Value at Risk Using the Factor-Arch Model

31 Pages Posted: 16 Aug 1998

Multiple version iconThere are 2 versions of this paper

Date Written: June 30, 1998


This paper presents a methodology for calculating the Value at Risk (VaR) of portfolios of financial assets using a multivariate ARCH model. A slight modification of the factor-ARCH model proposed by Engle, Ng, and Rothschild (1990) is used to calculate VaR of portfolios of Danish zero-coupon bonds. The proposed method combines the ARCH model with a factor model.

The factors are constructed as orthogonal portfolios of the bonds (denoted factor representing portfolios) using principial components analysis. The returns of the factor representing portfolios are assumed to be described by the univariate GARCH(1,1) model. From the findings of the factor representing portfolios, the model for the returns of the individual zero-coupon bonds is determined. Models with one and two factors are estimated, and the two-factor model is shown to outperform the one-factor model. Hence, the VaR is estimated solely in the two-factor setting for a number of zero-coupon bond portfolios, and these VaR quantities are compared to actual losses. The calculated VaR amounts are also compared to the RiskMetrics estimates. At the 1% VaR it is not conclusive whether the factor-ARCH approach or RiskMetrics is most effective in estimating VaR for the analyzed data. However, for the 5% VaR RiskMetrics is preferred to the factor-ARCH model.

JEL Classification: C32, G19, G31

Suggested Citation

Christiansen, Charlotte, Value at Risk Using the Factor-Arch Model (June 30, 1998). Available at SSRN: or

Charlotte Christiansen (Contact Author)

Aarhus University - CREATES ( email )

Fuglesangs Alle 4
Aarhus V, DK 8210

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