Stochastic modelling (insurance)

Last updated February 11, 2024

This page is concerned with the stochastic modelling as applied to the insurance industry. For other stochastic modelling applications, please see Monte Carlo method and Stochastic asset models. For mathematical definition, please see Stochastic process.

"Stochastic" means being or having a random variable. A stochastic model is a tool for estimating probability distributions of potential outcomes by allowing for random variation in one or more inputs over time. The random variation is usually based on fluctuations observed in historical data for a selected period using standard time-series techniques. Distributions of potential outcomes are derived from a large number of simulations (stochastic projections) which reflect the random variation in the input(s).

Valuation

Like any other company, an insurer has to show that its assets exceeds its liabilities to be solvent. In the insurance industry, however, assets and liabilities are not known entities. They depend on how many policies result in claims, inflation from now until the claim, investment returns during that period, and so on.

So the valuation of an insurer involves a set of projections, looking at what is expected to happen, and thus coming up with the best estimate for assets and liabilities, and therefore for the company's level of solvency.

Deterministic approach

The simplest way of doing this, and indeed the primary method used, is to look at best estimates.

The projections in financial analysis usually use the most likely rate of claim, the most likely investment return, the most likely rate of inflation, and so on. The projections in engineering analysis usually use both the most likely rate and the most critical rate. The result provides a point estimate - the best single estimate of what the company's current solvency position is, or multiple points of estimate - depends on the problem definition. Selection and identification of parameter values are frequently a challenge to less experienced analysts.

The downside of this approach is it does not fully cover the fact that there is a whole range of possible outcomes and some are more probable and some are less.

Stochastic modelling

A stochastic model would be to set up a projection model which looks at a single policy, an entire portfolio or an entire company. But rather than setting investment returns according to their most likely estimate, for example, the model uses random variations to look at what investment conditions might be like.

Based on a set of random variables, the experience of the policy/portfolio/company is projected, and the outcome is noted. Then this is done again with a new set of random variables. In fact, this process is repeated thousands of times.

At the end, a distribution of outcomes is available which shows not only the most likely estimate but what ranges are reasonable too. The most likely estimate is given by the distribution curve's (formally known as the Probability density function) center of mass which is typically also the peak(mode) of the curve, but may be different e.g. for asymmetric distributions.

This is useful when a policy or fund provides a guarantee, e.g. a minimum investment return of 5% per annum. A deterministic simulation, with varying scenarios for future investment return, does not provide a good way of estimating the cost of providing this guarantee. This is because it does not allow for the volatility of investment returns in each future time period or the chance that an extreme event in a particular time period leads to an investment return less than the guarantee. Stochastic modelling builds volatility and variability (randomness) into the simulation and therefore provides a better representation of real life from more angles.

Numerical evaluations of quantities

Stochastic models help to assess the interactions between variables, and are useful tools to numerically evaluate quantities, as they are usually implemented using Monte Carlo simulation techniques (see Monte Carlo method). While there is an advantage here, in estimating quantities that would otherwise be difficult to obtain using analytical methods, a disadvantage is that such methods are limited by computing resources as well as simulation error. Below are some examples:

Means

Using statistical notation, it is a well-known result that the mean of a function, f, of a random variable X is not necessarily the function of the mean of X.

For example, in application, applying the best estimate (defined as the mean) of investment returns to discount a set of cash flows will not necessarily give the same result as assessing the best estimate to the discounted cash flows.

A stochastic model would be able to assess this latter quantity with simulations.

Percentiles

This idea is seen again when one considers percentiles (see percentile). When assessing risks at specific percentiles, the factors that contribute to these levels are rarely at these percentiles themselves. Stochastic models can be simulated to assess the percentiles of the aggregated distributions.

Truncations and censors

Truncating and censoring of data can also be estimated using stochastic models. For instance, applying a non-proportional reinsurance layer to the best estimate losses will not necessarily give us the best estimate of the losses after the reinsurance layer. In a simulated stochastic model, the simulated losses can be made to "pass through" the layer and the resulting losses assessed appropriately.

The asset model

Although the text above referred to "random variations", the stochastic model does not just use any arbitrary set of values. The asset model is based on detailed studies of how markets behave, looking at averages, variations, correlations, and more.

The models and underlying parameters are chosen so that they fit historical economic data, and are expected to produce meaningful future projections.

There are many such models, including the Wilkie Model, the Thompson Model and the Falcon Model.

The claims model

The claims arising from policies or portfolios that the company has written can also be modelled using stochastic methods. This is especially important in the general insurance sector, where the claim severities can have high uncertainties.

Frequency-Severity models

Depending on the portfolios under investigation, a model can simulate all or some of the following factors stochastically:

Number of claims
Claim severities
Timing of claims

Claims inflations can be applied, based on the inflation simulations that are consistent with the outputs of the asset model, as are dependencies between the losses of different portfolios.

The relative uniqueness of the policy portfolios written by a company in the general insurance sector means that claims models are typically tailor-made.

Stochastic reserving models

Estimating future claims liabilities might also involve estimating the uncertainty around the estimates of claim reserves.

See J Li's article "Comparison of Stochastic Reserving Models" (published in the Australian Actuarial Journal, volume 12 issue 4) for a recent article on this topic.

Related Research Articles

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. The name comes from the Monte Carlo Casino in Monaco, where the primary developer of the method, physicist Stanislaw Ulam, was inspired by his uncle's gambling habits.

Financial economics is the branch of economics characterized by a "concentration on monetary activities", in which "money of one type or another is likely to appear on both sides of a trade". Its concern is thus the interrelation of financial variables, such as share prices, interest rates and exchange rates, as opposed to those concerning the real economy. It has two main areas of focus: asset pricing and corporate finance; the first being the perspective of providers of capital, i.e. investors, and the second of users of capital. It thus provides the theoretical underpinning for much of finance.

Actuarial science is the discipline that applies mathematical and statistical methods to assess risk in insurance, pension, finance, investment and other industries and professions. More generally, actuaries apply rigorous mathematics to model matters of uncertainty and life expectancy.

Mathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure theory.

Monte Carlo methods are used in corporate finance and mathematical finance to value and analyze (complex) instruments, portfolios and investments by simulating the various sources of uncertainty affecting their value, and then determining the distribution of their value over the range of resultant outcomes. This is usually done by help of stochastic asset models. The advantage of Monte Carlo methods over other techniques increases as the dimensions of the problem increase.

Network traffic simulation is a process used in telecommunications engineering to measure the efficiency of a communications network.

In mathematical finance, a Monte Carlo option model uses Monte Carlo methods to calculate the value of an option with multiple sources of uncertainty or with complicated features. The first application to option pricing was by Phelim Boyle in 1977. In 1996, M. Broadie and P. Glasserman showed how to price Asian options by Monte Carlo. An important development was the introduction in 1996 by Carriere of Monte Carlo methods for options with early exercise features.

Catastrophe modeling is the process of using computer-assisted calculations to estimate the losses that could be sustained due to a catastrophic event such as a hurricane or earthquake. Cat modeling is especially applicable to analyzing risks in the insurance industry and is at the confluence of actuarial science, engineering, meteorology, and seismology.

Dynamic financial analysis (DFA) is a simulation approach that looks at an insurance enterprise's risks holistically as opposed to traditional actuarial analysis, which analyzes risks individually. Specifically, DFA reveals the dependencies of hazards and their impacts on the insurance company's financial well being such as business mix, reinsurance, asset allocation, profitability, solvency, and compliance.

In insurance, an actuarial reserve is a reserve set aside for future insurance liabilities. It is generally equal to the actuarial present value of the future cash flows of a contingent event. In the insurance context an actuarial reserve is the present value of the future cash flows of an insurance policy and the total liability of the insurer is the sum of the actuarial reserves for every individual policy. Regulated insurers are required to keep offsetting assets to pay off this future liability.

The following outline is provided as an overview of and topical guide to finance:

Loss reserving is the calculation of the required reserves for a tranche of insurance business, including outstanding claims reserves.

A stochastic simulation is a simulation of a system that has variables that can change stochastically (randomly) with individual probabilities.

Retirement planning, in a financial context, refers to the allocation of savings or revenue for retirement. The goal of retirement planning is to achieve financial independence.

Asset/liability modeling is the process used to manage the business and financial objectives of a financial institution or an individual through an assessment of the portfolio assets and liabilities in an integrated manner. The process is characterized by an ongoing review, modification and revision of asset and liability management strategies so that sensitivity to interest rate changes are confined within acceptable tolerance levels.

At retirement, individuals stop working and no longer get employment earnings, and enter a phase of their lives, where they rely on the assets they have accumulated, to supply money for their spending needs for the rest of their lives. Retirement spend-down, or withdrawal rate, is the strategy a retiree follows to spend, decumulate or withdraw assets during retirement.

The Wilkie investment model, often just called Wilkie model, is a stochastic asset model developed by A. D. Wilkie that describes the behavior of various economics factors as stochastic time series. These time series are generated by autoregressive models. The main factor of the model which influences all asset prices is the consumer price index. The model is mainly in use for actuarial work and asset liability management. Because of the stochastic properties of that model it is mainly combined with Monte Carlo methods.

Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets.

A dynamic microsimulation pension model is a type of a pension model projecting a pension system by means of a microsimulation and generating the complete history of each individual in a data set. The results of such model offer both the aggregate and individual indicators of the pension system. Thanks to complexity of results, there is a possibility to investigate the distribution of pensions, poverty of pensioners, impact of the changes of the pension formula, for more examples see e.g.. Detailed individual set of (administrative) data should serve as a model input.

A Year Loss Table (YLT) is a list of historical or simulated years, with financial losses for each year. YLTs are widely used in catastrophe modeling, as a way to record and communicate historical or simulated losses from catastrophes. The use of lists of years with historical or simulated financial losses is discussed in many references on catastrophe modelling and disaster risk management, but it is only more recently that the name YLT has become standard.

References

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.