Risk-Adjusted Bond Returns Using Default Probability
Summary
The document considers how to rank live bonds using predicted default probabilities from a classifier trained with matured bond outcomes. It questions an individual-bond score that takes the logarithm of a return-to-default-probability ratio. The response argues that this score is not directly connected to the conventional Sharpe ratio, which compares excess expected return with return variability.
Instead, it models bond payoff over a fixed period as a Bernoulli outcome: the bond either avoids default or defaults, with the latter payoff reduced by loss given default. From these outcomes it derives expected reward and variance, then forms a Sharpe-style measure using the risk-free adjustment and default probability. The proposed framework depends on assumptions about horizon, payoff, and loss severity; the answer is explicitly presented as an opinion. It also raises the possibility that default timing and dependence among issuers matter, which a simple default indicator may not capture.
Key ideas
- A return divided by default probability is not automatically a Sharpe ratio.
- A Bernoulli default model can represent the bond’s payoff over a specified horizon.
- Expected reward depends on survival payoff, recovery after default, default probability, and a benchmark adjustment.
- Payoff variance rises with the gap between survival and default outcomes.
- Default timing and dependence may require modeling beyond a single probability estimate.
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Full text
# Risk-return ratio using ML default probability
# Risk-return ratio using ML default probability
I have access to a very large bond database (>20m rows) where 50% of the set are matured bonds for which a dummy variable identifies whether the bond defaulted or not. The remaining 50% are 'live' bonds.
I have already trained an ML classifier and predicted the probability of default for the 'live' component of the set. My endgame here is to evaluate the live dataset from a risk-adjusted return perspective. I have come up with the following augmented Sharpe ratio $\tilde{S_i}$ that can be applied on an individual bond basis:
$\tilde{S_i} = \log [ \frac{R_i}{\hat{p_i}(\text{Default})}]$
Is anyone aware of similar methods to evaluate the risk-adjusted expected return of debt securities? Does my method make sense to you? I could highly use some comments.
## Answer by Attack68 (score 1, accepted)
https://quant.stackexchange.com/a/44587
Disclaimer: these are just opinions, I do not necessarily have authoritative knowledge in this topic.
If you consider the traditional Sharpe definition:
$$S = \frac{reward}{risk}$$ where reward is the expected return (above risk free rate) and risk is the standard deviation of reward, it is not clear to me how your augmented Sharpe ratio is related to this. Instead, my instinctive approach would be to model the return of the bond under a benoulli distribution with $p_i$ the probability of default, i.e. when random variable $X_i=1$ and no default if $X_i=0$. Furthermore we need some consistency of time so I will measure over a time period $\Delta t$.
$$ reward_i = (100 + r_i) (1 - X_i) + 100(1-L_{gd})X_i - C_i$$
where $r_i$ is the return of the bond over $\Delta t$, $C_i$ is some standardisation, like subtracting risk free rate, and we have a loss given default factor. If you let $R_i=100+r_i$ and $L_i=100(1-L_{gd})$ then;
$$ E[reward_i] = R_i(1-p_i)+L_ip_i-C_i$$ and $$ Var(reward_i) = Var((-R_i+L_i)X_i)=(R_i-L_i)^2p_i(1-p_i)$$
So you end up with a formula which states that;
$$Sharpe(p_i; R_i, L_i, C_i) = \frac{R_i(1-p_i)+L_ip_i - C_i}{\pm(R_i-L_i)\sqrt{p_i(1-p_i)}} \;.$$
As a completely separate point I would suspect that your machine learning process would benefit by conditioning itself on the data of the timing of defaults. This was a failure of models based on copulas in the credit crisis if I recall. It might end up being quite a complicated formulation of a model that accounts for similarity based on timeliness of defaults.
Happy for harsh criticism.. don't hold back..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.