Non-linear actuarial models

Most actuarial work involves linear extrapolation of past experience, adjusted by actuarial judgement to allow for expectations of future experience.

Historical experience is analysed and a range of economic and demographic assumptions are selected.

It is often assumed (explicitly and/or implicitly) that the past is a suitable indicator of the future, with adjustments to past trends based on actuarial judgement being applied. The standard actuarial valuation approach becomes problematic when there are significant structural discontinuities.

Structural discontinuities can arise from external or internal force.omenological versus Axiomatic Modelli

**Phenomological versus axiomatic modelling **

In contrast to the actuarial emphasis on analysis of experience, economists often approach modelling from a more theoretical basis, deriving underlying principles which are used to project future experience.

Such axiomatic approaches assume that the world is too complex to model and distil the essential aspects of the system being studied into a number of assumptions from which conclusions can be derived.

For example, the Poisson distribution, which is commonly used to model claim numbers in general insurance, can be derived from three axioms, and it also can be developed using a heuristic approach.

No model is purely phenomenological or axiomatic, and "actuaries are primarily practitioners" (Huber & Verrall), more concerned with specific singular outcomes than over-arching theories, and tend to rely on phenomenological approaches with an underlying belief that professional actuarial judgement is often more accurate than simplifying economic models.

However, as Huber and Verrall point out, judgement while necessary should be exercised with caution, and "is most effective when used in conjunction with mathematical models".

In recent years, many actuaries have employed stochastic approaches for projecting future cash flows. With stochastic modelling, the exercise of actuarial judgement falls not to the selection of assumptions but to the choice of the statistical distribution from which random selections of assumptions are generated.

Each (randomly generated) selection determines a scenario (set of assumptions), and key outputs, such as future profit or solvency level, are produced for each scenario. The statistical distributions from which variables are selected are often derived from historical analysis.

The validity of data-based methods, deterministic or stochastic, is hard to assess as there is often no independent data to check results against. As Huber and Verrall point out, this is related to Hume's problem of induction, which asserts that "empirical evidence cannot be used to establish the truth of claims about the future".

**Indivisibility of the actuarial basis• Indivisibility of the Actuarial Basis **

When modelling in a dynamic environment where the future is hard to predict, the actuary may resort to experience and select assumptions by applying actuarial judgement based on a collateral source of data. Credibility theory provides a technique often employed for blending collateral and particular data (Hossack et. al.)

Such an approach requires judgement about the appropriateness of the collateral data, and as discussed above judgement should be exercised with caution and is best grounded in solid theory (Huber and Verrall).

In addition, there are technical difficulties with setting assumptions piece-wise based on a synthesis of actual data, experience and judgement.

A fundamental concept in actuarial modelling which may work against the applicability of such a phenomenological approach is the indivisibility of the actuarial basis.

This indivisibility means that the methodology employed and the assumptions used are part of a whole and any one assumption cannot be considered in isolation from other assumptions or the mechanism by which they are incorporated into a model.

This is analogous to the Duhein-Quine thesis from economic theory, which states that "factual statements only have meaning within systems of statements, so they can only be tested holistically" (Huber and Verrall).

For example, when projecting the future underwriting experience of an insurer, explicit or implicit assumptions need to made about future expense rates and loss ratios.

It may not be appropriate to reflect management assertions (perhaps supported by data) that the insurer intends to select better risks by adjusting downwards expected future loss ratios, without considering (for example) the possible expense increases arising from more selective underwriting.

Similarly, when projecting net of insurance results, it is not uncommon for exchange commission (amounts paid actually or notionally by a reinsurer to a direct insurer in recognition of the direct insurer's costs) to be modelled as a reduction in commission rates or acquisition expenses. This means that if it is assumed that retention rates will change in the future, commission/acquisition rates/expenses must also be assumed to change.

It is clear that economic systems are complex and non-linear, and that we are not able to specify precisely how system parameters are related.

The tools developed to model non-linear systems may provide opportunities to develop theoretically sound models which are consistent with both historical data and future possible outcomes.

The usual valuation method employed by actuaries is a data-driven approach which relies on the measurement of some outputs of the system where those outputs can be considered to be a solution to the equations (which we do not know) governing the system.

The system being modelled is a black box, and we use measurements of the output of the system to find the states of the system as some of the control parameters vary.

The actuarial approach often presumes that the system being modelled is in a stable steady state (after normalisation), which may be hard to describe precisely because of "noise", errors of measurement and inaccurate data rather than because it is intrinsically indefinable. Non-linear mathematics show that some systems may be intrinsically indefinable.

Non-linear analysis has developed methods which facilitate recognition of equilibrium points and the classification of systems into chaotic and non-chaotic states. Predictive propensities have much greater certainty in non-chaotic states.

For example, future projections of cash flows are dependent on the starting point of the projection. Non-linear analysis has developed ways of quantifying sensitivity to initial conditions, commonly known as Lyapounov exponents. Such techniques could be applied in actuarial modelling to assess the veracity of projections and the sensitivity of results to assumptions.

Scatter diagrams, a concept familiar from statistics, can provide useful information about dynamic systems. A phase plane can be plotted, with a measurable system variable d(t) as the abscissa and d(t + i ) as ordinate, where t is an independent variable and i a suitable positive number.

The value i must fall within a range so that d(t + i) is not closely correlated with d(t) (i too small), nor independent of d(t) (i too large). Such analyses can indicate whether or not the system being modelled is in a chaotic state.

In general, a (continuous) economic model can be considered to have the form dx/dt = F(a,x), where F is a function from R(l dimensions) x R (m dinemsions) into R (m dimensions,where **l, m, a, x ** and **F ** are unknown, and **x **cannot be measured directly. (Most economic models are discrete, and measurable outputs from the underlying system discrete rather than continuous variables. The real world has both discrete and continuous components.)

We are able to measure p data of the form **d(a,t) = g(a, x(t)) **at certain points of time for some unknown function g: **R( 1) x R (m) --> R (p) **. The challenge is to invert, i.e. find **F **from **d(a,t) **.

A lot of actuarial work relates to reserving for business about which some details are known or can be reliably estimated from past experience. M&As require an opinion on business to be written in the future about which a lot less is known, especially where significant structural changes have taken place recently or are expected to take place. The very nature of commercial transactions imply radical change may take place upon purchase.

Generalised Linear Models, which allow for non-linear relationships between system variables, are being widely used by actuaries. GLMs can be applied to both premium rating and reserving (Fry and Smith), two applications which in combination are similar to the modelling required for M&As. Conditional distributions and Bayesian approaches also provide techniques which may be useful where there is a lack of reliable experience.

In summary, the phenomenological approach and linear modelling techniques currently employed by actuaries in M&As can be augmented by models from non-linear mathematics to put actuarial analysis on a more theoretically sound basis and more closely reflect the complexity inherent in real world systems.