1. Investment Analysis and Portfolio Management
Lecture 5
Gareth Myles

2. 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

3. 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. 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

5. Choice with Risk
Event tree

6. 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

7. 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

8. 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 way
Example
Let , , ,
Then any investor who prefers a higher return to a lower return must rank p strictly preferable to q

9. 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

10. 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

11. Risk Aversion Consider receiving either An investor is risk averse if
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

12. Risk Aversion A risk averse investor will pay to avoid risk
The amount the will pay is defined as the solution to
U(M - r) = ½U(M[1 + r])
+ ½U(M[1 – r])
r is the risk premium
The more risk averse is the investor, the more they will pay

13. 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?

14. 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

15. Portfolio Choice
Possible wealth levels are illustrated on a “state-preference” diagram
Wealth in
bad state W
a = 0
a = W
W[1+rb]
Wealth in
good state W
W[1+rg]

16. 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

17. Portfolio Choice Wealth in bad state a = 0 W a* W[1+rb] a = W
good state W
W[1+rg]

18. 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?

19. 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?

20. 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

21. 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

22. Mean-Variance Preferences
Expected utility depends on just the mean and variance if either
= 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

23. Risk Aversion With mean-variance preferences
Risk aversion implies the indifference curves slope upwards
Increased risk aversion means they get steeper

24. 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

25. 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

26. Markowitz Model Less risk Optimal averse portfolio Expected return
Xa = 1, Xb = 0
Xa = 0, Xb = 1
More risk
averse
Standard
deviation
Choice with risky assets

27. Markowitz Model Less risk Optimal averse portfolio Expected return
Borrowing
Xa = 1, Xb = 0
More risk
averse
Xa = 0, Xb = 1
Lending
Standard
deviation
Choice with a risk-free asset

28. 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