Term life simulator notes

[MatthewCaseres]

Trying to come up with some notes that can serve as a reference for @serenity4. Tagging @alecloudenback in case he has input on this.

Motivations

Similar to this idea - https://github.com/mattheaphy/nyr2022_exp_study/blob/main/nyr2022_slides.pdf
Below is a knowledge dump of sorts. The main goal is just an end to end example of the actuarial modeling process.

Why simulated data makes sense

Actuaries want to build data driven lapse/mortality assumptions for financial models of term life insurance. The true distribution of the present value of net cashflows would be nice. In the real world we only get to observe the one outcome that actually occurs, and metrics in existing literature are things like "Actual/Expected" (which I believe tells us if our model matches the data it was trained on, and of course it always does).

With a data simulator, we can assess approaches to actuarial modeling while having access to the full underlying distribution of things, which is never the case in the real world.

How to use simulated data

1. Some configurations of the data simulator are set (lapse/mortality models etc.)
2. The data simulator is run and generates some data until we reach "the current date"
3. We use our knowledge of this data to make a guess about the probability distribution of future aggregate cashflows
4. The data simulator is able to run Monte Carlo simulations forward from the current date to approximate the true distribution of future aggregate cashflows

In the next section, we describe a relatively simple implementation. The goal is to use this simple implementation as described in the list above.

Modeling details

Mechanics

• A policy can leave due to death or lapse.
• Payments can happen annually, monthly, or quarterly.
  • We will likely use monthly timesteps even if maybe we can do annual timesteps. We will likely have constant force of decrement between timesteps and a uniform distribution of decrement within timesteps because I believe this is the easiest way to implement.
  • Monthly means starting_date + n * months_passed and not `(...((starting_date + Month(1)) + Month(1)) ....)
• Lapses happen on the date of a missed payment (in the real world there is something called a grace period, we ignore this for now)
• The probability of lapse is determined by a lapse rate model which takes the form of a logistic regression model. This logistic regression model is a function like f(duration, gender, face_amount, ...)
• Deaths happen on any date.

Assumptions

Mortality rates

• Use the 2017 Unloaded CSO Preferred Structure tables from MortalityTables.jl.
• The docs are good and there is some example in the Benchmarks here.
• The benchmarks use range 3299:3308 and we will use range 3341:3350

Lapse rates

• We are going to make up some toy model for lapse rates. It will be a logistic regression, as this is used in the 2021 U.S. Post-Level Term Lapse and
  Mortality Predictive Modeling report.

Post level term (PLT)

https://www.theactuarymagazine.org/better-with-age/#:~:text=Post%2Dlevel%20term%20(PLT),the%20premium%20will%20rise%20annually.

There are two terms

• Shock lapse (when the lapse rate increases after prices are no longer flat)
• Mortality deterioration (the people willing to pay the higher prices are not the healthiest as a group)

So higher prices -> higher lapses -> higher mortality -> higher prices. We will make no attempt to model some sort of infinite feedback loop.

We will be using the Dukes/MacDonald method 2 from this newsletter.

• Mortality in the PLT is a function of excess lapses
• excess lapses are a function of the premium increases
• Premium increases are hard-coded up front, just like in the real world the prices are known ahead of time