Investment Analysis and Portfolio Management

Lecture 5

Gareth Myles

Risk

An investment is made at time 0 The return is realised at time 1 Only in very special circumstances is the

return to be obtained at time 1 known at time 0 In general the return is risky The choice of portfolio must be made taking

this risk into account The concept of states of the world can be used

Choice with Risk

State Preference The standard analysis of choice in risky situations

applies the state preference approach

Consider time periods t = 1, 2, 3, 4, ... At each time t there is a set of possible events

(or "states of the world")

,4,3,2,1 ttttte

Choice with Risk

When time t is reached, one of these states is realized

At the decision point (t = 0), it is not known which

Decision maker places a probability on each event

The probabilities satisfy

,,,, 4321ttttt ppppp

11

E

i

itp

Choice with Risk

t = 0 t = 1 t = 2

10

11

21

12

22

32

42

52

Event tree

Choice with Risk

Each event is a complete description of the world

Let = return on asset i at time t in state e then

This information will determine the payoff in

each state Investors have preferences over these returns

and this determines preferences over states

,, 21tt eet rre

iet

r

Choice with Risk

Expected Utility Assume the investor has preferences over wealth in

each state described by the utility function

Preferences can be defined over different sets of probabilities over the states

WUU

W

U

U=U ( W )

Choice with Risk

Assume 1 time period and 2 states Let wealth in state 1 be W1 and in state 2 W2

Let p denote the lottery {p, 1-p} in which state 1 occurs with probability p

Lottery q is defined in the same wayExample Let , , , Then any investor who prefers a higher return to a

lower return must rank p strictly preferable to q

101 W 52 W 1.0,9.0p 9.0,1.0q

Choice with Risk

We now assume that an investor can rank lotteries 1. Preferences are a complete ordering 2. If p is preferred to q, then a mixture of p and r is

preferred to the same mixture of r and q 3. If p is preferred to q and q preferred to r, then

there is a mixture of p and r which is preferred to q and a different mixture of p and r which is strictly worse then q

The investor will act as if they maximize the expected utility function

2211 WUpWUpEU

Choice with Risk

This approach can be extended to the general state-preference model described above

For example, with two assets in each state

where ai is the investment in asset i Summary

Preferences over random payoffs can be described by the expected utility function

222

1212

212

1111 1111 raraUpraraUpEU

Risk Aversion

Consider receiving either A fixed income M A random income M[1 + r] or M[1 – r], each

possibility occurring with probability ½

An investor is risk averse if

U(M) > ½ U(M[1 + r]) + ½ U(M[1 – r]) The certain income is preferred to the

random income This holds if the utility function is concave

Risk Aversion

A risk averse investor will pay to avoid risk

The amount the will pay is defined as the solution to

U(M - ) = ½U(M[1 + r]) + ½U(M[1 – r]) is the risk premium The more risk averse is

the investor, the more they will pay

Wealth

Utility

10 hW 20 hW

20 hWU

10 hWU

EUWU 0

0W

Portfolio Choice

Assume a safe asset with return rf = 0

Assume a risky asset Return rg > 0 in “good” state

Return rb < 0 in “bad” state

Investor has amount W to invest How should it be allocated between the

assets?

Portfolio Choice

Let amount a be placed in risky asset, so W – a in safe asset

After one period Wealth is W - a + a[1 + rg] in good state Wealth is W - a + a[1 + rb] in bad state

A portfolio choice is a value of a High value of a

More wealth if good state occurs Less wealth if bad states occurs

Portfolio Choice

Possible wealth levels are illustrated on a “state-preference” diagram

W

W

W[1+rg]

W[1+rb] a = W

a = 0

Wealth ingood state

Wealth inbad state

Portfolio Choice

Adding indifference curves shows the choice Indifference curves from expected utility function

EU = pU(W - a + a[1 + rg]) + (1-p)U(W - a + a[1 + rb]) The investor chooses a to make expected utility

as large as possible Attains the highest indifference curve given the

wealth to be invested

Portfolio Choice

W

W

W[1+rg]

W[1+rb] a = W

a = 0

Wealth ingood state

Wealth inbad state

a*

Portfolio Choice

Effect of an increase in risk aversion What happens if rb > 0 or if rg < 0?

When will some of the risky asset be purchased?

When will only safe asset be purchased? Effect of an increase in wealth to be invested?

Mean-Variance Preferences

There is a special case of this analysis that is of great significance in finance

The general expected utility function constructed above is dependent upon the entire distribution of returns

The analysis is much simpler if it depends on only the mean and variance of the distribution.

When does this hold?

Mean-Variance Preferences

Denote the level of wealth by . Taking a Taylor's series expansion of utility around expected wealth

Here R3 is the error that depends on terms involving and higher

WE~

W~

WEWWEUWEUWU~~~

'~~

32 ~~~

''2

1RWEWWEU

W~

3 ~~WEW

Mean-Variance Preferences

Taking the expectation of the expansion

The expected error is

The expectation involves moments (mn) of all orders (first = mean, second = variance, third)

The problem is to discover when it involves only the mean and variance

3

2~~''

2

1~~REWWEUWEUWUE

WmWEUn

RE nn

n

~~!

1

33

Mean-Variance Preferences

Expected utility depends on just the mean and variance if either 1. = 0 for n > 2. This holds if utility is

quadratic

Or 2. The distribution of returns is normal since then all

moments depend on the mean and variance

In either case

WEU n ~

2~,

~~WWEUWUE

Risk Aversion

With mean-variance preferences Risk aversion implies

the indifference curves slope upwards

Increased risk aversion means they get steeper

pr

p

More risk averse

Less risk averse

Markowitz Model

The Markowitz model is the basic model of portfolio choice

Assumes A single period horizon Mean-variance preferences Risk aversion Investor can construct portfolio frontier

Markowitz Model

Confront the portfolio frontier with mean-variance preference

Optimal portfolio is on the highest indifference curve

An increase in risk aversion changes the gradient of the indifference curve

Moves choice around the frontier

Markowitz Model

MVP

MVPr

Standarddeviation

Expectedreturn

Optimalportfolio

Less riskaverse

More riskaverse

Xa = 1, Xb = 0

Xa = 0, Xb = 1

Choice with risky assets

Markowitz Model

Standarddeviation

Expectedreturn

Optimalportfolio

Less riskaverse

More riskaverse

Xa = 1, Xb = 0

Xa = 0, Xb = 1

Borrowing

Lending

Choice with a risk-free asset

Markowitz Model

Note the role of the tangency portfolio Only two assets need be available to achieve

an optimal portfolio Riskfree asset Tangency portfolio (mutual fund)

Model makes predictions about The effect of an increase in risk aversion Which assets will be short sold Which investors will buy on the margin

Markowitz model is the basis of CAPM