1. The Alpha and the Omega of Risk
1. See Freud (1915).
2. Portions of this and subsequent sections are based upon [6].
3. Insurance terminology makes a formal distinction between perils, which are specific causes of loss, and hazards, which are additional factors that can aggravate an underlying peril.
4. Portions of this section are based upon [13].
5. See Halley (1693).
6. This figure differs slightly from the supposed theoretical histogram because the 2001 CSO tables are intended to offer a “snapshot” of the insured U.S. population at a single point in time. This means that individuals with greater ages (x) were born further in the past and so tend to have shorter life expectancies than those with lesser ages (and in particular, a baby born today). In addition, insured individuals have passed through an underwriting selection process and so are expected to have longer life expectancies than people chosen at random from the U.S. population.
7. It should be emphasized that the mortality hazard rate specified here is discrete in the sense that it describes the probability of death within a fixed interval of time of positive length—in this case, one year. In practice, the term force of mortality is used to describe the limiting probability of death within a continuous interval as the interval’s length shrinks to 0.
8. In other words, the mortality hazard rate at age x is given by h(x) = a + becx, where e ≈ 2.71828 denotes the base of the natural logarithm, and the constants a, b, and c must be estimated statistically from empirical data. (See Gompertz, 1825; Makeham, 1860.)
9. See Martinez (1998).
10. This is reminiscent of the condemned prisoner’s plight in the “unexpected hanging” paradox. The prisoner is told by the authorities that: (1) he will be executed on one of the following seven days (Sunday through Saturday); but (2) he never will be able to deduce ahead of time on which day his death will occur. Since there is an endpoint to the week—that is, Saturday—the prisoner concludes that Saturday is impossible (because it would violate the second condition). Once Saturday is ruled out, the prisoner similarly knows that he cannot be hanged on Friday; once Friday is ruled out, he knows he cannot be hanged on Thursday, etc. Thus, by backward induction, the prisoner decides that conditions (1) and (2) are logically inconsistent and that therefore he cannot be hanged. However, on Wednesday morning he finds himself walking to the gallows with no violation of either (1) or (2). (See, e.g., Erickson and Fossa, 1998.)
11. See Clarke (1962).
12. Interestingly, Clarke revised his predictions in an updated version of the same book (see Clarke, 1984). The later table suggests a closer date of 2015 for suspended animation, but a more distant time—“beyond 2100”—for immortality.
13. See Lem (1987:90).
2. Into the Unknown
1. See Hume (1748).
2. Although the examples used in this chapter all involve sets of real numbers, there is no requirement that x be a number. For example, in tossing a standard coin, the two possible outcomes are actually the nonnumerical categories x = Heads and x = Tails (although statisticians often convert these categories to numbers using the convention that Heads = 1 and Tails = 0).
3. The words sum and add up are placed in quotation marks because those terms must be extended using concepts from the calculus to apply to a certain class of random variables (i.e., continuous random variables).
4. The latter probability distribution is a geometric distribution with parameter 0.0002, but is shifted to the left so that its sample space comprises the nonnegative integers, {0, 1, 2, 3, … }.
5. Mathematically, this may be shown by contradiction: Assume there is a uniform probability function, p(x) = k, where k is some positive constant. Then, summing up the probability function over all x yields p(1) + p(2) + p(3) + … = k + k + k + …, which equals infinity, rather than 1. Hence, no such uniform probability function can exist.
6. A googol is the fancifully large number 10100. Although of little practical use, it is often cited by mathematicians as an example of a “very large number” to which other large numbers may be compared. Those who find the googol too mundane may ponder the googolplex, which equals 10googol.
7. See Cantor (1915).
8. To unify these disparate approaches requires methods of measure theory, a branch of mathematical analysis that provides the technical foundations for probability theory.
9. Interestingly, it is not possible to select a point at random, with uniform probability, from an unbounded interval such as the set of positive real numbers. This is because making the interval unbounded introduces problems analogous to those encountered in attempting to select an element at random, with uniform probability, from the set of positive integers.
10. The circumflex, carat, or “hat” symbol (ˆ) is commonly used by statisticians to denote an estimate or forecast of a specified quantity.
11. This model, although purely hypothetical, was inspired by a game-theoretic model for the frequency of terrorism-related losses developed in [26].
12. Named after British mathematician Thomas Bayes (c. 1702–1761), whose work will be discussed in Chapter 5.
13. Portions of this section are based upon [17].
14. Possibility theory was formulated by Iranian American mathematician Lotfi Zadeh (b. 1921), the inventor of fuzzy logic, as an explicit alternative to probability theory. See Zadeh (1978).
15. See, for example, Derrig and Ostaszewski (1995).
16. See Lindley (1987:24).
3. The Shapes of Things to Come
1. See Fisher (1956).
2. Named after Italian economist Vilfredo Pareto (1848–1923).
3. This particular family is often called the Pareto II family to distinguish it from the similar, but not identical, two-parameter Pareto I family. The latter family is less useful for illustrative purposes because its sample space varies with one of its parameter values (and never consists of the entire set of positive real numbers).
4. Named after German mathematician Carl Friedrich Gauss (1777–1855).
5. See Mandelbrot (1963).
6. Named after French mathematician Paul Lévy (1886–1971).
7. In other words, if X is recorded in dollars, then Var[X] is recorded in dollars-squared (which is not easily interpretable), whereas SD[X] is recorded in dollars.
8. Sometimes the ratio of the standard deviation to the expected value, known as the coefficient of variation, provides a convenient risk measure that modifies the standard deviation to account for the scale, or likely size, of the underlying random variable.
9. The general rule for this distribution is that the kth moment is finite if and only if a is greater than k.
10. Portions of this section are based upon [24] and [28].
11. By independent, I mean that the failure (or survival) of one component has no impact on the failure (or survival) of any of the other components. The concept of statistical independence will be addressed in greater detail in Chapter 4.
12. See, for example, Nešlehová, Embrechts, and Chavez-Demoulin (2006) and [28].
13. The insurance industry’s “crisis” of insufficient asbestos and pollution liability reserves was and is a multidecade phenomenon that traces its roots to policies written in the mid-twentieth century and persists as a financial drain on certain segments of the industry today.
14. This is a simplified version of a model proposed in [8], in which traders also may hold short positions for single or multiple time periods.
15. See [24].
16. See, for example, Bidarkota and McCulloch (2004).
17. The inappropriate use of the Gaussian assumption in modern financial theory is discussed at some length by Taleb (2007).
18. For example, if these parameters were themselves drawn from an underlying gamma distribution, then the tails would follow a power law over time.
19. See [12].
20. See [25].
21. The symmetric Lévy-stable distribution with a = 1 is called the two-parameter Cauchy distribution (named after French mathematician Augustin-Louis Cauchy, 1789–1857).
4. The Value of Experience
1. As a member of the symmetric three-parameter Lévy-stable family, the Gaussian distribution is a member of the four-parameter Lévy-stable family a fortiori.
2. Portions of this section are based upon [23].
3. Denoting the value of the negative correlation by c, this is true because Var [(X1 + X2 + … + Xn)/n] = (1/n)2[nVar [X] + 2(n - 1) cVar[X]] = (Var [X]/n)[1 + 2(n - 1) c/n] < Var[X]/n, where the last term is the variance of the sample mean of a random sample of size n.
4. Somewhat coincidentally, I will return to this type of primitive hunting scenario when discussing Jean-Jacques Rousseau’s “stag hunt” in Chapter 15.
5. See Rubin (2003) for a discussion of the prominent role of constant-sum (or equivalently, zero-sum) economics in hunter-gatherer societies.
6. See Williams et al. (1987).
5. It’s All in Your Head
1. See Bayes (1764).
2. See Jaynes (2003) for a discussion of different approaches to Bayesianism.
3. Portions of this section are based upon [4].
4. See Bernoulli (1738).
5. See Menger (1934).
6. See Kahneman and Tversky (1979).
7. See Knight (1921:19–20).
8. Portions of this section are based upon [27].
9. Other commonly cited causes of uninsurability include: (1) unavoidable problems of adverse selection; (2) unavoidable problems of moral and/or morale hazard; and (3) an insurance company’s inability to hedge or diversify the risks under consideration.
10. In conventional expected-utility analysis, boundedness is frequently required of utility functions, as in Menger’s (1934) resolution of the St. Petersburg Paradox.
11. See, for example, [16] and [21].
12. See Allais (1953).
13. See Savage (1954:102–103).
6. Aloofness and Quasi-Aloofness
1. Portions of this and the following section are based upon [18].
2. See Nelli (1972).
3. Reinsurance is the insurance purchased by a primary (i.e., ordinary) insurance company to cover some portion of the company’s total losses.
4. In insurance terminology, the word policyholder refers specifically to the owner of an insurance policy, whether an individual or a firm. For simplicity, however, I will use this word to include both (1) the actual policy owner and (2) any other individual or firm afforded coverage by the relevant policy (i.e., any other insured).
5. The precise distinction between these two hazards will be explained in Chapter 8.
6. Portions of this and subsequent sections are based upon [11].
7. A catastrophe bond is a custom-made debt instrument whose interest and principal payments are subject to restructuring in the event of a specifically defined catastrophe.
8. Recall that these two concepts were introduced in Chapter 1.
9. See Shubik (2006:12).
10. Shubik’s (2006) reference is to Schumpeter’s work on the problem of how innovations are financed within an economy.
11. A random (or stochastic) process formed by a sequence of random variables propagating through time is usually written as a random variable with a time index, t (e.g., X(t) or Xt). In the present discussion, this index is suppressed because the various random processes are always evaluated at (or just prior to) the same generic instant t.
7. Trustworthy Transfer; Probable Pooling
1. See Parker (1928).
2. Portions of this section are based upon [18].
3. This four-step ERM paradigm is generally applicable to all organizations, not just insurance companies.
4. See, for example, Baranoff, Brockett and Kahane (2009).
5. This process also may be called risk retention.
6. See [32].
7. Portions of this section are based upon [30].
8. Most likely, the principal reason that pacification tends to be ignored is the (mistaken) intuition that no individual or firm can benefit by offering XIn in exchange for XOut if SD [XIn] is less than SD [XOut] and the overall expected value of the portfolio remains unchanged (i.e., E[XIn] equals E[XO]). However, it is quite possible for an individual or firm to benefit from such an exchange if XOut possesses statistical properties that the individual or firm finds useful in hedging.
9. Diversification and hedging take place simultaneously if Corr[XKeep, XOut] is positive and Corr[XKeep, XIn] is negative. Unfortunately, this can be a source of confusion. For example, if an insurance company with an investment portfolio consisting exclusively of stock in a company that makes raincoats sells a portion of those shares and uses the proceeds to purchase stock in a company that makes suntan lotion (whose profits presumably are negatively correlated with the raincoat maker’s), then some may think of this transaction as diversification, whereas others may consider it hedging. In fact, the transaction rightfully should be identified as both.
10. Portions of this section are based upon [2].
11. See Sears v. Commissioner, 96 T.C. 61 (1991), aff’d in part and rev’d in part, 972 F.2d 858 (7th Cir. 1992).
12. See Helvering v. LeGierse, 312 U.S. 531 (1941).
13. The case of Helvering v. LeGierse involved the application of real estate tax law to benefits from a life insurance policy in a case in which the policyholder had purchased the life insurance policy in conjunction with a life annuity that effectively offset the risk of death from both the insurance company’s and the policyholder’s perspectives.
14. See AMERCO v. Commissioner, 96 T.C. 18 (1991) Aff’d 979 F.2d 162 (9th Cir. 1992), The Harper Group v. Commissioner, 96 T.C. 45 (1991) Aff’d 979 F.2d 1341 (9th Cir. 1992), the Sears case, and ODECO v. U.S., 24 Cl. Ct. 714 (1991), 92–1 U.S.T.C. 50,018.
15. See Humana v. Commissioner, 88 T.C. 197 (1987), aff’d in part and rev’d in part, 881 F.2d 247 (6th Cir. 1989).
16. See [2].
17. This is explicitly mentioned in the discussion of diversification in the prior section of the present chapter.
8. God-Awful Guessing and Bad Behavior
1. Portions of this section are based upon [18].
2. In the European Union, Pillar 1 of the Solvency II program performs functions similar to those of the U.S. RBC system.
3. Portions of this and the following section are based upon [9].
4. Naturally, this expectation is based upon an assumption that insurance-company losses are sufficiently light-tailed to permit benefits from diversification. Although this assumption is too broad to be universally true, it will be accepted as reasonable for the purposes of the present and following sections.
5. For purposes of studying the financial leverages of insurers, it is more appropriate to aggregate individual companies into their corporate groups, which have some ability to share financial resources internally. However, apart from the immediate description of the data presented in Figures 8.1 and 8.2, I will continue to frame the discussion in terms of insurance companies, rather than insurance groups.
6. The data shown in the scatter plot represent all U.S. property-liability groups (formed of both primary insurers and reinsurers) operating in calendar year 2009 for which A. M. Best recorded—on a consolidated basis—both a positive net written premium and a positive surplus.
7. This type of error can be subdivided into two separate components—model-selection error and parameter-estimation error—based upon two technical steps in the actuarial analysis used to set prices.
8. The Peter Principle is “any of several satirical ‘laws’ concerning organizational structure, especially one that holds that people tend to be promoted until they reach their level of incompetence.” (Named after Canadian educator Laurence J. Peter, 1919–1990. See the Random House Dictionary, 2011.)
9. At times (e.g., during the late 1980s), Philadelphia policyholders have had to pay some of the highest automobile insurance premiums in the United States.
9. The Good, the Bad, …
1. See Phelps (1895).
2. Portions of this section are based upon [18].
3. Portions of this section are based upon [29].
4. One possible substitute is a schedule of varying contract benefits/prices that permits different types of policyholders to “classify” themselves (i.e., reveal their individual risk characteristics) by selecting different contract terms in market equilibrium (e.g., low-risk policyholders might purchase higher deductibles than high-risk policyholders; see Rothschild and Stiglitz, 1976).
5. This is a simplification. In common practice, premium loadings for profits and expenses may be subdivided into a number of separate components, some of which are not directly proportional to expected losses (but rather are fixed on a per-policy basis or proportional to the final premium itself).
6. The new law’s requirement that all individuals purchase health insurance is scheduled to become effective in 2014. However, numerous weaknesses of this provision, including the availability of certain exemptions (e.g., for religious beliefs), the relatively small fines for noncompliance, and even the provision’s possible unconstitutionality, may undermine the economic logic of the entire health care law.
7. Note that the specific positions assigned to the various insurance lines in Figure 9.1 are based entirely upon the author’s subjective appraisals.
8. Portions of this and subsequent sections are based upon [31].
9. See [31].
10…. And the Lawyerly
1. See Breyer (1993).
2. Portions of this section are based upon [14].
3. See Shapley (1953).
4. 105 is the total number of ways the seven bottles can be permuted without distinguishing among the various bottles of any given color (i.e., 71/(41211!) = 105).
5. See Aumann and Shapley (1974).
6. Note that this formulation assumes that the box tips at exactly five bottles, rather than at some arbitrary point between five and six bottles. However, the exact tipping point is immaterial to the argument in the continuous case.
7. Portions of this section are based upon [3].
8. See Keeton and O’Connell (1965).
9. Portions of this section are based upon [5].
10. In [5], the three compensation principles actually were subdivided into four principles.
11. A Philadelphia lawyer is “a lawyer of outstanding ability at exploiting legal fine points and technicalities.” (See the Random House Dictionary, 2011.)
11. What Is Randomness?
1. Portions of this and the following section are based upon [19].
2. See Heisenberg (1956).
3. German physicist Albert Einstein famously expressed his aversion to this fundamental randomness by declaring, “I am convinced that He [God] does not play dice.” (See letter to Max Born, December 4, 1926, in Einstein, Born, and Born, 1969:130.)
4. See Hume (1748), Section VI.
5. See Hume (1748), Section VIII, Part I.
6. See Solomonoff (1964a, 1964b); Kolmogorov (1965); Chaitin (1969).
7. See Chaitin (1974). Perhaps Chaitin’s result should not be viewed as too surprising. After all, to show that an infinitely long sequence is incompressible seems to presuppose the ability to describe, or at least apprehend, the sequence in intimate detail; but it also seems reasonable to believe that such familiarity is possible only if the sequence is compressible.
8. Portions of this section are based upon [20].
9. In Claude Shannon’s information theory, the expected value of the surprise function denotes the entropy of the associated random variable and may be interpreted as the expected number of bits of information available in one random observation. (See Shannon, 1948.)
10. For more on the Gompertz-Makeham law, see Chapters 1 and 12.
11. See [20] for the mathematical details.
12. There is actually a second, mirror-image sequence that is just as unsurprising, which is formed by exchanging the 0s for 1s and vice versa in the specified sequence (i.e., 100010110 … ).
13. See Hume (1748), Section VIII, Part I.
14. This quote is frequently attributed, without precise citation, to either British writer G. K. Chesterton or American writer William F. Buckley, Jr.
12. Patterns, Real and Imagined
1. See von Leibniz (1686).
2. Portions of this section are based upon [13].
3. See Bode (1772).
4. This distance is expressed in astronomical units (AU), where 1 AU equals the average distance of Earth from the sun.
5. Specifically, the distance from the sun to planet x is given by d(x) = 0.4 + 0.3 x 2x-2. Letting a = 0.4, b = 0.15, and c=ln(2) ≈ 0.69315, the TBL can be rewritten as a + becx, which is mathematically identical to the GML.
6. See Gompertz (1825).
7. See Makeham (1860).
8. These ten planets include Ceres and Pluto, now formally designated dwarf planets.
9. The GML estimates are based upon the equation h(x) = a + becx, with parameter values chosen to mimic the approach of the TBL. First, I selected a = 0.00022 judgmentally to match the mortality hazard rate at age x = 10 (comparable to the choice of a = 0.4 for Mercury’s orbit in the TBL), and then solved for the other two parameters, b = 0.000034 and c = 0.09, by statistical estimation (using semi-loglinear regression for simplicity).
10. Note that the estimate for Mercury’s orbit (0.55) is actually different from that originally proposed by Titius and Bode (0.40) because the original estimate was obtained by effectively defining 2x-2 = 2-1 = 0 in the case of Mercury, for which x = 1. However, this step is essentially numerology. To fit Mercury’s orbit exactly within the framework of the TBL, one must introduce a fourth parameter (i.e., an additional parameter beyond a, b, and c).
11. In insurance, the combined ratio is an inverse measure of profitability (i.e., a higher [lower] value indicates less [more] profitability). It is computed by dividing the sum of incurred losses and earned expenses by earned premiums.
12. See Venezian (2006); Venezian and Leng (2006:Tables I and II).
13. The term hybrid refers to the automobile’s propulsion system rather than its fuel source.
14. See National Highway Traffic Safety Administration (2009).
15. The insightful reader may note that the proposed randomized controlled study fails to address one confounding variable introduced by the study’s design itself: The possibility that a driver’s knowledge of which type of automobile he or she has been assigned will affect his or her driving behavior (e.g., that a hybrid driver will make a special effort to drive carefully to improve the hybrid vehicles’ overall collision statistics). This is, undeniably, a valid source of concern, and one that cannot be addressed by imposing a double-blind configuration—in which drivers do not know which type of automobile they have been assigned—because it is not possible to disguise a vehicle’s propulsion system. However, it does seem rather unlikely that a driver’s behavior over the long (five-year) course of the research study would be more heavily influenced by the abstract motivations of the study than by the routine incentives imposed by law enforcement penalties, insurance policy surcharges, and possible civil litigation.
16. To some extent, the authors of the NHTSA’s hybrid study did acknowledge the limitations of their analysis, writing: “This study is exploratory in nature and aims to guide researchers when designing pedestrian and bicyclist crash prevention research.” However, the authors appeared to be concerned primarily with problems of small sample sizes, rather than the presence of potential confounding variables.
13. False Choices and Black Boxes
1. See Shubik (2002).
2. In fact, Property Claim Services—a division of ISO (the Insurance Services Office)—currently defines a catastrophe in the United States as a single event generating at least $25 million in insured property losses.
3. See Wells (1920).
4. There is a growing economic literature on the “willingness to pay” for mitigative efforts in the context of GCC. However, as shown by Johnson and Nemet (2010), the heterogeneity of approaches taken by researchers in this area—in terms of populations sampled, timings of surveys, stated effects of mitigation, etc.—has led to an extremely wide range of average annual dollar amounts that subjects would be willing to sacrifice (specifically, $22 to $3,623 per household). This empirical work thus affords little evidence either to confirm or to contradict my assertions regarding the random variable X.
5. I recognize that these views are at odds with the general tenor of the Intergovernmental Panel on Climate Change’s (2007) report on the likely impacts of GCC and thus may indicate a certain degree of thermophilia on my part.
6. Portions of this section are based upon [7], [10], and [26].
7. The largest commercial risk-analysis firms are Risk Management Solutions, Applied Insurance Research, and Eqecat.
8. One notable exception is the Florida Public Hurricane Loss Model, funded by the Florida Office of Insurance Regulation, whose technical transparency serves as a model to be emulated.
9. Again, the state of Florida provides an example worthy of emulation. Currently, the Florida Commission on Hurricane Loss Projection Methodology—an independent entity created by the state legislature—must review and approve all hurricane models used in computing relevant property insurance premiums in the state.
10. See http://typhoon.atmos.colostate.edu/forecasts/.
11. The 1992 hurricane season was the earliest for which Gray made a December forecast.
12. See <http://astro.temple.edu/~powersmr/#forecasts>.
13. This is true whether or not the 2005 spike is included. The small estimated increase is clearly relevant in assessing whether or not GCC is responsible for increases in hurricane frequency, especially when it is observed that the number of recorded hurricanes in earlier years (i.e., prior to the weather satellite era) may have been underreported.
14. See [10] for the modeling details.
15. The simplest such test is the nonparametric sign test using only the fifteen years for which one forecast is better than the other. Under the null hypothesis, H0: “The two forecast methodologies are equally good,” each year’s pair of forecasts yields an independent “0-1” trial, with probability 1/2 that the GK forecast is better (or worse) than the author’s. For this test, the two-tailed p-value is approximately 0.30.
16. The principal changes were that in the TRIEA: (1) several lines of business, such as commercial automobile and surety, were deleted from the covered list; (2) the insurance company deductible rate was raised from 15 percent in 2005 to 20 percent in 2007; and (3) the federal share of insured losses exceeding the deductible was decreased.
14. Nullifying the Dull Hypothesis
1. See Born (1949).
2. Portions of this and the following section are based upon [15].
3. See Radin (1997) for discussions of the restrictive standards—in terms of experimental conditions as well as choices of a—required of parapsychology studies.
4. Portions of this section are based upon [16].
5. See Hume (1748), Section IV, Part II.
6. See Hume (1748), Section V, Part I.
7. See Kant (1781).
8. To my knowledge, this particular argument was not made explicitly by Kant, but rather is a presumptive interpretation of my own.
9. See Fisher (1956:107).
10. These techniques are often called credibility methods because weights are assigned to the different sources of information based upon their relative credibilities, or intrinsic levels of believability to the decision maker. This terminology reflects a Bayesian orientation.
15. Games and the Mind
1. See Rousseau (1755).
2. See Dubey and Shubik (1978, 1980).
3. See, for example, [1].
4. Regarding the Colonel Blotto game, see, for example, [26].
5. Naturally, a more realistic model would account for all venues within the city, but the principles would be the same as in the simpler analysis.
6. Note that this structure is not necessarily appropriate for every such game; for example, the terrorist might well reverse the order of the third-best and worst outcomes if he or she is more interested in conserving resources through a smaller engagement than in gaining publicity through a larger engagement.
7. See Cournot (1838).
8. See Nash (1951).
9. See, for example, [26].
10. Portions of this section are based upon [22].
11. See Jervis (1978). Powers and Shubik (1991) provide an alternative motivation for this game in which a young man and a young woman whose romance is new and tentative have to decide whether or not to send each other a Valentine’s Day card. In this formulation, both young people would prefer the outcome in which each sends a card to the other, but each would be embarrassed if he or she were to send an unreciprocated card.
12. See [22].