# Why the increase of presumed recovery rate will increase the implied default probability?

From Hull's paper, the implied default probability (lambda) = credit spread/(1-recovery rate).

Therefore, we can infer that, as the recovery rate goes up, the lambda will also go up. Why is that? I think it does not obey the intuition.

The credit spread is observable. So is the risk-free interest rate.

The recovery rate (1 - loss given default) is not observable until after the event happens, but can be assumed to be zero for some instruments.

The risk-neutral probability of default depends on the recovery assumption. It also has term structure and is non-decreasing with time.

As a numerical example, suppose we can observe in the market the yields of two zero-coupon bonds (ZCB) maturing in 1 year. Suppose for simplicity that a risk-free ZCB has yield 0, i.e. the time value of money plays no role in this simplistic example, and the time to default does not matter either. Suppose that the market prices a risky ZCB at 90% of its face value. With probability $$p$$, the risky ZCB will default and pay only $$R<1$$, and with probability $$1-p$$, it will pay 1. Solving for $$p$$, we see that it depends on $$R$$:

If $$R=0$$, then $$p=10\%$$ because $$90\% = 10\% \times 0\% + 90\% \times 1$$.

If $$R=20%$$, then $$p=12.5\%$$ because $$90\% = 12.5\% \times 20\% + 87.5\% \times 1$$.

If $$R=50%$$, then $$p=20\%$$ because $$90\% = 20\% \times 50\% + 80\% \times 1$$.

If $$R=80%$$, then $$p=50\%$$ because $$90\% = 50\% \times 80\% + 50\% \times 1$$.

So, increasing the value of the risky ZCB under the recovery scenario while keeping the price constant has to make this scenario more likely.

However the interplay between these parameters can sometimes get counter-intuitive. For example, if the interest is "high", rather than 0, making the present value of future cash flows "low", then it's possible to be more beneficial for the lender to have a default sooner and to receive only recovery, rather than to wait for the scheduled cash flows. But this is very unusual.

Edit: here is an exaggerated example of the latter that sounds contrived, but I've actually seen such weird things a few times. Suppose the risk-free ZCB is trading at $$d$$ (discount factor for time value of money). Suppose the risky (and well collateralized) ZCB is priced higher than the risk-free one because the market thinks that the risky bond's issuer is likely to (cross-)default soon and pay $$R>d$$ sooner than maturity. For concreteness, let riskless bond price $$d=50$$ and risky bond price $$=60$$. If $$R=75\%$$ then $$p=40\%$$. But if $$R=90\%$$ then $$p=25\%$$. In this rare case, increasing $$R$$ and keeping the prices constant causes $$p$$ to decrease.

• Thanks for your answer. So, in short, if the market prices of a risky asset remain constant, the increase of recovery rate (good thing) must coordinate to a increase of default probability (a bad thing) to keep the price constant. Is that right? Jun 3, 2021 at 15:24
• Yes, exactly! Increasing the recovery assumption would increase the risky instrument's price (=decrease the credit spread), so it needs to be offset by the incrase in the probability of default (except in bizarre extreme cases). Jun 3, 2021 at 16:42