Actuarial sound simulation using life/mortality tables?

I was directed to GoActuary based on a question I asked in Reddit’s r/actuary community. As the question below will reveal, I am not an actuary, or even a competent statistician, so I apologize in advance for the ignorance of the question.

I am writing a simulation that utilizes mortality tables to determine, at each step of the simulation, the members of the simulation that should die (or “expire”, if you prefer). At the beginning or initiation of the simulation, I am given a population of members, and then at each step of the simulation I would like to determine the members of the simulated population who should die. A simulation step corresponds to the advancement of time t=x to t=x+1, where a simulation step is one year.

I believe this would be easy to do using the appropriate male or female probability of death, q(x), provided by the mortality table, if I was simulating a population of homogenous members of the same gender and age. For a homogenous population, I believe I could just multiply q(x) by the size of the population and then remove that many members from the population? (I would also be interested in an acceptable approach to handling fractional deaths; i.e. the simulation needs to work with discrete members, so, for example, it cannot mark the 0.23% of the population in 3.23% of the population dead.)

Unfortunately, I am not working with a homogenous population. Instead, the populations I am working with are composed of men and women of both genders and various ages. So, the crux of my question is, given an individual from my simulated population at a specific simulated time step, and, therefore, the q(x) for individuals matching that individual’s attributes, is it possible to and is there an accepted/preferred method for determining if that individual should be marked as dead?

While I’m most interested in guidance of answering the above question, if there is a resource that explains the rationale for this guidance, I would appreciate being pointed at it (although, my knowledge in this space may prevent me from understanding it).

I was told I should make sure to tag @meep on this post.

I think understanding the purpose of your “simulation”, the size of your total population as well as subgroups you want to adjust for, and your need for precision might be helpful in getting answers, but I’ll throw out a few initial observations.

The search term you are looking for to find documentation/education materials is “life contingencies.” Life insurance is primarily the domain of the Society of Actuaries, but I’m not sure how much of their syllabus material is available for free (here is the SOA exam page that covers life contingencies https://www.soa.org/education/exam-req/edu-exam-ltam-detail/).

The Casualty Actuarial Society (which deals with property and casualty insurance) is much more forthcoming with free syllabus materials. Here is a short study note on LC https://www.casact.org/library/studynotes/ExamS-Contingencies-Study-Note-0115.pdf

The biggest issue I see with how I am interpreting what you are describing is your desire/inability to deal with fractional results but your interest in applying a mortality rate against a single individual (such that you can “remove them from the population.”) Yes, in reality you are either alive or dead, but the likelihood of a single person dying at any given point is generally very small. This might be ok if your population subgroups are in the 10s or 100s of thousands such that removing one person results in a population reduction reasonably close to the q(x), but trying to convert a 3%, 10% or even 75% probability of a single individual dying to 0% or 100% doesn’t make sense and will significantly skew any results.

Yup, it does help to understand the purpose of the simulation.

In the meantime, here is an article I wrote some years back about simulating 2000 40-year-old (American) women, and the distribution for the number of deaths:

And the spreadsheet behind that one:

Oh, and while that’s a population where everybody is the same age & sex, there are aspects that can be used in there.

In specific, you’re trying to do a Monte Carlo simulation, but the issue is that with very young ages, mortality rates tend to be very low. When you do a Monte Carlo simulation with very low probabilities, you’re going to need to think how big your confidence intervals will need to be. There are a variety of ways to approach that, but one can also bootstrap to get an estimate.

@Kenny & @meep, thanks for your responses!

I think understanding the purpose of your “simulation”, the size of your total population as well as subgroups you want to adjust for, and your need for precision might be helpful in getting answers […]

I can’t give all of the details of the simulation, but I’ll try to give enough so that you have an understanding of the simulation. I think it’s sufficient to think of this simulation as simulating the financials for an enterprise from which a group of people have purchased an annuity. I believe this simulation would usually be done using numerical methods; however, for reasons I cannot get into, we want to be able to use the actual system to capture simulated outcomes, rather than simulating those outcomes in a separate system using numerical methods. Hopefully, this explanation helps.

In regards to needed precision. I don’t really know how to classify that. Right now it’s more important to be able to describe the degree of error than control it. My general experience has been that in such situations you can control the error through more computationally expensive simulations.

The populations will generally start with around 10,000 individuals, and get smaller over time as individuals die.

I’ll take a look at the material you posted @meep. Hopefully, that might allow me to describe things more accurately or ask more informed questions.

So this thread has nothing to do with assigning mortality factors to musical notes?

Yeah, you’re likely going to have an issue with a range of results. If you do a straightforward Monte Carlo simulation, where you generally have a very small annual mortality rate (q_x) for each until much older ages, 10K people may not be sufficient to get reasonably close to the “real” distribution of outcomes.

It’s more that you need to read about determining how many people/scenarios you need to try to capture whatever you’re trying to capture. If it’s not an average result, but a distribution or even tail results, you need even more trials then.

Here’s a simplified version for averages: https://www.valuationresearch.com/wp-content/uploads/kb/SpecialReport_MonteCarloSimulationTrials.pdf

Something more technical:

Anyway, while it’s okay just to do a “coin flip” with probability q_x that person age x dies in the one year of simulation, there are Monte Carlo techniques used to reduce the number of simulations you have to do to get your error bars sufficiently small. That gets complicated very quickly – stratified sampling, antithetic variates, importance sampling – and it really depends on what you consider your result (average, percentile, entire distribution of final outcomes) what sorts of approaches work best.

And if all of this gets overwhelming, you might consider hiring an actuary on a consulting basis. that gets the job done right away, since there’s only investigation and implementation, not learning. Probably there’s people right in this thread you could talk to.

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This.

And if all of this gets overwhelming […]

Not overwhelming, but many aspects of this are new and unfamiliar, but I like that, as it means I am learning new things.

[…] you might consider hiring an actuary on a consulting basis.

That is absolutely the long-term plan. Actually, having an actuary on staff will be necessary, as questions like this one—and more complicated ones—will come up on a regular basis. However, as we are a pre-seed stage startup, we can only offer equity options as compensation. With that said, while I am not the sole decision maker on such topics, if there are people on this thread who would be willing and interested in consulting under such an arrangement, please send me a private message, and we can see if an agreement can be reached.

Unfortunately, as is sometimes the nature with startups, there are some problems that must be at least addressed early on—even if the early solution is suboptimal and must be replaced. This is one of those situations.

You’d be better offering the task and a pay scale than requesting someone reach out to find you’re paying \$50/hour for actuarial consulting. I don’t imagine anyone will reach out without any sort of definition.

Probably not your doing, but startups seem to commonly think they can sell hype instead of cash for work provided. I have a startup, and I work (well, did until last march) in a business incubator full of tech startups. They often pay their staff like crap, and bill it as ‘we pay competitively’.
I’m not a fan. If you need a job done, pay for it. If you’ve got staff, pay them well and treat them well. Just because the business isn’t making money doesn’t excuse grinding the income of employees and consultants. that’s on the business owner, not the employees.
As a startup owner, those costs are mine to bare, and when I ‘exit’, the profits are mine to bare. Employess shouldn’t be taking on business risk, except as an added bonus.

yes, /rant :).
Speaking of startups, I’m over to the annoyed thoughts thread.

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So, my point is that yes, you can do a Monte Carlo with the initial approach you mention, but I highly recommend keep running scenarios until the results are stable to the order of magnitude you’re interested in. That’s the very non-technical way of saying “keep in mind that when you are trying to simulate low-probability events, you have to generate a LOT of them to get reliable results.”.

I don’t know exactly what you’re trying to do, but it is likely that you really need an expert in this type of modeling. It need not necessarily be a life actuary, but many actuaries do have this knowledge/skill set.

A different sort of rant from @SpaceLobster – companies have hired “data scientists” and statisticians to do number-crunching re: all sorts of things where said people have statistical technical skills but no knowledge of the underlying business problem. It can end up with absurdities in useless advice. The one that keeps sticking out in my mind was the useless info that policy year xxxx was extremely profitable… so you should get a time machine and write more year xxxx business.

Well, it’s allegedly an annuity problem, so the events (surviving) are actually high-probability.

Probably just need to Monte-Carlo each person’s sample mean age at death based on current age and gender, and calculate the sample standard deviation.
Gender-specific SOA Mortality tables by age should suffice.
How tough could it be?
(Disclaimer: I don’t think I’ve ever done this before.)
And if the annuities differ by person or by future year (say, they are inflation-based), yeah, throw that in as well.
And if you try to PV those annuities to get an overall mean PV and standard deviation, well, good luck.
I mean, sure, I might do it for, say, \$500/hour. But, note the disclaimer. Cash only, please. I will not give you my real name. Should take about 600 hours, so do the math. Do not look for me in Costa Rica afterward.

startups seem to commonly think they can sell hype instead of cash for work provided.

Unfortunately, there are a lot of over-hyped ideas and opportunistic behavior in the startup space. I don’t disagree with you there.

Just because the business isn’t making money doesn’t excuse grinding the income of employees and consultants. that’s on the business owner, not the employees.

I completely agree with you!

With that said, anyone who would be willing to work for equity should think of themselves as a business owner (which is in actuality what they would be), not an employee. There are too many facets to being an owner, not employee, to discuss here, but anybody contemplating doing this should evaluate not only the idea, but the partners they would be working with, and the degree to which they feel they would be made a partner and could be a partner. This is ultimately, an evaluation of the quality of people you are working with, which is more important than the idea.

You’d be better offering the task and a pay scale than requesting someone reach out to find you’re paying \$50/hour for actuarial consulting. I don’t imagine anyone will reach out without any sort of definition.

Partially for the reasons given above, I can’t imagine this would work well or be very enjoyable as a series of one-off tasks: there is accumulated knowledge that ideally needs to be threaded through the work. But I could be wrong about that.

In regards to rate, I’ve seen consulting rates of between \$150 and \$350/hour, and that range seems perfectly reasonable.

Thank you very much for all of the material you have shared. I’m still digesting the details and have many questions, but the material you have shared has given me an idea of where to start.

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