Graduating Mortality Rates: Parametric Formulas

The Logic of Parametric Mortality Laws

Parametric graduation imposes a specific mathematical shape on the mortality curve. The idea is to assume that the true, underlying force of mortality, ${\mu }_x$, follows a particular function of age $x$, but with some unknown parameters. We then use the observed data (deaths and exposures) to estimate the values of these parameters.

The general form of one of the most famous mortality laws, the Makeham law, is shown in the image below.

This structure is appealing because it has a clear biological and practical interpretation, which we will now explore in more detail.

Classic Mortality Laws: Gompertz and Makeham

The earliest actuaries and demographers sought to find a "law of mortality" that could describe the pattern of human death. Two of the most enduring results of this search are the Gompertz and Makeham formulas.

1. Gompertz Law (1825)

Benjamin Gompertz observed that, for adult ages, the force of mortality increases at an exponential rate. This implies that the logarithm of the force of mortality is a linear function of age.

The Gompertz law is expressed as:
${\mu }_x=B{c}^x$
where $B>0$ and $c>1$ are parameters.

From your statistical background, you will recognize that taking the logarithm transforms this into a linear model:
$\ln ({\mu }_x)=\ln (B)+x\ln (c)=\alpha +\beta x$
This log-linear relationship captures the rapid acceleration of mortality risk at older ages, often attributed to biological senescence.

2. Makeham's Law (1860)

William Makeham extended Gompertz's work by adding a constant term to account for causes of death that are independent of age, such as accidents.

Makeham's law is given by:
${\mu }_x=A+B{c}^x$
where $A>0$ is the new parameter representing this constant, age-independent force of mortality. The term $A$ is often called the "Makeham constant," while $B{c}^x$ is the "Gompertz component."

The addition of the $A$ parameter often provides a much better fit to mortality data, especially at younger and middle adult ages where accidental deaths are a more significant proportion of total mortality.

The image below illustrates how the shapes of these curves change as the parameters are varied. In these plots, "ETA" corresponds to the Makeham constant $A$, and "ZETA" is related to the Gompertz parameter $c$.

Fitting Parametric Models to Data

Now for the central task: how do we find the best values for the parameters (e.g., $A$, $B$, and $c$ in Makeham's law) given our crude data of deaths $d_x$ and exposures $E_x^c$? The standard approach is Maximum Likelihood Estimation (MLE).

As we touched upon in a previous module, we model the number of deaths at each age, $D_x$, as a random variable following a Poisson distribution:
$D_x\sim \text{Poisson}({\lambda }_x)\text{where}{\lambda }_x=E_x^c\cdot {\mu }_x$
In parametric graduation, we replace ${\mu }_x$ with our chosen formula. For example, under Makeham's law, the model becomes:
$D_x\sim \text{Poisson}(E_x^c\cdot (A+B{c}^x))$
The goal is to find the parameters $\theta =(A,B,c)$ that maximize the likelihood of observing our actual death counts, $d_x$. This is done by maximizing the log-likelihood function, which, assuming independence across ages, is:
$l(\theta )={\sum }_x\left[d_x\ln ({\mu }_x(\theta ))+d_x\ln (E_x^c)-E_x^c{\mu }_x(\theta )-\ln (d_x!)\right]$
Since the terms $d_x\ln (E_x^c)$ and $\ln (d_x!)$ do not depend on $\theta$, maximizing the full log-likelihood is equivalent to maximizing the simplified kernel:
$l(\theta )\propto {\sum }_x\left[d_x\ln ({\mu }_x(\theta ))-E_x^c{\mu }_x(\theta )\right]$
This maximization problem does not have a closed-form solution and is solved using numerical optimization algorithms.

The following reading provides a detailed walkthrough of this statistical framework.

Generalizations and Model Selection

The Gompertz and Makeham laws are simple and effective, but sometimes mortality data exhibits more complex shapes. To handle this, actuaries use more flexible families of curves. A widely used extension is the Generalized Gompertz-Makeham (GM(r,s)) family, which replaces the constant and linear terms with polynomials:

${\mu }_x=\left({\sum }_{i=0}^{r-1}{a}_i{x}^i\right)+\exp \left({\sum }_{j=0}^{s-1}{b}_j{x}^j\right)$

• Gompertz is a GM(0,2) model: ${\mu }_x=\exp (b_0+b_{1}x)$.
• Makeham is a GM(1,2) model: ${\mu }_x=a_0+\exp (b_0+b_{1}x)$.

This flexibility creates a new problem: which model should we choose? A GM(1,3) model has four parameters and will almost certainly fit the data better than a two-parameter Gompertz model. But is the improvement in fit genuine, or are we just overfitting to noise?

This is a classic model selection problem. As you know from your statistics training, we can use information criteria like Akaike's Information Criterion (AIC) and the Bayesian Information Criterion (BIC) to make this decision. These criteria balance goodness-of-fit (measured by the maximized log-likelihood) against model complexity (the number of parameters).

• AIC: $-2l(\hat{\theta })+2K$
• BIC: $-2l(\hat{\theta })+K\ln (N)$

Where $l(\hat{\theta })$ is the maximized log-likelihood, $K$ is the number of parameters, and $N$ is the number of data points (ages). The model with the lower AIC or BIC is preferred.

The following reading demonstrates this entire process, from defining various laws to fitting them and using AIC/BIC for selection.

Conclusion

In this lesson, you have learned the first major technique for graduation: fitting parametric formulas. We have seen how historical laws like Gompertz and Makeham provide an elegant and interpretable structure for modeling mortality, and how they can be generalized to fit more complex patterns.

Key Takeaways:

• Parametric Graduation: This method assumes the force of mortality follows a specific mathematical formula (e.g., Gompertz, Makeham) whose parameters are estimated from data.
• Classic Laws:
  • Gompertz: ${\mu }_x=B{c}^x$, models an exponential increase in mortality with age.
  • Makeham: ${\mu }_x=A+B{c}^x$, adds an age-independent component of risk.
• Fitting Process: Parameters are estimated using Maximum Likelihood Estimation, typically by assuming the number of deaths at each age follows a Poisson distribution.
• Model Selection: When comparing multiple parametric models (e.g., Gompertz vs. Makeham vs. a more complex GM(r,s) model), we use AIC or BIC to find the best balance between adherence to data and model simplicity.

Preview of the Next Lesson

While powerful, parametric graduation has a key limitation: it forces the mortality curve into a predefined shape. What if the true mortality pattern doesn't conform to any simple formula? In our next lesson, we will explore an alternative approach: graduation by reference to a standard table. This semi-parametric method offers more flexibility by assuming that our population's mortality is a simple modification of a well-established, pre-existing mortality table.