Deriving Mortality & Morbidity Assumptions from Experience Data

Introduction

In our previous lesson, we surveyed the various data sources available to an actuary, from a company's own experience to broad population statistics. We established that the core challenge lies in balancing the relevance of the data against its statistical credibility. Now, we move from the source of the data to the process of transforming it into a core actuarial assumption.

This lesson focuses on the methods for deriving mortality and morbidity assumptions. This is a two-stage process that should align well with your statistical background. First, we'll explore graduation, the statistical technique of smoothing raw historical data to uncover the underlying pattern of mortality. Second, we will address how to make an allowance for future trends, projecting these historical patterns forward in time. By the end of this lesson, you will understand the practical and theoretical steps involved in constructing a robust mortality or morbidity basis.

Part 1: Graduation – From Raw Data to a Baseline

Crude mortality rates, calculated simply as deaths divided by exposure, are often volatile due to random statistical fluctuations, especially at ages with little data. Graduation is the process of smoothing these crude rates to produce a reliable underlying set of mortality rates that are free from this random noise. Think of it as a signal processing problem: the crude rates are a noisy signal, and graduation is the filter we apply to uncover the true, underlying signal.

There are several ways to perform this "filtering," but they generally fall into two categories: fitting a mathematical formula or applying a smoothing algorithm.

Graduation by Mathematical Formula (Parametric Approach)

Given your background in statistics, you'll recognize this approach as a form of non-linear regression. The idea is to fit a specific mathematical function, known as a "mortality law," to the observed data. This is a powerful technique because it summarizes the entire mortality pattern across a wide range of ages with just a few parameters.

A comprehensive guide from the Society of Actuaries, "A Practitioner's Guide to Statistical Mortality Graduation," provides an excellent statistical framework for this process. We will use it to explore the key concepts.

[PDF] A Practitioner's Guide to Statistical Mortality Graduation - SOA.org

This paper details the statistical theory and practice behind graduating mortality data using mathematical formulae, a method that is both robust and intuitive.

Please read the following sections to understand this powerful technique: Introduction: Begin with the introduction. This section frames the task as a modeling exercise and introduces the idea of fitting a simple parametric model to complex mortality data. A Statistical Approach to Graduation: Read this section. It outlines the statistical foundation. As you read, focus on how it formalizes the problem: The assumption that death counts follow a Poisson distribution. The use of Maximum Likelihood Estimation (MLE) to find the best-fitting parameters for the model. The use of information criteria like AIC and BIC to compare different models, which balances goodness-of-fit with model complexity. Generalizing the Functional Form: Now, read the section on generalizing the form. This introduces the "mortality laws" themselves. Pay attention to the common forms: Gompertz Law: A simple exponential growth model. Makeham Law: Adds a constant to the Gompertz law, often interpreted as an accident-related component of mortality. Perks and Makeham-Beard Laws: Logistic forms that better capture the slowing of mortality growth at very high ages, which can be linked to population heterogeneity or "frailty".

As the paper highlights, the parametric approach is elegant. It turns a noisy data problem into a regression exercise where we estimate a few meaningful parameters (e.g., a baseline mortality level, the rate of increase with age). This provides a smooth curve and allows for intuitive extrapolation to ages with sparse data.

The paper also briefly mentions other methods like P-splines and Whittaker-Henderson smoothing (Section 7). These are non-parametric approaches that offer more flexibility to fit complex patterns but come at the cost of having more parameters and less inherent explanatory power. For the CP1 exam, it is sufficient to know that these alternatives exist and that graduation by mathematical formula is a primary and powerful method.

Part 2: Allowance for Future Trends

The graduated table you produce represents the best estimate of mortality based on past data. However, mortality rates are not static; they change over time, a phenomenon known as mortality improvement. For long-term products like annuities or life assurance, failing to account for future improvements can lead to significant under- or over-reserving.

Historically, mortality rates have been falling, meaning people are living longer. The process of setting an assumption for future mortality improvements involves analysing these historical trends and making a judgment about how they will continue.

A thought-provoking paper from Willis Towers Watson (WTW) critiques simplistic extrapolation and proposes a more structured framework for thinking about future mortality.

[PDF] A new framework for future mortality - WTW

This paper challenges actuaries to think more deeply about the drivers of future mortality, rather than blindly extrapolating past trends. It provides a robust conceptual framework for setting improvement assumptions.

Please read the following sections to understand this forward-looking approach: A New Framework: Start with the section "Different time horizon, different approach". This introduces the core idea of splitting the future into three distinct periods. Short-term trends: Read the section on short-term trends. Note that this period is often based on an extrapolation of recent experience. Medium-term improvements: Read the section on medium-term improvements. This is where a driver-based approach becomes crucial, considering factors like medical pipelines, disease prevalence, and lifestyle changes. Long-term assumption setting: Finally, read the section on the long-term view. This is the most subjective element, relying on expert judgment about fundamental drivers like economic growth and healthcare spending.

The WTW framework highlights that setting a future trend assumption is not a single calculation but a structured process of judgment. It combines:

Extrapolative models (like the CMI models common in the UK), which are most useful for the short term.
Explanatory or driver-based models, which attempt to link mortality to its underlying causes (e.g., changes in smoking rates, obesity, or medical technology) and are powerful for the medium term.
Expert judgment, which is indispensable for the long term, where historical trends may no longer be a reliable guide.

Deriving Morbidity Assumptions

The principles for deriving morbidity (sickness or disability) assumptions are similar to those for mortality: you start with historical data, graduate it to remove random noise, and then make an allowance for future trends. However, morbidity is significantly more complex for several reasons:

Multiple States: Mortality is a simple binary transition (alive to dead). Morbidity involves multiple states (e.g., healthy, temporarily sick, permanently disabled, recovered) and multiple transition probabilities (incidence, recovery, mortality from a sick state).
Data Scarcity and Quality: Reliable, long-term morbidity data is often harder to obtain than mortality data.
Definition Dependency: Morbidity rates are highly sensitive to the definition of "disabled" or "sick," which can change over time due to medical advances, social norms, or policy wording.
Subjectivity: Claims for conditions like stress or back pain can be influenced by economic conditions (e.g., unemployment rates) and the general attitude of doctors and claimants.

A concept that bridges mortality and health status is select mortality. When an individual undergoes a medical examination for an insurance policy, they are "selected" as being in good health. For a period after this selection (the "select period"), their mortality is expected to be lower than that of the general population of the same age.

Chapter 3 The Life Table

This video from MJ the Fellow Actuary provides a simple introduction to life table concepts. We'll watch a short clip on select mortality, which directly connects an individual's health status to their mortality assumption.

Please watch the clip from this discussion on select mortality. Notice how the assumption depends not just on age, but also on the duration since underwriting. This illustrates how health information is explicitly incorporated into an assumption.

After the select period, the effect of the initial underwriting is assumed to wear off, and the individual's mortality merges back into the "ultimate" mortality rate for the general insured population. This same principle of selection and its wearing-off effect is fundamental to modelling morbidity.

Conclusion

Deriving mortality and morbidity assumptions from experience data is a sophisticated statistical exercise. It is a two-step process that moves from analysing the past to projecting the future. The first step, graduation, uses statistical techniques to smooth raw data and establish a baseline table of current rates. The second step, projection, involves making a reasoned allowance for future trends, considering short, medium, and long-term drivers of change.

While morbidity follows the same principles, it presents greater challenges due to its multi-state nature and the higher subjectivity and scarcity of data. The key takeaway is that setting these assumptions is not about finding a single "correct" number but about applying a robust, documented, and statistically sound process to blend data, models, and expert judgment.

In our next lesson, we will shift our focus to another critical non-financial assumption: how to set and maintain expense assumptions for a financial product.

Key Takeaways & Cheatsheet

Here is a framework summarizing the process of deriving a mortality assumption.

Phase	Step 1: Graduation (Analysing the Past)	Step 2: Projection (Forecasting the Future)
Goal	To remove random noise from historical data and find the underlying mortality pattern.	To forecast how the baseline mortality rates will change over time.
Input	Crude rates ( $d_{x} / E_{x}$ ) from own, industry, or population data.	Baseline mortality table from Step 1, historical trend data, driver analysis, expert opinion.
Methods	Parametric: Fit a mathematical "mortality law" (e.g., Gompertz, Makeham) using statistical methods like MLE. Non-Parametric: Apply a smoothing algorithm (e.g., P-splines, Whittaker-Henderson).	Extrapolative: Project past improvement trends forward (e.g., using CMI models). Explanatory: Model the impact of underlying drivers (e.g., medical advances, smoking). Structured Judgment: Use a framework (e.g., Short/Medium/Long-term views).
Pros of Method	Parametric: Few parameters, interpretable, good for extrapolation. Non-Parametric: Highly flexible to fit complex data shapes.	Extrapolative: Objective and simple. Explanatory: More robust to changes in trend, better captures causality.
Cons of Method	Parametric: Can be too rigid if the true shape is unusual. Non-Parametric: Many parameters, risk of overfitting, less explanatory power.	Extrapolative: Can fail when past trends do not continue. Explanatory: Complex and data-intensive.
Output	A baseline (or static) mortality table representing current mortality ( $q_{x}$ ).	A set of improvement factors or a dynamic mortality table that varies by age and time ( $q_{x, t}$ ).

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