Category Archives: Risk

Arrow-Pratt method vis–à–vis the “exact” method for calculating risk premiums

I received an email from a Finance 4335 student earlier today asking for further clarification of the two methods for calculating risk premiums which we covered in class last Thursday. Under the so-called “exact” method, one 1) calculates expected utility, 2) sets expected utility equal to the utility of the certainty-equivalent of wealth, 3) solves for the certainty-equivalent of wealth, and 4) obtains the risk premium by calculating the difference between expected wealth and the certainty-equivalent of wealth.  On the other hand, the Arrow-Pratt method is an alternative method for calculating the risk premium which is based upon Taylor series approximations of expected utility of wealth and the utility of the certainty equivalent of wealth (the derivation for which appears on pp. 16-18 of Both of these approaches for calculating risk premiums are perfectly acceptable for purposes of Finance 4335.

The value added of Arrow-Pratt is that it analytically demonstrates how risk premiums depend upon two factors: 1) the magnitude of the risk itself (as indicated by variance), and 2) the degree to which the decision-maker is risk averse. For example, we showed in class on Thursday that the Arrow-Pratt coefficient for the logarithmic investor (for whom U(W) = ln W) is twice as large as the Arrow-Pratt coefficient for the square root investor (for whom U(W) = W.5); 1/W for the logarithmic investor compared with .5/W for the square root investor. Thus, the logarithmic investor behaves in a more risk averse than the square root investor; other things equal, the logarithmic investor will prefer to allocate less of her wealth to risky assets and buy more insurance than the square root investor. Another important insight yielded by Arrow-Pratt (at least for the utility functions considered so far in Finance 4335) is the notion of decreasing absolute risk aversion (DARA). Other things equal,  an investor with DARA preferences become less (more) risk averse as wealth increases (decreases).  Furthermore, such an investor increases (reduces) the dollar amount that she is willing to put at risk as she becomes wealthier (poorer).

How Do Energy Companies Measure the Temperature? Not in Fahrenheit or Celsius

Instead of Fahrenheit or Celsius, a metric called “degree days” is used to capture variability in temperature. The risk management lesson here is that this metric makes it possible to create risk indices which companies can rely upon for pricing and hedging weather-related risks with weather derivatives.

How Hurricane Florence Could Move Insurance Markets

Hurricane Florence provides a particularly timely and compelling case study of the economic consequences of natural catastrophes; specifically, the nexus of direct and indirect effects upon property insurance markets, reinsurance markets, alternative risk markets (e.g., catastrophe bonds), and public policy.

Some hurricanes are worse than others — both for people in the way and the insurance industry that tries to understand storms and put a price on their risks.

Confirmation bias in the form of “information avoidance”

This article from the Wall Street Journal provides an interesting follow-up to yesterday’s behavioral finance discussion. “Information avoidance” represents a particularly strong (and potentially deadly) form of confirmation bias!

Getting past information avoidance to deal with health issues, financial difficulties and other worries.

A brief synopsis of Finance 4335 course content to date

  1. The most important concept covered in Finance 4335 so far centers around the notion that people vary in terms of their preferences for bearing risk.  Although we focused most of our attention in upon modeling risk-averse behavior, we also considered examples of risk neutrality (where you only care about expected wealth and are indifferent about riskiness of wealth) and risk loving (where you actually prefer to bear risk and are willing to pay money for the opportunity to do so).
  2. Related to point 1: irrespective of whether you are risk averse, risk neutral, or risk loving, the foundation for decision-making under conditions of risk and uncertainty is expected utility.  Given a choice amongst various risky alternatives, one selects the choice which has the highest utility ranking.
  3. If you are risk averse, then E(W) > {W_{CE}} and the difference between E(W)  and {W_{CE}} is equal to the risk premium \lambda. Some practical implications — if you’re risk averse, then you are okay with buying “expensive” insurance at a price that exceeds the expected value of payment provided by the insurer, since (other things equal) you’d prefer to transfer risk to someone else if it’s not too expensive to do so. On the other hand, you are not willing to pay more than the certainty-equivalent for a bet on a sporting event or a game of chance.
  4. If you are risk neutral, then E(W) = {W_{CE}} and \lambda = 0; risk is inconsequential and all you care about is maximizing the expected value of wealth.
  5. If you are risk loving, then E(W) < {W_{CE}} and \lambda > 0; you are quite willing to pay for the opportunity to (on average) lose money.

Your Tolerance for Investment Risk Is Probably Not What You Think

This (year-old) WSJ article is authored by Professor Meir Statman, the Glenn Klimek Professor of Finance at Santa Clara University. Professor Statman’s research focuses on behavioral finance, which is a very important topic in decision theory that I plan to cover during next Tuesday’s meeting of Finance 4335.

Your Tolerance for Investment Risk Is Probably Not What You Think

The questions financial advisers ask clients to get at the answer actually measure something completely different—often leading to misguided investment strategies.

On the Determinants of Risk Aversion

Four years ago, The Economist published a particularly interesting article about the determinants of risk aversion, entitled “Risk off: Why some people are more cautious with their finances than others”. Here are some key takeaways from this article:

  1. Economists have long known that people are risk-averse, yet the willingness to run risks varies enormously among individuals and over time.
  2. Genetics explains a third of the difference in risk-taking; e.g., a Swedish study of twins finds that identical twins had “… a closer propensity to invest in shares” than fraternal ones.
  3. Upbringing, environment, and experience also matter; e.g., “…the educated and the rich are more daring financially. So are men, but apparently not for genetic reasons”.
  4. People’s financial history has a strong impact on their taste for risk; e.g., “… people who experienced high (low) returns on the stock market earlier in life were, years later, likelier to report a higher (lower) tolerance for risk, to own (not own) shares and to invest a bigger (smaller) slice of their assets in shares.”
  5. “Exposure to economic turmoil appears to dampen people’s appetite for risk irrespective of their personal financial losses.” Furthermore, a low tolerance for risk is linked to past emotional trauma.

More on the St. Petersburg Paradox…

During last Thursday’s class meeting, we discussed (among other things) the famous St. Petersburg Paradox. The source for this is Daniel Bernoulli’s famous article entitled “Exposition of a New Theory on the Measurement of Risk“. As was the standard practice in academia at the time, Bernoulli’s article was originally published in Latin in 1738. It was subsequently translated into English in 1954 and published a second time that same year in Econometrica (Volume 22, No. 1): pp. 22–36. Considering that this article was published 280 years ago in an obscure (presumably peer-reviewed) academic journal, it is fairly succinct and surprisingly easy to read.

Also, the Wikipedia article about Bernoulli’s article is worth reading. It provides the mathematics for determining the price at which the apostle Paul would have been indifferent about taking the apostle Peter up on this bet. The original numerical example proposed by Bernoulli focuses attention on Paul’s gamble per se and does not explicitly consider the effect of Paul’s initial wealth on his willingness to pay. However, the quote on page 31 of the article (“… that any reasonable man would sell his chance … for twenty ducats”) implies that Bernoulli may have considered Paul to be a millionaire, since (as shown in the Wikipedia article) the certainty-equivalent value of this bet to a millionaire who has logarithmic utility comes out to 20.88 ducats.