Why Higher Assumed Recovery Can Raise Implied Default Probability
Summary
The note explains why an implied default probability can rise when the assumed recovery rate increases, even if the observed risky-bond price stays fixed. Under a simple one-period setup, the bond's price is the probability-weighted value of its payment after default and its payment without default. If the recovery payment becomes larger, default must be more likely to preserve the same observed price. The examples use a zero-coupon bond and show this relationship across several recovery assumptions.
This is a risk-neutral inference tied to market prices and model assumptions, not a direct forecast of real-world default frequency. Recovery is uncertain before default, and the simplified examples ignore features such as timing and, initially, discounting. The note also describes a less common case with substantial interest-rate discounting, where early default and recovery can be more valuable than waiting for maturity; in that setup, raising assumed recovery can instead lower the inferred default probability. The direction therefore depends on cash-flow timing and pricing assumptions.
Key ideas
- Implied default probability is inferred jointly with an assumed recovery rate from market prices.
- With the risky-bond price fixed in a simple one-period model, higher recovery implies a higher default probability.
- The inferred probability is risk-neutral and depends on modeling assumptions.
- Discounting and default timing can reverse the relationship in unusual cases.
Tags
Full text
# Why the increase of presumed recovery rate will increase the implied default probability? # 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. ## Answer by Dimitri Vulis (score 2) https://quant.stackexchange.com/a/64375 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.
Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.