Financial Products and Markets

Lecture 5

Investment choices and expected utility

• The investment techniques are based on a system of rules that allows to rank securities and portfolios.

• This system of rules is at the root of what is known as expected utlity theory. According to this theory, risky alternatives (lotteries) can be ranked comparing the expected value of a function, called utility function.

• If A and B are two risky alternatives, expected utility theory states that

A B E[U(A)] < E[U(B)]

where denotes preference of B wrt A and function U(.) is the utility function

Choice btw risky alternatives

• Expected utility: choice btw lotteries A and B,

A < B (B is preferred to A) iff E(U(A)) < E(U(B))• Function U(.) is increasing (prefer more to less)

and it isd concave in case of risk aversion. • The correspondence of preference and expected

utility rankings is based on a set of axioms. Particularly relevant is the independence axiom:

A < B A +(1- )C < B +(1- )C

Equivalent Probability

• Assume a lottery giving values WH and WL.

• The probability of WH is p.

• An agent is risk averse if

pU(WH)+ (1 – p) U(WL) < U(pWH+ (1 – p)WL)

• Consider a change of probability from p to q

qU(WH)+ (1 – q) U(WL) = U(pWH+ (1 – p)WL)

Certainty Equivalent

• Assume a lottery giving values WH and WL

• The probability of WH is p.

• An agent is risk averse if

pU(WH)+ (1 – p) U(WL) < U(pWH+ (1 – p)WL)

• The certainty equivalent WCE is such that

pU(WH)+ (1 – p) U(WL) = U(WCE)

Risk aversion implies WCE < E(W)

Expected utility and risk aversion• Assume lottery W, with expected value E(W).An agent is called

risk neutrale if he is indifferent to play the lottery or take an amount E(W) for sure

E[U(W)] = U(E(W)) • An agent is risk averse if he prefers the sum E(W) to the lottery W

E[U(W)] < U(E(W))• Using Jensen’s inequality, this deifnition implies that the utility

function has to be concave in the risk aversion case • The degree of risk averion can be measured directly by

determining a valur π such that E[U(W)] = U(E(W) – π )

• Using a Taylor expansion we can compute π = ½ (– U’’/U’)Var(W)

where U’ and U’’ denote first and second derivatives.

Risk Aversion Measures

• The term – U’’/U’ is a local measure of concavity of the utility function and is known as Arrow-Pratt absolute risk aversiion (ARA) measure.

• Another measure of risk aversion is proportiional to wealth defining the so-called relative risk aversion

RRA = W*ARA• Another measure, of opposite sign, is risk tolerance,

denoted by RT = 1/ARA.• Different utility functions represent different behaviours of

risk aversion with respect to wealth.

Utility functions

• Quadratic

• CARA

• CRRA

– Logarithmic

• HARA

• U(W) = W – b W2

• U (W) =a – exp (– b W)

• U(W) = [W – 1 ]/

• U(W) = ln(W)

•

1

1 WWU

Utility functions

• Different utility functions show different behaviour of the risk aversion measures. – Quadratic utility (easy to use but with two problems:

non-monotone preferences, risky assets inferior goods)– Exponential Utility or CARA (constant absolute risk

aversion)– Power utility, or CRRA (constant relative risk

aversion): log-utility as special case– HARA (hyperbolic absolute risk-aversion): (the most

general case, with risk tolerance linear in wealth)

Prospect theory

• Kahneman and Tversky proposed a new appproach to utility theory

• The main principles are– Existence of a “reference point” to ditinguish

between profits and losses– Probability deformation, different for profits

and losses – Aversion to loss (losses are weighted more

than profits )

Reference point

• Risk attitude may change depending on whether the loss could be below or above a reference “reference point”.

• What is the “reference point”?– For returns from investment it can be zero return

(cash), or a risk free reference return, or a benchmark.

– For a general lottery, it can be measured by average income or similar proxies. (“house money”)

The uttility function

• “Prospect theory” suggests the following general shape for the utility function

U(r) + w+(p) (U(WH) – U(r))

– w–(1 – p)(U(r)–U(WH)) with

– r il “reference point” – w+(p) and w–(1 – p) probability distorsions “loss aversion”

Probability distortion

• Tversky and Kahneman proposed the following probability distortion function

/11 pp

ppw

Expected utility: no loss aversion

0

2

4

6

8

10

12

14

16

18

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Prospect Theory

Power Utility

Expected utility: loss aversion

-17

-12

-7

-2

3

8

13

18

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Prospect Theory

Power Utility

Risck and (Knightian) uncertainty

• Knight, was an economist at the University of Chicago in the 1920s.

• Risk is when we know the probabilities of succes and failure. Uncertainty is when probabilities are not known (Knightian uncertainty)

• How do people behave in front of uncertainty? Ellsberg’s paradox is referred to preferences between ambiguous and unambiguous lotteries. This addresses the role of information in the decision making behaviour.

Ellsberg paradoxB < Z?...

Probability Z A B

State H 1/3 0.6 0 0

State M ? 0 0.6 0

State L ? 0 0 0.6

… 0.5Z + 0.5A < 0.5B + 0.5A?

Probability 50% A50% Z

50% A50% B

State H 1/3 0.3 0

State M ? 0.3 0.3

State L ? 0 0.3

Financial puzzles

• Home bias: – Investors hold a disproportionately large share of their portfolio in

domestic securities.

• IPO underpricing– Stock listed for the first time give a return systematically higher

than the market return.

• Seasoned securities– Bonds that are not heavily traded give a higher return than others.

• Closed-end funds: the NAV is typically higher than the sum of of the fund quotes.

Expected utility and mean variance

• In the most famous model of expected utility, the portfolio is allocated using only the first two moments of the distribution, that is mean and variance This is the so-called mean-variance model.

• The mean-variance model is a precise representation of the expected utility problen only if– The utility function id quadratic – The distribution of returns is gaussian

• In other cases the representation in terms of mean and variance is an approximation of the expected utility function given by a Taylor expansion up to the second order.

The mean-variance model

• In a mean-variance model one has to define 1. The efficient set, that is the set of best possible

return given the least possible risk, measured by variance (efficient portfolio curve)

2. The set of pairs of rxpected returns and variance giving the same level of expected utility

• The optimum portfolio will be given by the pair of expected return and variance that gives the highest possible utility while remaining in the feasible set.

Building the efficient frontier

• Goal: – Efficient portfolio: it has lowest possibile

volatility P

• Constraint:– All wealth must be invested in financial assets

– The portfolio must yield the expected return P.

Risk-return trade-off

• Goal– The highest level of expected utility

• Constraint:– The portfolio must be part of the efficient

frontier.

One risky and one risk-free asset

• Assume that the investment set is made of two securities:

– A risk-free asset with return i and zero volatility

– A risky asset with expected return E(r ) and volatility

• Compute the expected return and the volatility of a strategy of portfolio allocation, consisting of a percentage of wealth and the remaining percentage (1 - ) in the risk-free asset:

p

p rEi1

Mean variance level curve

• Compute the expansion of expected utility around the expected value of wealth

E(U(W)) = U()+0.5 U’’()2

where is the mean of wealth and 2 is the variance. The level curve is given by

dE(U(W)) = U’()d + U’’()d = 0from which

d/ d = – (U’’()/ U’()) = ARA

Efficient frontier

• In the model with a risky and a risk-free asset the efficient frontier is

• Since the slope of the level curve of the utility function is ARAP the optimal portfolio is at:

pp

irEi

irEARA p

Optimal portfolio

• The optimal portfolio in the tangency point of the level curve of the expected utility and the effficient frontier is

…and the investment in the risky asset is lower the higher the ARA index of risk aversion.

ARA

irE2

*

Rendimento e rischio del portafoglio ottimo

Model with two risky assets

• Assume that the investment set consists of two assets i = 1,2, with expected return E(ri) and volatility i

• Compute the expected return and risk of a portfolio obtained by investing a share in the first risky asset and the remaining part in the second one:

2122

221

2

21

121

1

p

p rErErE

Perfect correlation