The calculation of the standard error of the mean for repeated measurements is easily carried out on a data set; however, this method for determining error is only viable when the data varies as if drawing a name out of a hat. In other words, the data should be completely random, and should not show a trend or pattern over time. Therefore, accurately determining the standard error of the mean depends on the presence of chance.
Stochastic Modeling
"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).
In order to understand stochastic modeling, consider the example of an insurance company projecting potential claims. Like any other company, an insurer has to show that its assets exceed 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.
A stochastic model, in the case of the insurance company, 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. 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 center of mass of the distribution curve (formally known as the probability density function), which is typically also the mode of the curve. Stochastic modeling builds volatility and variability (randomness) into a 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 .
Monte Carlo Simulation
Monte Carlo simulation (10,000 points) of the distribution of the sample mean of a circular normal distribution for 3 measurements.
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,
Percentiles
This idea is seen again when one considers percentiles. 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 are assessed appropriately.