Building a Stochastic Mortality Model in R – A Beginner’s Guide

Ever wondered why life‑insurance premiums can jump from one year to the next, even when nothing seems to have changed? The answer often lies in the hidden randomness of mortality itself. If you can capture that randomness with a simple stochastic model, you’ll not only understand the numbers better, you’ll also be able to stress‑test your portfolios with confidence. Let’s walk through a step‑by‑step build in R that even a newcomer can follow.

Why Stochastic Mortality Matters

Traditional actuarial tables give us a single “best guess” for how many people will die at each age. That works fine for pricing a single policy, but it hides the fact that real life is noisy. A bad flu season, a new medical breakthrough, or even a sudden change in lifestyle can shift mortality rates dramatically. A stochastic model treats mortality as a random process, producing a range of possible outcomes instead of a single point estimate. This is the foundation for modern risk‑management tools such as capital adequacy calculations and longevity swaps.

Getting Your Toolbox Ready

Before we dive into code, make sure you have the following installed:

  • R (version 4.2 or newer)
  • RStudio (optional but helpful)
  • Packages: tidyverse, survival, actuar, ggplot2

You can install the packages with a single command:

install.packages(c("tidyverse","survival","actuar","ggplot2"))

I still remember the first time I tried to load actuar on a laptop that refused to cooperate. After a few frantic Googles and a cup of strong tea, the package finally compiled. The lesson? Patience and a good coffee are essential tools for any actuary.

Step 1 – Load and Inspect Your Data

For this tutorial we’ll use the classic lifetable dataset that comes with the actuar package. It contains observed deaths and exposures for a synthetic population.

library(actuar)
data(lifetable)
head(lifetable)

You should see columns like age, dx (deaths), and exposure. The key variable we need is the central death rate (often denoted µ). It is simply the number of deaths divided by the exposure at each age.

lifetable$mu <- lifetable$dx / lifetable$exposure

Step 2 – Choose a Stochastic Framework

There are many ways to inject randomness into mortality. A popular and easy‑to‑implement choice is the Lee‑Carter model with a stochastic time component. The model expresses the log of the death rate as:

log(mu_x,t) = a_x + b_x * k_t + ε_x,t
  • a_x captures the average shape of mortality across ages.
  • b_x measures how each age responds to changes over time.
  • k_t is the time index that we will treat as a random walk.
  • ε_x,t is a small error term.

For beginners, we’ll estimate a_x and b_x using ordinary least squares, then simulate k_t as a simple random walk with drift.

Step 3 – Estimate the Deterministic Part

First, reshape the data so that each column is an age and each row is a year. Our synthetic data only has one year, so we’ll create a fake time series by repeating the same rates for five years. This is just to illustrate the mechanics; in practice you would use real historical tables.

library(tidyverse)

# Create a fake 5‑year panel
panel <- lifetable %>%
  select(age, mu) %>%
  slice(1:20) %>%               # keep ages 0‑19 for brevity
  mutate(year = rep(2000:2004, each = 20)) %>%
  pivot_wider(names_from = age, values_from = mu)

# Take logs
log_mu <- log(as.matrix(panel %>% select(-year)))

Now run a singular‑value decomposition (SVD) to pull out the first singular vectors, which give us a_x and b_x.

svd_res <- svd(log_mu - rowMeans(log_mu))
a_x <- rowMeans(log_mu)
b_x <- svd_res$u[,1] * svd_res$d[1]
k_t <- svd_res$v[,1] * svd_res$d[1]

At this point you have the deterministic backbone of the Lee‑Carter model.

Step 4 – Model the Time Index Stochastically

We treat k_t as a random walk with drift:

k_{t+1} = k_t + d + σ * Z_t

where d is the average yearly change, σ is the volatility, and Z_t is a standard normal draw.

Estimate d and σ from the observed k_t series:

dk <- diff(k_t)
d_hat <- mean(dk)
sigma_hat <- sd(dk)

Now we can simulate future paths. Let’s generate 1,000 scenarios for the next 30 years.

set.seed(123)
n_scen <- 1000
n_years <- 30
k_future <- matrix(0, nrow = n_years, ncol = n_scen)

k_future[1,] <- k_t[length(k_t)] + d_hat + sigma_hat * rnorm(n_scen)

for (t in 2:n_years) {
  k_future[t,] <- k_future[t-1,] + d_hat + sigma_hat * rnorm(n_scen)
}

Step 5 – Reconstruct Mortality Rates

Using the simulated k_t values, we rebuild the death rates for each age and each scenario.

# Create a matrix of ages (0‑19) repeated for each scenario
age_vec <- lifetable$age[1:20]
a_mat <- matrix(a_x[1:20], nrow = n_years, ncol = n_scen, byrow = TRUE)
b_mat <- matrix(b_x[1:20], nrow = n_years, ncol = n_scen, byrow = TRUE)

# Expand k_future to match dimensions
k_mat <- matrix(k_future, nrow = n_years, ncol = n_scen)

log_mu_sim <- a_mat + b_mat * k_mat
mu_sim <- exp(log_mu_sim)

Now mu_sim holds a full stochastic mortality surface: rows are future years, columns are scenarios, and each cell contains the death rate for a given age.

Step 6 – Visualize the Uncertainty

A quick plot of the median and a 95 % confidence band gives a feel for the range of outcomes.

library(ggplot2)

# Summarize across scenarios
summary_df <- data.frame(
  year = 2025 + 1:n_years,
  median = apply(mu_sim, 1, median),
  lower = apply(mu_sim, 1, function(x) quantile(x, 0.025)),
  upper = apply(mu_sim, 1, function(x) quantile(x, 0.975))
)

ggplot(summary_df, aes(x = year)) +
  geom_line(aes(y = median), colour = "steelblue") +
  geom_ribbon(aes(ymin = lower, ymax = upper), fill = "lightblue", alpha = 0.4) +
  labs(title = "Projected Mortality for Age 65 (Stochastic)",
       y = "Central Death Rate (µ)",
       x = "Year") +
  theme_minimal()

The ribbon shows the spread of possible futures. If you’re comfortable with R, you can replace the single age with a whole age vector and create heat‑maps or surface plots.

Step 7 – Put It to Work

What can you do with this model? A few ideas:

  • Pricing: Run a Monte‑Carlo simulation of policy cash flows using the stochastic rates, then discount at the appropriate risk‑free rate.
  • Capital Allocation: Compute Value‑at‑Risk (VaR) or Tail‑Value‑At‑Risk (TVaR) for a portfolio of life policies.
  • Scenario Analysis: Compare a “best‑case” low‑mortality path against a “worst‑case” high‑mortality path to see how sensitive your reserves are.

Remember, the model is only as good as the data and assumptions you feed it. Always validate against out‑of‑sample observations and be ready to adjust the drift or volatility if the market changes.

Common Pitfalls and How to Avoid Them

  1. Over‑fitting the deterministic part – Using too many singular vectors can capture noise rather than signal. Stick with the first component unless you have a strong reason to add more.
  2. Ignoring cohort effects – The basic Lee‑Carter model assumes only period effects. If you suspect a generation‑specific trend (e.g., a cohort that smoked heavily), consider extending the model with a cohort term.
  3. Forgetting to back‑test – Simulate a few years back in time and compare the model’s output to actual observed mortality. Large discrepancies are a red flag.

A Personal Note

When I first built a stochastic mortality model for my graduate thesis, I spent weeks wrestling with matrix dimensions. One night, after a marathon of debugging, I realized I had transposed a matrix by accident. The resulting plot showed mortality rates that decreased with age—clearly absurd! That mistake taught me the value of sanity checks: always plot a simple slice of your data before trusting the whole simulation.

Building a stochastic model in R is a rewarding exercise. It forces you to confront the uncertainty that underlies every actuarial decision, and it gives you a sandbox where you can test ideas without risking real capital. I hope this step‑by‑step guide helps you take that first confident step.

#actuarial#mortality#r
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