Expected utility theory; Expected Utility Theory; risk aversion and utility functions

Transcription

1 ; Expected Utility Theory; risk aversion and utility functions Prof. Massimo Guidolin Portfolio Management Spring 2016

2 Outline and objectives Utility functions The expected utility theorem and the axioms of choice Properties of utility functions: non-satiation and risk preferences Absolute vs. Relative risk aversion The effects of the investment horizon 2

3 Generalities Modelling preferences will help us to exactly pin down the optimal portfolio selected on the efficient set by each investor Thus far we have discussed that to solve a portfolio problem we have to define the opportunity set and a preference function Under a few assumptions, we focused only on the efficient set We now turn to defining preferences (for or against risk) Interestingly, we have appealed already to some properties of such preferences to build the very efficient set o For instance, ptfs. inside the minimum variance cloud can be ignored because for any given level of risk, other portfolios have a higher mean; for a given mean ptf. return, other ptfs. carry lower risk Consider the following three investments: 3

4 Utility functions A utility function of wealth converts wealth outcomes in subjectively perceived value, i.e., investors satistifaction or happiness An intuitive approach consists of converting the outcomes in the value that these carry to investors o Of course, if the outcomes were the same as monetary payoffs, then value and outcome may correspond o However, it is easy to imagine reasons for paying more attention to the lowest outcomes in the worse states over the average/best ones Suppose the investor uses the following utility function to assign values to outcomes: which is a quadratic function A utility function of wealth (money) is a cardinal object that converts wealth outcomes in subjectively perceived value, i.e., investors satisfaction or happiness Because different outcomes come with an associated probability distributions, different investments will be indexed by their expected utility 4

5 Utility functions The expected utility of the three investments are computed as each of the outcomes times the value of probabilities Thus, an investor with a quadratic utility function, would select investment A. Cardinal utility functions of this type (i.e., functions that attribute a meaning to the U(W) happiness possess a key property 5

6 Utility functions Utility functions are unique up to monotone increasing linear transformations This means that A + bv(w) with b > 0 will lead to the same portfolio choices as v(w) o E.g., assume that preferences are V(W) = W (3/10)W 2 o The only difference between the two functions is the addition of the number 2 and the multiplication by 3 o Thus, the value of each outcome would be increased by two times the probability of the outcome plus 3 times wealth multiplied by the probability of the outcome o But if some choice gives a wealth W* such that U(W) = 4W (1/10)W 2 > U(W*) = 4W* (1/10)(W*) 2 then V(W) = 2 + 3W (3/10)W 2 = 2 + 3U(W) > V(W*) = 2 + 4W* (3/10)(W*) 2 = 2 + 3U(W*) If the investor obeys certain postulates, then the choice of preferred investment, using the expected utility theorem, is identical to the choice made by examining the investment directly 6

7 The expected utility theorem and the axioms of choice Utility functions are unique up to monotone increasing linear transformations The expected utility theorem (EUT) can be developed from a set of postulates concerning investors behavior If an investor acts in according to these axioms, her behavior is indistinguishable from taking decisions on the basis of the EUT The axioms are: Comparability. An investor can state a preference among all alternative outcomes; thus, if the investor has a choice of outcome A or B, a preference for A to be or of B to A can be stated or indifference btw. them can be expressed Transitivity. If an investor prefers A to B and B to C then she will have to also prefer A to C o I.e., investors are consistent in their ranking of outcomes o Although this seems reasonable, considerable experimental evidence displays stark cases of violations 7

8 The expected utility theorem and the axioms of choice o The difficulty occurs because some situations are sufficiently complex that the investor is unable to understand all of the implications of their decisions o In experimental situations, when the presence of irrational intransitivies are pointed out, subjects tend to revise their decisions Independence. Consider the certain prospects X and Y and assume the investor is the investor is indifferent btw. them; independence implies that the investor will also be indifferent btw: X with probabilty P and Z with probability 1 - P, and Y with probabilty P and Z with probability 1 - P, o If a person is indifferent btw. winning a Panda or a 500, then she will also be indifferent between a lottery ticket for 10 euros that gave a 1 in 500 chance of winning a Panda and a different lottery ticket for 10 euros that gave a 1 in 500 chance of winning a 500 Certainty Equivalent (continuity). For every gamble, there is a value (called certainty equivalent, CER) that makes the investor indifferent btw. the gamble and the CER 8

9 The expected utility theorem and the axioms of choice Using these axioms, we can derive the expected utility theorem o Consider a security G with two possible outcomes o Let C be the amount that would make the investor indifferent btw. gamble G and receiving C, the CER; clearly C depends on the prob. h o From axiom 4, C must exist; if we vary h, then a different value of C would be appropriate o If we varied h over a large number of values and then plotted all values of h versus C, we may have the following diagram o The investor's preference curve separates combinations of C and h for which the investor prefers the risky gamble from points where the investor prefers the certain amount o Points above the curve are points where the gamble is preferred and points below the curve are points where the CER is preferred 9

10 The expected utility theorem and the axioms of choice o Now consider a portfolio of securities S 1 with N possible outcomes: o Each W i is a known payoff and since C i s exist for all W i s, S 1 is equivalent to o Since for every C i there exists an equivalent lottery, we can represent an equivalent lottery as you can see on the right o If the investor declares btw. the C i s and each lottery, then S 1 S 2 10

11 The expected utility theorem and the axioms of choice o For instance, if outcome i occurs, if the investor selects S 1 then C i is received; if the investor selects S 2, then the investor receives b with prob. h i, and 0 with prob. 1 - h i o However investor has indicated in the construction of the preference curve an indifference btw. C i and this lottery; but W i is equal to C i so that the investor is indifferent btw. W i and this lottery o Thus, security 2 is equivalent to security 1 o From axiom 3 the investor does not change preference simply because the alternatives are part of a lottery o As the picture shows, S 2 has only 2 possible outcomes, b and 0 o We can equivalently write S 2 has payoff b with prob. and 0 with prob. 1 - o Utilizing this technique with any portfolio, that can be therefore reduced to two outcomes, b and 0, with known probabilities o How do we choose between these portfolios? We only need to consider the probability of receveing b 11

12 The expected utility theorem and the axioms of choice o Define ; then if H K > H L, security K is to be preferred to security L o This leads directly to the EUT: earlier we have replaced W i with C i, to which h i was associated o Call the function that relates W i to h i a utility function, h i = U(W i ) and note that H i = i P i h i = i P i U(W i ) o But i P i U(W i ) is simply expected utility and ranking securities on the basis of H i is equivalent to using expected utility Having an investor make choices between a series of simple investments, we can attempt to determine the weighting (utility) function that the investor is implicitly using Applying this weighting function to more complicated investments, we should be able to determine which one the investor would choose o A number of brokerage firms have developed programs to extract the utility function by confronting investors with simple choices o These have not been particularly successful 12

13 Properties of utility functions: non satiation Standard utility functions (of terminal wealth) are monotone increasing and represent non-satiated preferences Many investors do not obey all the rationality postulates when faced with a series of choice situations, even though they may find the underlying principles perfectly reasonable Investors, when faced with more complicated choice situations, encounter aspects of the problem that were not of concern to them in the simple choice situations What are the properties we expect of reasonable utility functions? The first restriction placed on a utility function is that it be consistent with more being preferred to less This attribute, aka nonsatiation, simply says that the utility of more (W + 1) dollars is higher than the utility of less (W) dollars Equivalently, more wealth is always preferred to less wealth If utility increases as wealth increases, then the first derivative of utility, with respect to wealth, is positive, U (W) >0 13

14 Properties of utility functions: risk aversion A fair gamble is a gamble priced at its expected value; a risk-averse investor will always reject a fair gamble in favor of its mean value o Earlier lectures discussed opportunity sets in terms of returns rather than wealth, but there is no substantive difference as W t+1 = (1 + R P t+1)w t The second restriction concerns preferences for risk Risk aversion, risk neutrality, and risk-seeking behaviors are all defined relative to a fair gambe A fair gamble is one that it is priced at its expected value, e.g., (1/2)(2)+(1/2)(0) = 1 in the example below o The position of the investor may be improved or hurt by taking the investment, but the expectation is of no change Against this background, risk aversion means that an investor will always reject a fair gamble 14

15 Properties of utility functions: risk aversion o In our example the investor prefers $1 for sure to the chances to win $2 with a ½ prob. Mathematically, risk aversion implies that the U( ) function is concave; if U( ) is differentiable, then U ( ) < 0 o If an investor prefers not to invest, then the expected utility of not investing must exceed the one of investing, or U(1) > (1/2)U(2) + (1/2)U(0) o Multiplying both sides by 2 and re-arranging, U(1) U(0) > U(2) - U(1) which means that for the same unit increase in wealth, the utility function changes less and less as the initial wealth to which the increase applies grows o Functions that exhbit this property are said to be concave Economically, risk aversion means that an investor will reject a fair gamble because the dis-utility of the loss is greater than the utility of the gain in the case of a good outcome Risk-seeking behavior obtains in the opposite case: the investor always likes a fair gamble 15

16 Properties of utility functions: risk neutrality and seeking A risk-loving investor will always accept a fair gamble over its mean value; a risk-neutral investor is indifferent to fair gambles Mathematically, risk-seeking preferences imply that the U( ) function is convex; if U( ) is differentiable, then U ( ) > 0 o If an investor prefers to take the fair gamble, then the expected utility of investing must exceed the one of not investing, or U(1) < (1/2)U(2) + (1/2)U(0) o Multiplying both sides by 2 and re-arranging, U(1) U(0) < U(2) - U(1) which means that for the same unit increase in wealth, the utility function changes more and more as the initial wealth to which the increase applies grows Risk neutrality means that an investor is indifferent to whether or not a fair gambe should be undertaken Risk neutrality implies a linear utility function; ; if U( ) is differentiable, then U ( ) = 0 These conditions are summarized in the following table 16

17 Properties of utility functions: risk preferences The figures shows preference functions exhibiting alternative properties with respect to risk aversion The leftmost figure shows utility functions in the wealth space, the rightmost in the mean-variance space 1= Risk-seeking 2= Risk-neutral 3 = Risk-averse 17

18 Properties of utility functions: absolute risk aversion Investors who can state their feelings toward a fair gamble can significantly reduce the set of risky investments they consider o E.g., risk-averse investors must consider only the efficient frontier when choosing among alternative portfolios The third property of utility functions that is sometimes presumed is an assumption about how the investor's preferences change with a change in wealth If the investor's wealth increases, will more or less of that wealth be invested in risky assets? If the investor increases the amount invested in risky assets as wealth increases, then the investor is said to exhibit decreasing absolute risk aversion If the investor's investment in risky assets is unchanged as wealth changes, then she is said to exhibit constant absolute risk aversion Finally, if the investor invests fewer dollars in risky assets as wealth increases, then she is said to exhibit increasing absolute risk aversion 18

19 Properties of utility functions: absolute risk aversion The index that can be used to measure an investor s absolute risk aversion is: Then A (W) is an appropriate measure of how absolute risk aversion changes w.r.t. changes in wealth The table below summarizes the relevant properties The final characteristic that is used to restrict the investor's utility function is how the percentage of wealth invested in risky assets changes as wealth changes 19

20 Properties of utility functions: relative risk aversion The coefficient of absolute (relative) risk aversion governs how the total (relative, percentage) amount invested in risky assets changes as wealth changes When the percentage invested in all risky assets does no change as wealth changes, the investor s behavior is said to be characterized by constant relative risk aversion Relative risk aversion is related to absolute risk aversion but RRA refers to the change in the percentage investment in risky assets as wealth changes, ARA to dollar amounts invested in risky assets 20

21 Quadratic utility functions and their limitations Quadratic utility functions may be satiated and imply problematically increasing absolute and relative risk aversion coefficients While there is general agreement that most investors exhibit decreasing absolute risk aversion, there is much less agreement concerning relative risk aversion Often people assume constant relative risk aversion The justification for this, however, is often one of convenience rather than belief about descriptive accuracy We are now ready to inquire about the properties of the quadratic utility function previously postulated, U(W) = W bw 2 o U (W) = 1 2bW, U (W) = -2b, A(W) = 2b/(1-2bW), o This investor is satiated iff 1 2bW >0, or W < 1/(2b) = bliss point o Below the bliss point, A(W) > 0 but it is monotone increasing o Below the bliss point, because R(W) = WA(W), clearly RRA increases as W increases 21

22 Logarithmic utility functions Logarithmic utility functions imply a decreasing absolute risk aversion, constant relative risk aversion, and are non-satiated In spite of its problems, quadratic utility takes a special place in mean-variance analysis because it is perfectly consistent with it However, considerable literature has shown that a quadratic utility function may provide an excellent approximation to another, more robust utility function, the logarithmic one: U(W) = lnw o U (W) = 1/W, U (W) = -1/W 2, A(W) = 1/W, R(W) = WA(W) = 1 o Therefore this utility function exhibits decreasing absolute and constant relative risk aversion o See the Appendix for details on the approximation argument The utility functions reviewed so far are, in general, based upon investor choice over a single-time horizon In reality, investors confront a multiperiod choice problem Any asset allocation chosen today can be undone tomorrow 22

23 The effects of the investment horizon under log utility Let us return to the log-utility function Suppose a log-utility investor with $1000 faces a multiperiod investment opportunity, she has the choice to invest in a risky asset for two periods, one period, or not to invest at all The risky asset is forecast to either double or halve in value each period, with equal probability The expected utility calculation for the risky investment in the first period would be U(one period) = (1/2)ln($2000) + (1/2)ln($500) = If she simply held cash, the utility would be ln($1000) = Thus, this investor is indifferent between investing for one period or not investing at all $1000 is called the certainty equivalent for the risky investment because it is the certain value that will make you indifferent between taking and not taking a gamble 23

24 The effects of the investment horizon under log utility Under constant relative risk aversion preferences and with uncorrelated (IID) risky returns, the investment horizon doesn t matter Under the same conditions for a second investment period, the log-utility investor will also be indifferent Calculating the expected utility for the four potential outcomes in period 2 (i.e., two doublings in a row, two halvings in a row, a doubling and then a halving, and a halving then a doubling), the expected utility is U(two period) = (1/4)ln($4000) + (1/2)ln($1000) + +(1/4)ln($250) = Thus, for the log-utility investor (for all classes of utility functions that display constant relative risk aversion for multiplicative investments), the horizon of the investment will not affect choices One important exception to this rule is when the returns of the risky asset are correlated over time o When, for example, asset prices tend to go down after a rise, or up after a fall 24

25 The effects of the investment horizon under log utility In this case risky investments are more attractive (less risky) in the long run than they in the short run The proportion invested in risky assets will increase as the horizon gets longer Obviously this is just a limit result, a sort of paradox When asset returns are dependent over time, the horizon will affect optimal portfolio choices If investors don't exhibit time indifference, how do they act? It turns out that investors who have some tolerance for very large losses may be willing to invest in risky assets over long horizons, but not necessarily over short horizons This increased willingness to invest in the risky asset as the investment horizon grows suggests that the asset allocation choice of many investors can depend on how long they expect keep their money invested before using it for retirement, education, or to meet other future liabilities 25

26 Myopic portfolio choice: canonical case The risky share should equal the risk premium, divided by conditional variance times the coefficient capturing aversion to risk Suppose the conditional mean of a single risky portfolio is E t [r t+1 ] and the conditional variance is 2 t The investor only cares for conditional mean and conditional variance, The problem has the classical solution: The portfolio share in the risky asset should equal the expected excess return, or risk premium, divided by conditional variance times the coefficient that represents aversion to variance Mean-variance portfolio choice 26

27 Myopic portfolio choice: canonical case If we define the Sharpe ratio of the risk asset as: then the MV solution to the problem can be written as: The corresponding risk premium and Sharpe ratio for the optimal portfolio are as follows: Hence all portfolios have the same Sharpe ratio because they all contain the same risky asset in greater or smaller amount Mean-variance portfolio choice 27

28 Myopic portfolio choice: multivariate case These results extend straightforwardly to the case where there are many risky assets o We define the portfolio return in the same manner except that we use boldfaced letters to denote vectors and matrices o R t+1 is now a vector of risky returns with N elements. o It has conditional mean vector E t [R t+1 ] and conditional covariance matrix t Var t [R t+1 ] o We want to find the optimal allocation w t The maximization problem now becomes Vector of 1s repeated N times with solution Mean-variance portfolio choice 28

29 Myopic portfolio choice: multivariate case A straightforward generalization of the single risky asset case o o The single risk premium return is replaced by the vector of risk premia and the reciprocal of variance is replaced by -1 t, the inverse of the covariance matrix of returns Investors preferences enter the solution only through the scalar Thus investors differ only in the overall scale of their risky asset portfolio, not in the composition of that portfolio o This is the two-fund, separation theorem of Tobin (1958) again The results extend to the case where there is no completely riskless asset: we call still define a benchmark asset with return R 0,t We now develop portfolio choice results under the assumption that investors have power utility and that asset returns are lognormal We apply a result about the expectation of a log-normal random variable X: Mean-variance portfolio choice 29

30 The power utility log-normal case o The log is a concave function and therefore the mean of the log of a random variable X is smaller than the log of the mean, and the difference is increasing in the variability of X Assume that the return on an investor s portfolio is lognormal, so that next-period wealth is lognormal Under of power utility, the objective is Maximizing this expectation is equivalent to maximizing the log of the expectation, and the scale factor 1/(1 - ) can be omitted Because next-period wealth is lognormal, we can apply the earlier result to rewrite the objective as The standard budget constraint can be rewritten in log form: (*) Mean-variance portfolio choice 30

31 The power utility log-normal case Under power utility and log-normal portfolio returns (hence, terminal wealth), a mean-variance result applies under appropriate definition of the mean of portfolio returns relevant to the portfolio choice Dividing (*) by (1 - ) and using the new constraint, we have: Just as in mean-variance analysis, the investor trades-off mean against variance in portfolio returns Notice that this can be further transformed as: so that Mean-variance portfolio choice 31

32 The power utility log-normal case o o o o o The appropriate mean is the simple return, or arithmetic mean return, and the investor trades off the log of this mean linearly against the variance of the log return When = 1, under log utility, the investor selects the portfolio with the highest available log return (the "growth optimal portfolio) When > 1, the investor seeks a safe portfolio by penalizing the variance of ln(1 + R p t+1) When < 1, the investor actually seeks a riskier portfolio because a higher variance, with the same mean log return, corresponds to a higher mean simple return The case = 1 is the boundary where these two opposing considerations balance out exactly Problem: To proceed further, we would need to relate the log ptf. return to the log returns on the underlying assets But while the simple return on a ptf. is a linear combination of the simple returns on the risky and riskless assets, the log ptf. return is not the same as a linear combination of logs Mean-variance portfolio choice 32

33 Summary In this lecture we have learned a number of things 1. That preferences for bundles of goods and services may be represented by utility functions 2. That this result, under appropriate conditions extends to the case of uncertainty, when the Expected Utility Theorem holds 3. That under the EUT, the notion of being risk averse is intuitive and corresponds to concavity of the utility function, U( ) 4. That degree of aversion to risk may be measured through the coefficients of absolute and relative risk aversion 5. That CARA and CRRA preferences induce special results when it comes to comparative statistics of portfolio choice 6. That mean-variance preferences and portfolio decisions may also represent approximation to more classical and better behaved utility functions 33

34 Appendix: the link between expected quadratic utility and mean-variance wealth preferences 34