And it feels like something went wrong—that someone should be held accountable.
The Works section has been updated with my most recent publication: Risk, Rates, and Reality.
I argue that stakeholders sometimes misinterpret losses as proof of actuarially unsound rates, yet capitation rates are set prospectively based on what the actuary knows at the time. Even after following a legitimate forecasting process, insurance losses can happen because of the forecasts' probabilistic nature, not due to forecasting errors.
Searching The Math
Let's lightly explore an idea the article does not: What's the binary probability of an error, given there was a loss?
$$
P(Error \mid Loss) = \frac{P(Loss\mid Error) \cdot P(Error)}{P(Loss \mid Error) \cdot P(Error) + P(Loss \mid \neg Error) \cdot P(\neg Error)}
$$
- \( P(Error) \): Prior hypothesis or belief before seeing a loss: 5% (Sorry, it's not 0%; I have yet to shed my mortal failings and merge with the singularity.)
- \( P(Loss \mid Error) \): Likelihood of observing \( Loss \) given there is an \( Error \): 80% (An erroneous calculation could result in higher profits. But let's be skeptical and assume the actuary would mess up in a way that disadvantages the insurance companies the majority of times.)
- \( P(Loss \mid \neg Error) \): Likelihood of observing a loss given no error: 33% (From the article's footnote No. 3.)
- \( P(Error \mid Loss) \): Posterior probability, updated belief, hypothesis, or the probability of \( Error \) after seeing a \( Loss \).
$$ P(Error \mid Loss) = \frac{0.80 \cdot 0.05}{(0.80 \cdot 0.05) + (0.33 \cdot 0.95)} $$
Simplifying: $$ P(Error \mid Loss) = \frac{0.04}{0.04 + 0.3135} = 0.1132 $$
Thus, the posterior probability, \( P(Error \mid Loss) \), is about 11%. Under skeptical conditions, observing a loss doubles the relative probability of an error but doesn't change the absolute probability of an error that much—it's 1:8 odds. Want to bet on that?
Finding Clarity
However, this exercise is less about the calculated probability when working with rough priors. Instead, the value lies in the discipline, the intentionality, of identifying each component and making explicit assumptions about them. It's less about the calculated probability and more about the clarity from thinking it through.
The editor didn't include my standard disclaimers, so I'll say it here for both this post and the article:
This website reflects the author's personal exploration of ideas and methods. The views expressed are solely their own and may not represent the policies or practices of any affiliated organizations, employers, or clients. Different perspectives, goals, or constraints within teams or organizations can lead to varying appropriate methods. The information provided is for general informational purposes only and should not be construed as legal, actuarial, or professional advice.
