Any institution responsible for the allocation of resources over time faces both risk and uncertainty. Risk can be defined as the probability of a set of events taking place where characteristics of the underlying probability distribution are unknown. Under uncertainty, these probabilities are unknown and one must rely on subjective probabilities to arrive at a decision. For general purposes we will use the term risk to refer to both types of probability, drawing distinctions where understanding and applications are critical to the process and outcome.

There are any types of risk: political, economic, and financial, for example, with sub-categories for each. Assessing these risks can be done as long as past events are typical of what future events will be. Responding to these risks is a function of the degree of risk aversion by decision-makers. Since the marginal cost of perfect information is likely to approach infinity, no one is willing to insure against all contingencies, and thus the choice that decision-makers face is one of deciding what is the prudent level of risk exposure.

Measuring risk requires the adoption of two basic simplifying assumptions. One is that events occur with the same probability over time, that is, they display stationarity in the conditional variance, or distribution of probabilities over time. Another is that the distribution of probabilities is normal, that is, that there is neither skewness nor kurtosis (differences in the height) in a distribution. In reality, these conditions often may not hold, thereby creating bias in the measurement of risk as well as in the selection of an appropriate risk-management instrument, namely, a specific form of insurance.

Once one has adopted specific assumptions regarding the stationarity and normality of a probability distribution, the next step is to link these properties to the risk preference of individual agents. While agents may be risk averse, risk neutral, or risk loving, risk aversion is the most frequently observed pattern of behavior. Given the degree of risk aversion, one can then select the degree of insurance coverage that corresponds the allocation of a given resource.

We can illustrate some of these properties in reference to the above graph. It shows four probability density functions, or distributions, namely, A, B, C, and D. All four are symmetric, which sets aside the question of skewness so that we can concentrate on other properties of risk. We can define risk as the distribution of probabilities, which is portrayed by the shape of the curves. In cases A and B, both have the same mean, but with different probabilities around the mean. Distribution A, which we can characterize as the Gaussian standard normal distribution, one can measure risk as the dispersion of probabilities around the mean. The standard deviation serves this function, and is measured as the square root of the sum of absolute deviations from the mean for each probability in the distribution. However, when one wants to compare the degree of risk for one asset in comparison to another it is useful to have a measure of relative risk. Relative risk can be defined in terms of the coefficient of variation, which is the ratio of the standard deviation to the mean of a distribution.

Let us now look at distribution B. It is more narrowly grouped around the mean, and thus has a smaller standard deviation and a smaller coefficient of variation, i.e., it has a smaller degree of absolute and relative risk. Given two events associated with these distributions, a risk averse agent will always prefer distribution B to distribution A. The question is how much is a risk averse agent willing to pay for the choice of a lower degree of risk.

We now look at distributions C and D. Both have the same means, and these means are higher than for the iso-means of distributions A and B. Given the choice between C and D, a risk averse agent will always prefer distribution C to D since it will have a smaller standard deviation and coefficient of variation.

Finally, we link these four distributions together in asking the question of what preference ordering an agent will display. A risk averse agent may be willing to accept either C or D over A or B as long as the outcome associated with the probability distribution is positive, e.g., winning the lottery, maximizing economic gain, or some positive incentive event.

Given the properties of a probability density function, the next step in modeling risk is to link the underlying probabilities to the value of alternative outcomes. These outcomes may be positive, negative, or mixed. The choice of an agent is thus to maximize the expected value of a choice, which is the sum of the associated products of the respective probabilities to their corresponding outcomes. The challenge for policymakers is to determine the extent to which asset pricing incorporates fully the degree of underlying risk, and to develop incentive compatible contracting systems to produce risk inclusive pricing in financial and economic markets.

Aaron, Henry J. and John B. Shoven. Should the United States Privatize Social Security? (Cambridge: MIT Press, 2000).

Aaron, Henry J., editor. Behavioral Dimensions of Retirement Economics. (Washington, D.C.: Brookings Institution, 1998).

Alexander, Carol. Market Models: A Guide to Financial Data Analysis. (New York: John Wiley and Sons, 2001).

Allen, Franklin, and Douglas Gale. Comparing Financial Systems. (Cambridge: MIT Press, 2000).

Auerbach, Alan. J., Laurence J. Kotlikoff, and Willi Leibritz. Generational Accounting Around the World. (Chicago: U. Chicago Press and NBER, 2000).

Baker, Dean and Mark Weisbrot. Social Security: The Phony Crisis. (Chicago: U. Chicago Press, 2000).

Bazley, John D., Loren A. Nikolai, and Hugh D. Grove. Financial Accounting: Concepts and Uses, third edition. (Cincinnati, Ohio: South-Western Publishing Company, 1995).

Bernstein, Peter L. Against the Gods: The Remarkable Story of Risk. (New York: John Wiley and Sons, 1998, 1996).

Binswanger, Mathias. Stock Markets, Speculative Bubbles and Economic Growth. (Northampton, Massachusetts: Edward Elgar Publishing Company, 1999)

Björk, Tomas. Arbitrage Theory in Continuous Time. (New York: Oxford University Press, 1998).

Block, Stanley B. and Geoffrey A. Hirt. Foundations of Financial Management, 9th edition. (New York: Irwin/McGraw-Hill Publishers, 2000).

Bradford, David F. Taxation, Wealth, and Saving. (Cambridge, Mass.: MIT Press, 2000).

Brealey, Richard A. and Stewart C. Myers. Principles of Corporate Finance, 6th edition. (New York: Irwin/McGraw-Hill, 2000).

Brunnermeier, Markus K. Asset Pricing under Asymmetric Infgormation: Bubbles, Crashes, Technical Analysis, and Herding. (New York: Oxford University Press, 2001).

Buchanan, James M. and Richard. A. Musgrave. Public Finance and Public Choice: Two Contrasting Visions of the State. (Cambridge: MIT Press, 2000).

Campbell, John Y., Andrew W. Lo, and A. Craig MacKinlay. The Econometrics of Financial Markets. (Princeton, New Jersey: Princeton University Press, 1997).

Chanellor, Edward. Devil Take the Hindmost: A History of Financial Speculation. (New York: Farrar , Straus, and Giroux, 1999).

Cole, Gerald E, "An Explanation of Pension Plans" Employee Benefits Journal 24:2 (June 1999), pp. 3-13.

Corrado, Charles J. and Bradford D. Jordan. Fundamentals of Investments: Valuation and Management. (New York: Irwin/McGraw-Hill, 2000).

DeSoto, Hernando. The Mystery of Capital: Why Capitalism Triumphs in the West and Fails Everywhere Else. (New York: Basic Books, 2000).

Farmer, Roger E.A. The Macroeconomics of Self-Fulfilling Prophecies. (Cambridge, Mass.: MIT Press, 1993).

Fischer, David Hackett. The Great Wave: Price Revolutions and the Rhythm of History. (New York: Oxford University Press, 1996).

Froot, Kenneth A., editor. The Financing of Catastrophe Risk. (Chicago, Illinois: The University of Chicago Press for the National Bureau of Economic Research, 1999).

Gollier, Christian. The Economics of Risk and Time. (Cambridge, Mass.: MIT Press, 2001).

Gup, Benton E., editor. International Banking Crises: Large-Scale Failures, Massive Government Interventions. (New York: Quorum Books, 1995).

Haugen, Robert A. The New Finance: The Case Against Efficient Markets. (Upper Saddle River, New Jersey: Prentice-Hall Publishers, 1995).

Helfert, Erich. Techniques of Financial Analysis, 10th edition. (New York: Irwin/McGraw-Hill, 2000).

Houthakker, Hendrik S. and Peter J. Williamson. The Economics of Financial Markets. (New York: Oxford University Press, 1996).

Hull, John C. Fundamentals of Futures and Options Markets, fourth edition. (Upper Saddle River, New Jersey: Prentice-Hall, 2002).

Hull, John C. Introduction to Futures and Options Markets, third edition. (Upper Saddle River, New Jersey: Prentice-Hall, 1998).

Hull, John C. Options, Futures, and Other Derivatives., 3rd. edition. (Upper Saddle River, New Jersey: Prentice-Hall Publishers, 1997).

Jacobs, Bruci I. Capital Ideas and Market Realities: Option Replication, Investor Behavior, and Stock Market Crashes. (New York: Basil Blackwell, 1999).

Johnson, Hazel. Global Financial Institutions and Markets. (New York: Basil Blackwell, 1999).

Jorion, Philippe. Financial Risk Management. (New York: Basil Blackwell, 1998).

Jorion, Philippe. Value at Risk: The New Benchmark for Controlling Derivatives Risk. (New York: McGraw-Hill Publishing Company, 1997).

Kindleberger, Charles P. Manias, Panics, and Crashes: A History of Financial Crises, third edition.. (New York: John Wiley and Sons, 1996).

Kolb, Robert W. Futures, Options and Swaps, third edition. (New York: Blackwell Publishers, 1999).

Kolb, Robert W. Options: An Introduction, 2nd edition. (New York: Basil Blackwell Publishers, 1995).

Kolb, Robert W. Financial Derivatives. (Miami, Florida: Kolb Publishing Company, 1993).

Kolb, Robert W. Understanding Futures Markets, third edition. (Miami, Florida: Kolb Publishing Company, 1991).

Laffont, Jean-Jacques. Essays in the Economics of Uncertainty. (Cambridge, Massachusetts: Harvard University Press, 1980).

Levy, Haim and Marshall Sarnat. Portfolio and Investment Selection: Theory and Practice. (New York: Prentice-Hall, 1984

Lo, Andrew W. and A. Craig MacKinlay. A Non-Random Walk Down Wall Street. (Princeton, New Jersey: Princeton University Press, 1999).

Lowenstein, Roger. When Genius Failed: The Rise and Fall of Long-Term Capital Management. (New York: Random House, 2000).

Macho-Stadler, Inés, and David Pérez-Castrillo. An Introduction to the Economics of Information: Incentives and Contracts. (New York: Oxford University Press, 1997).

Mackay, Charles. Extraordinary Popular Delusions and the Madness of Crowds. (New York: Three Rivers Press reprint of 1852, 1841 original editions, 1980).

Magill, Michael and Martine Quinzii. Theory of Incomplete Markets, vol. I. (Cambridge, Mass.: MIT Press, 1998, 1996).

Malkiel, Burton. A Random Walk Down Wall Street, 7th edition. (New York: W.W. Norton & Company, 1999).

Mariger, Randall P., "Social Security Privatization: What Are the Issues?" National Tax Journal LII: 4, pp. 783-802.

Maskus, Keith E. Intellectual property Rights in the Global Economy. (Washington, D.C.: Institute for International Economics, 2000).

Masuyama, Seiichi, editor. East Asia's Financial Systems: Evolution and Crisis. (Singapore: Institute of Southeast Asian Studies, 1999).

McInish, Thomas H. Capital Markets: A Global Perspective. (New York: Basil Blackwell Publishers, 1999).

Mills, Terence C. The Econometric Modelling of Financial Time Series, second edition. (Cambridge, Mass.: Cambridge University Press, 1999).

Milne, Frank. Finance Theory and Asset Pricing. (New York: Oxford University Press, 1999, 1995).

Mishkin, Frederic and Stanley Eakins. Financial Markets and Institutions, third edition. (New York: Addison-Wesley Publishers, 2000).

Molho, Ian. The Economics of Information: Lying and Cheating in Markets and Organizations. (New York: Blackwell Publishers, 1997).

Montes, Manuel F. and Vladimir V. Popov. The Asian Crisis Turns Global. (Singapore: Institute of Southeast Asian Studies, 1999).

Munnell, Alicia H. "Reforming Social Security: The Case Against Individual Accounts", National Tax Journal LII:4, pp. 803-817.

Neftci, Salih N. Introduction to the Mathematics of Financial Derivatives, second edition. (New York: Academic Press, 2000).

Neilsen, Lars Tyge. Pricing and Hedging of Derivative Securities. (New York: Oxford University Press, 1999).

Rose, Peter S. Money and Capital Markets, 7th edition. (New York: Irwin/McGraw-Hill, 2000).

Salanié, Bernard. Microeconomics of Market Failures. (Cambridge, Mass.: MIT Press translation of Microéconomie: Les défaillances du marché (1998), 2000).

Salanié, Bernard. The Economics of Contracts: A Primer. (Cambridge, Mass.: MIT Press translation of Théorie des Contrats (1997), 1997).

Samwick, Andrew A. "Social Security Reform in the United States", National Tax Journal LII: 4, pp. 819:842

Saunders, Anthony. Financial Institutions Management, third edition. (New York: Irwin/McGraw-Hill, 2000).

Saunders, Anthony. Credit Risk Measurement: New Approaches to Value at Risk and Other Paradigms. (New York: John Wiley and Sons, 1999).

Seidman, Laurence S. Funding Social Security: A Strategic Alternative. (New York: Cambridge U. Press, 1998).

Shiller, Robert J. Irrational Exuberance. (Princeton, New Jersey: Princeton University Press, 2000).

Shleifer, Andrei. Inefficient Markets. (New York: Oxford University Press, 1999).

Spencer, Bruce D. Statistics and Public Policy. (New York: Oxford University Press, 1997).

Stiglitz, Joseph E. Economics of the Public Sector, third edition. (New York: W.W. Norton & Co, 2000).

Taleb, Nassim Nicholas. Fooled by Randomness: The Hidden Role of Chance in the Markets and in Life. (New York: Texere Press, 2001).

Valdès-Prieto, Salvador, editor. The Economics of Pensions: Principles, Policies, and International Experience. (New York: Cambridge U. Press, 2000).

Viscusi, W. Kip. Rational Risk Policy. (New York: Oxford University Press, 1998).

Williamson, Oliver E. Markets and Hierarchies: Analysis and Antitrust Implications. (New York: The Free Press, 1983, 1975).