We address two empirical issues related to the long end of the yield curve based on euro swap rates. First, for maturities longer than 20 years we find evidence for an ‘excess’ downward slope that cannot be explained by convexity. Second, volatility at the very long end of the yield curve is larger than predicted by no-arbitrage models. We construct a model-based arbitrage-free extrapolation of the yield-curve and compare it to the regulatory discount curve. Because of near-zero mean reversion, there is no convergence towards an ‘ultimate forward rate’ and convexity effects cause the arbitrage-free extrapolations to have slightly downward sloping curves. The low level of mean-reversion also implies that the volatility of long-term rates does not decline relative to the 20-year volatility. Therefore, we conclude that the mean-reversion and resulting smoothing adopted by the regulatory curve is much too strong.

我们基于欧元掉期利率，解决了两个与收益率曲线的长端相关的经验问题。首先，对于超过20年的到期日，我们找到了无法用凸度解释的“过度”下降斜率的证据。其次，收益率曲线很长端的波动率大于无套利模型所预测的波动率。我们构建了收益率曲线的基于模型的无套利外推法，并将其与监管折现曲线进行比较。由于均值回归几乎为零，因此无法趋向于“最终正向汇率”，而凸面效应会导致无套利外推的曲线略微向下倾斜。均值回归水平低也意味着长期利率的波动率相对于20年波动率并未下降。所以，