We document that the credit spread on consumer unsecured debt exhibits a persistent, hump-shaped response to an increase in the charge-off rate. This stylized fact poses a significant challenge for a standard model of consumer default in which lenders have rational expectations and, therefore, the credit spread continuously adjusts to reflect the true default incentives of each borrower. In an effort to explain this feature of the data, we construct a model of consumer default with countercyclical income risk in which lenders learn about default risk over time by observing the history of repayment decisions, as is the case in practice. In addition to matching credit spread dynamics, allowing lenders to learn about default risk substantially improves the model’s ability to generate realistic business cycle fluctuations in the consumer unsecured credit market and match the cross-sectional distribution of unsecured debt and dispersion of interest rates observed in the data.

The liquidity premium on corporate bonds has been high on the agenda of Solvency regulators owing to its potential relationship to an additional discount factor on long-dated insurance liabilities. We analyse components of the credit spread as a function of standard bond characteristics during 2003–2014 on a daily basis by regression analyses, after introducing a new liquidity proxy. We derive daily distributions of illiquidity contributions to the credit spread at the individual bond level and find that liquidity premia were close to zero just before the financial crisis. We observe the time-varying nature of liquidity premia as well as a widening in the daily distribution in the years after the credit crunch. We find evidence to support higher liquidity premia, on average, on bonds of lower credit quality. The evolution of model parameters is economically intuitive and brings additional insight into investors’ behaviour. The frequent and bond-level estimation of liquidity premia, combined with few data restrictions makes the approach suitable for ALM modelling, especially when future work is directed towards arriving at forward-looking estimates at both the aggregate and bond-specific level.

This paper was written by the Derivatives Working Party, a permanent working party of the Life Research Committee of the Institute and Faculty of Actuaries. Our aim is to consider how life assurers may use, or may wish to use, derivatives, and if their use is unduly constrained, e.g. by regulation. This paper focuses on credit derivatives. We provide an overview of the credit derivatives market, and the strong growth in this market over recent years. We then focus on the two main traded credit derivative instruments — Credit Default Swaps (CDSs) and Collateralised Debt Obligations (CDOs). We explain how these instruments work and are priced, and clarify some of the more complex topics involved, such as the settlement of CDSs, basis risk and the relevance of implied correlation in pricing CDOs. We then consider how life insurers could make use of credit derivatives, for example to provide more efficient investment management in taking exposure to credit risk, or to hedge credit exposures, and consider the regulatory implications of so doing. Finally, in the Appendix, we discuss the credit spread puzzle, and the existence or otherwise of a liquidity premium in corporate bond spreads, with implications for the valuation of illiquid liabilities.

In this paper we consider a structural form credit risk model with jumps. We investigate the credit spread, the price, and the fair premium of the zero-coupon bond for the proposed model. The price and the fair premium of the bond are associated with the Laplace transform of default time and the firm's expected present market value at default. We give sufficient conditions under which the Laplace transform and the expected present market value of a firm at default are twice continuously differentiable. We derive closed-form expressions for them when the jumps have a hyperexponential distribution. Using the closed-form expressions, we obtain numerical solutions for the default probability, the credit spread, and the fair premium of the bond.

We study defaultable bond prices in the Black–Cox model with jumps in the asset value. The jump-size distribution is arbitrary, and following Longstaff and Schwartz (1995) and Zhou (2001) we assume that, if default occurs, the recovery at maturity depends on the ‘severity of default’. Under this general setting, the vehicle for our analysis is an integral equation. With the aid of this, we prove some properties of the bond price which are consistent numerically and empirically with earlier works. In particular, the limiting credit spread as time to maturity tends to 0 is nonzero. As a byproduct, we show that the integral equation implies an infinite-series expansion for the bond price.

The Capital Asset Pricing Model arises in an economy where agents have exponential utility functions and aggregate consumption is normally distributed, and gives the prices of assets with payoffs which are jointly normal with consumption. Such assets have normal marginal distributions and have dependence with consumption characterised by a normal copula. Wang has derived a transform which extends the CAPM by allowing pricing of assets in such an economy which have non-normal marginal distributions but still are normal-copula with consumption.

Here we set out the stochastic discount factors corresponding to this version of the CAPM and to Wang’s transform, and show how to calculate stochastic discount factors and hence asset prices for assets whose dependence with consumption is non-normal. We show that the impact of dependency structure on asset prices is significant.