## Abstract

We derive the credit default swap (CDS) premium a financial institution requires to assume the default risk of fixed income instruments and the maximum premium a CDS buyer is willing to offer. These premiums are functions of the institution’s capital and current risk exposure. In most cases, an institution requires an increasing premium to assume additional risk. However, we show that an under-capitalized institution that already has substantial default risk exposure would engage in risk-shifting and assume more risk at lower CDS premiums. Consistent with this, prior to the 2008 financial crisis credit default swap issuance increased substantially, as did the volume of the underlying mortgage-backed securities, but the data suggests that required CDS premiums remained constant or declined.

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## Notes

- 1.
The ABX Index represents 20 subprime residential mortgage-backed securities and is used as a financial benchmark of the overall value and performance of the subprime residential mortgage market. See Wachter (2018) for further discussion.

- 2.
*See*Susan M. Wachter, Credit Risk Transfer, Informed Markets, and Securitization, Economic Policy Review 24(3), 2018. In January 2006, Markit Group, in collaboration with a group of major banks, launched the ABX.HE (the ABX), linked to the pricing of twenty specific home equity RMBS deals, including some of the largest deals during this period. The overall index incorporated a basket of indexes, differentiated by credit risk rating. - 3.
The literature identifies the mispricing of the put option embedded in nonrecourse lending (Pavlov & Wachter, 2006; Gomes, Grotteria, & Wachter, 2018), deriving from bank managers and shareholders exploitation of mispriced deposit insurance. In the model we develop here, insurance is mispriced but not due to government backing but rather due to profit maximizing firm behavior. In a related paper, Gollier, Koehl, and Rochet (1997) shows that the optimal exposure to risk of firms with limited liability is larger than firms with unlimited liability.

- 4.
Two solutions exist, negative and positive. When there is a perturbation of the capital stock away from the steady state, a negative h suggests a concave value function and will bring the capital stock back to the steady state, while the positive solution is unstable.

- 5.
Note that the CDS premium is not risk premium. It is akin the premium charged on any insurance policy.

- 6.
In our illustration, the offer rate is lower than the ask rate in all cases. This is intuitive because the issuer needs to be compensated for putting the underlying business at risk, while the buyer needs to be compensated for the credit risk of the issuer. If taken literally, our results suggest that there would be no CDS transactions. We note, however, that from a modelling point of view this can be easily overcome. For instance, the buyer may have some hedging demand for the CDS and be willing to pay a premium in excess of the one depicted in Figure 6. Or, the buyer may be risk-averse, which would also induce them to offer a higher premium than what we compute.

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## Appendix

### Appendix

### Differential equation with a general dividend policy

The risk-neutral, no-arbitrage condition for the value of the lender, *V*, under a general dividend policy, δ, is:

The Ito’s Lemma for the jump-diffusion processes remains unchanged:

This provides the following differential equation for the lender value, *V*:.

This system of equations can only be solved numerically.

### Valuation of the CDS Buyer’s Claim

Applying Ito’s Lemma for jump-diffusion processes while the issuer is in business, >*ξ*, we obtain:

Note that the second part above has (-*W*) because the CDS buyer receives a payout but the claim expires in case of a jump. Substitute this into Equation (7) to obtain the following differential equation for the value of the CDS claim, *W*, while the issuer is in business, *X* > *ξ*:

Simplify:

The solution to this equation is:

where *B*_{1} and *B*_{2} are arbitrary constants, *ξ* is the abandonment boundary as determined by the issuer optimization, and

Determine *B*_{1}, *B*_{2}, and *B*_{3} with value-matching and smooth-pasting at *X* = *Cg* + *ξ* and at value-matching to zero at the abandonment boundary, *X* = *ξ*:

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Pavlov, A., Schwartz, E. & Wachter, S. Price Discovery Limits in the Credit Default Swap Market in the Financial Crisis.
*J Real Estate Finan Econ* **62, **165–186 (2021). https://doi.org/10.1007/s11146-020-09747-8

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### Keywords

- Credit insurance
- Credit default swaps
- Price discovery