Loss Distributions

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

Overview

One 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.

Applications

This 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 Risk

In 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 various counterparts, and understanding probabilistic and statistical concepts and rationale perhaps first introduced to them in an undergraduate level statistics course. In addition, actuaries focus their attention on loss distributions using a variety of methodologies through which many authors suggest they need an all-inclusive method to be the most effective.

The literature review for this essay points to opportunities for other important refinements. For instance, there is real need for a formal everlasting distinction between operations management and operational risk management. O'Brien (2007) suggests that the difference between those two functional areas corresponds to the difference between expected and unexpected losses. By extension, a comparable interpretation suggests that operations management personnel focus their energies and resources on the subject of expected losses whereas operational risk management personnel focus on unexpected losses.

Challenges & Perspectives in Loss Estimation

While some scholars concern themselves with the previously discussed refinements, others examine the capacities and methods of financial professionals to estimate the appropriate amount of capital to be set aside for covering an unexpected loss. One complication among many is that a low risk of loss translates into a small amount of capital reserves, thereby making financial resources available for other purposes.

Dutta & Perry (2007) and other authors point out that bank regulations advance the capital reserve as a method to mitigate risks. They also advance the notion of integrating loss frequency and loss severity into a single model for risk assessment. More importantly, their comprehensive analysis highlights the significance of examining the fit between probability distributions and loss data. Most of the literature reviewed by the author of this essay makes it clear that risk managers both today and into the future need to understand the various types of probability distributions, the shapes of those distributions, and the basics in their construction.

Hayne and associates (2005) draw some attention to the work of accountants that appears to be at odds with the work of actuaries. Evidently, what some accountants consider appropriate for inclusion as a reserve on the financial statements or books frequently drives what actuaries methodically assess as a probable risk. One interpretation is that drive needs to occur in reverse because so much more is at stake when given the apparent sequence. Nonetheless, as a back-up or recourse measure, prospective reconciliation can occur when actuaries examine the likelihood that future payments differ from the book amount and calculate the expected financial consequences of the booked number. Hayne and associates (2005) conclude that book value determination processes of those in the accounting profession stand to gain a lot from the leadership and wisdom of those in the actuarial profession.

Casting issues of appropriateness aside for the moment, let us move deeper into the domain of actuaries and statisticians while highlighting the absence of a single vocabulary.

Statistical Dimensions of Loss Distributions

An underlying problem stems from the need to specify with some certainty a single point estimate of the distribution of future payments when a range is more appropriate given the inherent existence of uncertainty about contemplating future status. Whether one employs phrases such as "the range of reasonable estimates" or the "distribution of future payments" there is a statistical concept universal in its scope, application, and/or definition. In general, the concept draws heavily from statistics terms including the mean and the standard deviation.

Readers need to keep those statistical terms in mind as they progress through this essay as well as their undergraduate course work. Those terms provide a foundation for the distribution of probabilities. A probability distribution describes a range of possible outcomes. Furthermore, objective models for estimating those ranges of distribution are readily available. Moreover, there are two requirements for a probability distribution. First, the sum of the probabilities of all events within the sample space must equal one. Second, the probability of each event in the sample space must be between zero and one. The sample space, by definition, is the set of all possible outcomes from a probability experiment.

A variable is a characteristic or attribute that can assume different values. A random variable is a variable specifically associated with a probability distribution. The value of a random variable is determined by chance. Let us pause for a moment, before moving into a discussion of the types of probability distributions, to recognize that variables may be discrete or continuous. Discrete variables such as the die and the coin have countable values. In contrast, continuous variables are variables that can assume all values in between any two given values. Continuous random variables provide data that are measurable as opposed to countable. By extension, these properties determine whether the probability distribution is continuous or discrete.

Calculations of the mean, the variance, and the standard deviation for a probability distribution differ from those for samples. As a first step, the former calculation involves recording the outcomes of an infinite number of samples randomly drawn from a population in constructing the probability distribution. The second step is to multiply each possible outcome by its probability and then add all the products. That resultant mean of the random variable describes the theoretical average, which is called the expected value. That average or expected value is a good start, but we need more information that informs us about the spread of values within the probability distribution. For guidance on how to calculate the variance and the standard deviation, the author of this essay refers readers to textbooks for statistics courses such as Bluman (2003) and others.

A summary statistic of the distribution is a specific value that conveys some information about the entire distribution. The mean along with the mode and the median are summary statistics known as measures of central tendency. As was discussed above, the mean is the average or expected value. The mode is the most likely value and the median is the middle value. The median is also the 50^th percentile because one-half of all possible values are above it and the other half of all possible values are below it. In addition, the 25^th percentile is the point halfway between the lowest value and the median and the 75^th percentile is the point halfway between the highest value and the median. Furthermore, one-half of possible values reside between the 25^th percentile and 75^th percentile, which by definition is the inter-quartile range (IQR).

Relevance of Normal Probability Distribution

With some certainty, analysts know that the mean or the expected value will occur somewhere within the IQR. Certainty and uncertainty, as natural components of estimation processes, play an important role in statistical analyses. Hayne and associates (2005) inform us of three sources of uncertainty. Roughly speaking, process uncertainty arises from attempts to predict the outcomes of a roll of die; parameter uncertainty arises from attempts to claim the mean or the standard deviation in a sample will be equal to that of the population; model uncertainty arises from attempts to recognize future patterns as an extension of past patterns. All these uncertainties exert undue influence on the work of analysts. Recognizing them when estimating loss values and ranges may be most effective when communicated in terms of its association with a probability distribution. In brief, analysts need to form ranges and convey the degree of confidence that the mean lies within the estimated interval; more on confidence intervals will follow a brief discussion of two major types of probability distributions.

Binomial and normal probability distributions are relevant to understanding and articulating loss distributions. On the one hand, the binomial distribution contains all the probabilities for each of two possible outcomes. Comprising this distribution are events that actually have only two outcomes such as success or failure; it is permissible to collapse other events with more than two outcomes into two categories. On the other hand, the normal distribution contains all the probabilities for an infinite number of possible outcomes. Many continuous variables are normally distributed. The normal distribution is bell shaped.

When the mean, mode, and median assume the same numeric values, one can be certain that the normal distribution is symmetrical in its shape and appearance. The symmetry can encompass those bell shapes with a short height and wide base to those with a tall height and narrow base. Likewise, the top-middle portion of the normal or bell-shaped probability distribution is higher in some instances than other instances. Furthermore, movement away from that middle segment to the right or to the left toward the outer fringes of the base takes one into the area known as the tail of the distribution. Measures of variation describe the area between the outer fringes with reference to the middle section. The range, the variance, and the standard deviation are measures of dispersion that indicate the relative horizontal distance between the outside edges of the bell and the mean.

The normal distribution can also be asymmetrical in shape. In the case of asymmetry, there will be different values among the mean, the median, and the mode. That inequality is observable in normal distributions that are left skewed wherein the mean is less than the median and both are less than the mode. It is also observable in those that are right skewed wherein the mean is greater than the median and both are greater than the mode. Approximately 60 to 70 percent of bank operational losses arise from a few large events according to O'Brien (2007). This evidence suggests a left skewed distribution because the largest probability or area under the normal curve appears in the left side of the distribution.

Among the several properties of a normal distribution, the total area under the normal distribution curve is equal to one and separation of the area expressed in terms of standard deviations. Approximately 68 percent of the area is within one standard deviation of the mean; 95 percent is within 2 standard deviations; and, 99 percent is within three standard deviations. Looking at these segments from a different angle, using the three standard deviation approach, 99 percent of the probability values reside within three standard deviations from the mean and so on. Conversely, approximately 1 percent of the values occupy the tails; one-half of it is in the right-hand tail and one-half of it is in the left-hand tail.

These segments are very useful whether one refers to the range of reasonable estimates, the distribution of future payments, or the distribution of losses. Utilization of these deviations and the area they represent allow the analyst to make statements about his or her confidence in the estimate. For example, the normal distribution informs us that 95 percent of the possible values are within 2 standard deviations of the mean; that is, two standard deviations below the mean and two standard deviations above the mean. By extension, the actuary is able to form a statement such as the mean true loss will likely be somewhere within that range 95 percent of the time. In other words, an opportunity exists to include on the balance sheet the dollar value of a potential liability greater than or less than the calculated mean. If one subscribes to the convention of conservative estimation thereby overstating expenses and liabilities and understating revenues and assets when permissible, then the book value would equate to the highest possible amount, which is two standard deviations above the mean.

With the information from O'Brien (2007) with respect to a left-skewed loss distribution, the amount may be smaller than it sounds because few values will exist in the upper tail area. The area in the tail of the curve is an important consideration receiving brief mention here, but is otherwise beyond the scope of this essay. Readers should keep in mind that the area under the normal distribution curve is often also separated into columns of varying width to suit specific inquiries. In general, each column of the area represents a specific amount of probability.

A literature review will produce references to the vertical height of the normal distribution. In fact, several articles on loss distribution refer to the vertical dimension or thickness of its tails. The probability density function for a given value of variable is the height of the curve as opposed to the size of an area. It is a measure of the vertical distance between the tail portion of the curve and the horizontal axis.

This wraps up the presentation of key characteristics of the normal distribution, as we head toward conclusion of this essay. The normal distribution is one among many types of probability distributions. Analyses of the risk of loss suggest that there is a need to incorporate multiple probability distributions. At the most basic level, the binomial distribution can accommodate the probability of a claim filing and/or whether it will exceed a specific amount. At a more complex level, probability distributions other than the normal probability distribution may be required to handle combinations such as loss frequency and loss severity because some analysts may be unwilling to assume that losses follow a normal distribution. Readers who are comfortable with the normal probability distribution are better equipped to examine alternative forms. Their cursory review of some recent works will most likely find conclusions that losses follow the Chi-square, the lognormal, and other advanced distributions.

Conclusion

Some recent reports assert that the actuarial profession needs a single all-inclusive method for estimating the distribution of future payments for property and casualty liabilities. The primary aim of this essay is to enhance an analyst's ability to evaluate each alternative model they may engage in for estimating future payments and loss distributions. The essay covered a few applications of the concepts and the methods found in a recent scan of the literature. Readers who desire more breadth and depth are encouraged to consult academic journals and trade publications. Certainly, a reader will find a diverse array of information that will challenge and satisfy the needs of that individual. Some articles will be more helpful and comprehensive than others.

Terms & Concepts

Binomial Distribution: The outcomes and probabilities of an event in which there are only two possible outcomes such as success or failure.

Credit Risk: The risk of nonpayment by a borrower to a banker/creditor.

Expected Value: The population mean when derived by taking several random samples from a population.

Inter-quartile Range: The range of values between the 25^th percentile and the 75^th percentile or the 1^st quartile and the 3^rd quartile.

Market Risk: The risk to bankers of unfavorable fluctuations in the domestic and international financial sectors of an economic market.

Mean: The sum of values divided by the total number of values.

Median: The midpoint of a data array; the same number of values are above it as are below it.

Mode: The value that occurs most often in a data set.

Normal Distribution: A continuous, usually symmetric, bell-shaped distribution of a variable.

Operational Risk: The risk to financial institutions of fraud, failure, or incompetence in its operations; see definitions provided by Basel Committee on Banking Supervision.

Probability Density Function: The vertical position of a probability distribution that determines the thickness of the tail segment of distribution curve.

Probability Distribution: The values a random variable can assume and the corresponding probabilities of those values.

Random Variable: A variable with values determined by chance.

Sample Space: The sample of all possible outcomes of a probability event.

Bibliography

Bernardi, M., Maruotti, A., & Petrella, L. (2012). Skew mixture models for loss distributions: A Bayesian approach. Insurance: Mathematics & Economics, 51(3), 617-623. Retrieved November 15, 2013, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=83160135&site=ehost-live

Bluman, A.G. (2003). Elementary statistics (2nd ed.). Boston: McGraw-Hill Higher Education.

Brazauskas, V., & Kleefeld, A. (2011). Folded and log-folded-t distributions as models for insurance loss data. Scandinavian Actuarial Journal, 2011(1), 59-74. Retrieved November 15, 2013, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=58618867&site=ehost-live

Gzyl, H. (2011). Determining the total loss distribution from the moments of the exponential of the compound loss. Journal of Operational Risk, 6(3), 3-13. Retrieved November 15, 2013, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=67279241&site=ehost-live

Hayne, R., Khury, C., Pulis, R., Leise, J., Kumar, R., Sanders, D. et al. (2005, Fall). The analysis and estimation of loss & ALAE variability: A Summary report. Retrieved December 3, 2007, from http://www.casact.org/pubs/forum/05fforum/05f29.pdf

O'Brien, P. (2007, October 30). Operations management vs operational risk management. Retrieved January 17, 2008, from www.openpages.com

Ramaswamy, S. (2005). Simulated credit loss distribution. Journal of Portfolio Management, 31(4), 91-99. Retrieved December 1, 2007, from EBSCO Online Database Business Source Premier. http://search.ebscohost.com/login.aspx?direct=true&db=buh&AN=17840345&site=ehost-live

Suggested Reading

Dimakos, X., & Aas, K. (2004). Integrated risk modeling. Statistical Modeling: An International Journal, 4(4), 265-277. Retrieved December 1, 2007, from EBSCO Online Database Business Source Premier. http://search.ebscohost.com/login.aspx?direct=true&db=buh&AN=15344820&site=ehost-live

Neil, M., Fenton, N., & Tailor, M. (2005). Using Bayesian networks to model expected and unexpected operational losses. Risk Analysis: An International Journal, 25(4), 963-972. Retrieved December 1, 2007, from EBSCO Online Database Business Source Premier. http://search.ebscohost.com/login.aspx?direct=true&db=buh&AN=18613712&site=ehost-live

Merton, Robert C. (1974). On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance, 29(2), 449-470. Retrieved December 1, 2007, from EBSCO Online Database Business Source Premier. http://search.ebscohost.com/login.aspx?direct=true&db=buh&AN=4656328&site=bsi-live

Peters, G. & Terauds, V. (n.d.). Quantifying bank operational risk. Retrieved December 3, 2007, from http://web.maths.unsw.edu.au/~peterga/index*#95;files/Papers/garethnventas.doc