Rare Disasters, Credit and Option Market Puzzles. Online Appendix Peter Christo ersen Du Du Redouane Elkamhi Rotman School, City University Rotman School, CBS and CREATES of Hong Kong University of Toronto September, 015 1
Online Appendix: Robustness Analysis In the empirics we have assumed a xed jump size. In the rst part of this online appendix we assess the impact of stochastic jumps and of changing the habit speci cation. In the second part, we study the impact on credit spreads when varying some of the key parameters in our model. OA.1 Stochastic Jump Size and Habit Formation We further examine our model by relaxing the assumption that the consumption jump size, J c, is constant. To this end, we instead assume it is normally distributed with mean J = J c = 10% and variance J = 1 j Jj = 5% inducing fairly volatile jump magnitudes. The top left panel of Figure OA.1 plots the model-implied Baa-Treasury credit spreads as the function of S; and compares it to the case with constant jump sizes. The top right panel of Figure OA.1 shows the state-dependent default-risk premia which varies little when allowing for stochastic jumps. Figure OA.1 shows that credit spreads and default risk premia are largely unchanged following the addition of randomness in consumption jumps. Note that idiosyncratic default does not carry a risk premium but still helps generating credit spreads. The credit spread di erences in Figure OA.1 are thus somewhat larger than the di erences in default-risk premia. Allowing for stochastic jump sizes has little impact on implied credit spreads due to the Menzly, Santos and Veronesi (00; MSV) habit formation used in this paper which imposes a speci c structure on the risk compensation for jumps. In the bottom left panel of Figure OA.1 we plot the state-dependent jump-risk premia which vary little when allowing for stochastic jump sizes. This result is con rmed when plotting the implied option volatility smirks in the bottom right panel of Figure OA.1. We conclude that our assumption of constant jump sizes is not overly restrictive in the context of our modeling framework. We also investigate if our benchmark results are driven by the speci c MSV-habit that we assume. Speci cally, we consider the SV-habit speci cation by Santos and Veronesi (010) which is close to the original Campbell and Cochrane (1999) habit, and which has one extra parameter compared with MSV habit. Given that our model is not tted to bond prices, matching credit spreads in an out-of-sample exercise may actually be more di cult with a richer parameterization. We solve for all asset valuations in the SV habit framework and details of the results are available upon requests. We nd that the SV-habit augmented with a consumption disaster component is also able to match empirical credit spreads in terms of their levels, term structure and time series dynamics. Consequently, we conclude that our results are not sensitive to changes in the habit speci cation.
OA. Parameter Sensitivity Analysis We nally investigate the sensitivity of model-implied spreads to some key parameters. We focus on and L which control the response of habit and loss rates to consumption innovations. We also consider jj c j which captures the absolute log consumption jump size. These parameters play important roles in the three key features of our model, namely, consumption disasters, time-varying risk aversion, and countercyclical loss rates. We vary each of the three parameters by up to 0% in both directions from their calibrated values in Table 1, and plot the implied average and standard deviation of model-implied 5-year Baa-Aaa/Aa spreads in Figure OA.. The direction of changes in spreads and volatilities are as expected. By comparison, jj c j has the largest impact. This result re-emphasizes the importance of accounting for consumption disasters, when explaining investment grade credit spreads. The importance of the jump component can also be inferred from the long-run risk literature. Chen (010) generates a realistically high average credit spread for 10-year Baa bonds. As in our model, Chen emphasizes the e ect of a large jump risk premium on the pricing of defaultable bonds. In particular, by shutting down jumps, Chen reports a roughly 50% reduction in the generated spread, which is very much in line with our numerical results. References Santos, T., and P. Veronesi, 010. Habit formation, the cross section of stock returns and the cash- ow risk puzzle. Journal of Financial Economics 98, 85 1.
Jump Risk Premium (%) Implied Volatility (%) Credit Spread (bps) Default Risk Premium (%) Figure OA.1. Impact of Stochastic Consumption Jump Size 50 Baa T reasury Spread 10 Default Risk Premium 00 50 0 00 150 100 0.01 0.0 0.0 0.0 0.05 6 8 J =0 J =5% 0.01 0.0 0.0 0.0 0.05 Jump Risk Premium 6 Volatility Smirk.5.5 5 1.5 1 0.5 1 0.01 0.0 0.0 0.0 0.05 0.9 0.9 0.9 0.96 0.98 1 Moneyness Notes to Figure: We compare the case with a constant 10% jump size (solid lines) to that with stochastic jump size (dashed lines) using a standard deviation of 5%. The top left panel shows the Baa credit spread over Treasury plotted against the surplus ratio. The top right panel shows the corresponding default risk premium and the bottom left panels shows the contribution of the jump risk premium. The bottom right panel plots the option implied volatility smirk against moneyness.
Basis Points Basis Points Figure OA.. Model-Implied Spread Sensitivities 10 Baa Aaa/Aa Spread (bps) 11 10 9 8 7 L J c 0 0 10 0 10 0 0 Percentage Deviation from Baseline Calibration.5 10 Volatility of Baa Aaa/Aa Spread (bps).5.5 L J c 0 0 10 0 10 0 0 Percentage Deviation from Baseline Calibration Notes to Figure: The top panel plots the sensitivity of the average model-implied 5-year Baa- Aaa/Aa spread level to changes in three key parameters: The sensitivity of habit and loss to consumption shocks ( and L ), and the absolute consumption jump size (jj c j). The bottom panel plots the sensitivity of the volatility of the 5-year model-implied Baa-Aaa/Aa spreads to changes in the same three parameters. 5