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# Practical Applications Of Statistics And Operational Research For Actuaries

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RECORD OF SOCIETY OF ACTUARIES 1984 VOL. 10 NO, 4B PRACTICAL APPLICATIONS OF STATISTICS AND OPERATIONS RESEARCH FOR ACTUARIES
Moderator: DAVID 114,HOLLAND, Panelist: ROBERT _ CLANC_ JAMES C HICKMAIV, EDWARD L, ROBBINS. Recorde_ JAMES L. SWEENEY
(I) - A nontechnical introduction to Applied Statistics and Operations Research (2) - Practical applications and examples for financial reporting and asset management (3) - Overview of the potential use in actuarial applications of these management tools This session is sponsored by the Education and Examination Committee of the Society. MR. DAVID M. HOLLAND: There was an old fellow named Harry, Who, of statistics, was wary. He felt that statistics were used by sadistics, and not the real actuary. This is a discussion of existentialism. Namely, does the existence of Applied Statistics and Operations Research (OR) on the Society syllabus precede any practical essence or is there a practical essence which precedes their inclusion on the syllabus? Stated more simply, there is a common complaint that statistics and OR are tested on the early syllabus but they have very little to do with real actuarial practice. The purpose of today's session is to point out how useful these topics are to the practicing actuary. The person on the front line who has to answer some difficult and important questions for management needs to be aware of the advantages of these tools. Our profession has evolved beyond the ques- tion of "have the reserves been correctly calculated" to "is the reserve correct". There is a big difference between a calculation procedure and the correct level of reserves. Classical actuarial models offer demonstrations over long periods of time but an actuary may live or die based on projec- tions of short range economic financial events. Actuaries construct mor- tality tables and set assumptions, but it is equally important for the actuary to be able to monitor and manage these assumptions on a timely basis. For instance, there is a lot of interest in smoker/nonsmoker mor- tality and it has had an influence on pricing. We need to be able to statistically monitor the mix of business that is being developed to deter- mine the validity of our original assumptions. As background, you may be interested in the principles which guided the Task Force on Operations Research and Applied Statistics, chaired by Mr. Jim Tilley, which is the basis for the current material on the syllabus. This report stated: "The mathematical syllabus for the associateship examinations should be
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viewed as an introduction to selected topics in a number of disciplines with emphasis on fundamental principles and concepts. The educational goal should be to present a wide range of topics in mathematics which have, or potentially have, useful applications to practical actuarial problems, or which help the actuary to communicate more effectively with those in allied professions. . . We wish to stress that our work was guided by the principle that topics included on the syllabus should have demonstrable applicability to actuarial problems. Moreover, our recom- mendations concerning the extent to which any topic is developed in the course of reading, recognize our belief that the Society's educational role relating to the development of mathematical skills is to train gen- eralists and not specialists." The point is that a practicing actuary may face a number of problems. The syllabus, particularly as it relates to Applied Statistics and Operations Research, does not necessarily make you an expert in these areas but it gives you an awareness of tile tools that are available. So if you encounter a problem which has to do with maximization or projections, etc., you will know that an area of expertise exists and there is already a well defined body of methodology. Then you can pursue it appropriately for your pro- ject. Another factor that came after Mr. Tilley's Task Force in developing the syllabus, was something called the "Strategic Premise for Actuarial Educa- tion". This was set forth to the Board of Governors from the Education & Examination Committee. Mr. Michael Cowell was then the Chairman. This Strategic Premise is both a goal and plan for actuarial education. I would like to share part of this with you to give you a flavor of some of the thinking which went into the development of the mathematical concepts on the syllabus. "In a broad sense, the mathematical elements of actuarial education in the associateship syllabus prepare the actuary to 'measure' the impact of contingent events on financial arrangements while the practice oriented subjects in the fellowship syllabus prepare the actuary to 'manage' that impact and those arrangements, and to 'communicate' their predicted out- come in a dynamic environment. To that end, a strategic premise for actuarial education and the ongoing development of the evolution of the syllabus could be summed up as: (i) To provide the actuary with an understanding of fundamental mathematical concepts and how they are applied, with recogni- tion of the dynamic nature of these fundamental concepts and that they must remain consistent with developments in mathe- matical knowledge. (2) To provide the actuary with an accurate picture of the socio- demographical, political and legal, and economic environments within which financial arrangements operate, along with an understanding of the changing nature and potential future di- rections of these environments. (3) To expose a broad range of techniques that the actuary can recognize and identify as to their application and as to their inherent limitations, with appropriate new techniques intro- duced into this range as they are developed. (4) To expose a broad range of relevant actuarial practice, includ- ing current and potential applications of mathematical concepts

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and techniques to the various and specialized areas of actuar- ial practice. (5) To develop the actuary's sense of inquisitiveness so as to en- courage exploration into areas where traditional methods and practices do not appear to work effectively." Basically, these are the goals that direct us in the development of the syl- labus. We think that we have created tools which will be very useful to the actuary but it is as though we have invented the car but told no one where they are trying to go. We hope that this session will give people a per- spective of how these topics can be practical and useful in their day to day responsibilities as actuaries. Our panelists today are Jim Hickman, who is Professor of Statistics of the University of Wisconsin, Ed Robbins who is Manager for Peat, Marwick, Mitchell & Company in Chicago and Bob Clancy who is Associate Actuary in the Treasury Department at John Hancock Mutual Life Insurance Company. Jim will go first and will discuss actuarial applications of Applied Statistics in general. Ed will discuss a specific statistical example with emphasis on the design of samples for auditing and actuarial modeling. Then Bob will discuss some very important actuarial applications of Operations Research. MR. JAMES C. HICKMAN: Dave has set such a very high standard by starting out with a limerick. I sat there and thought "what on earth can I do?" The only limerick I could think of was that old one that goes, "There was an odd fellow from Trinity who solved the square root of infinity, counting up the digits gave him such fidgets that he chucked math and took up divinity". I hope that what we do today will not cause you to chuck Applied Statistics and Operations Research, although perhaps divinity could use your talents. Applied Statistics has had an increasing role in our educational efforts. In the last few years there have been several movements to bring statistics and stochastic models into actuarial education. Part 2 has definitely moved, over the last decade, to a greater emphasis of ideas useful in data analysis, and not just those cute things that solve combinatorial problems. Applied Statistics actually entered Part 3 as a separate topic in 1983. A more statistical approach was introduced in llfe contingencies in 1984. And in Part 5, statistical models have become important. (They have always been there.) But in 1983, came survival models and in 1984 new notes on gradua- tion were written for Part 5, all of which were rooted in statistical ideas. I would like to talk about the reasons for these changes. One of the rea- sons is intellectual. This has been a century in which stochastic models have entered first physics, then agriculture, economics and many other fields. Today the physical, biological, social and management sciences all make extensive use of probabilistic rather than mechanistic or deterministic models. To keep in the intellectual mainstream, it was necessary that actu- aries bring more of these models into our syllabus. Another reason is edu- cational coherence. The idea is that the basic concepts that you learn in Part 2 are now repeated in Part 3 - Applied Statistics, Part 4 - Life Contingencies, and Part 5 - Risk Theory and the Construction of Tables. This

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leads to coherence and reinforcement. There is another good reason for in- troducing Applied Statistics on the syllabus from the viewpoint of educa- tional policy. We compete in the world of ideas. We like to view ourselves as being preeminent in the area of creating models for insurance systems, but there are a lot of other people who also believe themselves to be com- petent in that area. In order to maintain that preeminence, we must be able to talk their language and to beat them on their own grounds. What are we talking about? The topics actually on Part 3 under the topic of Applied Statistics are really three in number. These three are: Analysis o[ Variance, Regression, and Auto-Regressive Moving Average Time Series Models. These are the three topics that form the bulk of the Applied Sta- tistics section on Part 3 and amazingly they are closely related. As a matter of fact, they may all be summarized in simple expression. A. General Model Response = Fit + Error YT = F(XI,T' ....XP,T) + ET Where Xi, T i = 1,2, ....P are explanatory variables at time T. Example: Loss Ratio = Function of earlier for Quarter T loss ratios and general economic variables + Error The error term should appear to be "white noise", containing no in- formation about the process generating the response. All of the topics fit into this model. There is a response variable that we would like to understand and we find a function that may be a function of a lot of explanatory variables. In the equation, the '_" is the response variable observed at time "T" which is a function of a lot of "X"'s or ex- planatory variables plus an error term, an "E" term, at the end. For ex- ample, that response might be a loss ratio for quarter "T" and those "X"'s might be a bunch of economic variables in quarter "T", or they might be values of a loss ratio of earlier quarters plus, of course, that error term. That error term, after we have completed our modeling, should contain no information. It should appear as "white noise", which, if you like to play stereos, is that sound which contains no music or message. Our chore in applied statistics is to squeeze out all information except white noise. Now, let's take a look at that general model and those three specific ex- amples. First let us look at the analysis of variance. Now, we were ad- monished to suppress equations and keep to words, so that is what I am going to try to do. B. Analysis of Variance In the analysis of variance, that fit (remember response is equal to fit plus error) is of the form of some kind of overall constant plus the effect of a factor or factors which may exist at several levels.

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Fit = Overall mean + Effect of a factor(s) which exists at several levels Example: Yijk = M + Li + B.j+ Eijk Claims = Overall + Deviations + Deviations + Error incurredby mean for for driver insured auto K territories classes in a fixed period Chang, L., and Fairley, W. (1979), "Pricing Automobile Insurance Under a Multivariate Classification," Journal of Risk and Insurance, 46, 75-93. As an example, suppose that the response function that you are interested in was claims incurred by insured auto "K". K is a label that we will put on a particular auto and let's suppose that the factors for determining the re- sponse function may be territorial division or driving record divisions. The analysis of variance is a device, as in this example, for analyzing the response variable that is made up of a constant and other terms that can bump around at several different levels. C. Regression The second of those models is regression which is probably a non-informative name. It goes back to Francis Galton, one of those remarkable Victorians who seemed to be able to dabble successfully in everything. The difference between regression and analysis of variance is really very small. The main point is that those explanatory variables no longer have to be confined to just a few levels. Those explanatory variables can take on a continuum of values. An example is loss ratios for autos in various groups which might be a constant, plus terms that have to do with horsepower of the automobile, the size of the hometown and the driving record. And, as a matter of fact, there is a paper by Hilary Seal in the eighteenth Transactions of the International Congress of Actuaries addressing this topic. Fit = Function of several variables where the value of the variables need not be confined to a few levels

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Example: Loss Ratio = Constant + Terms for horsepower, + Error of Auto mileage, size of home Insurance city, driving record, etc. for Groups Seal, H. L. (1964), "The Use of Multiple Regression in Risk Classification Based on Pro- portionate Losses," 18th TICA, 2, 659-669. or transformed: Yi = logf _= BO +_BjX j + ei Loss ratios do not fit well into the regression model but that is alright since you can transfo_n them. Transfo_nation is one of the ways that you can expand the range of applicability of these models. In his paper, Mr. Seal actually takes a look at loss ratios which have been transfomned by logistic transformation. Logistical transformation is used to fit loss ratios into a regression model. Why go to the extra effort? Well, we know much about the regression model. There are many canned programs and tech- niques by which we can study these models. So, rather than punt, we will transform our variables so that we can use them. In this example, our transformed function consists of a constant term and those "X"'s, or explan- atory variables, which might be terms for horsepower, mileage, size of home- town, etc., which will contribute explanatory information. We will model until what is left is white noise. D. Time Series The third type of model studied on Part 3 is the time series model. In time series models the function, or the fit, is a function of earlier values of the same variable rather than other explanatory variables. For example, if you are interested in hog prices, you try to predict tomorrow's hog prices based on today's and yesterday's and a few days' before that, so that the basis of the function is earlier values of the same response variable plus perhaps earlier realizations of error terms. As an insurance example, av- erage paid claims in a quarter might be a function of average paid claims in earlier quarters plus an error term. Fit = Function of earlier values of the response variable and earlier random errors

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Example: Average paid = Function of average + Error claim in paid claim in earlier
quarter T quarters Cummins, J. D. and Powell, A. (1980) "The Performance of Alternative Models for Forecasting Automobile Insurance Paid Claim Costs," ASTIN Bulletin, iI, 91-106. If you are interested in forecasting average paid claims, you will want to read a paper by Dave Cummins and Alyn Powell. In that paper they examine several time series models in which the fit is the time series type. They also examine some regression models where the response variable is the av- erage paid claim cost, but the explanatory variables, instead of being earlier values of the same variable, are various economic variables like the CPI component for auto repair costs. It is an interesting paper and I hope it gives you some idea of the applicability of these time series models. Time series models are the third of this general class of models that we are studying now in Part 3. Now, there is a certain commonality in the approach to all of these models. Here are some examples. Moving Average - First Difference Yt - Yt - 1 = 8 et _ i + et Here we look at the first difference of a response variable, Y, at time "t" and time "t - i" which may be a function of an earlier error term and a pa- rameter, e, which is a constant. This is called moving average because we are averaging, in a certain way, previous error terms. That model happens to be of some note in some business areas. It is the basic model for what is called exponential smoothing. It is only one of a broad class of time series models that are now studied in Part 3. Auto-Regressive 2 Yt = _lYt - 1 + _2Yt-2 + et This model, where today's response is a function of yesterday's and the day before plus a random error term, is called an auto-regressive 2 model. This is called auto-regressive because it looks like a regression equation except the '_"'s are earlier values of the same variable, automatically regressed onto itself. These are simply examples of the classes of models, all of which fit into that same general model that we set out in the beginning. Now, probably more important than the models themselves is the general phil- osophy of applied statistics. That philosophy starts with a goal. The goal is to learn about a real world process for the purpose of prediction or con- trol. We try to have a systematic and coherent method for attempting to learn about that process. Now, often the difference between prediction and control is very great; you and I as private citizens or as businessmen can- not control that greater world. If we were on the Federal Reserve Board or

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actually complete the input for the audit sample. The input is then used by a computer program and the output is an audit selection set of instructions. The other by-product that comes out of having a formalized sample selection procedure is that it reduces the exposure to litigation substantially. You minimize the possibility of an auditor being on the witness stand and having an attorney ask him: "You selected a sample of 30 in this case, yet in ex- actly the same situation two years ago, you selected a sample of 100. Why?" We are in a situation where we are exposed to an awful lot of potential lit- igation over time, so that if by establishing a standardized sample basis we have minimized this potential litigation factor, that is extremely impor- tant. The reason I am going into our firm's organizational chart to some extent is that I am emphasizing that whenever you audit a large population of inde- pendent elements, there are some very real improvements that you can realize in precision and in cost from proper sampling technique. The major ac- counting firms recognize this. If the actuary knows how to sample well, the results should be good estimates at a relatively low cost. There can easily be 40 or 50 times better precision by going fr_n a poorly drawn sam- pie to a well chosen one at about the same sample size. The important thing for the actuary to recognize is that these situations arise frequently in the life insurance environment and these techniques are not used to their capacity. Take the example of the home office actuary on February 26, who is about to finish all of his assembling and filing of the annual statement data and is about to send it out to the states and sign the actuarial opinion. There is really very little time to do a review of any depth at that time. What a typical actuary might do at that time is simply arrange for a summer project to spot check his reserves in some manner. His basic motivation is "How do I get this spot check out of the way quickly, and with the least possible cost and yet discover that which should be dis- covered?" It is a tough question and there are techniques for improving your ability to do this. In our firm, it is not the actuaries that do the job of sample selection. However, in many other instances the independent consulting actuary or the home office actuary may have to organize the sam- ple selection and testing process himself. So it is a good idea to have a little background in some reasonably easy statistical techniques that can be applied in this setting. Just a word about the different types of tests during audits. Audit tests are generally divided into two types. Attributes testing is the most com- mon. That is the "yes-no" type of test. Is there an error or not? The other is the variables test. For example, you have a new computer run and you want to estimate the total on that computer run. The computer run is 80 pages long and you cannot do a tab total on it. So you do a sampling. You take the total sum from the sample and you gross that up to an estimate of the total population size. Another example would be if you have discovered an error of principle in the deferred premiums. You can determine the total error in the entire population based on a sampling of some of those deferred premiums. Thus attributes testing and variables testing are the basis of the two major types of formulas that have been worked upon by the Statis- tical Audit Division of our firm. Leaving the audit function for a moment, let's take a traditional actuarial management tool, modeling and model offices. They have been around for a long time. Modeling is really nothing more than an attempt to get a

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representative sample in order to estimate a number or an array of numbers. Put into statistical terminology, modeling is an attempt to get the expected value of something given an extremely stratified sample. What is a strati- fied sample? Leaving precise definitions aside, it is a sample where pains are taken to be sure that the sample is representative of each major cate- gory in the population. Typically, when an actuary does modeling, he makes an effort to select sample elements so that the sample is drawn proportion- ately from the major population categories. Efforts are then made to gross up the model into total portfolio numbers. If, for example, the model is to be used for projections or for GAAP reserves it is important for the actuary to have a high comfort level with the model. By tying into portfolio to- tals, such as reconciling with statutory reserves, the actuary can achieve that comfort level. I have discussed auditing and modeling as two prime applications of good statistical sampling techniques. Let's jump into the techniques themselves. There has been little quantifying of the statistical sampling processes employed by most actuaries. The sampling has mostly been designed by feel instead of by more formal techniques. Now this is not all bad. Research is expensive and an actuary who is well grounded in some of the principles in- volved may have gut feelings as to the sample selection which may give pret- ty good results. It certainly saves a lot of time and there are many con- cepts which defy quantification. Otherwise you might be doing too much re- search for the accuracy you get. Certain rules of thumb of good sampling are the following: (i) Cost and time should be minimized while representativeness of the sample should be maximized. This is something like smoothness and fit. They tend to pull in opposite directions. (2) Once an element is being sampled, as many attri- butes of that element as possible should be tested. In other words, once you have an application folder out, test as many relevant things in that application folder as would be useful to you. (3) Over-emphasize your sam- piing effort in categories or blocks where the degree of internal variation is likely to be high. For example, if you are doing an actuarial model of two blocks where one block of business has a issue year range of 20 years and the other has a issue year range of 5 years, then you want to sample the first of the two blocks more. There are some basic mathematical formulas that bear out these precepts and that also give you an actual quantification of these three precepts. First, let me define stratified sampling; it is breaking your population into two or more components, each of which is more homogeneous than the population at large. A proportionate stratified sample is the selection from each com- ponent of a number of sample elements in the same proportion that the com- ponent bears to the total population. For example_ you have a stratum A and a stratum B that make up the entire population. If the population consists 60% of stratum A and 40% of stratum B, then your sample should also be cho- sen in that proportion. Generally, the degree of imprcvement going from a random sample of the popu- lation to a stratified sample can be very significant if two things occur: (I) the strata have stratum means which are significantly apart from each other and (2) the sampling is random within each stratum and proportionate to the size of the population strata. Expressed mathematically, the degree

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of improvement in the variance of the estimate, moving from random sampling to stratified sampling is: 1 _E/_2 _ - 2]
= nL' h_ (E(Xh_
, where: _h = Sample mean in stratum h. n = Number of elements in entire sample. E(x) = Expected value of x. Thus the improvement is proportionate to the variance of the stratum sample means about the total sample mean. This expression is the variance of your stratum means around your estimate of the population mean multiplied by I/n. I have attached the proof of this (Exhibit i). Now the question is, "How do you select your stratum size?" Proportionate stratification is not the best kind, let alone the only kind of stratifica- tion. It can be proved (Exhibit 2) that the following expression is the optimum method of choosing your stratum size. The formula that selects the optimum number of elements to he drawn from a stratum once you know the total sample size, is: nh = n • NhSh , where:
_hNhSh
n = Total sample size nh = Sample size of stratum h. Thus, _hnh = n. N = Total population Nh = Total size of stratum h.
Thus,_N h = N.
What this shows is that your stratum sample size varies with the number of the population in your stratum, "Nh". It also varies with the standard deviation of an element in a particular stratum, stratum h. Your total sam- ple size, "n", times that weighting factor, gives your stratum sample size "nh"- You could go a step further and say, "Some things cost me more than others to sample." One example is a complicated reserve vs. a simple reserve in the audit selection situation. You want to minimize your total cost and you

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also want to maximize your precision. Then how do you calculate your opti- mum stratum sample size? Suppose that your total cost is both an out-of- pocket cost and an implicit cost. Then Total Cost = a +_chn h + Var(est.) where a = Fixed cost ch = Marginal cost of sampling stratum h Vat(est.) = Variance of the estimator of the implicit cost If this is your cost function, a neat expression for your optimum stratum sample size falls out. You add one additional term to the formula above. Total cost is minimized when
nh= INhSh+ hSh
where Sh = _ ch = Cost of sampling an element in stratum h. Thus, nh is proportionate to NhSh and NhSh _h is the weighting factor. See Exhibit 3 for the proof. So now you can see that the larger the stratum size, the larger the vari- ation

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A. If total out-of-pocket cost is given: C = a + _ chn NhSh ?--_- NhSh
h
I NNSh Thus n = (C - a) h _h _" NNEN_ h B. If the variance of the estimator is given:
ZSh sL_
n = 11 N h N_JC h' Var(est.) + h_N_ Sh2 See Exhibit 4 for the proof. You can obtain the variance using the out-of-pocket cost equations for a given value of "n" and vice versa. The variance and costs obviously vary inversely with respect to each other. You can try differing values of "n" and iterate to get the most desirable blend of the estimated variance and the total cost. In other words, you can get your total sample size by working through these two formulas. In the real world you may get your total sample size from method A. Given the resulting out-of-pocket cost, you can plug that back in and get the degree of variance of your estimator and see if you like those two numbers. You iterate back and forth until you find your optimum sample size. When an actuarial method is chosen by an actuary's intuition, it may be a pretty good method if he has some of these precepts regarding sampling choices in mind. However, it is nice to be able to support our impressions with demonstrations. The frequent problem with impressions is that they give you a direction without giving you a value. By using these methods we give you a means of getting that value. I am going to conclude with two practical examples. Let's say an error of principle was found in the deferred premiums. The typical manner by which an actuary might go through the deferred premiums to determine the total amount of error might be to take every nth policy of the in force. This method by its very nature gives a stratified sample, a proportionate stratified sample, so it is going to give better results than a random sample. But, you can improve that tremendously by something that actuaries have intuitively done. For example, all deferred premiums over \$5,000 might be tested 100%, or a lot more intensively than by simply sampling every 50th policy or so. In this way, you have done better than proportionate stratification. Implicitly the variance among the larger amounts of deferred premiums is expected to be larger. This bears out one of the precepts that we have been talking about earlier in this discussion.

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project scheduling, queuing theory and simulation. 1 will try to touch on most of these topics by giving examples of applications. Since we are talking about making decisions, perhaps the best place to start is the subject of decision analysis. I think one of the most under-utilized and useful Operations Research techniques is the decision tree. If it has ever been true that a picture is worth a thousand words, then a decision tree is it. Here is an example of how a decision tree can be used in the underwriting process. (See Figure i.) In this diagram, the rectangles represent decisions to be made. In this simple example, they involve the decision whether or not to get an inspec- tion report and whether to accept or reject an insurance application. The circles in the diagram represent chance events that can occur. In this ex- ample, they represent the possibilities of a favorable or an unfavorable inspection report and the possibilities of ending up with a normal risk or a bad risk. The decision tree shows that we can decide to get a_ inspection report and the result can be either favorable or unfavorable. Depending on the result that cones back, we can then make the decision to accept the application or reject it. Should we choose to accept it, we will be insuring what will eventually be either a normal risk or a bad risk. Now for each combination of decisions and chance events that can occur, we can compute an outcome. In this simple example, the outcomes represent the profits or losses from insuring a normal or bad risk, adjusted for the cost of the inspection re- port. I find a decision tree allows management to quickly get a feel for a problem, or to see the big picture, and to understand how combinations of their decisions and chance events can conspire to produce good or bad re- suits. Typically one would enter the appropriate probabilities into the decision tree taking into account preceding events. For example, presumably the probability of getting a normal risk given that a favorable inspection re- port was received is greater than the probability of ending up with a normal risk given that an unfavorable report was received. By factoring in the appropriate probabilities and costs, one can derive an optimal decision strategy for a problem like this. Underwriting research departments have effectively been using this technique for years although their decision making process may not be done in quite this over-simplified way. For anyone interested in a more detailed discus- sion of this application of the decision tree, I refer you to a discussion by Don Jones in the Transactions, Volume 22, part 2, starting on page 440. Another extremely useful application of decision trees is the analysis of tax strategies. The various decisions in the diagram could include whether or not to set up a subsidiary, whether or not to involve the national office of the IRS, whether or not to appeal a ruling_ etc. The various outcomes could range from gaining some desired tax savings to losing your shirt. Now, lawyers rarely give you probabilities for any of these events. So, it will be difficult, if not impossible to actually quantify the optimal de- cision strategy. Nevertheless, I have found such diagrams to be very use- ful, even without the probabilities, in allowing management to quickly see the big picture and to cut through some of the legalese. Therefore man- agement can understand certain potential ramifications of their decisions.

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Figure i
>
OUTCOMES
Profit from "Normal Risk" - Cost "Normal J RiS_ __n J'"Bad p0 •"_ao Loss from "Bad Risk" - Cost > Accept f Risk" ,-_ ApPI_��n Reject App|lcstlon m LoseCost P= J "Normal _ Profit from "Normal Risk" - Cost Favorable. . Ris_ - Accepz _ > Report ;_ f Application "_ (Cost) _ "Ba-'_'_. Loss from "Bad Risk" - Cost Get _'-Un, avor able __,_. Inspection Report Reject Lose Cost Appllcati��n"Normal Profit from "Normal Risk" Report Application "Bad-'_ Loss from "Bad Risk" No Inspection _ Risk" Report No Profit or Loss "_ Application

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Figure 2
OUTCOMES NEW
Product / Big Promotion
PRODUCT -7" Fired DESIGN Go/-Produc,
ForIt Loses Product Good_Early Wins f
Big Promotion
Results Go
J Fired
" Product Loses _Z Aggressive
No Promotion for Long Time
Design Losses
, "Nominal" Promotion
Conservative Design

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STATISTICS/OPERATIONS RESEARCH 2089
One final comment on decision analysis relates to a subject called utility theory. Now, utility theory is very useful for quantifying results that are not easily quantifiable. For example, consider an _bitious young actuary who has been assigned to develop a new product. The actuary is debating with an aggressive or conservative product design. The actuary might con- struct a decision tree to choose whether to go with an aggressive product design assuming they would monitor the early results of promoting of the product. (See Figure 2.) If the early results come back looking bad for the aggressive product then the company can decide at that point to cut their losses. Or they could decide to "go for it" by marketing the product extensively, and hope that it turns out to be a winner as opposed to a loser. If the early results are good, chances are the actuary would decide to "go for it" and hope that the product turns out to be a winner. Note that the actuary has computed the outcomes, not so much in terms of profitability for the company, but in terms of career ramifications. In this case, the career ramifications are getting a big promotion, getting fired, getting no promotion for a very long time, or getting a nominal pro- motion. Now these outcomes may not seem very quantifiable but they are quantifiable relative to one another. Utility theory achieves this by asking a series of questions such as: would you rather have a job in which you have a guarantee of getting a nominal promotion or would you rather have a high risk job that gives you a 50% chance of getting a big promotion and a 50% chance of getting fired? By playing around with the probability of getting fired, utility theory can allow an individual to quantify his feeling about a nominal pro_:,otion relative to the other possible outcomes. Once these outcomes are quantified relative to one another they can be en- tered into the decision tree and an optimal decision strategy can be de- rived. In short, utility theory can be combined with decision analysis to determine optimal decisions even when the results of such decisions are not easily quantifiable. Some of the potential wide range of decisions for which this technique could be used include the use by doctors to help people decide whether or not to undergo certain medical procedures such as surgery. Now clearly, one's feelings about potential surgical complications are not easy to quantify. This technique presumably would have plenty of potential applications in the insurance business where little emotion is involved. Linear programming is another Operations Research topic with widespread ap- plications especially for determining investment strategies. Consider an analysis for a new interest sensitive product. Here we consider three pos- sible investments: a 5 year bond, a 15 year bond or a 30 year mortgage with three possible interest rate scenarios.

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2090 PANELDISCUSSION
SURPLUS POSITION (30 YEARS) PER \$I OF INITIAL INVESTMENT Initial Investment 81% 5 Yr. and 5 Year 15 Year 30 Year 19% 15 Yr. Scenario Bond Bond Mortgage Bonds I 5.85 5.85 5.85 5.85 2 13.92 5.78 7.30 12.37 3 -.33 1.41 -2.20 0.00 This represents a simple type of C-3 risk analysis. For each combination of investment and scenario we compute the surplus position. In this case, it happens to be at the end of 30 years. All results are computed per dollar of initial investment. Suppose that we would like to find a good combination of investments and we would like not to have to do a lot of trial and error analysis. In par- ticular, we would like to investigate an initial investment strategy whereby we do not lose any money under any scenario. Since the surplus numbers are expressed per dollar of initial investment, an initial investment strategy which is a combination of th_ investments shown will be a linear combination of those surplus results. For example, an initial investment strategy that places 30% of the initial investment in five year bonds, 30% in 15 year bonds and 40% in 30 year mortgages will produce a surplus at the end of 30 years under scenario number two of 30% of \$13.92 plus 30% of \$5.78 plus 40% of \$7.30. This totals to \$8.83 per dollar of initial investment. This problem can be formulated as a linear programming problem. Note that the only scenario in which there is any chance of losing money is the third scenario. Let X5 = Fraction Invested in 5 Yr. Bond _15 Fraction Invested in 15 Yr. Bond x30 Fraction Invested in 30 Yr. Mortgage Then we want:
(1) x5 + x15+ X30= 1
(2) Surplus Under Scenario # 3 _ 0, or -.33 . X5 + 1.41 • XI5 - 2.20 - X30 _ 0 We can come up with some expcessions to impose the realistic constraints and also the goals we are trying to achieve. For example, we want the sum of our investments to equal the whole which gives rise to the first constraint. We also want the surplus under each scenario to be equal to or greater than zero. As I pointed out, we only have to worry about that in scenario number three. So the second expression constrains our surplus under scenario num- ber three to be equal to or greater than zero. This begins to look like a linear programming problem.

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STATISTICS/OPERATIONS RESEARCH 2091
GENERAL LINEAR PROGRAMMING PROBLEM i. Find XI, X2, • . . XN such that: C1 • X1 + C2 • X2 + . CN" XN is Minimized Subject to a Set of Linear Constraints, Such as: al,I X1 + al,2 X2 + . al,N XN _ b1 a2,1 XI + a2,2 • X2 + . a2,N • XN _ b2 a X1 + • X2 • XN > bm m,l am,2 + " am,N -- Xi_ 0 i = 1,2, . . . N 2. In Equivalent Notation: Min C • Subject to A - X > b We have N decision variables here, X , X2 to XN, such that we want some • • 1 .... linear combination of those decision variables to be minimized subject to a number of linear constraints. The linear expression that we are trying to minimize is called the objective function. The linear expressions with the inequalities are referred to as the constraints. For those of you who are more comfortable thinking in terms of matrix notation, a shorthand statement of the same problem is given as number 2 above. Linear programming techniques allow us to solve these types of problems. The investment strategy problem that we are trying to solve begins to look like a simple linear programming problem with three decision variables• In the • i i! paper_ "The Matching of Assets and Liabilit es , in the 1980 Transactions, Jim Tilley presented the framework for solving these types of problems by formulating the objective function and the constraints as a linear program- ming problem. He also provided the software for solving the problem. Using the techniques from that paper, we find that there is an investment strategy that meets our goals. An initial investment strategy that places 81% of our initial investment in five year bonds and 19% in 15 year bonds will not lose money under any of the three scenarios and, in particular, under scenario number three. Now, this problem may seem fairly simple and you could probably do it with- out some knowledge of Operations Research techniques. However, if the num- ber of scenarios were more realistic and if the number of possible invest- ments were much greater, the problem would not be a trivial one to do by hand and you would definitely want some software capabilities to handle it. For the record, if these scenarios were such that every investment strategy entailed a loss under at least one scenario then we could still use a linear programming approach where we would alter the objective function so as to

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2092 PANEL DISCUSSION
use maxi-min criteria. All this means is that we could select the invest- ment strategy which minimized our loss assuming that the worst possible scenario actually happened. Another popular application of linear programming is the construction of dedicated portfolios. Dedicated portfolios are designed to match the cash flows from a set of assets against a specified set of liabilities. Pension plan sponsors have used dedicated portfolios to take care of projected annu- ity payments for a block of retired lives. Some insurance companies have used dedicated portfolios to match their assets with their GIC liabilities and/or structured settlements. Consider the following simple example where we have a one year bond and a two year bond. The one year par bond has a 12% annual coupon and the two year par bond has a 13% annual coupon. We know that for any _maount that we invest in the one year bond we will get back 112% of our investment at the end of one year, We know that for any amount we choose to invest in the two year bond at the end of one year we will get back 13% of our initial investment and at the end of two years we will get back 113% of our initial investment. Let: X1 = Amount to be Invested in ] Year Bonds X2 = Amount to be Invested in 2 Year Bonds Then, Asset Flow in 1 Year = 1.12 • X1 + .13 • X2 Asset Flow in 2 Years = 1.13 • X2 Cost of Portfolio = X1 + X2 If we define X and X as the amounts to be invested in one and two year bonds, respectlvely, _hen 1 we can compute expressions for the amount of money we will get back at the end of one year and also at the end of two years. Those expressions are above. We also know that the cost of our portfolio would be the sum of the two amounts invested. Well, suppose we want to match our asset flows from a portfolio against the following liability flows.

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STATISTICS/OPERATIONS RESEARCH 2093

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Figure 3
SIMULATION RESULTS
Percent
11.00% INITIAL GUARANTEE RATE >
Frequency _Z
40 N Insured Insured
_trategy #t 30 20 10 -2_-1_ 0_ 1_ 2_ 3_ -2_-1_ 0_ 1_ 2_ 3_ Earning Spread

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STATISTICS/OPERATIONS RESEARCH 2095
Liability Year Flow 1 \$1.38 Million 2 \$2.26 Million Then we want: Min X1 + X2 Subject to: 1.12 • XI + .13 • X2 _ 1,380,000 1.13 • X 2 _ 2,260,000 Then: X2 = \$2,000,000 X 1 \$i,000,000 We would like to be able to pay off a liability of \$1.38 million at the end of one year and \$2.26 million at the end of two years. We would like to find the cheapest portfolio which will generate at least \$1.38 million of cash at the end of one year and \$2.26 million of cash at the end of two years. Well, that gives rise to the linear programming problem shown above. In solving this problem, we find that a \$2 million investment in two year bonds and a \$i million investment in one year bonds does the trick. Now this is an extremely simple problem and you may have been able to do it in your heads. But if we were trying to match a large number of liability flows and if we were looking at a large number of possible investments then the problem is no longer trivial. Again, you would want a linear program- ming software package to handle it. Even for a more complicated problem the format would be exactly the same. We would want to find the portfolio that has the least cost which will generate cash flows of at least the specified amounts at each specified point in time. Another topic of growing interest in this era of interest sensitive products is the use of simulation techniques. Many problems, including actuarial ones, do not have readily available analytical solutions. Simulation models can help provide a solution in such circumstances. In addition, rapid ad- vances in computer science in recent years have further enhanced the use- fulness of simulation models. At the New York meeting last spring, we heard time and time again that simulation was an important part of the product pricing, product design and investment strategy process for developing in- terest sensitive products. Simulation provides important input to the AIM triangle where AIM stands for Actuarial, Investment and Marketing. In order to do a simulation for an interest sensitive product, we will want an inter- est rate model. One will probably want to use some time series or regres- sion analysis similar to those described here to estimate the form and parameters of the model. The results of a set of simulations can be sum- marized in chart or graph form. (See Figure 3.) Shown in Figure 3 are the summarized results for two synthetic option strat- egies supporting a single premium deferred annuity product. I do not want to get into synthetic option strategies, but let me say that if we ware to change the investment strategy, or change the interest guarantee, or change some of the product design features we would end up with graphs that look

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PERT CHART
S T

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STATISTICS/OPERATIONS RESEARCH 2097

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2098 PANEL DISCUSSION

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STATISTICS/OPERATIONS RESEARCH 2099

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2100 PANEL DISCUSSION
certain level. One other application that I left out entirely is dynamic programming. That has a very wide range of applications also, but the ex- amples tend to get fairly complicated. I left it out because of the com- plicated examples, not because it was an inappropriate topic for discus- sion. MR. HICKMAN: This is perhaps the kind of thing that is difficult to bring up in an exam, but we all should recognize it. There is no intellectual difference between queuing theory and collective risk theory. If you re- place the random number of claims in a fixed time period, with the random number of arrivals in a fixed time period and replace the service time with claim amount, you will see that they are exactly the same model. It is one of those amazing coincidences that early in the century, separated by the Skayerrak and Katteyat, Lundberg was doing collective risk theory and Erlang was doing queuing theory. Erlang was a telephone engineer and Lundberg was an actuary and they were really doing exactly the same thing. It took the rest of us 50 years to figure out that the two subjects were the same. The interrelationship between collective risk theory and queuing theory is an important one and should, I think, be used to reinforce each other. Although there are some technical differences, if you start out at the end of the branches of a decision tree diagram and roll back by taking either the expected values or the maximizations, the process is essentially a dy- namic programming technique. Perhaps the most important single application of dynamic programming in actuarial science is that it provides a systematic way to "solve" a tree. MR. HOLLAND: Thank you for pointing out the bridge between queuing theory and risk theory. Our time is just about up. I would like to thank our panelists for their participation and thank you for your participation.

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STATISTICS/OPERATIONS RESEARCH 2101
EXHIBIT 1
Comparison of Variance of Sampling Estimates
X 6
Random Sample (n independent elements out of a population of N). Estimate = c:, -A-
VS.
Proportionate Stratified Sampling (h strata of nh independent elements each }
_L
Estimate = _- : --_'_ _ E _ Theorem: In such case, the degree of improvement in the sampling estimate is proportionate to the variance of the stratum sample means about the total sample
mean.
Standard error of Random Sampling estimate = _ VA_ (K;) (') Standard error of Stratified Sampling estimate =
h
, = i _
_. V,,,.(_x,.)
(I)- (z.) _ -ft. _- -_
r
" __ -- _ 4- 5 "_ -_-( _-:, _

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2102 PANEL DISCUSSION EXHIBIT 2 Theorem ; Optimum sample size for stratum h (_h) varies directly with stratum size and stratum variance. Proof; Define Variance of the estimate as Var (n_st), Where xi independent. and modify it via: - the Finite Population Correction ( /Vj,7,¢_-"nh ) - LaGrange Multiplier, L._,applied to a zero term
i,'v (_A I"
_V'_ * N=
. _ A/_ s. f/V_ 4}, 7 _N O .£._
Source: "Sampling Techniques," W. A. Cochran; 3. Wiley & Sons,Inc. Copyright 1953.

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STATISTICS/OPERATIONS RESEARCH 2103 EX/-I/BIT 3 CF = N-_ _ _ "_
d CF /vz " ,'v_.r_ "d% o / _ + ACh. , %
and 91h is proportionate to _ 54
[bid,pp.75,76.

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2104 PANEL DISCUSSION EXHIBIT 4 A. Obtaining Sample Size_ n_ if you know your total Out-o_-pocket cost. Total Out-of-Pocket Cost = C =a + _ C_k ( d +C_ /V. E,. _ /V_ c_ _ B. Obtaining Sample Size niI the tolerable Variance o[ your estimator is a given item:
7.
-- / £_
/% _.
E-N's'I _ N_s,
/V _
i _
k A, _ h _ /V_
t _- N_ .s-k
N _
Ibid, p 76.