Portfolio Managment 3-228-07 Albert Lee Chun

1. Portfolio Managment3-228-07Albert Lee Chun Construction of Portfolios: Introduction to Modern Portfolio Theory Lecture 3 16 Sept 2008

2. Course Outline Sessions 1 and 2 : The Institutional Environment Sessions 3, 4 and 5: Construction of Portfolios Sessions 6 and 7: Capital Asset Pricing Model Session 8: Market Efficiency Session 9: Active Portfolio Management Session 10: Management of Bond Portfolios Session 11: Performance Measurement of Managed Portfolios

3. Portfolio Risk as a Function of the Number of Stocks in the Portfolio 7-2

4. Portfolio Diversification 7-3

5. Two-Security Portfolio: Return w1 = proportion of funds in Security 1 w2 = proportion of funds in Security 2 r1 = expected return on Security 1 r2 = expected return on Security 2 7-4

6. Two-Security Portfolio: Risk 12 = variance of Security 1 22 = variance of Security 2 Cov(r1,r2) = covariance of returns for Security 1 and Security 2 7-5

7. Covariance 1,2 = Correlation coefficient of returns 1 = Standard deviation of returns for Security 1 2 = Standard deviation of returns for Security 2 7-6

8. Correlation Coefficients: Possible Values Range of values for1,2 + 1.0 >> -1.0 If = 1.0, the securities would be perfectly positively correlated If= - 1.0, the securities would be perfectly negatively correlated 7-7

9. Three-Security Portfolio 7-8

10. Generally, for an n-Security Portfolio: 7-9

11. Review of Portfolio Statistics

12. Today’s Lecture • Utility Functions, Indifference Curves • Capital Allocation Line • Minimum Variance Portfolios • Optimal Portfolios in a 2 security world (1 risk-free and 1 risky) 2 security world (2 risky) 3 security world (2 risky and 1 risk-free) N security world (with and without risk-free asset)

13. Utility Functions

14. Risk Aversion • Given a choice between two assets with equal rates of return, risk-averse investors will select the asset with the lower level of risk. • Risk-averse investors need to be compensated for holding risk. • The higher rate of return on a risky asset i is determined by the risk-premium: E(Ri) – Rf.

15. W1 = $150 Profit = $50 p = .6 Risky Investment W2= $80 Profit = -$20 1-p = .4 100 T-bills Profit = $5 Expected return: (50%)(.6) + (-20%)(.4) = 22% Risk Premium = E(Ri) – Rf = 22% - 5% = = 17% Example: Risk Premium

16. Measure of Investor Preferences • A utility function captures the level of satisfaction or happiness of an investor. • The higher the utility, the happier the investors. • For example, if investor utility depends only of the mean (let µ= E(R)) and variance (2) of returns then it can be represented as a function: • The locus of portfolios that provide the same level of utility for an investor defines an indifference curve. U = f ( µ, )

17. Example: An Indifference Curve U = 5 (Rp) U = 5 The investor is indifferent between X and Y, as well as all points on the curve. All points on the curve have the same level of utility (U=5).

18. Expected Return Direction of Increasing Utility Standard Deviation Direction of Increasing Utility U3 U3 > U2 > U1 U2 U1

19. Two Different Investors Expected Return U3’ U2’ U1’ Which investor is more risk averse? U3 U2 U1 Standard Deviation

20. Quadratic Utility • The utility of the investor is quadratic if only the mean and variance of returns is important for the investor. • A is a constant that determines the degree of risk aversion: it increases with the risk-aversion of the investor. (Note that the 1/2 is just a normalizing constant.) • Note that A > 0, implies that investors dislike risk. The higher the variance the lower the utility.

21. Indifference Curves Let’s look at an example of points on an indifference curve for an investor with a quadratic utility function. Note that higher variance is accompanied by a higher rate of return to compensate the risk-averse nature of the investor.

22. Certain Equivalent • The certain equivalent is the risk-free (certain) rate of return that offers investors the same level of utility as the risky rate of return. • The investor is indifferent between a risky return and it’s certain equivalent. • Example: Suppose an investor has quadratic utility with A = 2. A risky portfolio offers an E(R) equal to 22% and standard deviation 34%. The utility of this portfolio is: U = 22% - ½×2×(34%)² = 10.44% • The certain equivalent is equal to 10.44% because the utility of obtaining a certain rate of return of 10.44% is U = 10.44% - ½ × 2×(0%)² = 10.44%

23. E(RP) U4 U3 U2 U1 sP Neutral attitude toward risk. Investor is indifferent between different levels of standard deviation. Risk-Neutral Indifference Curves U3 > U2 > U1

24. Slope of the Indifference Curve • A steep indifference curve coincides with strong risk-aversion. • The slope of the indifference curve captures the required compensation for each unit of additional risk. • This compensation is measured in units of expected return for each unit of standard deviation. • High risk-aversion implies a high degree of compensation for taking on an additional unit of risk and is represented by a steep slope.

25. U4 U3 E(RP) U2 U1 sP Risk-Averse Indifference Curves Expected Return U3 > U2 > U1 Standard Deviation

26. Two Different Investors Expected Return Less risk averse U3’ U2’ More risk averse U1’ Which investor is more risk averse? U3 U2 U1 Standard Deviation

27. Stochastic Dominance Prefers any portfolio in Z1 to X. The rankings between portfolios in Z2 or Z3 and X, depends on the preferences of the investor! Prefers X to any portfolio in Z4.

28. Imagine a world with 1 risk-free security and 1 risky security

29. 1 Risk-Free Asset and 1 Risky Asset Suppose we construct a portfolio P consisting of 1 risk-free asset f and 1 risky asset A: Note: The variance of the risk-free asset is 0, and the covariance between a risky asset and a risk free asset is naturally equal to 0.

30. 1 Risk Free Asset and 1 Risky Asset Suppose WA = .75 E(rA) = 15% A E(rP) = 13% P rf = 7% f  P =16.5% A =22% 0 E(rP)= .25*.07+.75*15=13%p=.75*.22= 16.5%

31. Capital Allocation Line (CAL) Equation of CAL Line E(rA) Slope of CAL A E(rp) P rf f Intercept  0 p A

32. Maximize Investor Utility • In our world with 1 risk free asset and 1 risky asset, if an investor has quadratic utility, what is the optimal portfolio allocation? Utility: Expected return and variance: Goal is to Maximize utility. How?

33. Normally a Bear Lives in a Cave, that is Concave, A concave function has a negative second derivative. then to find the top of the cave (i.e. or to maximize a concave function), take the first derivative and set it equal to 0:

34. However, if the Bear is Swimming in a Bowl, that is Convex, A convex function has a positive second derivative. Then to find the bottom of the bowl (i.e. or to minimize a convex function), take the first derivative and set equal to 0:

35. Maximize Investor Utility Take derivative of U with respect to w and set equal to 0: w* is the optimal weight on risky asset A

36. Example 1 Supppose E(rA) = 15%; (rA) = 22% and rf = 7% and we have a Quadratic investor with A = 4, then w* = (0.15-0.07)/[4*(0.22)2] = 0.41 His optimal allocation is: 41% of his capital in the risky portfolio A and 59% in the risk-free asset. E(rp) = 0.59*7%+0.41*15%=10.28% and (rp) = 0.41*0.22=9.02%

37. Example 2 Supppose E(rA) = 15%; (rA) = 22% and rf = 7% and we have a less risk-averse Quadratic investor with A = 1, then w* = (0.15-0.07)/[1*(0.22)2] = 1.65 > 1 This investor should place 165% of his capital in A. He needs to borrow 65% of his capital at the risk free rate of 7%. E(Rp) = 1.65(0.15) + -0.65(0.07)= 20.2% (rp) = 1.65*0.22= 0.363 = 36.3% His utility is: U = 0.202 – 0.5*1*(0.3632) = 0.1361

38. Graphical View The optimal allocation along the capital allocation line depends on the risk-aversion of the agent. Risk-seeking agents with w* greater than 1 will borrow at the risk-free rate and invest in security A E(r) A Ex2: Borrower 7% Ex1: Lender  A = 22% The optimal allocation is the point of tangency between the CAL and the investor’s utility function.

39. E(r) A 9% 7%  A = 22% Different Borrowing Rate • What if the borrowing rate is higher than the lending rate? w* = (0.15-0.09)/[1*(0.22)2] = 1.24 1.24 < 1.65

40. Different Borrowing Rate Supppose E(rA) = 15%; (rA) = 22% and rf = 7% lending rate, and a 9% borrowing rate, Quadratic investor with A = 1, then w* = (0.15-0.09)/[1*(0.22)2] = 1.24 1.24 < 1.65 This investor should place 124% of his capital in A. He needs to borrow 24% of his capital at the risk free rate of 9%. This is less than what he would borrow at a 7% borrowing rate. E(Rp) = 1.24(0.15) + -0.24(0.09)= 16.44% (rp) = 1.24*0.22= 27.28% Increasing the borrowing rate, lowers his utility from before: U = 0.1644 – 0.5*1*(0.27282) = .1272 < .1361

41. Imagine a world with 2 risky securities

42. Expected Return and Standard Deviation with Various Correlation Coefficients 7-41

43. Portfolio Expected Return as a Function of Investment Proportions 7-42

44. Portfolio Standard Deviation as a Function of Investment Proportions 7-43

45. Returning to the Two-Security Portfolio and , or Question: What happens if we use various securities’ combinations, i.e. if we vary r? 7-44

46. Portfolio Expected Return as a function of Standard Deviation 7-45

47. Perfect Correlation E(R) E With two perfectly correlated securities, all portfolios will lie on a straight line between the two assets. Rij = +1.00 D With short selling

48. Perfect Correlation  = +1

49. Zero Correlation E(R) f f 2 g With uncorrelated assets it is possible to create a two asset portfolio with lower risk than either asset! h i j Rij = +1.00 k 1 Rij = 0.00

50. Zero Correlation  = 0