Modelling the Dynamic Relationship between Systematic Default and Recovery Risk

Default correlation modelling is becoming the most popular problem in the field of credit derivatives pricing. An increase in default risk would cause the recovery rate to change correspondingly. Correlation between default and recovery rates has a noticeable effect on risk measures and credit derivatives pricing. After an introduction, we review the most recent literature covering default correlation and the relationship between default and recovery rates. We adopt the copula methodology to focus on estimating the default correlations rather than focus on modelling probabilities of default, we then use stress testing to compare the distributions of the probability of default under different copula functions. We develop a Gamma-Beta model to link the recovery rate directly with the individual probability of default, this is instead of an extended one factor model to relate them by a systematic common factor. One factor models are re-examined to explore correlated recovery rates under three distributions: the Logit-normal, the Normal and the Log-normal. By analyzing the results respectively obtained from these two classes of modelling scheme, we argue that the direct dependence (Gamma-Beta) model behaves better, in estimating the recovery rate given individual probability of default and in suggesting a better indication of their relationship. Finally, we apply default correlation and the correlated recovery rate to portfolio risk modelling. We conclude that if the recovery rates are independent stochastic variables, the expected losses in a large portfolio might be underestimated because the uncorrelated recovery risks can be diversified, so the correlation between default rate and recovery risk can not be neglected in the applications. Here, we believe the first time, the recovery rate depends on individual default probability by means of a closed formula

Hard X-ray emission cutoff in anomalous X-ray pulsar 4U 0142+61 detected by INTEGRAL

The anomalous X-ray pulsar 4U 0142+61 was studied by the INTEGRAL observations. The hard X-ray spectrum of 18 -- 500 keV for 4U 0142+61 was derived using near 9 years of INTEGRAL/IBIS data. We obtained the average hard X-ray spectrum of 4U 0142+61 with all available data. The spectrum of 4U 0142+61 can be fitted with a power-law with an exponential high energy cutoff. This average spectrum is well fitted with a power-law of $\Gamma\sim 0.51\pm 0.11$ plus a cutoff energy at $128.6\pm 17.2$ keV. The hard X-ray flux of the source from 20 -- 150 keV showed no significant variations (within 20$\%$) from 2003 -- 2011. The spectral profiles have some variability in nine years: photon index varied from 0.3 -- 1.5, and cutoff energies of 110 -- 250 keV. The detection of the high energy cutoff around 130 keV shows some constraints on the radiation mechanisms of magnetars and possibly probes the differences between magnetar and accretion models for these special class of neutron stars. Future HXMT observations could provide stronger constraints on the hard X-ray spectral properties of this source and other magnetar candidates.Comment: 9 pages, 5 figures, 2 tables, figures are updated, new data are added, conclusion does not change, to be published in RA