Credit risk Q&A with a Finance 4335 student

Yesterday afternoon, I received the following email from a Finance 4335 student about credit risk:

Date: Friday, November 20, 2020 at 3:40 PM
To: James Garven <>
Subject: Question for Problem Set 9

Doctor Garven,

I just have a question for part 3 of the problem set 9 which is due next Tuesday. I’m confused as to why the fair premium of the insurance corresponds to dollar value of the limited liability put option. And if the firm purchases the insurance, will its credit risk premium decline to 0?

Here’s my answer:

Perhaps I should more clearly define “fair” premium. In this setting the “fair” insurance premium corresponds to the premium paid by the firm such that it is indifferent between buying and not buying credit insurance.

Consider the following numerical example. Suppose the riskless rate of interest is zero, and a firm promises to pay $100 one year from now. Then the yield to maturity of the debt is also zero, and the present value is $100. Next, suppose an otherwise identical and uninsured firm issues debt with a promised repayment of $100, but it has a high probability of default so the current market value of the debt is only $80; in this latter case, the yield to maturity and the credit risk premium are the same, since YTM = r + credit risk premium and r = 0. Thus, YTM = credit risk premium = 25% (here I am assuming annual compounding for the sake of simplicity, so FV = PV(1+YTM) ==>100 = 80(1.25)). Now, suppose this uninsured firm changes its mind and purchases full insurance coverage for the “fair” premium of $20 (which corresponds to the value of the limited liability put option). Since the insured firm’s debt is no longer risky to investors, investors currently value the firm’s debt issue at its par value of $100, which implies that the credit risk premium falls to 0. However, from the standpoint of the firm, it can only expect to net $80 from its bond issue, irrespective of whether it purchases credit insurance, so it is indifferent between buying and not buying credit insurance.

In the real world, credit risk enhancement is a viable financial services business because so-called “prudent-person” rules often severely limit the marketability of sub-investment grade credit to institutional investors. Thus, credit enhancement to investment grade can add more value for firms, NGO’s, and government organizations by substantially expanding the market for potential investors in such credit issues.


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