expected utility theory; expected utility theory; risk...

Expected utility theory; Expected Utility Theory; risk aversion and

utility functions

Prof. Massimo Guidolin

Portfolio Management

Spring 2016

Outline and objectives

2Expected utility theory

Utility functions

The expected utility theorem and the axioms of choice

Properties of utility functions: non-satiation and riskpreferences

Absolute vs. Relative risk aversion

The effects of the investment horizon

Generalities


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 seto 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:

Modelling preferences will help us to exactly pin down the optimal portfolio selected on the efficient set by each investor

Utility functions


An intuitive approach consists of converting the outcomes in the value that these carry to investorso Of course, if the outcomes were the same as monetary payoffs, then

value and outcome may correspondo 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

A utility function of wealth converts wealth outcomes in subjectively perceived value, i.e., investors’ satistifaction or happiness

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

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) = 2 + 12W – (3/10)W2

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 thatU(W) = 4W – (1/10)W2 > U(W*) = 4W* – (1/10)(W*)2 then

V(W) = 2 + 3W – (3/10)W2 = 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



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 Co I.e., investors are consistent in their ranking of outcomeso Although this seems reasonable, considerable experimental evidence

displays stark cases of violations

Utility functions are unique up to monotone increasing linear transformations



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, andY 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



Using these axioms, we can derive the expected utility theoremo 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 separatescombinations 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



o Now consider a portfolio of securities S1 with N possible outcomes:

o Each Wi is a known payoff and since Cis exist for all Wi’s, S1 is equiva-lent to

o Since for every Ci there existsan equivalent lottery, we can represent an equivalentlottery as you can see on the right

o If the investor declares btw. the Ci’sand each lottery, then S1 S2


Expected utility theory

o For instance, if outcome i occurs, if the investor selects S1then Ci is received; if the investor selects S2, then the investor receives b with prob. hi, and 0 with prob. 1 - hi

o However investor has indicated in the construction of the preference curve an indifference btw. Ci and this lottery; but Wi is equal to Ci so that the investor is indifferent btw. Wi and this lottery

o Thus, security 2 is equivalent to security 1o From axiom 3 the investor does not change preference simply

because the alternatives are part of a lottery o As the picture shows, S2 has only 2 possible outcomes, b and 0o We can equivalently write S2 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



o Define ; then if HK > HL, security K is to be preferred to security L

o This leads directly to the EUT: earlier we have replaced Wi with Ci, to which hi was associated

o Call the function that relates Wi to hi a utility function, hi = U(Wi) and note that Hi = iPihi = iPiU(Wi)

o But iPiU(Wi) is simply expected utility and ranking securities on the basis of Hi 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 chooseo A number of brokerage firms have developed programs to extract

the utility function by confronting investors with simple choiceso These have not been particularly successful

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Properties of utility functions: non satiation


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

Standard utility functions (of terminal wealth) are monotone increasing and represent non-satiated preferences

13

Properties of utility functions: risk aversion


o Earlier lectures discussed opportunity sets in terms of returns rather than wealth, but there is no substantive difference as Wt+1 = (1 + RP

t+1)Wt

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 belowo 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

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

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Properties of utility functions: risk aversiono 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’’(·) < 0o 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 investoralways likes a fair gamble 15Expected utility theory

Properties of utility functions: risk neutrality and seeking

Mathematically, risk-seeking preferences imply that the U(·) function is convex; if U(·) is differentiable, then U’’(·) > 0o 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

A risk-loving investor will always accept a fair gamble over its mean value; a risk-neutral investor is indifferent to fair gambles


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-seeking2= Risk-neutral3 = Risk-averse

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 considero 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 18Expected utility theory

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 riskaversion 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


Properties of utility functions: relative risk aversion

When the percentage invested in all risky assets does no change aswealth 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

The coefficient of absolute (relative) risk aversion governs how the total (relative, percentage) amount invested in risky assets changes aswealth changes


Quadratic utility functions and their limitations

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 – bW2

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 pointo Below the bliss point, A(W) > 0

but it is monotone increasingo Below the bliss point, because

R(W) = WA(W), clearly RRA increases as W increases

Quadratic utility functions may be satiated and imply problematicallyincreasing absolute and relative risk aversion coefficients


Logarithmic utility functions

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) = lnWo U’(W) = 1/W, U’’(W) = -1/W2, A(W) = 1/W, R(W) = WA(W) = 1o Therefore this utility function exhibits decreasing absolute and

constant relative risk aversiono 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

Logarithmic utility functions imply a decreasing absolute riskaversion, constant relative risk aversion, and are non-satiated


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) = 6.9077 If she simply held cash, the utility would be ln($1000) = 6.9077 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


The effects of the investment horizon under log utility

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) = 6.9077

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 timeo When, for example, asset prices tend to go down after a rise, or up

after a fall

Under constant relative risk aversion preferences and with un-correlated (IID) risky returns, the investment horizon doesn’t matter


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


Myopic portfolio choice: canonical case

26Mean-variance portfolio choice

Suppose the conditional mean of a single risky portfolio is Et[rt+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

The risky share should equal the risk premium, divided by condi-tional variance times the coefficient capturing aversion to risk

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

Myopic portfolio choice: multivariate case


These results extend straightforwardly to the case where there are many risky assetso We define the portfolio return in the same manner except that we use

boldfaced letters to denote vectors and matriceso Rt+1 is now a vector of risky returns with N elements.o It has conditional mean vector Et[Rt+1] and conditional covariance

matrixt ≡ Vart[Rt+1]o We want to find the optimal allocation wt

The maximization problem now becomes

with solution

Vector of 1srepeated N times

Myopic portfolio choice: multivariate case


A straightforward generalization of the single risky asset caseo The single risk premium return is replaced by the vector of risk

premia and the reciprocal of variance is replaced by t-1, the inverse

of the covariance matrix of returnso 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 portfolioo 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 R0,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:

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:

(*)



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

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



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

o When = 1, under log utility, the investor selects the portfolio with the highest available log return (the "growth optimal“ portfolio)

o When > 1, the investor seeks a safe portfolio by penalizing the variance of ln(1 + Rp

t+1)o 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

o 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

Summary

33

In this lecture we have learned a number of things1. That preferences for bundles of goods and services may be

represented by utility functions2. That this result, under appropriate conditions extends to the case

of uncertainty, when the Expected Utility Theorem holds3. 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 aversion5. That CARA and CRRA preferences induce special results when it

comes to comparative statistics of portfolio choice6. That mean-variance preferences and portfolio decisions may also

represent approximation to more classical and better behavedutility functions


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


expected utility theory; expected utility theory; risk...

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