## Loss Distributions

(Research Starters)

A very small portion of events generates the majority (about two-thirds) of all financial losses. This essay devotes a larger amount of its text to distributions than it does to losses. Specifically, it directs most of its attention to the normal probability distribution. That distribution introduces the target readership, which is undergraduate college students, to basic elements in statistical analyses and prepares them for inquiries into sophisticated loss estimation processes. Toward that end, this essay demonstrates the relevance and importance of mastering undergraduate courses in statistics, economics, and finance. Another aim of this essay is to enhance the ability of college students, financial analysts, and prospective actuaries to evaluate each alternative model they may engage in for estimating future payments and loss distributions. With an overarching challenge related to calculating the appropriate dollar amount of a balance sheet reserve, the literature review finds a variety of publications concerned with defining the nature of risk management itself and with refining methods of loss estimation. A basic challenge stems from the need to specify with some certainty a single point estimate for the distribution of future payments and to attach an interval that expresses some degree of uncertainty about its variability. Certainty and uncertainty are natural components of estimation processes and statistical analyses.

Keywords Actuaries; Balance Sheet Reserve; Binomial Distribution; Credit Risk; Distribution of Future Payments; Expected Value; Inter-quartile Range; Loss Estimation; Market Risk; Mean; Median; Mode; Normal Probability Distribution; Operational Risk; Probability Density Function; Probability Distribution; Random Variable; Risk Management; Sample Space; Statistical Analyses; Uncertainty

### Actuarial Science: Loss Distributions

OverviewOne percent of all finance-related events create about two-thirds of all financial losses according to a recent report. What does this say, if anything, about the risk of loss for insurance, banking, or other organizations? Any attempt to answer this and similar questions needs to start by acknowledging the critical roles of financial institutions in regional, national, and global economies. Effective management of risk, therefore, will contribute to the success of financial institutions and other sectors of an economy.

Transactions between a seller and a buyer certainly require some form of bank involvement. In terms of the range of possibilities, one can imagine some economic consequences from a failure by the banking system to assess and manage its potential for loss. Estimation of that exposure may involve assessing whether any loss will occur, the frequency at which a loss will occur, and/or the amount of a loss beyond an expected value. Several articles tend to favor a comprehensive approach taking into account all those considerations. Authors of some key publications address many challenges in their endeavors to develop models of risk that can harness the compound nature of loss frequency and severity.

Some of those challenges arise from the need to separate expected losses from unexpected losses because of their natural inclusion in the distribution of future payments. Portraying risk as the difference between expected and unexpected losses presents the opportunity to delineate operations management and operational risk management. A broad scan of recent publications on the topic of loss distributions brought many issues to the forefront. A summary report, which was presented at a 2005 meeting of the Casualty Actuarial Society, calls attention to the following items: The lack of common definitions on key terms among risk managers; the problematic nature of the 'range of reasonable estimates' approach to reserve creation; and, the value of adopting a more objective method by which to estimate the 'distribution of future payments' for property and casualty losses.

On a much lighter note, recent commercials by a major insurance company seem to portray risk in a most effective manner as they succinctly inform television viewers of the pervasive and nocturnal nature of risk. Nevertheless, risk of loss is indeed a serious matter for a countless number of professionals whether they serve in the insurance or the banking field. Around the world, banking supervisors are also working collaboratively to integrate regulations and a superlative set of practices and methodologies pertinent to risk management arena. The fruits of their labors are evident in the recent series of regulatory frameworks many refer to as Basel I & II; the first was released in 1988 and the second in 2004. In short, those regulations challenge banks to measure and manage their risks against a common set of standards and to create and advance a global culture of risk awareness.

A shift in culture usually requires time and, more often than not, its realization evolves through new recruits and the fresh perspectives they bring to the field and the work place. Emergence of that culture suggests that current, as well as prospective, bank staff and bank supervisors will need to pursue additional opportunities for education and training in risk management. Whether one is a seasoned bank professional or an undergraduate business or mathematics major in college, a number of international certificate programs are now available to those who can demonstrate a basic knowledge of statistics and who seek a deeper understanding of banking risk as set forth by the Basel Committee on Banking Supervision.

Including references to a small yet critical portion of the larger Basel framework and intermingling examples from banking and casualty insurance, this essay directs attention to the interrelationship between loss probability distributions and liability reserve estimations. More precisely, the purpose of this essay is to convey various aspects of how statistical analysis, probability distributions, and model development serve as key tools and objective methods for estimating the amount of resources that organizations need to set aside in order to mitigate their potential losses. In essence, some banking regulations seem to call for the formation and the implementation of valid and reliable methods with which to estimate with better precision the amount of those financial reserves.

ApplicationsThis essay is about loss distributions but it devotes a larger amount of its text to distributions than it does to losses. Specifically, it directs attention to the normal probability distribution because it introduces the reader to basic elements in statistical analyses and it serves as a reference point for sophisticated estimation processes. As readers progress though this essay toward a discussion of the tools available for estimation purposes, it seems that the nature of risk in the financial domain is a good place to start.

Types of RiskIn general, the three types of bankers usually face are credit risk, market risk, and operational risk.

- Credit risk refers to the probability that a borrower will default on a bank loan. Banks exercise a significant amount of control over this risk by conducting reviews of applicant credit scores, income history, and the like.
- Market risk is a largely a function of current and anticipated economic states over which banks have minimal control. The author of this essay found a list of operational risks in all reports that cite the work of the Basel committee.
- Operational risk seems to be an area in which banks have the greatest control and the largest exposure so it receives more attention in this essay than does the other types.

In an effort to cover operational risks, regulatory agents expect banks to set aside funds from internal sources and/or to acquire insurance coverage. By taking these actions, they are demonstrating good faith efforts to stabilize economies and to comply with regulations governing risk capital requirements. In other words, bankers are responsible for gathering enough resources to accommodate the likelihood of a potentially hefty loss. The reader should also note that some banking officials devote more attention to unexpected losses than expected losses. Most importantly, banking supervisors and examiners expect to find valid and reliable statistical models and procedures for estimating the amount of capital reserves.

Quantitative skills are obviously critical for developing models of expected profitability and loss risk. Actuaries are the professionals who develop those models and who demonstrate admirable levels of understanding and appreciation for quantitative methods of inquiry. At a minimum, their work involves conducting price and risk analyses before contract execution, validating model assumptions and variables in collaboration with...

(The entire section is 3878 words.)