Investment Analysis and Portfolio Management

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 2W2 • 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 • 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 a = 0 W 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 Wealth in 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 • 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

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 averse Optimal 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 averse Optimal 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