Random Recovery Rate Research Articles

MEAN–VARIANCE INSURANCE DESIGN WITH COUNTERPARTY RISK AND INCENTIVE COMPATIBILITY

AbstractThis paper studies the optimal insurance design from the perspective of an insured when there is possibility for the insurer to default on its promised indemnity. Default of the insurer leads to limited liability, and the promised indemnity is only partially recovered in case of a default. To alleviate the potential ex post moral hazard, an incentive compatibility condition is added to restrict the permissible indemnity function. Assuming that the premium is determined as a function of the expected coverage and under the mean–variance preference of the insured, we derive the explicit structure of the optimal indemnity function through the marginal indemnity function formulation of the problem. It is shown that the optimal indemnity function depends on the first and second order expectations of the random recovery rate conditioned on the realized insurable loss. The methodology and results in this article complement the literature regarding the optimal insurance subject to the default risk and provide new insights on problems of similar types.

Mean-variance insurance design with counterparty risk and incentive compatibility

This paper studies the optimal insurance design from the perspective of an insured when there is possibility for the insurer to default on its promised indemnity. Default of the insurer leads to limited liability, and the promised indemnity is only partially recovered in case of a default. To alleviate the potential ex post moral hazard, an incentive compatibility condition is added to restrict the permissible indemnity function. Under the actuarial premium principle and mean-variance preferences of the insured, we derive the explicit structure of the optimal indemnity function through the marginal indemnity function formulation of the problem. It is shown that the optimal indemnity function depends on the first and second order conditional expectations of the random recovery rate. The methodology and results in this paper complement the literature regarding the optimal insurance subject to the default risk and provide new insights on problems of similar types. Moreover, we also apply the techniques in this paper to the case of additive background risk, which yields an alternative proof of the main result of Chi and Tan (2021).

Self-Remanufacturing or Outsourcing? Hybrid Manufacturing System With Remanufacturing Options Under Yield Uncertainty

Yield uncertainty widely exists in manufacturing industries. Because of the particularities of the recovery and production processes, remanufacturing is seriously affected by the uncertainty of the production process, which also affects the remanufacturing choice of enterprises. In this paper, we focus on the case where a make-to-order type manufacturer is confronted with the strategic option of obtaining core materials by self-remanufacturing or outsourcing new cores. Through a stochastic modelling approach, a two-echelon closed-loop supply chain is constructed to capture the manufacturer's sourcing strategy and production planning by considering yield uncertainty in both the remanufacturing and production stages. We first examine the case of a deterministic recovery rate. The results show that regardless of whether the firm chooses to remanufacture or buy new core materials, the expected profit of the (re)manufacturer is always concave. Then, in the case of a random recovery rate, we show that the distribution bounds are important and play key roles in deciding the strategic option of self-remanufacturing or outsourcing. By comparing the upper and lower bounds of the recovery rate with the procurement cost of new raw material and used product, three production/sourcing strategies, namely, using remanufactured cores only, outsourcing new material only and jointly using the two materials, are investigated. The optimal procurement quantity and the optimal production input are derived to maximize the (re)manufacturer's expected profits under the three scenarios. Finally, we perform sensitivity analysis of the impact of yield uncertainty on the optimal results. The results show that when the shortage cost increases, the collection quantity increases faster than the production input, which implies that a “bullwhip effect” exists in the proposed two-period collection and production system. The conclusions of this paper can be applied to remanufacturing in related enterprises to provide production and procurement decision support for managers and to promote the efficiency of closed-loop supply chains.

On the Optimal Risk Sharing in Reinsurance with Random Recovery Rate

This paper studies a Pareto-optimal reinsurance contract in the presence of negative statistical dependence between the insurance claim and the random recovery rate. In the context of symmetric information model and asymmetric information model, we investigate properties of the Pareto-optimal indemnity schedules. For risk neutral reinsurer with proportional cost and associated expense, we present possible forms of the Pareto-optimal indemnity schedule as well.

Sensitivity analysis for marked Hawkes processes: application to CLO pricing

This paper deals with a model for pricing Collateralized Loan Obligations, where the underlying credit risk is driven by a marked Hawkes process, involving both clustering effects on defaults and random recovery rates. We provide a sensitivity analysis of the CLO price with respect to the parameters of the Hawkes process using a change of probability and a variational approach. We also provide a simplified version of the model where the intensity of the Hawkes process is taken as the instantaneous default rate. In this setting, we give a moment-based formula for the expected survival probability.

Contact Processes with Random Recovery Rates and Edge Weights on Complete Graphs

In this paper we are concerned with the contact process with random recovery rates and edge weights on complete graph with $n$ vertices. We show that the model has a critical value which is inversely proportional to the product of the mean of the edge weight and the mean of the inverse of the recovery rate. In the subcritical case, the process dies out before a moment with order $O(\log n)$ with high probability as $n\rightarrow+\infty$. In the supercritical case, the process survives at a moment with order $\exp\{O(n)\}$ with high probability as $n\rightarrow+\infty$. Our proof for the subcritical case is inspired by the graphical method introduced in \cite{Har1978}. Our proof for the supercritical case is inspired by approach introduced in \cite{Pet2011}, which deal with the case where the contact process is with random vertex weights.

Sensitivity Analysis for Marked Hawkes Processes - Application to CLO Pricing

This paper introduces a model for pricing Collateralized Loan Obligations, where the underlying credit risk is driven by a marked Hawkes process, involving both clustering effects on defaults and random recovery rates. We provide a sensitivity analysis of the CLO price with respect to the parameters of the Hawkes process using a change of probability and a variational approach. We also provide a simplified version of the model where the intensity of the Hawkes process is taken as the instantaneous default rate. In this setting, we give a moment-based formula for the expected survival probability.

Portfolio credit risk model with extremal dependence of defaults and random recovery

The extremal dependence of defaults, and negative correlation between defaults and their recovery rates, are of major interest in modeling portfolio credit risk. In order to incorporate these two features, we propose a portfolio credit risk model with random recovery rates. The proposed model is an extension of the traditional t-copula model for the credit portfolio with constant recovery rates. A skew-normal copula model is adopted to represent dependent random recovery rates. In our proposed model, various types of dependency between the defaults and their recovery rates are possible, including an inverse relation. We also propose a conditional Monte Carlo simulation algorithm for estimating the probability of a large loss in the model, and an importance sampling version of it. We show that the proposed Monte Carlo simulation algorithm is relatively efficient compared with the plain Monte Carlo simulation. Numerical results are presented to show the performance and efficiency of the algorithms.

Phase Transition for the Large-Dimensional Contact Process with Random Recovery Rates on Open Clusters

In this paper we are concerned with the contact process with random recovery rates on open clusters of bond percolation on $$\mathbb {Z}^d$$ . Let $$\xi $$ be a random variable such that $$P(\xi \ge 1)=1$$ , which ensures $$\mathrm{E}\frac{1}{\xi }<+\infty $$ , then we assign i. i. d. copies of $$\xi $$ on the vertices as the random recovery rates. Assuming that each edge is open with probability p and the infection can only spread through the open edges, then we obtain that $$\begin{aligned} \limsup _{d\rightarrow +\infty }\lambda _d\le \lambda _c=\frac{1}{p\mathrm{E}\frac{1}{\xi }}, \end{aligned}$$ where $$\lambda _d$$ is the critical value of the process on $$\mathbb {Z}^d$$ , i.e., the maximum of the infection rates with which the infection dies out with probability one when only the origin is infected at $$t=0$$ . To prove the above main result, we show that the following phase transition occurs. Assuming that $$\lceil \log d\rceil $$ vertices are infected at $$t=0$$ , where these vertices can be located anywhere, then when the infection rate $$\lambda >\lambda _c$$ , the process survives with high probability as $$d\rightarrow +\infty $$ while when $$\lambda <\lambda _c$$ , the process dies out at time $$O(\log d)$$ with high probability.

Critical value for the contact process with random recovery rates and edge weights on regular tree

In this paper we are concerned with contact processes with random edge weights on rooted regular trees. We assign i.i.d weights on each edge on the tree and assume that an infected vertex infects its healthy neighbor at rate proportional to the weight on the edge connecting them. Under the annealed measure, we define the critical value \lambda_c as the maximum of the infection rate with which the process will die out and define \lambda_e as the maximum of the infection rate with which the process dies out at exponential rate. We show that these two critical values satisfy an identical limit theorem and give an precise lower bound of \lambda_e. We also study the critical value under the quenched measure. We show that this critical value equals that under the annealed measure or infinity according to a dichotomy criterion. The contact process on a Galton-Watson tree with binomial offspring distribution is a special case of our model.

Pricing credit default swaps with a random recovery rate by a double inverse Fourier transform

We evaluate the par spread for a single-name credit default swap with a random recovery rate. It is carried out under the framework of a structural default model in which the asset-value process is of infinite activity but finite variation. The recovery rate is assumed to depend on the undershoot of the asset value below the default threshold when default occurs. The key part is to evaluate a generalized expected discounted penalty function, which is a special case of the so-called Gerber–Shiu function in actuarial ruin theory. We first obtain its double Laplace transform in time and in spatial variable, and then implement a numerical Fourier inversion integration. Numerical experiments show that our algorithm gives accurate results within reasonable time and different shapes of spread curve can be obtained.

The large homogeneous portfolio approximation with a two-factor Gaussian copula and random recovery rate

In this paper we consider the large homogeneous portfolio (LHP) approximation with a two-factor Gaussian copula and random recovery rate. In addition, we assume that the earlier the default occurs, the less the asset recovers; in other words, random recovery rate and individual default times have a positive rank correlation. Under the LHP assumption, the conditional cumulative loss of the reference portfolio is approximated by the product of loss given default and conditional default probability. In order to derive semi-analytic formulas for the loss distribution and the expected tranche loss, we use a Gaussian two-factor model and assume that the recovery rate depends on one systematic factor. In addition, we consider stochastic correlation for a better fit to credit default swap index tranches. The derived semi-analytic formula only involves integration with respect to the standard normal density and can be computed by Gauss–Hermite quadrature. Numerical tests show that the two-factor model with stochastic correlation and random recovery fits iTraxx tranche premiums better than other correlation or recovery assumptions under the Gaussian LHP framework. We also apply our model to credit risk assessment such as value-at-risk of the loss distribution.

Credit Spreads, Recovery Rates and Bond Portfolio Risk Measures in a Hybrid Credit Risk Model

This paper presents a framework in which many structural credit risk models can be made hybrid by randomizing the default trigger, while keeping the capital structure intact. This produces random recovery rates negatively correlated with the default probability. The approach is implemented on a firm-by-firm basis using maximum likelihood and the unscented Kalman filter (UKF) on each of the 225 companies of the CDX NA IG and HY indices using weekly CDS data from December 2007 to January 2012. Adding the surprise element and the time-varying distribution of recovery rates has a large impact on credit spreads as it modifies both the level and shape of the curves. When a bond portfolio is considered, the presence of dependence among firm leverage ratios and between default probabilities and recovery rates produces clusters of defaults with low recovery rates. It has a major impact on standard risk measures such as Value- at-Risk and conditional tail expectation.

The Mean-Variance Hedging of a Defaultable Option with Partial Information

We consider the mean-variance hedging of a defaultable claim in a general stochastic volatility model. By introducing a new measure Q 0, we derive the martingale representation theorem with respect to the investors' filtration 𝔾. We present an explicit form of the optimal-variance martingale measure by means of a stochastic Riccati equation (SRE). For a general contingent claim, we represent the optimal strategy and the optimal cost of the mean-variance hedging by means of another backward stochastic differential equation (BSDE). For the defaultable option, especially when there exists a random recovery rate we give an explicit form of the solution of the BSDE.