The importance of parameter uncertainty for the yield spreads: An application of Markov Chain Monte Carlo simulation

Frederik Ougaard

Student thesis: Master thesis


This thesis focuses on the impact of parameter uncertainty on the yield spreads. Many researchers have shown that the yield spreads can not solely be explained by default probabilities. This is often referred to as the credit spread puzzle. If default probabilities can not explain the magnitude of the yield spread then something else has to contribute. This thesis seeks to nd the impact of uncertainty in the parameters; asset value and asset return volatility, which typically are not known with certainty by the investors. Mathematically, pa- rameter uncertainty drives debt value down due to concavity of a debt function. It is shown that the impact of parameter uncertainty depends on the magnitude of parameter uncertainty and the shape of the debt function. The idea and intuition about this is investigated through the famed Merton model(1974). As the main model of debt value, the Leland (1994b) model is used. This is a structural model with an endogenous default boundary. Other models, including the original Leland model(1994), assume that debt is perpetual. However, the model we use incorporates a maturity structure. We derive closed form solutions for debt value and equity values as a function of asset value. The Leland model is tted using monthly equity values from 55 rms in the period 1994 to 2010. This gives us 935 observations of yield spread implied by the Leland model with comparable yield spreads calculated with parameter uncertainty. The parameters of the Leland model are estimated with the technique; Markov Chain Monte Carlo Simulation (MCMC). The thesis is built upon the work of Polson and Korteweg on estimating a Leland model using MCMC in their article Corporate Credit Spreads under Parameter Uncertainty, (2009). We show how MCMC is based on Bayesian Inference and proof the Metropolis-Hastings algorithm i.e. how the algorithm constructs a Markov chain that converges to the required distribution, from which we sample. The samples constitute the posterior distributions of the estimated variables; as- set value and volatility which enables us to quantify the e ect of parameter uncertainty. Our results show that uncertainty raises the yield spread modestly. In absolute values the 99% quaintile is 7.5 bps. We analyse uncertainty, which is the estimated standard deviation of the parameters asset value and volatility, across time, rating, maturity and proxies for uncertainty. We nd that lower rated rms are more uncertain than higher rated rms and that uncertainty is correlated with a general index for uncertainty, the VIX-index. A shorter time to maturity is shown to give a higher e ect of parameter uncertainty, especially for the low credit rated rms. We nd that proxies for uncertainty, Property, Plant and Equipment(PPE), log size, log age and Market to Book value of equity(M/B) are negatively correlated with uncertainty. That is, old, large rms with tangible assets and a high market to book value tends to have relatively little uncertainty. The e ect of uncertainty is divided into asset value uncertainty and asset return volatilty uncertainty, and hereby is the marginale e ects of uncertainty investigated. It is shown that if the asset value is held x and thereby only the uncertainty in asset volatility is considered , more uncertainty in asset volatility can lead to a lower yield spread in contradiction to our intuition. That is we show that the debt function can be convex in asset volatility. In the analysis several assumptions are made by the researcher which will be discussed. It is shown that the researcher can in uence the results by set- ting the prior distributions and the number of iterations in the Markov Chain Monte Carlo simulations. The results in this thesis are compared to Polson's and Korteweg's results and it is discussed why the e ect of parameter uncer- tainty in this thesis is neglected compared to Polson's and Korteweg's results.

EducationsMSc in Mathematics , (Graduate Programme) Final Thesis
Publication date2011
Number of pages127