Stochastic modelling
Encyclopedia
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
.
Its application initially started in physics. It is now being applied in engineering, life sciences, social sciences, and finance. See also Economic capital
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.
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.
Based on a set of random outcomes, 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.
). 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:
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 flow
s.
A stochastic model would be able to assess this latter quantity with simulations.
). 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.
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 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.
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.
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.
Monte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
and Stochastic asset models. For mathematical definition, please see Stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
.
Stochastic model
"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).Its application initially started in physics. It is now being applied in engineering, life sciences, social sciences, and finance. See also Economic capital
Economic capital
-Finance and Economics:In financial services firms, economic capital can be thought of as the capital level shareholders would choose in absence of capital regulation....
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 outcomes, 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
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
) 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 methodMonte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
). 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 meanMean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
of a function, f, of a random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
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 flow
Discounted cash flow
In finance, discounted cash flow analysis is a method of valuing a project, company, or asset using the concepts of the time value of money...
s.
A stochastic model would be able to assess this latter quantity with simulations.
Percentiles
This idea is seen again when one considers percentiles (see percentilePercentile
In statistics, a percentile is the value of a variable below which a certain percent of observations fall. For example, the 20th percentile is the value below which 20 percent of the observations may be found...
). 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 reinsuranceReinsurance
Reinsurance is insurance that is purchased by an insurance company from another insurance company as a means of risk management...
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
Stochastic investment model
A stochastic investment model tries to forecast how returns and prices on different assets or asset classes, vary over time. Stochastic models are not applied for making point estimation rather interval estimation and they use different stochastic processes. Investment models can be classified...
, including the Wilkie Model
Wilkie investment model
The Wilkie investment model or 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...
, 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.