Much of the asset-modeling literature is directed towards pricing derivatives, but modeling for risk management is a bit different. A discussion of methodology for testing the set of scenarios generated by an ESG from a risk-management perspective can be found in "Testing Distributions of Stochastically Generated Yield Curves"  from the May 2004 ASTIN Bulletin, a winner of the Bob Alting von Geusau Memorial Prize.

Personal lines have managed to gain better control of the cycle than commercial lines. Some theories of that include a potential oligopolistic market structure, shorter policy terms, which shorten the information lags, and more direct management control of pricing. Improving information systems in the commercial lines have the potential for better monitoring of underwriter pricing adjustments, which could help match the pro-gress made in the personal lines.

One of the more difficult issues to model in the commercial cycles is the degree of price elasticity and cor-responding brand loyalty, as well as how much having a strong balance sheet can attract new business when a tight market arises. The role of an acquisition strategy tied to the cycle is also worth exploring The Manage the Cycle Series at GC Capital Ideas provides a magazine-level discussion of some of the cycle issues.

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Robust estimation has approaches for these problems. Some alternative estimators to MLE are nearly as efficient as MLE with well-behaved data but are not as distorted by outliers. For instance when fitting distributions, doing MLE on every subset of 4 observations and then taking the median of those estimators is a reasonably good robust estimator, although more computationally intensive.

Another part of robust estimation is measuring the impact of the individual observations on the parameters. If a few observations are having a very large impact, alternative models can be tried to find some without that problem. This is the approach taken in "Robustifying Reserving,"  from the fall 2008 CAS Forum, which applies this aspect of robust estimation to building loss reserving models. It appears that models that are better overall can often be found by trying to avoid large impacts of individual cells in a triangle.

Several studies have found that there is a liquidity premium in the prices of many securities, such as corporate bonds, but to a lesser degree credit default swaps and even some treasury notes. Ignoring the liquidity risk can make it appear that arbitrage is possible, when in fact the positions can be quite risky.

There are a number of measures for quantifying the liquidity of various securities and these can be accumulated to estimate the liquidity position of the firm or even of the economy.

For a discussion of quantification and management of liquidity risk, see "Modeling and Managing Liquidity Risk" from the SOA essay series and the presentation, Liquidity Risk.

This is a special case of a more general objection: there are instances where reducing risk can increase expected earnings. For insurance companies, carrying too much risk can turn off potential customers. Building up financial strength is costly but in many cases can increase earnings or at least prevent a decrease.

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This can end up showing too little risk for these lines, as there are loss elements that do not diversify the way the collective risk model assumes. For instance if the price is wrong, you do not make that up by adding volume. Uncertain inflation is one of the sources of parameter risk that does not diversify with volume. Such risks can actually be the largest source of uncertainty for large companies, particularly if the catastrophe risk has been reinsured. Many internal models understate the parameter risk component.

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Whether associating lines of business, asset risks, credit risks, or whatever, besides the strength of the association, another critical factor is where in the probability distribution the association takes place. Some lines might be largely independent of each other but could both be hit by the same catastrophic events, for in-stance. Bonds might be largely uncorrelated except in hard economic times, etc. In both these cases the asso-ciation is strongest in the tails of the distributions. Failure to model the potential for high correspondence in extreme events can, and has, severely understated the tail of the distribution of the sum of the risks.

Scalar measures like tail association coefficients are an attempt to quantify where the correlation is strong, but they are still single numbers, which can only give a limited picture of the linkages. Descriptive functions that show properties of the relationships across the whole probability spectrum provide more powerful analysis.

"Tails of Copulas" from the 2002 PCAS, introduces copulas and gives examples of a number of bivariate copulas and illustrates how descriptive functions can help decide among the copulas. This paper won the Dorweiler prize.

"Quantifying Correlated Reinsurance Exposures with Copulas", a Ferguson prize paper from the 2003 CAS Spring Forum, discusses the t-copula, which is a multivariate copula that provides complete flexibility for the correlation matrix as well as allowing for control over the strength of association in the tails. Multivariate correlation coefficients and descriptive functions are introduced and examples given.

However the t-copula strongly links the tail association with the correlation coefficient. This is not always appropriate with real data, so more choices for multivariate copulas are needed. A step in that direction is provided in "Multivariate Copulas for Financial Modeling" from Variance 1:1. A more flexible version of the t-copula is introduced and some other multivariate copulas are reviewed. Still, more work on finding useful multivariate copulas is called for.

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Probably the biggest weakness of using age-to-age factors is that they miss calendar-year specific effects, such as years with high inflation, or years with a slowdown in claims processing, etc. Regression methodology applied to the loss reserving problem is capable of modeling such effects. However actuaries tend to have had little practical experience in building regression models. Typical modeling issues like choice of explanatory variables, eliminating insignificant parameters and other parameter-reduction techniques make quite a difference in the goodness of fit and the standard errors of the model. The paper "Refining Reserve Runoff Ranges," from the 2007 Spring Forum, illustrates how these regression methods can produce more parsimonious reserve models that reduce runoff ranges by reducing parameter uncertainty.

An emerging topic in reserving is simultaneous estimates of outstanding losses using both paid and incurred development triangles. Formalized methods have been designed for this problem. The paper "Distribution and Value of Reserves Using Paid and Incurred Triangles," from the 2008 Fall Forum, illustrates the application of regression model-building to this problem, as well as carrying the results into estimation of runoff ranges.

Even with good models of the underlying reserving process, the risk of future inflation on the runoff is not readily captured by the parameters fit. Another model of the impact of future inflation risk usually has to be superimposed on the runoff. However care must be taken to understand the inflation impacts that are already in the model, which are often not explicit, in order to avoid double counting. See "Stochastic Trend Models in Casualty and Life Insurance," from the 2009 ERM Symposium for a discussion of some of the methodology involved.

However this simplified approach to credibility also simplifies some of the generalizations. For instance when estimating a vector of correlated variables, like frequency of the four most serious injury types for workers compensation, using the correlations improves the estimates, and a fairly simple formula that looks like credibility but with covariance matrices can be derived directly by minimizing the estimation variance. This theory is worked out in "Structured Credibility in Applications - Hierarchical, Multi-dimensional and Multivariate Models," in ARCH, 1985. Jose Couret and I apply this to excess workers compensation pricing to get excess prices to a class level instead of just a hazard group level in "Using Multi-Dimensional Credibility to Estimate Class Frequency Vectors in Workers Compensation," in ASTIN 2008 . There it is shown that selecting risks based on this methodology can produce a preferred risk group.

In the original 1990 CAS textbook "Foundations of Casualty Actuarial Science" I included some generalizations of the standard theory in the credibility chapter. Much of this was already printed in the 1987 Fall Forum.

One generalization was to look at the case where individual risk variance does not decrease inversely with volume. This had been found years earlier by Charles Hewitt, summarized in the maximum that a large risk is not an independent combination of smaller risks. Basically risk conditions change and there are common influences, producing dependence across an organization and increasing risk. The implication for credibility is that full credibility is never achieved. Related size effects result if large risks display less variability from the average. The text worked out the formulas in the case of a single year of observation, as they got very messy otherwise. Howard Mahler later worked out all the gory details of the multi-year case.

The comparison to empirical Bayes approaches, worked out earlier by Morris and Van Slyke, was also extended. Credibility is regarded as non-parametric but uses the minimization of squared error, which is the parametric result of normal distribution assumptions. Empirical Bayes just assumes normality to begin with. In practice the insistence on non-parametric methods creates a problem for credibility theory when estimating the parameters. As Jose Garrido once aptly described it,  the methodology first pretends you know the variances and finds an unbiased estimator of the credibility factor in that case. Then you plug in unbiased estimators of the variances. The problem is that one of those variances is in the denominator, and the reciprocal of an unbiased estimator is not an unbiased estimator of the reciprocal. This yields biased credibility formulas, which empirical Bayes avoids by parametric methods.

The adjustment in the normal case is to increase 1 - Z by (N-1)/(N-3) where N is the sample size. I argue that for other distributions the correction is likely to be even greater than this, so being non-parametric is not a valid excuse for not doing it. However there are cases where the fact that 1 - Z would be capped to stay in [0,1] reduces the bias and makes the correction unnecessary. This could probably use further research.

Another connection made in that chapter is the relationship of credibility to regression. When you estimate an individual mean as Z times its sample value plus 1 - Z times an overall mean, that is a regression for the next observation. You could redo the regression after the next observations come in, and compare the credibility regression based on structural assumptions to the actual regression after the data is available. In that chapter I do this for an empirical Bayes example with batting averages.

The textbook chapter contained a section on why credibility does not work for heavy-tailed distributions and shows that taking logs first helps a lot in that case. Think of workers comp rates that vary from 10¢ per $100 of payroll for architects to $100 per $100 for window washers on the skyscrapers they design. Credibility weighting with an average rate of $5 is not going to be that useful, but in logs you are minimizing squared relative errors, which would be more meaningful. That portion did not make it into the Forum article, but survives today in the Loss Models textbook. In the 3rd edition, examples 20.29 and 20.30 are based on this work, but have actually worked out some exact distributions that I had simulated.

A credibility approach to loss reserving is found in my 1989 IME paper "A Three-way Credibility Approach to Loss Reserving." However subscriber access to IME is needed for this. Basically it credibility weights the pegged incurred from ratemaking with both the chain ladder and Bornheutter-Ferguson estimates. It is a generalization of Ira Robbin’s similar paper on claim counts, which I had partially generalized in my review in the 1986 PCAS.

One application of credibility is experience rating. In a 1987 article "Experience Rating - Equity and Predictive Accuracy,"  in NCCI Digest, I argue that it is fair to charge insureds for past experience to the extent it is predictive of future experience. An empirical example is found in "A Comparative Analysis of Most European and Japanese Bonus-malus Systems: Extension," in the 1991 Journal of Risk and Insurance. Basically I find that the auto bonus-malus plans studied actually did not give enough weight to past accident experience.

All of that is from the least-squares approach to credibility. My paper "Classical Partial Credibility with Application to Trend," (Dorweiler Prize winner) from the 1986 PCAS, addresses a number of issues in the limited fluctuation approach, including a probabilistic interpretation of the square-root rule, adjusting credibility for the NP instead of the normal approximation, and application of the paradigm to trend credibility.

Later on, Rodney Kreps reparameterized these distributions in a way that made the distribution functions more awkward but the interpretation of the parameters more meaningful. Where Loss Models would have the GB2 density f(x) expressed with the incomplete beta function as β(τ, α; 1/u), with u = 1 + (θ/x)γ, Rodney had it as β(τ/γ ,α/γ ; 1/u) with the same u. What this does is make the range of k for which the kth moment exists =   (-τ, α). Thus α now by itself gives the tail strength, and τtells how many negative moments exist. The number of negative moments turns out to determine the basic shape of the distribution. This is all laid out in my 2003 Winter Forum paper "Effects of Parameters of Transformed Beta Distributions." Besides these effects of α and τ, γmoves around the middle of the distribution and θis a scaling parameter.        

Kreps parameterization also makes it easy to find the limiting distribution when γ→ ∞, which I had missed and McDonald may have too. This limit turns out to be a power curve that meets an inverse power curve (i.e., simple beta meets simple Pareto). It also makes it clear that the simple Pareto is a limiting case of the GB2, but that can also be seen by taking a limit of the inverse transformed gamma.

A practical problem in fitting distributions to data is that actuaries typically have data excess of a number of different retentions. We have been taught how to adjust the likelihood function in this case, basically by dividing each observation’s probability by the survival function at its retention. This procedure can be improved upon, however, if we also know the exposures at each retention. Then a joint likelihood for frequency and severity can incorporate the additional information. This is presented in my 2003 Winter Forum paper "MLE for Claims with Several Retentions."  There may be typos in the formulas however.

When there is little or no data, actuaries have often used improvised rules for increased limits factors. A classic is Riebesell’s method, which uses geometrically increasing factors. Mack and Fackler found that this is consistent with an almost Pareto severity. The paper "Riebesell Revisited" lays out some of the issues. (Link to - will send)

There are problems with using utility theory for corporations, however, including the fact that they tend to have diverse and diversified ownership. In 1991 I applied the arbitrage-free pricing approach of using means under transformed probabilities for pricing purposes. The paper "Premium Calculation Implications of Reinsurance without Arbitrage," seemed to arose some controversy, although it was hardly the first actuarial paper to suggest such an approach.

One of the objections was that the insurance market is incomplete (you can’t short others’ policies, for instance) so you do not have to worry about arbitrage. This is incorrect, however. In a complete market the no-arbitrage condition uniquely determines prices, which it does not in an incomplete market, but vigorous competition in itself is enough to prevent arbitrage from being widely available.

Another objection was that arbitrage-free prices are additive, and so violate the benefits of pooling risk. I would counter, however, that in a competitive situation, the benefits of pooling go mostly to the customers. Imagine, for instance, that a fleet of 100 cars gets less profit load than a portfolio of 100 cars built up individually. Then the insurer can reinsure the portfolio completely for an arbitrage profit. If this were possible, soon the profit loads on the individual policies would be eaten away to get the arbitrage, eventually resulting in no possible arbitrage. However, to the extent that pooling makes the insurer more financially sound, the customers are getting better risk protection, and the insurer should get some of the benefit of pooling for providing a more valuable product.

A third objection, attributed eventually to Thomas Mack, was that when you move probabilities towards the more adverse outcomes to get a higher mean, you are moving it away from the lower losses, and you can thus construct contracts that would get negative risk loads. The buyback of a franchise deductible turns out to be such a contract. Although not likely to be a problem in practice, it is a bit disconcerting to have a methodology that is always faced with some negative risk loads. This problem has now been solved by doing joint transforms of frequency and severity probabilities, discussed below. Basically, mean frequency is increased more than any severity probability is decreased, so no negative loads are possible.

My paper used scale transforms as an example, but that is not a good transform for pricing. Shaun Wang did a number of papers introducing other transforms, such as the proportional hazards transform and the Wang transform. A paper by Møller discussed joint transforms of frequency and severity, such as the minimum martingale transform and the minimum entropy martingale transform. In the 2004 PCAS my discussion paper "Discussion of Distribution-Based Pricing Formulas Are Not Arbitrage-Free," based on a paper of David Ruhm, reviewed the joint transforms as well as discussed a point that Ruhm had helped clarify: you do not get arbitrage-free prices by transforming the distribution of each contract’s results - rather you have to transform the underlying distribution of basic losses, then apply the transformed probabilities to each contract.

In the 2004 AFIR Colloquium paper "Market Value of Risk Transfer: Catastrophe Reinsurance Case", the minimum martingale and minimum entropy martingale transforms are compared to actual reinsurance pricing. The minimum entropy transform seems to work a bit better. Still market price is not the end of the story, as company reservation prices might be different, especially because specific risk is important in insurance. That is why I recommend calibrating the probability transforms to company risk, not market risk, as discussed in "Strategic Planning, Risk Pricing and Firm Value", from the 2009 Astin Colloquium.