What Are the Odds?

By Matt Easley

Product Matters!, February 2024

As actuaries, how many times do we hear some version of the following:

“I thought you said that was never going to happen.”

“1/100 means I don’t have to worry about that, right?”

“Why does it seem like something is always going wrong?”

Let’s think about the problem like an exam question.

You are the product actuary for XYZ Life Insurance Company and you just experienced a 1/100 event on one of your products. The CEO is saying that he needs to understand what happened, but you suspect that he really means that you need to figure out how you screwed up. You have reviewed the work and cannot see any gross errors. The overall profitability appears to be in line across the overall product line. The one thing that troubles you is that the cause of the excess claims is not a variable that you considered explicitly in your risk analysis. How do you answer the CEO’s challenge?

Most of our pricing deals with expected results. We certainly look at volatility and reflect it in our risk charges. However, many of those risk charges are given to us through standard formulas. We are expected to make a return on capital based on one or more approved formulas. There is a certain discipline to that, but it means that work on actual pricing volatility is less practical.

Let’s suppose for the moment that your pricing is correct and that your estimate of the given outcome is also correct at 1/100. What does that mean in terms of your results? I remember learning something about fighter pilots that is relevant to this situation. If a pilot goes on 100 missions with a 1% chance of being shot down each time, what are the chances that he never gets shot down? The correct answer is 36.6%, which means that there is a 63.4% chance that he experienced a 1/100 event.

This is the dark side of diversification. If you take a lot of different risks, it means that some of them will go wrong as a normal part of your business. Let’s take a look at how that might play out. First, let’s assume that you have identified 10 important variables each of which is 1/100 to have a bad outcome. That means that we have a 99% chance of a good outcome on each variable in a given year, which translates to 90.4% chance of all good outcomes. That is nearly a 10% chance of having something to explain to the CEO! But you do business in more than one year. Over five years, the chance of a bad outcome rises to 39.5%. That is enough to require a ready explanation.

But there is more to the story. What about those variables that didn’t make your list? The ones that were too hard to price, you didn’t know the distribution, or that you didn’t anticipate being a problem.  They still exist even if they are not included explicitly in the pricing. These can be harder to explain precisely because you don’t have the same information from the original work. What happens to our probabilities if the number of variables that can go wrong is really 20 instead of 10, assuming the same 1/100 level of bad outcome? Now we are looking at an 18.2% chance of a bad outcome each year and a 63.4% chance of a bad outcome over a five-year period. (Note that this is the same number as our fighter pilot above.)

We talk about 1/100 losses as if we have great clarity about the loss distributions of our key variables. Some variables are subject to nice loss distributions that we understand. Others are not symmetrical and we may not have enough historical information to know the standard deviation let alone the actual distribution. For this article, let’s just look at the possibility that the actual probability is 2% instead of 1%. Now we expect a bad outcome 1/3 of the time in a given year and almost 87% of the time over five years. What happens if the probability was actually 3% is left to the student. Remember the lesson of The Black Swan—the tails of distributions are usually fatter than we think.

You look around the table at the rest of the product team. There are four other people looking back at you. Each of them has a product portfolio with similar risks. This is the CEO perspective. Now the odds of a bad outcome in a given year is nearly 87% and rounds to 100% over five years. His feeling that something is always going wrong is correct. Across a broadly diversified portfolio of liabilities you fully expect to have something going wrong. The ability to describe this and demonstrate that this is consistent with being successful as a company is not simple.

The other thing to remember is that we just imagined a company with the potential for 100 different variables that can go wrong. (20 per product and five products) Not all bad outcomes have equal weight. Also, half of them are risks that we assumed were not deeply analyzed in the pricing and risk process. What does that mean? I suggest the following practical responses:

1. It is critical to identify risks that can have a material impact on the business. Understanding these risks and the actual loss distributions is critical. This means that you need to benignly neglect the less important variables to make more time available for the big ones.
2. Even risks that are hard to measure are important to describe. An imperfect estimate of the right thing is better than limiting your analysis to things that you can model fully. Drag those variables into the light and think about them.
3. Watch new products closely for evidence that you may have a problem with your best estimates. It is not easy to separate statistical variation from a problem with the mean, but you want to get on top of a problem before you sell too many policies.
4. The same goes for emerging risks. Get them into your analysis early and keep improving your estimates.
5. For the major risks, it is important to understand how they can go wrong in addition to the probability. Understanding the mechanism can lead to loss mitigation plans. Discussions with your team about bad scenarios can make these risks real to them in addition to preparing for the day when something goes wrong. You might even adjust your design or underwriting to prevent some of the worst outcomes.
6. Be ready to explain to the CEO what happened. When that day comes, it will go better if you have this type of analysis completed and an action plan for what to do in response. Remember that it just a matter of time until it is your turn.

Statements of fact and opinions expressed herein are those of the individual authors and are not necessarily those of the Society of Actuaries, the editors, or the respective authors’ employers.

Matt Easley, FSA, is an independent actuary residing in Nokomis, Florida.  He can be reached at easleyadv@gmail.com.