I'm calculating expected loss on fixed-income using actual default probabilities and risk-free rate as the discount factor. I understand this is not theoretically correct. In absence of risk-neutral probabilities, are there any alternative rates I can use (besides risk-free rate) as the discount factor?


Yes, I solve this in a paper I am about to submit. It is important that you understand what it means to say "actual probabilities." Most people use Pearson-Neyman type statistics and they do not and cannot give rise to actual probabilities except in the narrow case of "exact" tests when the null is actually true.

The reason is that Pearson-Neyman methods use minimax distributions. Minimax distributions are worst-case distributions, but they allow you to guarantee that you will not have false positives more than $\alpha$ percent of the time if your sample size goes to infinity.

The result is that any statistics generated are not coherent, with the technical definition of coherence being that a bookie offering to buy risks could not be gamed into taking a sure loss by a crafty actor or set of actors. In essence, you can gamble on these probabilities and they will be the true probabilities, given the information available. If you would use Pearson-Neyman probabilities then someone who is crafty would be capable of gaming your solution. This is very surprising, but also true.

When Pearson and Neyman put together their system of thinking, which rolls into economics, they did not intend it to be used for gambling, but instead to direct behavior. Should you keep the lot coming off the assembly line or reject the lot based on a sample? It is a wonderful system when used as intended.

Let us begin by estimating the default rate and to simplify the argument we will drop things like logistic regression because they nicely add dimensionality, but detract from the simplicity of the explanation. Note that the only thing that would change in this discussion is the form of the likelihood function and the dimensionality of the prior distribution. How you would do it would change, but the logic is unchanged.

Let us also narrowly define what a default is, acknowledging that a wide definition would merely add dimensionality to the problem. So, for example, in a real model, you would separately define the probability of a temporary suspension of payments, the probability of a permanent suspension with full recovery of principal, the probability of a permanent suspension with partial recovery and the probability of permanent suspension with no recovery. For partial recoveries, either of interest or principal, then you also need to model the likely recovery given the type of default happened.

For our narrow discussion, we will pretend we live in an all or nothing world, wherein default results in no form of recovery. In essence, this is now a Bernoulli trial. Let us also make this a zero coupon bond so that we do not need to model separate defaults on each coupon.

I know I just shrank the problem down tremendously but there is no loss of generality.

You would then use Bayesian decision theory and there is no issue of either risk neutrality, risk aversion, loss aversion or risk loving preferences. The first issue is to set the prior distribution. A flat or non-informative prior is not a true credible solution as it would imply that the chance of default for a bond is equally 99% and 0.1%. This would provide a 50% chance of default as the prior expectation. That is ridiculous.

There are two ways to set a prior. The first would be to observe actual historical rates. Since we are dropping logistic regression we are assuming true independence of defaults rather than dependence on accounting values or the state of the economy. In this pretend world we can get away with that. Let us imagine that you found a large data set of one thousand observations and in that set, there was one default. That would create a prior probability for $\theta$ where $\theta$ is the probability of default equal to $999(1-\theta)^{998}$ by assuming the prior is a beta distribution. This may not be true, but for our purposes is very convenient mathematically. If you have a better way to construct the prior, then don't do this.

To understand, this density is not the probability of default, but rather the probability $\theta=k,\forall{k}\in(0,1),$ where $\theta$ is the true chance of default.

A second solution could be to work backward from rates using relationships such as $i=r^*+p_h+p_m+p_d,$ where you would factor out inflation, maturity and the Fisher effect from bond rates. There are other solutions too. Let us stick with the simple one above though.

Now lets assume you buy a portfolio of 100 bonds and none of them default. We are also going to assume you are not collecting outside data though the process would be the same as no one cares where the data came from. Random sampling is not required here. The binomial density is $(1-\theta)^{100}$. The beta distribution above and the binomial likelihood would be multiplied together and normalized to a density. The resulting density of $\Pr(\theta|data)=\frac{1}{1099}(1-\theta)^{1098}$ is the posterior density of $\theta.$

You are not interested in the location of the parameter, but rather, the probability there will be zero defaults, one default, two defaults and so forth. To do this, you would integrate out any unknowns leaving only the knowns, which are the data and the prior knowledge. This is often expressed as $\Pr(\tilde{x}|X)$, where $X$ is the data.

To solve this prediction you integrate out the uncertainty as follows: $\Pr(\tilde{x}=k|X)=\int_\theta\Pr(\tilde{x}|\theta)\Pr(\theta|X)\mathrm{d}\theta$. In this specific case, the solution is analytic and well known. It is called the beta-binomial distribution. If $\alpha$ are defaults, in this case, 1, and $\beta$ is surviving bonds, in this case, 1099, and $n$ is the future number of bonds purchased and $k$ is possible default levels, then you can construct a discrete mass function for each level and take the expectation.

The analytic solution is $\Pr(default=k|\alpha, \beta) = \frac{\Gamma(n+1)}{\Gamma(k+1)\Gamma(n-k+1)} \frac{\Gamma(k+\alpha)\Gamma(n-k+\beta)}{\Gamma(n+\alpha+\beta)} \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)},$ where $\Gamma$ is the gamma function for the mass function and then the expectation would be trivial to take as the chance of two defaults is very small. You would then discount at the institution's marginal cost of funds, which is the deposit rate in the case of a bank. There need only be a subjective cost of funds. You end up with $$\frac{E(k)}{1+i}$$ where $i$ is the marginal cost of funds for the actor. This is the present value of your expected losses, though not expressed as a rate.

The above outcome has a known solution in the literature. It is $$\frac{\frac{n\alpha}{\alpha+\beta}}{1+i}=100/1100/(1+i)\approx{.09}/(1+i).$$ So in one hundred future bonds you would expect to lose nine hundreths of one bond.

Now, if I were doing it, I would ignore everything I just told you as it would make computation very costly unless I wanted to write a do loop and calculate the factorials in logs and invert the process. For a large enough sample size, the normal approximation would be better to work with in theoretical work. If I were serious about the computation then I would use Bayesian logistic regression because it does not have the assumption violations so well documented in Nwogugu at:

Nwogugu, Michael; Decision-making, risk and corporate governance: A critique of methodological issues in bankruptcy/recovery prediction models; Applied Mathematics and Computation; 2007;(185); pp. 178-196.

Still, you would take the expectation of the Bayesian predictive distribution divided by the subjective cost of funds. It would just now be very complex due to the true dimensionality of the problem and you would almost certainly be forced to use a Metropolis-Hastings algorithm to solve the problem.

If you are not risk neutral, then you would not use quadratic loss as it assigns equal quadratically growing costs to the risk of mislocating the point estimate of $E(k)$ downward as upward. If you were loss averse then you would underweight underestimates of $k$ and overweight overestimates of $k$. This is why I said it may be easier to do with the normal distribution. It may be easier to solve as a traditional risk aversion problem with something like CARA or CRRA.

There is a good book on both Pearson-Neyman and Bayesian decision theory at:


I do apologize that this is so long, but you would likely ask the question in a different manner if you were accustomed to thinking in terms of decision theory. Also, you wanted to know about the actual default probabilities and not abstracted ones if the parameters were truly known at a risk-free rate.

In the real world, I used Bayesian methods to test 78 models of bankruptcy and found two that worked, although all are statistically significant. All significance tells you is that the results are unlikely to be due to chance if the null is true, it does nothing to tell you how useful it is.

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