Risk and Choice





Richard MacMinn










Keynote Speech Presentation at

The International Risk Management and Insurance Conference




July 1999


Risks are commodities that may be exchanged.  The corporation, long viewed as a nexus of contracts, may also be viewed as a nexus of risks.  The corporation may be described as a composite commodity or bundle of risks that may be separated.  “Indeed, the history of the development of risk instruments is a tale of the progressive separation of risks, enabling each to be borne in the least expensive way,” (Kohn 1999).  An economy may achieve an efficient allocation of risks as well as resources through separation and trading, i.e., see (Arrow 1963) or (Debreu 1959).  Risks have traditionally been categorized as speculative or pure. 

The speculative risk may yield a gain or a loss.  Aristotle provides an early example of a speculative risk in Book I of Politics.  He tells a story about the philosopher Thales of Miletus.   According to the story, Thales was chided because he was poor and that was taken to be evidence that philosophy is of no practicalvalue.  Thales demonstrated the foolishness of the reproach.

Thales had exceptional skill in reading the stars.  One winter he foresaw that the autumn olive harvest would be much larger than normal.  He took the little money he had saved up and paid quiet visits to all the owners of olive presses in the area, placing small deposits with each of them to guarantee him first claim on the use of their presses when fall arrived.  He was able to negotiate low prices, for the harvest was still nine months off, and, anyway, who could know whether the harvest would be large or small.  The story ends as you may have guessed: “When the harvest-time came, and many [presses] were wanted all at once and of a sudden, he let them out at any rate he pleased, and made a quantity of money.  Thus he showed the world that philosophers can easily be rich if they like, but that their ambition is of another sort.’ [1]

This is an early example of an instrument that has become known as an option.[2]  The call options written on tulip bulbs on the Amsterdam exchange in the 1630’s provide another early but colorful example of such a risk. 

The pure risk may yield only a loss.  In the Middle Ages it was possible to retire into a monastery and obtain subsistence for the remainder of one’s life in exchange for an advance payment.  This is an early example of a life annuity.  Later it became possible for a merchant to buy life annuities.  Guilds provided a way to pool and spread risk, i.e., insure.  Guilds provided an early form of life insurance.  “Gilds established funds, maintained by periodic subscription, to support the widows and orphans of members and to extend loans to brethren in need.  Gilds and communes also provided insurance against fire, shipwreck, ransom, and other misfortunes: in the event of a loss on the part of one member, the others were assessed a contribution proportional to the value of their property.”[3]

The line between speculative and pure risks has never been clear.  The sea loan or foenus nauticum was in use in Genoa in the twelfth century but its origins go farther back; the sea loan is mentioned in the Hammurabi Code in 2250BC.[4]  The sea loan represented an exchange of risk.  This contract allowed an entrepreneur to purchase and equip a ship and the loan was repaid with interest only if the ship returned safe; otherwise the loan was forgiven.  The sea loan is an example of a debt instrument used to finance maritime commerce.  Such a loan is an example of a package of contracts including a debt contract and an insurance contract.  It was an imperfect attempt to separate the casualty risk from the business or credit risk, or equivalently, the pure risk from the speculative risk.  The creation of marine insurance in the 1300s represents another and more successful attempt to separate risks, i.e., casualty and credit.[5]

The line between speculative and pure risk has also historically been the line between the finance and insurance disciplines.  The line between pure and speculative risk has been blurred by recent innovations in contracting.  As the line has been blurred, the claim that risk is a commodity has been strengthen by those same innovations.[6]  In 1998 the first catastrophe (CAT) bond was issued; this type of bond is designed to provide a return to the issuer in the event of catastrophic losses due, for example, to hurricane damage.  The money used to purchase catastrophe bonds is invested in securities and the bondholders receive a return from the portfolio of securities in the event that a catastrophe does not occur.  The catastrophe bond is just one example of a securitization process that has begun and that will continue to blur the lines between risks and disciplines.

In this talk I want to provide a brief historical sketch of the development of the insurance discipline and then a discussion of some of the research challenges that we in the discipline face.

History of the Science


The revolutionary idea that defines the boundary between modern times and the past is the mastery of risk: the notion that the future is more than a whim of the gods and that men and women are not passive before nature.  Until human begins discovered a way across that boundary, the future was a mirror of the past or the murky domain of oracles and soothsayers who held monopoly over knowledge of anticipated events.[7]



In 1494 Paccioli posed the following problem.  How do you divide the stakes of an unfinished game of chance between two players when one is ahead?  This became known as the problem of the points.  The problem was not solved for almost two hundred years.  The resolution of this problem marks the beginning of probability theory and the quantification of risk.[8]  The quantification of risk is one of the necessary steps in valuing that risk.

Consider the following example of the problem.  Player A flips a coin.  If it comes up heads then A gets a point; otherwise B gets a point.  The first player to get three points wins the game.  Both players contribute fifty dollars and the winner takes all.  Player A is called away from the game when A has two points and B has one point.  The problem of the points is how to divide the one hundred dollars in prize money.  The solution method is contained in Chu Shih-chieh’s Precious Mirror of the Four Elements or, as it has become known, Pascal’s Triangle.[9]  The game will be over in two more coin tosses and three of the four include one or more heads for player A.  In this example, player A receives seventy-five dollars since three of the four outcomes yield a win for that player.

The method for quantifying risks with expected values found general agreement.  Daniel Bernoulli noted the use of expected value but thought that using it as a description of how people made decisions was flawed.  In order to motivate his expected utility theory by providing an application of it that solves the Saint Petersburg Paradox. 

Nicolaus proposes a game to be played between Peter and Paul, in which Peter tosses a coin and continues to toss it until it comes up heads.  Peter will pay Paul one ducat if heads comes up on the first toss, two ducats if heads comes up on the second toss, four ducats on the third, and so on.  With each additional throw the number of ducats Peter must pay Paul is doubled.  How much should someone pay Paul-who stands to rake in a sizable sum of money-for the privilege of taking his place in this game?[10]

The paradox arises because the expected value of the game is infinite.  Daniel Bernoulli introduced the idea that “[The] utility resulting from any small increase in wealth will be inversely proportionate to the quantity of goods previously possessed.”  His notion of utility and decreasing marginal utility provided a necessary distinction between the risk and the risk taker and a solution to the paradox.  He also provided a decision-making framework for choices in the face of risk.

How good or representative is the information that we have?  Daniel’s uncle Jacob Bernoulli provided another pillar of the discipline in answering that question.  Jacob Bernoulli provided us with the Law of Large Numbers, which tells us that increasing the sample size increases the probability that the sample mean will vary from the mean by less than a stated amount no matter how small.


Roots of Convergence

A convergence is occurring in the insurance and finance and disciplines.  There is also a convergence taking place in the financial services industry.  The imminent demise of the Glass-Steagall Act and the merger of Travelers and Citicorp are just two indications of the changing conditions and markets.  According Henri Louberge, “In 1973, the insurance/banking interface was a sensitive subject.  It was generally not well considered, in the insurance industry, to state that insurance was a financial claim and that insurers and bankers performed related functions in the economy.  Twenty-five years later, and after numerous recent experiences of mergers and agreements between banks and insurers, the question is not whether the two activities are closely related, but where do they differ.”[11]  The point is well taken.

In addition to a reliance on the foundations of the discipline, the roots of the convergence go back to key results contained in papers including (Von Neumann and Morgenstern 1944), (Arrow 1951; Arrow 1963; Arrow 1963), (Arrow 1974), (Markowitz 1952), (Modigliani and Miller 1958), (Debreu 1959), (Borch 1962), (Pratt 1964), (Rothschild and Stiglitz 1970; Rothschild and Stiglitz 1976).   Arrow’s “The Role of Securities in the Optimal Allocation of Risk Bearing”[12] and Debreu’s Uncertainty chapter in the Theory of Value helped set the stage by showing how risk could be incorporated in a general equilibrium model;  both used a contingent claims economy but Arrow went on to show that such an economy could be replaced with a stock market economy and achieve the same Pareto optimal allocation.  The allocations now included an optimal allocation of risk bearing as well as resources. 

Borch showed that Arrow’s general equilibrium model could be applied to the risk sharing problem among reinsurers; the result was far more robust, however, than originally noted.  The argument in Borch’s model is that, in an economy of risk-averse individuals, only non-diversifiable, equivalently systematic risks matter. Diversifiable risks do not because they can be eliminated using insurance, e.g., the reinsurance pool in Borch's analysis.  The systematic risks have to be shared.  Borch's optimal risk exchange theorem shows that systematic risks will be pooled and that risk averse individuals will share the pooled risk.[13]

Some of the early work like that of Arrow, Borch, and Debreu show how risks may be viewed, pooled and shared.  The models show that the relationships between the risks matter and that the exchange of the risks allows the economy to achieve efficiency.  The generality of Borch’s contribution should convince even the agnostic that the boundaries between insurance and finance are without merit; risk is a commodity that may be exchanged and, in that exchange, the relationship of the risk to other risks is the critical focus.  The relationship of the risk to others determines how the risk is handled and priced in an exchange.

Arrow and Pratt, i.e., (Pratt 1964; Arrow 1974), defined the measures of risk aversion and risk premia that provide part of the basis for understanding risk sharing and risk pricing.  Arrow defined a measure of absolute risk aversion that allowed a quantification and comparison of risk aversion to be made.  Similarly, Pratt defined a risk premium as that dollar amount which when added to the expected value of a risk makes the risk averse individual indifferent between that expected value plus the premium and holding the risk.  Pratt’s theorem shows that greater risk aversion is equivalent to a greater risk premium. 

A better understanding of risks and risk comparisons was still needed.  Rothschild and Stiglitz, i.e., (Rothschild and Stiglitz 1970), provided a perspective for comparing risks.  They showed the equivalence of three definitions of increasing risk.  Roughly put, a risk may be described as smaller than another risk, equivalently, its expected utility is larger than that of another risk; it may be described as having less weight in the tails of its distribution than another risk; the other risk may be described as this risk plus noise.  Rothschild and Stiglitz showed that these comparisons are equivalent.  While this risk comparison is useful and has provided the basis for many comparative static results, it is also very limited.  Little recognition is given to the character of risks or the processes that generate the risks.  Much of the literature simply distinguishes between that risk which is diversifiable and that which is not.  Such a simplification is useful but ignores the historical imperative.  The development of risk instruments and markets reveals the continued separation of risks so that each can be borne at the least cost.  The theory required to understand this behavior requires a more fundamental notion and understanding of risks.


Future of the Science

Doherty provides an interesting perspective on the changing nature of the science.

The new focus on managing corporate risk is not entirely due to the availability of hedging instruments.  It derives in part from changes in the intellectual climate.  Prior to the Capital Asset Pricing Model, the conventional wisdom was that risk was costly to the firm’s owners (and to other stakeholders) and, given risk aversion, it was held that its removal was beneficial.  This climate boded well for insurance: it followed routinely that insuring risk would enhance firm value.  But the CAPM message was that investors can diversify, thus there was no advantage to a firm hedging risk, when investors could achieve the same results in the management of their portfolios.  This caused a re-evaluation as to why firms still displayed a marked aversion to risk, even when it was diversifiable.  Under the new paradigm, risk is costly because it reduces the expected value of cash flows (e.g., it enhances incentive conflicts between the firm’s stakeholders, because it increases the costs of financial distress and because it can increase taxes when tax functions are non-linear).[14]

The 1958 Modigliani-Miller theorem, (Modigliani and Miller 1958), and the CAPM, (Sharpe 1964; Mossin 1966), have indeed had an impact on the intellectual environment.  The message of both is that, ceteris paribus, hedging does not increase value.  A literature has been generated that deals with the issue of value, e.g., (Jensen and Meckling 1976; Mayers and Smith 1982; Green 1984; Myers and Majluf 1984; MacMinn 1987; MacMinn 1987; MacMinn 1993; MacMinn and Garven 1993).  One important implication of these theoretical constructs is that risk management is more about the preservation and creation of value than it is about the elimination or reduction of risk.  Risk is not a “bad” to be eliminated rather it is a commodity to be created, managed and exchanged. 

The intellectual climate is important and causal but the historical imperative has not changed and has not been altered by the intellectual climate.  The risk markets are changing so rapidly that it seems that those changes are currently having more of an impact on the intellectual climate and activity than the reverse.  Part of the challenge for economists studying insurance and finance is generalize the notion of risk so that process of separating risks and allocating risks to minimize cost becomes more transparent.  The challenge includes the choice of instruments and markets.  What contractual forms and markets should be used for the risks?  Will financial markets replace some insurance markets?  Should some risks be securitized? 

Catastrophe Risks

The Law of Large Numbers implies that insuring risks is possible by constructing a large pool or portfolio of independent risks.  This, however, is not always possible.  Some risks have large positive correlation coefficients.  This makes insuring difficult but not impossible.

In recent years, the magnitude of catastrophic property-casualty disaster risks has become a major topic of debate.  The insurance industry now regularly discusses potential U.S. earthquake or hurricane losses of $50-$100 billion, a magnitude of lass that was unthinkable ten years ago.  The disasters of Hurricane Andrew and the Northridge Earthquake alone totaled over $45 billion in 1997 dollars, with the insured component running to almost $30 billion.  This compares with cumulative insured losses from natural catastrophes in the decade prior to those events (roughly 1980-92) of only about $25 billion (according to data from Property Claims Services).[15]

Catastrophic risks will continue to grow since, for example, the population of hazard prone states like California and Florida have grown at rates two or three times that of the national average.  An event loss will exceed $50 billion at some point.  Estimates from A. M. Best place the capital and surplus of U. S. insurers at about $239 billion but this capital and surplus is for all risks, not just catastrophes.  Reinsurance provides little more help since its capital and surplus is also small relative to the potential losses, i.e., $26.7 billion for U.S. reinsurers, $6.5 billion for Bermudan reinsurers, $7 billion for German reinsurers, and $16.8 billion for others.[16]  The estimates are all in 1997 dollars.

In 1992 the United Services Automobile Association (USAA) experienced a $600 million loss when Hurricane Andrew hit south Florida; the industry loss was $16.5 billion but a storm the size of Andrew about forty miles north would have resulted in insured losses of over $50 billion.  A variety of options are available to handle the catastrophe (CAT) losses.  The insurers can expand reinsurance, issue catastrophe bonds, purchase catastrophe options, engage in catastrophe or basis swaps, etc..  Reinsurance has some advantages because of the relationship that exists between insurer and reinsurer; this type of contracting can reduce the moral hazard and adverse selection problems but the capacity of the reinsurers raises questions about credit risk.

The CAT bond represents a securitization of the risk.  Securitization is the process of aggregating similar instruments into a negotiable security; this process is well known in real estate where the instruments are loans.  In 1998 USAA issued the first catastrophe (CAT) bond.  This type of bond is designed to provide a return to the issuer in the event of catastrophic losses due, for example, to a hurricane or an earthquake.  The money used to purchase catastrophe bonds is invested in securities and the bondholders receive a return from the portfolio of securities in the event that a catastrophe does not occur.  Some CAT bonds provide principal protection and others, with higher interest rates to compensate for the risk, do not.  As in the USAA case the instrument may be designed to cover a particular layer of CAT losses in its book of business and that will affect the probability that loss event coverage is triggered.  The CAT bonds provide a good instrument for diversification because the losses they cover are not highly correlated with other financial instruments.  One of the biggest advantages, however, is the securitization.  The catastrophic losses may represent no more than a normal day’s fluctuation in U.S. equity values and the CAT bonds tap the much greater capacity of the capital markets.  Since the U. S. financial markets represent about $12 trillion, a $50-$100 billion event would only represent about 40-80 basis points of wealth.

A CAT swap represents a diversification of the risk.  The diversification benefits are easy to see.  Suppose Lc and Lj represent the random net losses for a book of business on California quakes and Japan quakes, respectively.  Suppose a fraction q of the California book is swapped for that proportion of the Japan book.  Then the new book is the portfolio (1 - q) Lc + q Lj.  If the net random losses are independent then the diversification advantages are clear.[17]  The swap represent a form of geographic diversification but it does introduce some credit risk that will have to be borne or transferred.

A CAT option is a derivative.  An industry index of losses I is constructed and the options are written on that index.  A call option would have the form max{0, I – i} and so would be in the money for all indexed losses exceeding the strike price, equivalently, the loss i.  This instrument taps the capital markets for coverage and avoids some of the credit risk problems of the CAT swap.  This type of instrument, however, introduces the potential for basis risk because the loss experience of the firm may be different from that of the index.  If L represents the random loss of the firm then the basis risk would be the difference (I – L).  One of the advantages of the CAT option is that the use of an index eliminates the moral hazard and adverse selection problems that hamper the risk markets. 

It is possible to view the decision for or against the CAT option as a tradeoff between the agency costs and the basis risk costs.  It is also possible for the firm to select a CAT option and then eliminate the basis risk with a basis swap.  With a basis swap the firm trades the call option for loss coverage at a price.  Such a swap would reintroduce the agency costs that the CAT option eliminate but those costs would be reduced because they would be on the value of the basis risk rather than the risk on the whole book of business, i.e., any deterioration in underwriting standards due to the moral hazard problem would have to be reflected in the swap price but to a lesser extent.


Other Risks

There are many other risks that firms may choose to bear, transfer, or hedge.  The risks include property risk, casualty risk, interest rate risk, foreign exchange risk, commodity risk, credit risk, and weather risk.  Some risk management tools, e.g., insurance and leverage, can have either a direct or indirect impact on the consequences of several of these risks and hence may not be the lowest cost method for managing the risks.

The weather derivatives provide one recent example of an instrument designed to manage a risk that does not have an immediate and direct impact on other risks.  These derivative are written on an index of the weather conditions, e.g., cooling degree days or heating degree days, and can take forms such as that of calls, puts, swaps, caps, collars, or floors.  The development of these instruments provides an interesting example of the convergence in the insurance and finance markets.  It is also an example of the historical trend of the development of instruments that separate risk that will lower the total risk bearing cost; if this is not the case then the market for the instrument will eventually disappear.

Finally, there is an instrument that plays a central role in much of the new development of risk management.  Most of the instruments noted here can only succeed if the contracts are honored.  Hence, credit derivatives must play a central role in the new risk management.  The notional volumes have grown from $5-$10 billion in 1995 to between $50 and $100 billion in 1996 to over $250 billion in 1997.  Like other derivatives, the credit derivative can take a variety of forms.  The simplest is the credit swap.  Suppose firms C and J enter a CAT swap and, as above, suppose that Lc and Lj are the random net losses on the books of business for C and J, respectively.  Each firm incurs some credit risk.  Firm C may default when a California quake makes the net loss positive.  To protect itself against this credit exposure, J enters a deal with firm A in which J pays A a fixed amount and in return A agrees to assume C’s obligations to J in the event that C defaults.  This reduces J’s credit exposure and J only suffers a loss on the California quake exposure if both A and C default.



While the historical imperative of separation and cost minimization continues to hold, the pace of that separation of risks seem far more rapid.  The first option pricing models could rely on no arbitrage conditions and a security that was already valued in the financial market.  Some of the new derivative instruments are being written on indices for which there are no underlying assets that are traded, e.g., weather derivatives.  One of the many challenges for a new risk management is the development of a model sufficiently robust that it allows the risks to be valued and so the choices for cost minimization to be made.  That model will probably be a synthesis of insurance and finance. 



Arrow, K. J. (1974). Essays in the Theory of Risk Bearing, North Holland.

Doherty, N. (1997). “Corporate Insurance:  Competition from Capital Markets and Financial Institutions.” Assurances 65(1): 63-94.

Bernstein, P. L. (1992). Capital Ideas: The Improbable Origins of Modern Wall Street. New York: Free Press

Bernstein, P. L. (1996). Against the Gods: the Remarkable Story of Risk. New York: John Wiley & Sons.

Froot, K. A., Ed. (1999). The Financing of Catastrophe Risk. Chicago, The University of Chicago Press.

Green, R. C. (1984). “Investment Incentives, Debt and Warrants.” Journal of Financial Economics 13: 115-36.

Jensen, M. and W. Meckling (1976). “Theory of the Firm:  Managerial Behavior, Agency Costs and Ownership Structure.” Journal of Financial Economics 3: 305-60.

Kohn, M. (1999). Risk Instruments in the Medieval and Early Modern Economy. Working Paper 99-07

Louberge, H. (Forthcoming). Developments in Risk and Insurance Economics: The Past 25 Years. Handbook of Insurance. G. Dionne.

MacMinn, R. D. (1987). “Forward Markets, Stock Markets, and the Theory of the Firm.” Journal of Finance 42(5): 1167-85.

MacMinn, R. D. (1987). “Insurance and Corporate Risk Management.” Journal of Risk and Insurance 54(4): 658-77.

MacMinn, R. D. (1993). On The Risk Shifting Problem and Convertible Bonds. Advances in Quantitative Analysis of Finance and Accounting.

MacMinn, R. D. and J. Garven (1993). “The Under-investment Problem, Bond Covenants and Insurance.” Journal of Risk and Insurance.

Mayers, D. and C. Smith (1982). “On the Corporate Demand for Insurance.” Journal of Business 55: 281-96.

Modigliani, F. and M. H. Miller (1958). “The Cost of Capital, Corporation Finance and the Theory of Investment.” American Economic Review.

Mossin, J. (1966). “Equilibrium in a Capital Asset Market.” Econometrica 34: 768-783.

Myers, S. C. and N. S. Majluf (1984). “Corporate Financing and Investment Decisions When Firms Have Information That Investors Do Not Have.” Journal of Financial Economics 13: 187-221.

Pratt, J. W. (1964). “Risk Aversion in the Small and in the Large.” Econometrica 32: 122-36.

Rothschild, M. and J. E. Stiglitz (1970). “Increasing Risk:  I. A Definition.” Journal of Economic Theory 2: 225-43.

Sharpe, W. F. (1964). “Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk.” Journal of Finance 19(3): 425-442.



Suppose the firm purchases index option on catastrophic risks.  Let I be the random index loss; this may be an industry or region average loss.  Suppose options are written on the index.  Let i denote the option strike price.  The payoff on the call is max{0, I – i} and firm payoff is

                                                        P – L + max{0, I – i}

where P is the business risk.  In the absence of basis risk L - I.  If a loss occurs and there is basis risk then the payoff is

                                             P – L + max{0, I – i} = P - i + I – L

The difference I – L represents the gain or loss on the basis risk.

Now suppose the firm uses a swap to eliminate the basis risk.  The firm delivers the call option with the payoff max{0, I – i} in exchange for L – p where p is the price of the swap.  If a loss occurs then the net payoff is (i – p) - (I – L) and the firm payoff is P - p.  If the swap price is less than the call strike price less the value of basis risk then the swap would be preferred.


[1] (Bernstein 1992), pp. 203-4.

[2] An option is a contract that gives its owner the right to take a specified action under conditions indicated and agreed to in advance in the contract.  A call option, for example, gives its owner the right to purchase a risk at a previously agreed price.

[3] See p. 13 of (Kohn 1999).

[4] (Kohn 1999), pp. 2-5.  Also see Borch, (Borch 1974), who mentions the bottomry bond; it is a special case of the sea loan.

[5] (Kohn 1999), pp. 5-10.  According to Kohn the earliest surviving explicit insurance contract was written in Palermo in 1350 and it specified a premium of 54 florins to cover a possible loss of 300 florins. 

[6] The claim that history is the tale of a progressive separation of risks is not supported by this development. The CAT bond, however, is similar to the sea loan and so may represent an intermediate step in the development of a new contract.

[7] See (Bernstein 1996) p. 1.  Much of the following is based on Bernstein.

[8] This is an important insight due to Bernstein, p. 43.

[9] See (Bernstein 1996), p. 64.

[10] See (Bernstein 1996), p. 106.

[11] See (Louberge Forthcoming)

[12] This originally appeared as "Le rôle des valeurs boursières pour la répartition la meilleure des risques," in Econométrie, CNRS, Paris, 1953, 41-47

[13] It may be noted that Borch’s theorem contains the Capital Asset Pricing Model (CAPM) as a special case.  Louberge (Louberge Forthcoming) provides a more complete discussion of the Borch theorem.  Also see (Doherty 1997).

[14] See (Doherty 1997), p. 65.

[15] (Froot 1999), p. 1.

[16] (Froot 1999), p. 2.

[17] See (Samuelson 1967) and (MacMinn 1984).


Modification Date:  Tuesday, 29 May 2007 12:06 -0700
Comments to: Richard MacMinn