Gary Venter - Academia.edu (original) (raw)

Drafts by Gary Venter

unpublished draft, 2024

This is a brief look at how the measurement problem can be addressed using a mathematical view of... more This is a brief look at how the measurement problem can be addressed using a mathematical view of complementarity. The measurement problem briefly stated is how a wave function can become much less dispersed when there is nothing in the apparatus that would concentrate it.

Here two viewpoints on quantum mechanics are reviewed: (a) it is purely formulas for calculations... more Here two viewpoints on quantum mechanics are reviewed: (a) it is purely formulas for calculations and predictions vs. (b) the formulas describe the behavior of real physical objects. What makes the latter a challenge to accept is such objects would not fall easily within the common paradigm of materialism-particles moved around by forces as governed by cause and effect. If it is accepted, also including consciousness in the quantum field becomes a relatively small addition that can help build quantum-neurological models of conscious experience.

Regressions using variables categorized or listed numerically, like 1 st one, 2 nd one, etc.-such... more Regressions using variables categorized or listed numerically, like 1 st one, 2 nd one, etc.-such as age, weight group, year measured, etc., are often modeled with a dummy variable for each age, etc. Cubic splines are used to smooth the fitted values along age curves, year curves, etc. This can give nearly as good a fit as straight regression but with fewer variables. Spline smoothing adds a smoothing constant times a smoothness measure, often the integral of the curve's squared-second derivative, to the negative loglikelihood, which is then minimized. The smoothing constant is estimated by cross validation. Picking the knots (curve-segment connection points) is a separate estimation. Here we look at using simpler measures of curve smoothness like sum of squares of the needed parameters. This gives very similar curves across the fitted values, both in goodness of fit and visual smoothness. It also allows the fitting to be done with more standard fitting methods, like Lasso or ridge regression, with the knot selection optimized in the process. This helps modelers incorporate spline smoothing into their own more complex models. It also makes it possible to smooth using Bayesian methods. That is slower but it gives distributions for each fitted parameter and a direct estimate of the probability distribution of the smoothing constant. Cross-validation is a good method to compare models but has problems if used for estimation, discussed. Also linear splines can be modeled this way as well, and after smoothing look similar to cubic splines and are often easier and faster to fit. Modern Bayesian methods do not rely on Bayesian interpretations of probability and can be done within frequentist random effects, liberally interpreted.

Quantum mechanics accounts well for behavior of objects, but what does it mean for the actual nat... more Quantum mechanics accounts well for behavior of objects, but what does it mean for the actual nature of things? That is difficult within physicalism. Panpsychism can help, and even provides possibilities for quantum-neurologic modeling of conscious experience.

Smoothing splines are splines fit including a roughness penalty. They can be used across groups o... more Smoothing splines are splines fit including a roughness penalty. They can be used across groups of variables in regression models to produce more parsimonious models with improved accuracy. For APC (age-period-cohort) models, the variables in each direction can be numbered sequentially 1:N, which simplifies spline fitting. Further simplification is proposed using a different roughness penalty. Some key calculations then become closed-form, and numeric optimization for the degree of smoothing is simpler. Further, this allows the entire estimation to be done simply in MCMC for Bayesian and random-effects models, improving the estimation of the smoothing parameter and providing distributions of the parameters (or random effects) and the selection of the spline knots.

Science is learning more about the brain activity necessary for consciousness, but has not identi... more Science is learning more about the brain activity necessary for consciousness, but has not identified any mechanisms for how it could actually arise through neural processes. Here I present ways to build consciousness into the mathematics of quantum mechanics for use in the growing area of quantum neurology. Quantum waves are not physical in the sense of forces acting on particles through cause and effect, and they are expressed using mathematical placeholders that do not have agreed-upon real-world representations. These can be used to represent consciousness. Quantum mechanics has shown how the traditional aspects of the physical world emerge from non-physical quantum information through a combination of mathematical, not causal, determinism, and stochastic interactions. The modeling methods here could help account for how the mental world could also emerge from information fields. Neural interactions for mental experiences are complex, so it is reasonable to expect that consciousness is an emergent property of neural networks. But emergent properties are not magic-they work through procedural mechanisms-in this case for how neural processes generate experiences. No steps for how consciousness could be manufactured in this way are apparent, and philosophers have strong arguments for such not being possible. An alternative is to model consciousness as part of quantum waves, so it is accessed, not created, by the brain. This is a form of neutral monism-the idea that the physical and mental worlds both come from a single underlying source-in this case quantum waves. The classical physical picture of particles and forces acting under cause and effect has developed into a belief system, not just a theory, and this has created difficulties in taking quantum mechanics itself at face value. The result has been the creation of numerous "interpretations" of quantum mechanics that seek to frame it within these philosophical presuppositions. The conceptual framework of Cartesian dualism provides a platform for analysis of the related philosophical issues of consciousness and of quantum mechanics itself.

Papers by Gary Venter

Fitting parameters on spline curves produces more parsimonious models, maintaining fit quality. S... more Fitting parameters on spline curves produces more parsimonious models, maintaining fit quality. Smoothing the splines reduces predictive variance. Individual splines are often fit by type of variable, e.g., in age-period-cohort models. Linear and cubic splines are most common. Several smoothing criteria have been used for parameter curves, with cubic splines fit by constraining the integral of splines' squared-second derivatives popular recently. Constraining the sum of squared second differences for linear splines is analogous. Generally the degree of smoothing is selected using cross-validation. Known spline dummy-variable matrices allow regression estimation of splines, with smoothing done via constrained regression. Smoothing criteria based on sums of squares or absolute values of parameters, as in ridge regression or LASSO, improves predictive accuracy and produces splines similar to smoothing by second-derivative constraints. Variables with very low t-statistics represent points where curve-shapes barely change. Eliminating those variables leaves knots concentrated where spline shapes change. A Bayesian version of this puts shrinkage priors on spline parameters. This yields realistic joint parameter distributions, avoids problems associated with using cross-validation for parameter estimation, and readily expands to non-linear modeling, such as interactions among variable types. Regularized regression and Bayesian spline methods are compared for two example datasets.

Social Science Research Network, 2017

Background: Mortality from drug overdose has increased sharply over recent decades [1]. The purpo... more Background: Mortality from drug overdose has increased sharply over recent decades [1]. The purpose of this study is to estimate the variation in overdose mortality across generational cohorts, particularly for white non-hispanic males-a group hit especially hard. Actuaries estimate mortality rates by year-of-birth cohorts for pricing life insurance and annuities. Their Lee-Carter plus cohorts model is applied here to estimate trend by year, pickup rate of trend by age, and cohort impacts on overdose mortality rates in the period 1999-2015 for ages 17-61. Findings: Overdose mortality increased most markedly for the older ages but also for the youngest in this period. Drug mortality for ages 54-61 increased as the pre-boomer generation moved out of this range. The early millennialsborn 1981-1990-have sharply higher mortality rates than any other generation after controlling for age and trend. The later millennials are showing much lower overdose mortality. After adjusting for cohort effects, the trend rate was higher in 1999-2007 than in the following years, but 2013-15 gets back towards the higher trend range. Conclusion: The Lee-Carter plus cohorts model suggests that as they move across the age groups the early millennial cohort will create a bulge in overdose mortality rates that will then subside substantially for the late millennials.

The Musical Quarterly, 2020

Science is learning more about the brain activity necessary for consciousness, but has not identi... more Science is learning more about the brain activity necessary for consciousness, but has not identified any mechanisms for how it could actually arise through neural processes. Here I present ways to build consciousness into the mathematics of quantum mechanics for use in the growing area of quantum neurology. Quantum waves are not physical in the sense of forces acting on particles through cause and effect, and they are expressed using mathematical placeholders that do not have agreed-upon real-world representations. These can be used to represent consciousness. Quantum mechanics has shown how the traditional aspects of the physical world emerge from non-physical quantum information through a combination of mathematical, not causal, determinism, and stochastic interactions. The modeling methods here could help account for how the mental world could also emerge from information fields. Neural interactions for mental experiences are complex, so it is reasonable to expect that consciousness is an emergent property of neural networks. But emergent properties are not magic – they work through procedural mechanisms – in this case for how neural processes generate experiences. No steps for how consciousness could be manufactured in this way are apparent, and philosophers have strong arguments for such not being possible. An alternative is to model consciousness as part of quantum waves, so it is accessed, not created, by the brain. This is a form of neutral monism – the idea that the physical and mental worlds both come from a single underlying source – in this case quantum waves. The classical physical picture of particles and forces acting under cause and effect has developed into a belief system, not just a theory, and this has created difficulties in taking quantum mechanics itself at face value. The result has been the creation of numerous “interpretations” of quantum mechanics that seek to frame it within these philosophical presuppositions. The conceptual framework of Cartesian dualism provides a platform for analysis of the related philosophical issues of consciousness and of quantum mechanics itself.

Social Science Research Network, 2017

Very similar modeling is done for actuarial models in loss reserving and mortality projection. Bo... more Very similar modeling is done for actuarial models in loss reserving and mortality projection. Both start with incompleted data rectangles, traditionally called triangles, and model by year of origin, year of observation, and lag from origin to observation. Actuaries using these models almost always use some form of parameter reduction as there are too many parameters to fit reliably, but usually this is an ad hoc exercise. Here we try two formal statistical approaches to parameter reduction, random effects and Lasso, and discuss methods of comparing goodness of fit.

Social Science Research Network, 2017

Actuaries use age-period-cohort (APC) models for mortality modeling and general insurance loss re... more Actuaries use age-period-cohort (APC) models for mortality modeling and general insurance loss reserving. Several recent papers have addressed simultaneously modeling related datasets, such as loss triangles for subsets of a class of business or mortality data across regions. This paper does joint modeling by shrinking the differences among the same parameters for different datasets. This could loosely be described as credibility weighting for triangles, but it comes more directly from statistical approaches such as ridge regression and lasso. Like credibility, these seek to reduce estimation and prediction error by various forms of shrinkage. The models discussed here already incorporate parameter reduction by smoothing linear spline slope changes. This is extended to also shrink the differences between the same slope changes for different datasets. Doing so can reduce prediction error, measured using penalized log-likelihood, by increasing model parsimony. Bayesian Markov Chain Monte Carlo (MCMC) estimation is used in an example to illustrate the method. A related classical approach based on random effects is introduced as an alternative. The example is a joint model of historical female mortality data for Spain and Japan-two of the world's longest lived populations.

The North American Actuarial Journal, 1997

Page 1. Financial Institutions Center Corporate Hedging in the Insurance Industry: The Use of Fin... more Page 1. Financial Institutions Center Corporate Hedging in the Insurance Industry: The Use of Financial Derivatives by US Insurers by J. David Cummins Richard D. Phillips Stephen D. Smith 96-26-B Page 2. THE WHARTON FINANCIAL INSTITUTIONS CENTER ...

Every 50 years or so a study of workers compensation mortality patterns is done, generally findin... more Every 50 years or so a study of workers compensation mortality patterns is done, generally finding that after medical stabilization-10 or more years after injury-mortality for seriously injured workers is comparable to that of the overall population. It has been about 25 years since the latest study, so we might be half way to the next one. But in the meanwhile there are trends in population mortality, and these impact loss reserve risk. Mortality data over time can be arranged in triangles, and models fit to such data are similar to those used in casualty loss development-particularly those that model trends in the three dimensions of calendar year of finalization, age at finalization, and origin year. We fit such models to U.S. population male and female mortality data for death (finalization) ages 55 to 89, with several distributions of residuals. The information matrix is used to estimate parameter standard deviations. Although there is an extensive literature on fitting these models, most of the papers do not address parameter significance through t statistics, etc. and doing so finds problems with the standard models. One problem is overparameterization, and a conclusion here is that parameter reduction methods such as smoothing should be used. Other authors have tried this, but a sticky issue is finding parameter reduction methods that actually produce improvements in goodness of fit, as measured by AIC, etc. This is an open problem as far as we know and a direction for future research. Typically the starting point for the distribution of model residuals is Poisson, but several authors have found that negative binomial fits better. Unfortunately, some of these have misinterpreted the derivation of the negative binomial as a gamma-mixed Poisson to conclude that the negative binomial arises because there are different subpopulations each with different Poisson distributions. But a sum of subpopulations each Poisson distributed is itself Poisson distributed. The mixture becomes interesting when you are drawing at random from a subpopulation whose parameter you do not know. Probably the negative binomial arises from other contagion effects, like weather, disease outbreaks, etc. Unfortunately, these also make residuals across cells not independent, and this effect has been found in other studies as well. A few alternative ways of parameterizing negative binomial residuals are discussed, and these are also applied to the Poisson-Inverse Gaussian distribution and its generalization, the Sichel. For females the negative binomial fits best but the male data is a bit more skewed than the negative binomial. However the Poisson inverse-Gaussian appears to be too skewed for this data. The Sichel is more flexible, with one more parameter, and fits best. Further insight into the shifts in mortality over time is provided by fitting Makeham-like curves to each year of death. One implication from this exercise is that male mortality trends at the older ages had a shift in 1988, possibly data related. Probably data older than that is not reliable, or at minimum comes from a different process. The overall conclusion is that more work is needed to come up with reasonable models for mortality trend, with parameter reduction a leading candidate. For trending, ARIMA models have often been fit to the calendar-year parameters after first differencing for stability. But since the parameters are estimated with error, differencing induces an autocorrelation, so the ARIMA models could be mostly fitting this artifact. Alternatives are discussed.

Many loss reserving models are over-parameterized yet ignore calendar-year (diagonal) effects. Ve... more Many loss reserving models are over-parameterized yet ignore calendar-year (diagonal) effects. Venter [1] illustrates techniques to deal with these problems in a regression environment. Venter [2] explores distributional approaches for the residuals. Gluck [3] shows that systematic effects can increase the reserve runoff ranges by more than would be suggested by models fitted to the triangle data alone. Quarg and Mack [4] show how to get more information into the reserve estimates by jointly using paid and incurred data. This paper uses the basic idea and data from [4] and the methods of [1] to build simultaneous regression models of the paid and incurred data, including diagonal effects and eliminating non-significant parameters. Then alternative distributions of the residuals are compared in order to find an appropriate residual distribution. To get a runoff distribution, parameter and process uncertainty are simulated from the fitted model. The methods of Gluck [3] are then applied to recognize further effects of systematic risk. Once the final runoff distribution is available, a possible application is estimating the market value pricing of the reserves. Here this is illustrated using probability transforms, as in Wang [5].

Capital allocation is never an end in itself, but rather an intermediate step in a decision-makin... more Capital allocation is never an end in itself, but rather an intermediate step in a decision-making process. Trying to determine which business units are most profitable relative to the risk they bear is a typical example. Pricing for risk is another. While the general topic is capital allocation, this article looks at methods of answering the questions that capital allocation addresses. The following four basic approaches will be reviewed: Selecting a risk measure and an allocation method, and using them to allocate all capital. Comparing the actual versus model pricing by a business unit. Computing the cost of the marginal capital needed for or released by target strategies. Evaluating the profitability in comparison to a leveraged mutual fund. Keywords: capital; allocation

Astin Bulletin, Nov 1, 1992

Journal of Risk and Insurance, Sep 1, 1991

Upper and lower constraints for the sensitivity of experience rating are discussed by Lemaire(198... more Upper and lower constraints for the sensitivity of experience rating are discussed by Lemaire(1988). A single widely advocated criterion predictive accuracy is shown here to provide both upper and lower constraints. This is illustrated by a simulation of the impact of four bonus-malus systems on the rating of a driver with long term consistent parameters. All four are found to be under-responsive to such a driver's individual experience. Lemaire's (1988) invited article in this Journal elucidated the functioning of several countries' bonus-malus rating systems for automobile insurance. This comment reviews some aspects of those systems from the perspective of the American actuarial tradition. A bonus-malus system (BMS) is a particular form of experience rating. Experience rating has been used in most lines of insurance in the US, beginning with workers' compensation in the years before World War I. It is generally used as a supplement to classification rating, in recognition that there are differences among risks, identifiable by individual risk experience, that are not accounted for by classification rates. The sensitivity of a rating plan has both upper and lower constraints. Lemaire introduces the concept of efficiency, which measures the incremental change in premium induced by an incremental change in an insured's true loss frequency. The ratio of these changes should be 100 percent for perfect efficiency. The sensitivity of the plans reviewed never reaches this level, due to the constraints of market resistance to high surcharges and a social reluctance towards having the plan financially encourage the non-reporting of accidents. Similar constraints have operated in the Western Hemisphere. For instance, Snader (1980) points out that the sensitivity of a 1940 US workers' compensation experience rating plan was determined in part by specifying that the smallest rated risk could be debited no more than 25 percent for a single accident. Gary G. Venter is President of the Workers' Compensation Reinsurance Bureau, New York, New York. This content downloaded from 157.55.39.17 on Wed, 06 Jul 2016 06:15:00 UTC All use subject to http://about.jstor.org/terms A Comparative Analysis of Most European And Japanese Bonus-malus Systems 543 The main criterion cited for optimal plan sensitivity, however, has been how well the plan estimates individual insured results. For instance, Meyers (1985) says "The purpose of experience rating is to estimate the expected loss ratio," Venter (1987) says "the degree to which an insured should be charged for past loss experience is the degree to which that experience is predictive of future loss experience," or Freifelder (1985) says "The accurate prediction of an insured's true loss potential is the goal of the ratemaking process." This note will explore BMS plan sensitivity from the viewpoint of predictive accuracy. Predictive accuracy provides both an upper and lower constraint on the sensitivity of a plan to risk experience. If the plan is not sensitive enough, high risk insureds will be undercharged and good risks overcharged. If the plan is too sensitive, the opposite will occur, as random fluctuations in experience will be accorded too much weight. This happens currently with experience rating of large commercial risks, which are sometimes given full credibility. Lemaire's concept of efficiency is related to predictive accuracy, in that a 100 percent efficient plan in actuarial balance should be highly accurate. Nonetheless, as will be seen below, plans with comparable measures of efficiency do not necessarily predict equally well. A direct measure of predictive accuracy could be developed by using a risk's relative charge from the BMS as an estimate of its relative frequency. By simulation, the estimated frequency could be compared to the true frequency, and the average of the squared errors, for example, could provide a measure of the predictive accuracy of the plan. An analytic result using the expected squared error approach to evaluating experience rating can be found in Freifelder (1985) where, rather than using BMS type rules, experience rating is based on Bayes estimation in the gamma-Poisson model, introduced by Greenwood and Yule (1920) and by Keffer (1929). Applying this approach to bonus-malus systems, a simulation was performed for four countries' BMS's at five assumed claim frequencies (X). Each risk kept its assumed frequency over time. For each risk, 20 years of simulated experience were put through the BMS rules to determine a rating level, and hence an implied estimated frequency level. At each claim frequency 10,000 such risks were simulated, and for each risk the estimated frequency was compared to the assumed frequency that generated the experience. The resulting mean squared errors for each assumed frequency are shown in Table 1.

unpublished draft, 2024

Social Science Research Network, 2017

The Musical Quarterly, 2020

Science is learning more about the brain activity necessary for consciousness, but has not identi... more Science is learning more about the brain activity necessary for consciousness, but has not identified any mechanisms for how it could actually arise through neural processes. Here I present ways to build consciousness into the mathematics of quantum mechanics for use in the growing area of quantum neurology. Quantum waves are not physical in the sense of forces acting on particles through cause and effect, and they are expressed using mathematical placeholders that do not have agreed-upon real-world representations. These can be used to represent consciousness. Quantum mechanics has shown how the traditional aspects of the physical world emerge from non-physical quantum information through a combination of mathematical, not causal, determinism, and stochastic interactions. The modeling methods here could help account for how the mental world could also emerge from information fields. Neural interactions for mental experiences are complex, so it is reasonable to expect that consciousness is an emergent property of neural networks. But emergent properties are not magic – they work through procedural mechanisms – in this case for how neural processes generate experiences. No steps for how consciousness could be manufactured in this way are apparent, and philosophers have strong arguments for such not being possible. An alternative is to model consciousness as part of quantum waves, so it is accessed, not created, by the brain. This is a form of neutral monism – the idea that the physical and mental worlds both come from a single underlying source – in this case quantum waves. The classical physical picture of particles and forces acting under cause and effect has developed into a belief system, not just a theory, and this has created difficulties in taking quantum mechanics itself at face value. The result has been the creation of numerous “interpretations” of quantum mechanics that seek to frame it within these philosophical presuppositions. The conceptual framework of Cartesian dualism provides a platform for analysis of the related philosophical issues of consciousness and of quantum mechanics itself.

Social Science Research Network, 2017

The North American Actuarial Journal, 1997

Astin Bulletin, Nov 1, 1992

Journal of Risk and Insurance, Sep 1, 1991

Insurance Mathematics & Economics, Feb 1, 2000

It is shown that the (over-dispersed) Poisson model is not the same as the distribution-free chai... more It is shown that the (over-dispersed) Poisson model is not the same as the distribution-free chain ladder model of Mack (1993) although both reproduce the historical chain ladder estimator for the claims reserve. For example, the true expected claims reserves, ignoring estimation issues, described by the two models are different. Moreover, the Poisson model deviates from the historical chain ladder algorithm in several aspects that the distribution-free chain ladder model does not. Therefore, only the latter can qualify to be referred to as the model underlying the chain ladder algorithm.

Encyclopedia of Actuarial Science, Sep 24, 2004

Social Science Research Network, 2019

Interest-rate risk is a key factor for property-casualty insurer capital. P&C companies tend to b... more Interest-rate risk is a key factor for property-casualty insurer capital. P&C companies tend to be highly leveraged, with bond holdings much greater than capital. For GAAP capital, bonds are marked to market but liabilities are not, so shifts in the yield curve can have a significant impact on capital. Yield-curve scenario generators are one approach to quantifying this risk. They produce many future simulated evolutions of the yield curve, which can be used to quantify the probabilities of bond-value changes that would result from various maturity-mix strategies. Some of these generators are provided as black-box models where the user gets only the projected scenarios. One focus of this paper is to provide methods for testing generated scenarios from such models by comparing to known distributional properties of yield curves. Typically regulators, security analysts, and customers focus on one to three-year timeframes for capital risk. This is much different than risk-management in other financial institutions, where the focus is on how much markets can move from one day's close to the next day's opening. Those institutions trade continuously when the markets are open, and manage risk with derivatives. P&C insurers, on the other hand, hold bonds to maturity and manage cash-flow risk by matching asset and liability flows. Derivative pricing and stochastic volatility are of little concern over the relevant time frames. This requires different models and model testing than what is common in the broader financial markets.

Encyclopedia of Quantitative Risk Analysis and Assessment, Sep 15, 2008

Asset–liability management for nonlife insurers is more complicated than quotidian duration-match... more Asset–liability management for nonlife insurers is more complicated than quotidian duration-matching analysis of changing interest rates. When stochastic liabilities are correlated with a number of financial variables, enterprise-wide models are needed to quantify the impacts of possible investment and hedging strategies. Keywords: assets; liabilities; ALM; duration; ERM; investments; portfolio; risk return; efficient frontier; reinsurance; hedging

Springer eBooks, Oct 24, 2021

The crisis caused by COVID-19 has had various impacts on the mortality of different sexes, age gr... more The crisis caused by COVID-19 has had various impacts on the mortality of different sexes, age groups, ethnic and socioeconomic backgrounds and requires improved mortality models. Here a very simple model extension is proposed: add a proportional jump to mortality rates that is a constant percent increase across the ages and cohorts but which varies by year. Thus all groups are affected, but the higher-mortality groups get the biggest increases in number dying. Every year gets a jump factor, but these can be vanishingly small for the normal years. Statistical analysis reveals that even before considering pandemic effects, mortality models are often missing systemic risk elements which could capture unusual or even extreme population events. Adding a provision for annual jumps, stochastically dispersed enough to include both tiny and pandemic risks, improves the results and incorporates the systemic risk in projection distributions. Here the mortality curves across the age, cohort, and time parameters are fitted using regularised smoothing splines, and cross-validation criteria are used for fit quality. In this way, we get more parsimonious models with better predictive properties. Performance of the proposed model is compared to standard mortality models existing in the literature.

Social Science Research Network, 2018

Maximum likelihood estimation has been the workhorse of statistics for decades, but alternative m... more Maximum likelihood estimation has been the workhorse of statistics for decades, but alternative methods are proving to give more accurate predictions. The rather vaguesounding term "regularization" is used for these. Their basic feature is shrinking fitted values towards the overall mean, much like in credibility. These methods are introduced and applied to loss reserving. Improved estimation of ranges is also addressed, in part by a focus on the variance and skewness of residual distributions. For variance, if large losses pay later, as is typical, the variance in the later columns does not reduce as fast as the mean does. This can be modeled by making the variance proportional to a power of the mean less than 1. Skewness can be modeled using the three-parameter Tweedie distribution, which for a variable Z has variance = φµ p , p ≥ 1. It is reparameterized here in a, b, p to have mean = ab, variance = ab 2 , and skewness = pa −1/2. Then the distribution of the sum of N individual claims has parameters aN, b, p, and cZ has parameters a, bc, p. These properties are both useful in actuarial applications.

Enterprise Risk Management Symposium Society of Actuaries, 2006

Insurance company issues that do not necessarily arise in banks are identified and utilized to de... more Insurance company issues that do not necessarily arise in banks are identified and utilized to develop insurer ERM modeling approaches that start from the work done for banks but respond to insurance-specific matters. Some of the approaches discussed could also be used to refine banking models.