including the dynamic versions of the Nelson-Siegel and Svensson models, to compute the parametric VaR of a portfolio composed of fixed income securities. The proposed approach provides additional modeling flexibility as it can accommodate alternative specifications of the yield curve as well as alternative specifications of the conditional heteroskedasticity in bond returns. An empirical application involving a data set with 15 fixed income securities with different maturities indicate that the proposed approach delivers accurate VaR estimates.

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Fig. 1

Notes

  1. 1.

    See Poon and Granger (2003) and Andersen and Benzoni (2010) for recent surveys on volatility forecasting.

  2. 2.

    All models are estimated using a recursive expanding estimation window. Departing from the first 500 observations, all models are estimated and their corresponding one-step-ahead VaR estimated are obtained using (17). Next, we add one observation to the estimation window and re-estimate all models and obtain another one-step-ahead estimate of the VaR. This process is repeated until the end of the data set is reached. In this way, we obtain 486 out-of-sample one-step-ahead VaR forecasts. All results discussed in Sect. 4.2 are based solely on out-of-sample observations.

  3. 3.

    A review of issues related to the estimation of univariate GARCH models, such as the choice of initial values, numerical algorithms, accuracy, and asymptotic properties are given by Berkes et al. (2003), Robinson and Zaffaroni (2006), Francq and Zakoian (2009) and Zivot (2009). It is important to note that even when the normality assumption is inappropriate, the QML estimator of univariate GARCH models based on maximizing the Gaussian likelihood is consistent and asymptotically normal, provided that the conditional mean and variance of the GARCH model are correctly specified, see Bollerslev and Wooldridge (1992).

  4. 4.

    The DI-futuro rate is the average daily rate of Brazilian interbank deposits (borowing/lending), calculated by the Clearinghouse for Custody and Settlements (CETIP) for all business days. The DI-futuro rate, which is published on a daily basis, is expressed in annually compounded terms, based on 252 business days. When buying a DI-futuro contract for the price at time \(t\) and keeping it until maturity \(\tau \), the gain or loss is given by:

    $$\begin{aligned} 100.000\left( \frac{\prod _{i=1}^{\zeta (t,\tau )}(1+y_i)^{\frac{1}{252}}}{(1+DI^*)^{\frac{\zeta (t,\tau )}{252}}}-1\right) , \end{aligned}$$

    where \(y_i\) denotes the DI-futuro rate, \((i-1)\) days after the trading day. The function \(\zeta (t,\tau )\) represents the number of working days between t and \(\tau \).

  5. 5.

    For further details and applications of this method, see Hagan and West (2006) and Hayden and Ferstl (2010).

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Caldeira, J.F., Moura, G.V. & Santos, A.A.P. Measuring Risk in Fixed Income Portfolios using Yield Curve Models. Comput Econ 46, 65–82 (2015). https://doi.org/10.1007/s10614-014-9438-7

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Keywords

  • Dynamic conditional correlation (DCC)
  • Dynamic factor models
  • Value-at-risk (VaR)
  • Yield curve