Decision Making Under Risk and Uncertainty

Decision Making under Risk and Uncertainty (Part 3 of 4)1Lecture #7: Decision Making Under Risk and Uncertainty (Part 3)Take calculated risks.That is quite different from being rash.General George S. Patton

#What We Learned Last TimeExpected utility theory enables us to model how investors make tradeoffs between risk and reward.Arrow-Pratt measures of risk aversionAbsolute risk aversion (RA(W) = -UWW(W)/UW(W)).By measuring the utility functions curvature, RA(W) indicates the investors degree of risk aversion at a given level of wealth. Relative risk aversion (RR(W) = WRA(W)).RR(W) measures how the proportion of wealth that the investor is willing to risk changes as wealth changes.2Lecture #7: Decision Making Under Risk and Uncertainty (Part 3)I.e., as investors grow wealthier, they typically are willing to put larger dollar amounts at risk, but the proportion that they are willing to risk stays roughly the same.

#What We Learned Last TimeEmpirically, investors typically exhibit decreasing absolute risk aversion (DARA) and constant relative risk aversion (CRRA).Risk premiums depend upon an objective factor (i.e., how risky the risk is, as measured by variance) and a subjective factor (how risk averse the risk taker is, as measured by the Arrow-Pratt absolute risk aversion coefficient). 3Lecture #7: Decision Making Under Risk and Uncertainty (Part 3)The DARA/CRRA observations explain why most of our analysis is done with logarithmic and power utility functions. Also, the U = -W^-1 function has these properties.

It is important to note that Arrow-Pratt is an approximation since it ignores the impact of other potentially important risk characteristics beyond variance (such as skewness and kurtosis); well take this issue up in todays lecture!

#Todays AgendaMore on using Arrow-Pratt to compare degree of risk aversionBroader Definitions of Risk and Risk PreferenceMean-Variance Analysis: a shortcut to the Expected Utility Model4Lecture #7: Decision Making Under Risk and Uncertainty (Part 3)We are interested in finding Broader Definitions of Risk and Risk Preference because in some cases, variance may not be a complete risk measure distributions may be skewed positively or negatively, and they may also have fat tails, which is yet another source of risk.#Using Arrow-Pratt to compare degree of risk aversion

5Lecture #7: Decision Making Under Risk and Uncertainty (Part 3)#

6Lecture #7: Decision Making Under Risk and Uncertainty (Part 3)Using Arrow-Pratt to compare degree of risk aversion I generally find that it helps to develop the calculus behind the FOC on the board so that students can better see whats going on.Note that for the natural log function, df/dx = 1/x.It is helpful to reference the math tutorial, where we discussed how to maximize a function. The idea is that the first order condition (FOC) gives us an equation from which we can solve for the optimal value of the decision variable. Since EU = .6U(W+B) + .4U(W-B), the derivative involves using the chain rule. From the chain rule, we know that if y = u(v(x)), then y = u(v(x))v. Therefore, the first derivative of EU with respect to B is .6U(W+B) + .4U(W-B)(-1).Working from the general form of the derivative, we rewrite this in terms of state contingent marginal utility for the log and power functions. #7Lecture #7: Decision Making Under Risk and Uncertainty (Part 3)Using Arrow-Pratt to compare degree of risk aversion What happens as the odds worsen (improve)?Heres what are two bettors would do, according to expected utility theory (p represents the odds of winning):

The original case is where pi = .6; in this table, we see the results that we have already obtained.The result for pi =.5 is quite interesting. This implies an actuarially fair bet; i.e., .5(B) + .5(-B) = 0. Although the bet is actuarially fair, neither bettor has any interest in betting. Logically, this is a corollary to the Bernoulli hypothesis in the theory of the demand for insurance. We show this in the upcoming insurance economics lecture, where we find that arbitrarily risk averse consumers will always purchase actuarially fair insurance.It is obvious that the log utility bettor is more risk averse. Irrespective of the odds, she always puts less wealth at risk than the power utility bettor.The negative values simply imply that the bettor is willing to offer bets at those unfair odds; the more money they can make on the bet, the more they are willing to "short sell" the bet.#Note that the power utility bettor is always willing to wager more, irrespective of the odds. This is not surprising since our earlier analysis showed that a person whose utility equals the square root of wealth has half of the risk aversion of a log utility individual.Furthermore, since both bettors have CRRA preferences, we know that this result holds irrespective of the level of initial wealth!8Lecture #7: Decision Making Under Risk and Uncertainty (Part 3)Using Arrow-Pratt to compare degree of risk aversion #9Lecture #7: Decision Making Under Risk and Uncertainty (Part 3)Broader Definitions of Risk and Risk Preference

Recall that for any function f(x), we can characterize the approximate value of that function evaluated at the point using a Taylor series expansion:

where Rn+1 is a remainder term.Using a Taylor series, we can characterize the value of U(W) around the expected value of wealth, E(W):

The analysis up to this point has implicitly focused on comparing means and variances of probability distributions; clearly, expected utility rewards expected value and penalizes volatility. However, expected utility is also sensitive to higher order moments, as indicated by the Taylor series.It is useful to explain to students how the second equation (approximating utility with the Taylor series) maps to the first equation (a general characterization of the Taylor series). f(x) corresponds to U(E(W)), dx corresponds to W-E(W), f(x+dx) corresponds to U(W-E(W) + E(W)) = U(W). #10Lecture #7: Decision Making Under Risk and Uncertainty (Part 3)Broader Definitions of Risk and Risk PreferenceSince we are interested in the expected utility of wealth, we compute this by applying the expected value operator to both sides of the previous equation (we also assume that R5 is negligible):

Thus expected utility is a function of expected wealth (E(W)), variance ( ),

Note that E(W-E(W)) = 0, which is why the term related to the first derivative drops out.Note that if there is no skewness or kurtosis (as in the case of the normal distribution, then expected utility only depends upon expected wealth and variance of wealth; thus we can use the mean variance model to evaluate risks (more on this toward the end of this lecture).See Eeckhoudt, L, 2012, Beyond Risk Aversion: Why, How and Whats Next?, Geneva Risk and Insurance Review, pp. 1-15. The equation on this page corresponds to Eeckhoudts equation 1 on page 3 of his paper.

#11Lecture #7: Decision Making Under Risk and Uncertainty (Part 3)Broader Definitions of Risk and Risk PreferenceExpected value, variance, skewness, and kurtosis provide us with the following information:Expected value represents the mean, or central value about which variable observations scatter;Variance/Standard Deviation indicate how far most of the variable observations scatter about the mean;Skewness indicates the lack of symmetry, or the degree to which variable observations pile up on either side of the mean; and Kurtosis indicates how far variable observations scatter from the mean.

The source for this characterization of mean, variance, skewness, and kurtosis is taken from The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century (@ http://amzn.to/jKk9Fa), p. 16.As we explained in the statistics lecture, variance traditionally is relied upon in finance and risk management theory as the primary measure of risk. However, skewness and kurtosis are also important aspects of risk, as we see here in our decomposition of the expected utility function.In the case of skewness, a positively (negatively) skewed distribution has more than half of the variable observations occurring to the right (left) of the mean. A symmetric distribution such as the normal is symmetric (not skewed), in the sense that half of the observations lie to the right and to the left of the meanKurtosis is generally interpreted as an indication of whether the probability distribution has fat tails. Compared with a thin tailed distribution such as the normal distribution, a fat tailed distribution has more extreme outcomes than does the thin tailed distribution.#12Lecture #7: Decision Making Under Risk and Uncertainty (Part 3)Broader Definitions of Risk and Risk PreferenceAn interesting question concerns how expected utility varies with respect to changes in expected wealth (E(W) ), variance, skewness, and kurtosis.

All utility functions which have diminishing marginal utility have these features i.e., positive expected wealth preference, negative variance preference, positive skewness preference, and negative kurtosis preference. A check of common utilities such as U(W) = ln W, U(W) = W.5, and U(W) = 1-exp(-aW) all reveal this to be the case. Differentiating with respect to E(W), we obtain marginal utility evaluated at E(W), which by definition must be positive.Differentiating with respect to variance, we obtain multiplied by the second derivative. Suppose U = Wn, where 0 < n < 1. then:UW = nWn-1 > 0;UWw = n(n-1)Wn-2 < 0;UWww = n(n-1)(n-2)Wn-3 > 0;UWWWW = n(n-1)(n-2)(n-3)Wn-4 < 0;Thus it follows that we obtain moment preferences which are positive for odd valued moments and negative for even valued moments. Consequently, we obtain positive E(W) and skewness preference, and negative variance and kurtosis preference.

#Lecture #7: Decision Making Under Risk and Uncertainty (Part 3)13Most of your previous analysis of reward and risk (prior to this course) has focused uponthe tradeoff between mean and variance. Other things equal, a higher mean return is preferred to a lower mean return, and a lower variance is preferred to a higher variance.EU theory encompasses mean-variance analysis as a special case while further challenging us to think more carefully about the nature of risk; while variance is an important risk attribute, so are other characteristics of probability distributions such as skewness and kurtosis. Broader Definitions of Risk and Risk PreferenceLecture #7: Decision Making Under Risk and Uncertainty (Part 3)14

Broader Definitions of Risk and Risk PreferenceIntuitively, this result makes sense because the positively skewed distribution exposes the investor to small probabilities of large positive returns, whereas the negatively skewed distribution exposes the investor to small probabilities of large negative returns. #Lecture #7: Decision Making Under Risk and Uncertainty (Part 3)15

Broader Definitions of Risk and Risk PreferenceThe fat-tailed (higher kurtosis) distribution hasmore observations in the tails andis thinner in the midrange than the normal distribution (lower kurtosis) with which it is shown. Thus higher kurtosis implies that more of the variance is the result of infrequent extreme deviations away from the mean, as opposed to frequent modestly sized deviations.From the investors perspective, the fat-tailed distribution is more risky than the thin-tailed distribution; thus given a choice between the two distributions shown above, she will prefer the thin-tailed distribution since it will provide her with higher expected utility.As far as kurtosis goes, investors dont like tails that are too fat because it increases the risk of a major loss. Leptokurtosis (fat tails) is characteristic of assets that are prone to price shocks (e.g., financial markets during the financial crisis). Kurtosis also appears to be a feature of markets for contingent claims that pay off in the event of catastrophes, which helps explain why there is such limited insurance for catastrophe risks (e.g., natural disasters and terrorism).

#16Lecture #7: Decision Making Under Risk and Uncertainty (Part 3)Broader Definitions of Risk and Risk PreferenceConsider the following numerical example: X and Y are both symmetrically distributed (i.e., skewness is zero). However, both are kurtotic; in fact, Y is significantly more fat-tailed than X. Even though X has higher variance than Y, it provides higher expected utility since its kurtosis is lower.X(s)p(s)MeanVarianceSkewnessKurtosisU1(X) = ln[1+X]U2(X) = X^.53.585850%1.79291.0000-1.41421.99991.52301.89366.414250%3.20711.00001.41421.99992.00342.532652041.76322.2131Y(s)p(s)MeanVarianceSkewnessKurtosisU1(Y) = ln[1+Y]U2(Y) = Y^.503%00.7500-3.750018.75000.00000.0000594%4.70.00000.00000.00001.79182.2361103%0.30.75003.750018.75002.39793.162351.5037.51.75622.1968This numerical example is taken from Brockett, P. and J. Garven, 1998, A Reexamination of the Relationship between Preferences and Moment Orderings by Rational Risk Averse Investors, Geneva Papers on Risk and Insurance Theory, Vol. 23, No. 2 (December), pp. 127-137. The utility functions are the exponential, logarithmic, and power functions. Note that 1 and 2 have the same mean and skewness. 1 has higher variance, but lower kurtosis than 2.

#EU Theory, MV Analysis, and SD AnalysisExpected utility (EU) theory is the foundation for decision-making under risk and uncertainty.Although EU theory is elegant, a more practical, scientific (i.e., data-based) framework would be quite helpful as we seek to implement the EU model; i.e., can we find useful shortcuts to EU?Two shortcuts come to mind:Mean-Variance (MV) AnalysisStochastic Dominance (SD) Analysis17Lecture #7: Decision Making Under Risk and Uncertainty (Part 3)#18Lecture #7: Decision Making Under Risk and Uncertainty (Part 3)Mean-Variance (MV) AnalysisMV analysis = EU analysis if variance is a complete risk measure and if:E(x) > E(y) and E(x) > E(y) and E(x) = E(y) and (mean preserving spread)However, watch out for differences in higher order parameters such as skewness and kurtosis!

19Lecture #7: Decision Making Under Risk and Uncertainty (Part 3)MV Analysis Example

20Lecture #7: Decision Making Under Risk and Uncertainty (Part 3)MV Analysis Example

21Lecture #7: Decision Making Under Risk and Uncertainty (Part 3)MV Analysis Example

The key point is that MV analysis is overly restrictive; i.e., circumstances often arise which make ranking ordering on this basis logically inconsistent with EU theory.# Suppose you have initial wealth of $W0 and are considering a gamble that has a 60 percent chance of winning $B and 40 percent chance of losing $B. How much money are you willing to wager on this bet?

To answer this question, we must determine the value of B that allows us to maximize expected utility; i.e., we must solve the following equation:

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First order condition: .

Suppose U(W) = ln W. Then, or B = .2W0; i.e., you are willing to bet 20% of your initial wealth.

Suppose U(W) =. Then, or B = .385W0; i.e., you are willing to bet 38.5% of your initial wealth. Since logarithmic and power utility imply CRRA, at these odds the logarithmic bettor will always wager 20%, and the square root bettor will always wager 38.5%, irrespective of the value for W0._1119027393.unknown

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Since can be negative, zero, or positive. Here are pictures of negatively skewed and positively skewed distributions:

EU theory implies if these two distributions have the same expected value, variance, and kurtosis, then the positively skewed distribution is preferred by an arbitrarily risk averse investor to the negatively skewed distribution. According to EU theory, if two riskshave the same expected value, variance, and skewness, then the more thin-tailed (lower kurtosis) riskis preferred by an arbitrarily risk averse investor to the fat-tailed(higher kurtosis) risk; e.g., consider the following picture:

Consider two risky prospects, Xl and X2, with payoffs given by:

and

Assume that your initial wealth (W0) is $0, and your utility U(W) = ._1265213696.unknown

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Which prospect is preferred according to the mean-variance rule?

SOLUTION:

According to the MV rule, Xl is preferred to X2 because E(X1) > E(X2) and

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and

EMBED Equation.DSMT4 Why the apparent conflict between MV and EU?

Note that X2 is highly positively skewed (i.e., it provides a small chance of a really large payoff), whereas X1 is symmetric about its mean.

Apparently positive skewness effect more than offsets its negative expected value, variance, and kurtosis effects.

Here, the MV rule is not appropriate because variance is not a complete risk measure. If one selected according to the mean variance rule, one would make the wrong choice!

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