Actuarial Science in the Wild: Risk Models in Everyday Life

here is a point in an actuary’s career when ordinary life begins to look different.

A few months ago, I found myself driving behind an old, visibly beat-up car that was weaving somewhat precariously through traffic. My immediate reaction was probably the same as anyone else’s: That tracks.

The car’s abused appearance showed some unfavorable historical loss experience, and the driver was currently providing fresh evidence consistent with it — but then I started thinking about the less convenient cases.

Suppose the car had been immaculate, yet the driver was swerving wildly. Would I revise my view of the risk despite the apparent absence of prior damage or losses? Conversely, suppose the car was dented and battered, but I watched its driver behave impeccably for miles. How much weight should I give the unfavorable history relative to what I was observing now? That tension is increasingly familiar in P&C practice. In modern P&C pricing and experience rating, actuaries must decide how historical loss experience should be weighed against new behavioral or predictive information.

At that point, an absent-minded observation of an ordinary defensive-driving decision I encountered had suddenly become an experience rating problem.

There is a distinctly Bayesian flavor to the question. We begin with some belief about an underlying, unobserved level of risk and update it as new evidence arrives. Possibilities that make the observed evidence more likely receive more weight in the posterior distribution; possibilities that explain it poorly receive less.

Traditional experience rating in commercial insurance approaches the same general problem from a different direction. Under a Bühlmann credibility framework, we do not need to specify a full prior distribution and likelihood. Instead, the weight placed on an individual risk’s experience depends on the relationship between expected process variance (“EPV”, how much observed loss experience can vary around a risk’s own expected level) and the variance of hypothetical means (“VHM”, how much the underlying expected loss level varies across risks).

So agreement or disagreement between past experience and new information is not itself the answer. The deeper question is how informative each source is about prospective risk.

Believe it or not, pay enough attention to your everyday life and actuarial questions like this begin to emerge almost on their own.

My dog, Mickey, has provided another question.

Anyone who owns a dog knows that some walks are really just a waiting game for nature’s call. Mickey, however, operates according to his own schedule. Five minutes may pass. Then ten. Eventually I find myself wondering: what is the probability that he will have taken care of his business by a particular point in the walk? That is, of course, a time-to-event problem.

If T represents the amount of time until the event occurs, then I can think about the probability that T is less than ten minutes, fifteen minutes, or twenty. After enough uneventful time, the more relevant question becomes conditional: given that nothing has happened yet, what is the probability that the event will occur in the next few minutes?

At that point, I have wandered into survival analysis and hazard rates.

Mickey remains unaware of any of this.

The P&C analogue is straightforward. Claims are reported after varying lags. Payments emerge over time. Claims close at different rates. At any valuation date, we observe part of the process and infer what remains unseen.

The same reasoning applies to actuarial models of claim settlement times. Does a claim that has remained open for six months become more or less likely to settle in the next thirty days? The answer depends on the distribution of settlement times, which may vary by claim type, severity, litigation status, and other characteristics. Survival analysis, using approaches such as Cox proportional hazards models or GLMs, can help actuaries identify these patterns and project future claim closures and payments.

A dog walk and an insurance claim settlement are obviously not the same problem. But both illustrate a fundamental idea in survival analysis: the longer we’ve waited for an event, the more informative that waiting time may become. For a memoryless distribution, such as the exponential, waiting longer tells us nothing about how much longer we’ll have to wait. For other distributions, the likelihood of an event occurring soon may increase or decrease with elapsed time. Understanding these relationships helps actuaries better predict claim settlement patterns and assess uncertainty in outstanding liabilities.

Another actuarial thought experiment emerged from a rather costly yet common household phenomenon: buying food or groceries that I fully intend to eat, only to rediscover them later in a condition that makes eating them seem like an increasingly poor decision.

This led me to wonder what would happen if I treated household food spoilage as a miniature self-insured loss program.

The actuarial structure is there. There is a frequency component: How often do I lose food to spoilage? There is a severity component: How much does each event cost? Given enough observations, I could construct an aggregate monthly loss distribution, estimate an expected loss cost, and contribute a corresponding amount into a dedicated fund.

Naturally, however, an expected value alone would not be enough.

Suppose I wanted the fund to remain adequate in 95% of months. I would need to consider the variability of aggregate spoilage losses and determine how much additional surplus or contingency funding was necessary above the mean. A particularly unfortunate month involving produce, leftovers, dairy products, and some poorly planned bulk purchase could generate what might reasonably qualify, by refrigeration standards, as adverse development.

The analogy begins to look suspiciously like reserve and capital adequacy.

The expected monthly spoilage cost is analogous to an expected loss provision. The additional money needed to absorb unfavorable variation plays a role similar to risk capital. If actual spoilage consistently exceeds expectations, I would eventually need to revisit my assumptions, contribution rate, or underwriting practices.

In this particular program, “underwriting practices” would mostly consist of meal planning and remembering what is already in the refrigerator — unless I decided to get more sophisticated with inventory apps, smart appliances, or other household IoT.

The underlying actuarial principle here is not exotic. Whether we are dealing with a household budget, a self-insured employer, or an insurance company, the question is not simply, “What do we expect to lose?” It is also, “How much variability surrounds that expectation and how much financial capacity do we need to absorb it?”

At one of my previous jobs, my commute took me down Sixth Street in Los Angeles through what seemed like an endless parade of traffic lights.

Somewhere along the route, usually while sitting at one of them, I would begin thinking about the total amount of time I spent waiting at red lights — and perhaps more importantly from a lateness-risk management perspective — what the tail of that distribution might look like on one of those especially unfortunate mornings.

Suppose the waiting time at each traffic signal was a random variable. In a highly simplified version of the problem, assume the waiting times are independent and identically distributed, each with mean μ and variance σ².

If I encountered enough lights, the central limit theorem would suggest that total waiting time could be approximated by a normal distribution, with mean nμ and variance nσ². It is a pleasant little model.

Unfortunately, traffic signals do not generally behave like identically distributed, independent draws from the same process.

Traffic lights may be synchronized. Catching one red light can affect the probability of catching the next. Congestion changes arrival times at downstream intersections. Different signals have different cycle lengths. The covariance terms therefore matter.

For total waiting time T,
Var(T) = ∑_i Var(X_i) + 2∑_i<j Cov(X_i, X_j).

Here, X_i and X_j represent waiting times at different traffic lights. The covariance terms capture dependence between pairs of lights, and together they form the off-diagonal elements of a covariance matrix for the entire route.

The broader point is familiar to P&C actuaries: Aggregate variability depends not only on the volatility of individual components, but also on how those components move together.

That same issue becomes far more consequential in catastrophe modeling. Individual insured locations may each have manageable expected losses, but a hurricane, wildfire, or earthquake can create losses across thousands of risks simultaneously. Treating those exposures as independent would badly misstate the distribution of aggregate loss.

My commute did not involve catastrophe risk — although on certain mornings it may have felt that way. The underlying lesson was the same: Aggregation is easy when risks are independent. Real life often declines to cooperate.

None of these are particularly sophisticated actuarial problems.

That is partly the point.

Actuarial science is often described through its technical machinery: probability distributions, credibility theory, reserving methods, generalized linear models, simulation, capital models, and increasingly, machine learning. All of those tools matter. But beneath them lies something more basic: a way of looking at uncertain events.

What information do we have? How credible is it? What remains unobserved? What distribution might plausibly describe the outcomes? What happens when risks are combined? How should our estimate change when new information arrives?

Those questions appear constantly in insurance. They also appear, perhaps more often than we notice, in everyday life.

I recently explored a version of this question in a presentation I delivered at the joint CAS/AICT seminar in Taipei, “Modernizing Experience Rating: Credibility, AI, and Regulatory Trust.” The underlying issue was much the same: How should traditional loss experience be combined with increasingly rich behavioral and predictive information when estimating prospective risk?

That brings me back to the story of the driver in the opening segment.

Traditional experience rating asks us to learn about a risk partly from what has happened to that risk before. Today, telematics can measure acceleration, braking, speed, cornering, and mileage. Cameras and other sensors can observe circumstances that previously disappeared as soon as they occurred. AI systems can interpret streams of behavioral information on a scale no human observer could approach. In a sense, technology is beginning to do systematically what I was doing informally behind that battered car — observe behavior and ask what it tells us about prospective risk.

One can imagine, in effect, an automated version of me sitting behind every insured vehicle — minus, presumably, the commentary.

That creates enormous opportunities, but it also returns us to the original question. Suppose historical loss experience suggests one thing, and current behavioral information suggests another. How much weight should each receive? How credible is the observed experience? How reliable is the model interpreting the new signal? Is the signal genuinely predictive, or merely correlated with something else?

The technology may be new. The underlying actuarial problem is not.

For practicing actuaries, that suggests a simple discipline: When historical experience and a new predictive signal point in different directions, make the tradeoff explicit. Ask how credible each source is, what evidence supports the new signal, and how much the resulting estimate would change if either source were given more or less weight.

Together, these examples illustrate one of the most appealing things about actuarial science. Its principles are not confined to insurance textbooks, spreadsheets, or sophisticated models. These principles are ways of reasoning with incomplete information and uncertain futures. Once you become accustomed to thinking that way, the same questions begin to appear in traffic, in household expenses, on a walk with the dog — whether you were looking for them or not.