Abstracts

Market consistent pricing of very long-dated liabilities is difficult due to the limited liquidity or absence of very long-dated market instruments. Life-insurance or pension fund liabilities can be as long as 100 years, whereas the available liquid instruments in the market have maturities of 20 years or less.
The purpose of the paper is to extrapolate the yield curve. Taking data from time series of observed yield curves up to 20 years, what are model implied yields for longer maturities? We use a standard Gaussian essentially affine model that we estimate by Bayesian methods. Our data consists of Euro swap rates from 2002 to September 2013. From the full posterior density of the model parameters we obtain the predictive density for all maturities conditional on the initial maturities up to 20 years.
With an uninformative prior we find that longer term yields beyond the 20 years point very quickly become highly uncertain. The posterior mean for such yields can easily be negative with a very large credible interval. The problem is due to the very low estimate of the mean reversion parameter in the risk neutral density. Credible intervals become smaller and more sensible when we use an informative prior based on a zero lower bound for nominal interest rates. Since the mean reversion parameter of the level factor remains very small under this prior, convergence of the yield curve to a constant ultimate forward rate is very slow and does not occur before maturities of 100 years.
Our extrapolation method uses a financial no-arbitrage model. When we compare our method to curve fitting methods such as Smith-Wilson extrapolation and Nelson-Siegel calibration, we find large differences. Since our Bayesian procedure also estimates the uncertainty around the mean estimate of the extrapolation, the alternatives are, however, mostly within our estimated credibility regions.