Stochastic Processes

There are two types of actuarial models that are commonly used to create scenarios that model future contingent events; they are stochastic and deterministic models. Deterministic models are built with constant values and therefore are not effective in projecting scenarios that are associated with random variables. Stochastic models and their associated processes are becoming popular with actuaries because the processes estimate a probability distribution in their outcomes. Stochastic processes use random variables as inputs and take into account the variable impact on the outcome over time. Stochastic processes are more complex to model than deterministic processes, but because they provide the user with quantitative data about the probability of scenarios, they are invaluable in the assessment of complex risk scenarios. Stochastic processes have proven to be valuable tools in modeling catastrophic risk which allow actuaries to develop scenarios about "what could happen" rather than just looking at historical data and what did happen. This essay investigates some of the business applications that are evolving around the use of stochastic processes. This article also discusses how stochastic models and processes are helping to project the probability of certain scenarios regarding the solvency of the U.S. Social Security system. Users of stochastic processes are able to evaluate a large number of possible outcomes having to do with a number of potential risks. The real value added that stochastic processes provide to users today is the ability to not only model complex risk scenarios, but also to assess the likelihood of the risk actually occurring.

Keywords Actuarial Uncertainty; Catastrophic Risk Models; Deterministic Model; Probabilistic Models; Random Variables

Actuarial Science > Stochastic Processes

Overview

In general, when discussing actuarial models, there are two distinct types of models that are used: Deterministic models and stochastic models. This article focuses on the advantages of using stochastic or probabilistic models to model risk scenarios. This topic would be incomplete if one were not to consider the varied reasons why stochastic models are replacing deterministic models in the areas of insurance and finance. Early deterministic models centered upon examining what the results of past events would look like if the event were to happen today (Boyle, 2002). The major drawback in using "a deterministic model is… that [it] takes no account of random variation and therefore gives a fixed and precisely reproducible result" ("Modeling biological systems," n.d.).

Deterministic models are not without their complexity; "deterministic models are often described by sets of differential equations, […] and also rely on numerical analysis and computer simulation" in their development. It is not the lack of complexity that puts deterministic models at a disadvantage for modeling today's complex risks, but rather that "deterministic models find statistical variations in the average behavior of the system that are relatively unimportant" ("Modeling biological systems," n.d.). To reiterate, deterministic models don't rely upon random events (random variables) in modeling their outcomes. According to Uday Virkud of Applied Insurance Research, "History is too short for an actual model to see what is possible" (Boyle, 2002). Since deterministic models rely on historical experience and look only at what has actually happened in the past, deterministic models don't adequately assess what events or outcomes could actually happen in the future (Boyle, 2002).

The following description of a stochastic model is taken from its use in the biological sciences, but explains nicely the importance of random events and random times as inputs into the probabilistic model. "Stochastic models should be used […] where there is reason to expect random events to have an important influence on the behavior of the system. It may also be necessary to take account of events occurring at random times. The essential difference between a stochastic and deterministic model is that in a stochastic model, different outcomes can result from the same initial conditions" ("Modeling biological systems," n.d.).

The nature and complexity of risk in today's insurance and financial markets requires that actuarial models keep pace in modeling complex risk scenarios. Stochastic models have been in use for several decades, but became popular initially as a means to model catastrophic risk. Stochastic models are needed to model complex catastrophic models because they meet the following three imperatives:

• They model the chances of an event happening.
• They help to define the upper and lower limits of a catastrophic (event) occurrence.
• Depending upon the intensity, these models can project what effect the event will have (in a given area).

Stochastic or probabilistic models don't predict when an event may occur, but instead use mathematical probabilities to help insurers and the insured assess what will happen if a given event or situation were to occur (Boyle, 2002).

There have been a number of 'defining' events in the insurance industry in the past 15 years that have reinforced the value of using stochastic models. Stochastic models and their use of random events provide a much more comprehensive view of possible risk outcomes. One industry analyst stated that prior to Hurricane Andrew in 1992, "no one could fathom a hurricane that could cost $13 billion" (Boyle, 2002). Suddenly, there was a realization that past experience was not enough to model risk; the industry needed to examine what a single event like Hurricane Andrew could do. Insurers quickly realized that stochastic models were the best option for actually gauging their risk.

Stochastic models incorporate data from many different sources, including: Census data, demographic data and past history of events. Actuaries who build stochastic models don't limit the data sources for their models; any and all data that may be relevant in determining actual risk is considered for use in the model. Stochastic models rely heavily upon variables that are site specific; local conditions regarding geology, urbanization and local weather hazards are all customized in the stochastic modeling process. Stochastic models provide valuable information to a number of users; local insurance agents, insurance companies, re-insurers, banks and corporate clients all benefit from the output of stochastic models.

Future Trends in Stochastic Modeling

There are a number of trends occurring in the use of stochastic models today. Third party vendors such as Risk Management Solutions (RMS) develop stochastic models that are rented to insurers to help with risk assessment. It is acknowledged that using more than one model helps to identify more possible risks and thus allows insurers to further diversify their risks. Not only is it important for insurers to use more than one model for risk assessment, but stochastic models cannot remain static. Stochastic models need to be upgraded constantly with new data and incorrect assumptions (variables) must be modified to improve the scenario outcomes.

Dynamic stochastic models incorporate the systematic process for revisiting the model in response to observed results. Models chart plausible future outcomes within a given framework. Stochastic models provide a range of possible future outcomes that in totality imply something about a reasonable range in which future actual results can be expected to lie ("The roles," 2006).

Stochastic models are widely used to model catastrophic risk scenarios, but the applications of stochastic models are spreading to other areas in the insurance and financial services industries. This essay includes a discussion of the use of stochastic processes as they are being applied to industry today.

Applications

Principals Based Approach to Capital Reserves

Increasing competition within the insurance industry is increasing the risk of insolvency for insurers, but there are other factors that increase the risk of ruin for today's insurance companies as well. Both consumers and distributors have been insisting on more assurances in their life and annuity products. In response to these demands, U.S. life insurers have been creating product designs with increased guarantee features (Friedman & Mueller, 2006). Insurers market their products as innovative, reasonably priced and with the guarantee that the policy or product will deliver on its promised benefits. These factors have led to increased scrutiny by regulators and the insurance industry in defining what minimum capital requirements are needed to support products and policy holders.

Risk Based Capital (RBC) requirements were first implemented in the early 1990s, but the requirements didn't address the volatility in the markets or the demands that customers and regulators were putting upon annuity and life products. Companies responded by implementing sophisticated hedging methods to mitigate risks. There's been increased concern that some insurance companies don't have adequate capital reserves and risk insolvency. Efforts to modify the static, formulaic methods to ensure that individual companies are addressing their specific risks have been investigated for a number of years. Insurers are more sophisticated (as are their product offerings) and this has added to uncertainty with regard to hedging risk. Companies haven't had good methods to calculate what they need in terms of capital to cover their risks and as a result, the insurance industry has resorted to the following strategies to mitigate risk (Friedman & Mueller, 2006):

• Reinsurance;

• Credit;

• Securitization.

The United States is moving towards a principals based approach (PBA) to determine appropriate levels of financial reserves and define minimum capital requirements for life insurance and annuity products. This principals-based approach will replace the fixed ratios approach to determining capital requirement with specific company-based requirements. The hope is that PBA will do a much better job of reflecting the risk of each individual company when determining its capital reserves requirements. The overall goal of the principals-based approach will allow for more comprehensive assessment of risk by product, type and company portfolio (Friedman & Mueller, 2006).

A stochastic methodology will be implemented because this type of model will help with the implementation of rules and guidelines that will customize risk profiles and move away from the one size fits all formula that has been in use for several decades. Stochastic models and processes will enable the modeling of varied risk across markets and products. Stochastic models and processes are essential in allowing insurance companies the flexibility to model risk in a changeable environment. Today, actuarial assumptions are much less conservative than they were in the past and actuarial judgment is critical in making projections. Today's insurance companies offer many more and diverse products and customer experience is important in winning and retaining policy holders. Lastly, it is important to re-iterate that insurers are responding to customers by introducing new and innovative products into the marketplace. Some products will incur more risk than others and therefore require higher levels of capital to guarantee them. Insurers will need to assess the risk of each individual product offering and ensure that adequate capital is in reserve to guarantee the product (Friedman & Mueller, 2006).

Rating Agencies

All major ratings agencies are integrating PBA capital requirements into their Enterprise Risk Management (ERM) and capital models. Rating agencies will be using these principals based models to evaluate insurance companies' capital adequacy.

The implications for companies moving to a PBA in determining capital reserves are numerous. There will be increased need for modeling complexity as well as flexibility in systems and the capability to provide audits. Other factors that companies will need to consider are (Friedman & Mueller, 2006):

• Some products will have lower capital reserves;
• Some products will require higher capital reserves;
• Complexity and cost of financial reporting will increase;
• Actuaries will need to have in-depth knowledge of capital market pricing and valuation assumptions;
• There will likely be more of a focus on risk management and hedging;
• Actuarial functions will need to be much more closely integrated with product development and pricing. Valuation expertise will be needed as well as more knowledge of risk capital management.

The outcome of the principals based approach to risk management through the implementation of stochastic processes will offer companies the ability to develop products with lower tail risk and lower capital requirements and expected costs. Of course, companies that are able to successfully implement the stochastic processes that are required of PBA will have a competitive advantage in the marketplace. Innovative products will be able to be priced lower than competitors and will be attractive to customers. The downside for small companies is that the increased expertise, costs and complexity will not allow them to compete with larger insurers. This is likely to lead to more consolidation in the insurance marketplace.

Issues

Applying Stochastic Processes: Social Security & Projecting Solvency

Financial projections for Social Security systems (around the world) depend upon demographic social and economic factors. Actuarial expertise is required to select the values associated with variables such as mortality, fertility, immigration and workforce trends. Interest rates, inflation and productivity are additional variables that must be considered when modeling possible scenarios that will affect the financial viability of a Social Security system (Buffin, 2007).

The solvency of the U.S. Social Security system has been questioned repeatedly by economists, actuaries and politicians. There really are two distinct questions regarding the solvency issues of the U.S. Social Security system. The questions are:

• Based on a number of variables and scenarios, how long will the Social Security system remain solvent and be able to deliver on promised or expected benefits to retirees?
• What steps can be taken to shore up, re-design or replace the current Social Security system.

Actuarial models are the tools being employed to make informed projections and possible outcomes in response to the first question above. Stochastic models are becoming a clear choice of actuaries in helping to model possible outcomes over time. It is important to understand that actuaries are not creating models that provide definitive answers that indicate a preferred course of action. Instead, actuaries and stochastic processes provide a number of scenarios and offer stakeholders realistic scenarios about what the risks of insolvency actually are and how likely a given scenario is likely to happen.

Variables that stress the Social Security system include: Declining fertility rates, declining mortality rates, and increasing pressure on the system from baby boomer retirees. There are secular trends in the U.S. that support long term trends for the above mentioned variables. Still, considering the known demographic changes that are occurring in the U.S. population, there is uncertainty surrounding the timing and extent to which these changes will occur. Other actuarial uncertainties that need to be modeled are the differing economic factors (variables) that influence the Social Security system's finances (Lee & Tuljapurkar, 1998).

Stochastic models require that an actuary select variables that define probability distributions which add to the complexity of the model. The actuary must also determine the inter-relationships (correlation) between different variables when building the stochastic model. There is a demonstrated correlation between interest rates, inflation and wage growth-history has shown that these factors don't move independently of each other over time (Buffin, 2007).

Historically, actuarial models used to project possible outcomes for Social Security had relied on deterministic models. These projections involved high, medium and low assumptions, but lacked probabilistic interpretation. Deterministic projections assigned a central value to each factor (variable); this value was held constant at each point in time when the scenario was run. The results of such (deterministic) models provide a range of outcomes, but the outcomes lack a "quantitative measure" (likelihood of the outcome being realized) (Buffin, 2007). In other words, deterministic projections give a range of possible outcomes but don't provide the user the probability associated with the outcomes.

Stochastic processes, on the other hand, offer an entire range of possible future outcomes and when properly designed, provide a measure of the likelihood that a particular scenario will actually occur (Buffin, 2007).

Stochastic projections play an important role in making "best estimates" in measuring long-term solvency of Social Security. For example, actuarial projections have been made for the next 75 years. Stochastic scenarios give the following time estimates for possible outcomes (Reno & Lavery, 2005):

• Employee and employer contributions will fully fund social security until 2018.
• Interest from Social Security will cover benefits until 2028.
• Between 2028-2042 reserves will be drawn down until the securities that fund the system will need to be redeemed to pay benefits.
• Social Security will continue to operate even after securities have been redeemed because workers will continue to pay into the system. The (then) current workforce will pay into the system but will only be able to fund 73% of benefit costs.

Conclusion

Actuarial projections provide policy makers with a wide range of possible scenarios regarding the financial viability of the U.S. Social Security system over time. Stochastic projections present possible scenarios and their likelihood of occurrence. The ability to assess not only the possibility of a given outcome, but also its probability, provides a comprehensive framework around which policy makers can plan a response.

Uncertainties about given variables are an inherent part of actuarial models. Models are built using variables that project possible outcomes of future contingent events. It is impossible to model scenarios with complete certainty because the environments in which the models are designed are dynamic and contain elements of "pure randomness." Risks vary as underlying risk factors change in dynamic manners over time (Buffin, 2007) Stochastic processes require the regular re-evaluation of projections to capture the extent of deviation of the model from actual experience (or original projections).

"Many… aspects of uncertainty could be examined using these stochastic forecasts" (Buffin, 2007). Specifically related to Social Security, "these simulations could also be used to evaluate the performance of policies designed to achieve various goals in the face of uncertainty; buffer the system's finances, reduce intergenerational inequities, reduce the uncertainty experienced by individual cohorts, or avoid rapid changes in taxes or benefits" (Lee & Tuljapurkar, 1998).

Terms & Concepts

Actuarial Uncertainty: Actuarial valuation cannot be interpreted as absolutely correct. Actuarial projections are based on uncertain criteria and therefore cannot be assumed to be 100% certain.

Annuity: An insurance contract that provides a series of payments to the contract holder at specific intervals over a fixed period of time.

Capital Adequacy (E) Task Force C-3: A National Association of Insurance Commissioners (NAIC) committee which aims to assess capital requirements for all insurers and suggest appropriate adjustments.

Conditional Tail Expectation: In risk analysis a conditional tail expectation refers to the anticipated potential risk with respect to other variables and given that the potential risk surpasses a minimum threshold value.

Contingency Risks C-risks: In the United States, four officially recognized categories of risk that the actuarial profession has identified as being vital to insurers. See also C-1 risk (asset risk), C-2 risk (pricing risk), C-3 risk (interest-rate risk), and C-4 risk (general management risk).

Deterministic Model: A mathematical model in which the parameter and variables are not subject to random fluctuations.

Monte Carlo Simulation: In actuarial applications, this simulation is often used to estimate loss distributions, particularly when the underlying processes are too complex for analytic manipulation. Carrying out repeated simulations allows one to get a picture of the central tendency, the dispersion, and outliers.

Risk Based Capital (RBC): A method used by the NAIC to determine the minimum capital needed by an insurance company to support its business operations. This capital threshold measurement is used to set capital requirements that account for the amount of risk undertaken by the insurer.

Probabilistic Models: A statistical analysis model used to determine the degree of risk involved in individual decisions, thereby allowing for predictions of the overall risk based on each decision made before reaching the final outcome.

Secular Trend: A non-periodic trend in a time series where there is a long-term upward or downward variation as opposed to a periodic trend where there are smaller cyclical variations.

Stochastic Model: A model that predicts through random process many alternate outcomes. As variables change some outcomes become more or less probably than others.

Tail Risk: Tail risk describes when the risk of an investment moving more than three standard deviations from its current price is higher than average as determined by a normal distribution.

Time Series Analysis: Time series analysis involves analyzing time series data to describe the nature of the observations and extract significant characteristics. Time series analysis can also be used for time series forecasting which uses the information gained from the data to predict future values of the time series variables.

Value-at-Risk: A risk measurement technique that uses historical price trend and volatility data to predict the amount of risk involved with a specific portfolio.

Bibliography

Boyle, C. (2002, February 25). Catastrophe modeling — Feeding the risk transfer food chain. Insurance Journal. Retrieved January 18, 2008, from http://www.insurancejournal.com/magazines/west/2002/02/25/features/18828.htm

Buffin, K. (2007, May). Stochastic projections methods for social security systems. International Actuaries Association. Retrieved January 17, 2008, from http://www.actuaries.org/PBSS/Colloquia/Helsinki/Papers/Buffin.pdf

Coppola, M., Di Lorenzo, E., Orlando, A., & Sibillo, M. (2011). Solvency analysis and demographic risk measures. Journal of Risk Finance (Emerald Group Publishing Limited), 12(4), 252-269. Retrieved November 15, 2013, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=64678144&site=ehost-live

De Nicola, R., Latella, D., Loreti, M., & Massink, M. (2013). A uniform definition of stochastic process calculi. ACM Computing Surveys, 46(1), 5-5:35.Retrieved November 15, 2013, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=91956724&site=ehost-live

Friedman, E. & Mueller, H. (2006). A principals-based reserves and capital standard. Emphasis, 2006(3), 10-13. Retrieved January 17, 2008, from Towers Perrin. http://www.towersperrin.com/tp/getwebcachedoc?webc=TILL/USA/2006/200608/PBA.pdf

Gozgor, G. (2013). The application of stochastic processes in exchange rate forecasting: Benchmark test for the EUR/USD and the USD/TRY. Economic Computation & Economic Cybernetics Studies & Research, 47(2), 225-246. Retrieved November 15, 2013, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=89707083&site=ehost-live

Hart, D., Buchanan, R. & Howe, B. (1996). Nature and operation. In Actuarial practice of general insurance(7^th ed.). Australia: Institute of Actuaries of Australia. Retrieved January 6, 2007, from http://www.actuaries.asn.au/NR/rdonlyres/8D07821C-0ED1-4578-9521-1B81E1E7253D/3178/1NatureandOperation16doc.pdf

Lee, R., & Tuljapurkar, S. (1998). Uncertain demographic futures and social security finances. American Economic Review, 88(2), 237-241. Retrieved January 17, 2008, from EBSCO Online Database Business Source Premier. http://search.ebscohost.com/login.aspx?direct=true&db=buh&AN=665321&site=ehost-live

Modeling biological systems tutorial. (n.d.) Basis: Biology of ageing e-science integration and simulation system. Retrieved January 18, 2008, from http://www.basis.ncl.ac.uk/tutorial.html

Reno, V. & Lavery, J. (2005) Options to balance social security funds over the next 75 years. NASI.org. Retrieved January 21, 2008, from http://www.nasi.org/usr%5fdoc/SS%5fBrief%5f18.pdf

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Uncertainty in social security's long-term finance: A stochastic analysis. (2001). Congressional Budget Office. Retrieved January 18, 2008, from http://www.cbo.gov/ftpdoc.cfm?index=3235&type=0&sequence=2

Suggested Reading

Fei, W. (2005). On the theory of (dual) projection for fuzzy stochastic processes. Stochastic Analysis & Applications, 23(3), 449-474. Retrieved January 22, 2008, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=17405640&site=bsi-live

Kaden, S. & Potthoff, J. (2004). Progressive stochastic processes and an application to the ito integral. Stochastic Analysis & Applications, 22(4), 843-865. Retrieved January 22, 2008, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=13919536&site=bsi-live

Pareto-improving social security reform when financial markets are incomplete!? (2006). American Economic Review, 96(3), 737-755. Retrieved January 17, 2008, from EBSCO Online Database Business Source Premier. http://search.ebscohost.com/login.aspx?direct=true&db=buh&AN=21794735&site=ehost-live

Toland, T. (2005). Risk management: You've come a long way, baby. Annuity Market News, 11(8), 1-11. Retrieved January 9, 2008, from EBSCO Online Database Business Source Premier. http://search.ebscohost.com/login.aspx?direct=true&db=buh&AN=18206543&site=ehost-live