Pulled-to-Par Returns for Zero-Coupon Bonds Historical Simulation Value at Risk

Abstract

Due to bond prices pull-to-par, zero-coupon bond historical returns are not stationary, as they tend to zero as time to maturity approaches. Given that the historical simulation method for computing value at risk (VaR) requires a stationary sequence of historical returns, zero-coupon bonds’ historical returns cannot be used to compute VaR by historical simulation. Their use would systematically overestimate VaR, resulting in invalid VaR sequences. In this paper, we propose an adjustment of zero-coupon bonds’ historical returns. We call the adjusted returns “pulled-to-par” returns. We prove that when the zero-coupon bonds’ continuously compounded yields-to-maturity are stationary, the adjusted pulled-to-par returns allow VaR computation by historical simulation. We firstly illustrate the VaR computation in a simulation scenario, and then, we apply it to real data on eurozone STRIPS.