OPTIMAL RISKY PORTFOLIOS- ASSET ALLOCATIONS. BKM Ch 7

Transcription

1 OPTIMAL RISKY PORTFOLIOS- ASSET ALLOCATIONS BKM Ch 7

2 ASSET ALLOCATION Idea from bank account to diversified portfolio Discussion principles are the same for any number of stocks A. bonds and stocks B. bills, bonds and stocks C. any number of risky assets BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/2010 2

3 DIVERSIFICATION AND PORTFOLIO RISK Market risk Systematic or nondiversifiable Firm-specific risk Diversifiable or nonsystematic BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/2010 3

4 FIGURE 7.1 PORTFOLIO RISK AS A FUNCTION OF THE NUMBER OF STOCKS IN THE PORTFOLIO BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/2010 4

5 FIGURE 7.2 PORTFOLIO DIVERSIFICATION BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/2010 5

6 COVARIANCE AND CORRELATION Portfolio risk depends on the correlation between the returns of the assets in the portfolio Covariance and the correlation coefficient provide a measure of the way returns two assets vary BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/2010 6

7 TWO-SECURITY PORTFOLIO: RETURN r r p D D E E P w r D w r E D E w r w r Portfolio Return Bond Weight Bond Return Equity Weight Equity Return E( r ) w E( r ) w E( r ) p D D E E BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/2010 7

8 TWO-SECURITY PORTFOLIO: RISK 2 w 2 2 w 2 2 w Cov r r P D D E E D E D E 2 (, ) 2 D 2 E = Variance of Security D = Variance of Security E Cov( r, r ) D E = Covariance of returns for Security D and Security E BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/2010 8

9 TWO-SECURITY PORTFOLIO: RISK CONTINUED Another way to express variance of the portfolio: w w Cov( r, r ) w w Cov( r, r ) 2 w w Cov( r, r ) 2 P D D D D E E E E D E D E BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/2010 9

10 COVARIANCE Cov(r D, r E ) = DE D E D,E = Correlation coefficient of returns D = Standard deviation of returns for Security D E = Standard deviation of returns for Security E BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

11 CORRELATION COEFFICIENTS: POSSIBLE VALUES Range of values for 1, > > -1.0 If = 1.0, the securities would be perfectly positively correlated If = - 1.0, the securities would be perfectly negatively correlated BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

12 TABLE 7.1 DESCRIPTIVE STATISTICS FOR TWO MUTUAL FUNDS BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

13 THREE-SECURITY PORTFOLIO E( r ) w E( r ) w E( r ) w E( r ) p p = w w w w 1 w 2 Cov(r 1, r 2 ) + 2w 1 w 3 Cov(r 1, r 3 ) Cov(r 2,r 3 ) + 2w 2 w 3 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

14 ASSET ALLOCATION Portfolio of 2 risky assets (cont d) examples BKM7 Tables 7.1 & 7.3 BKM7 Figs. 7.3 (return), 7.4 (risk) & 7.5 (trade-off) portfolio opportunity set (BKM7 Fig. 7.5) minimum variance portfolio choose w D such that portfolio variance is lowest optimization problem minimum variance portfolio has less risk than either component (i.e., asset) BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

15 TABLE 7.2 COMPUTATION OF PORTFOLIO VARIANCE FROM THE COVARIANCE MATRIX BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

16 TABLE 7.3 EXPECTED RETURN AND STANDARD DEVIATION WITH VARIOUS CORRELATION COEFFICIENTS BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

17 FIGURE 7.3 PORTFOLIO EXPECTED RETURN AS A FUNCTION OF INVESTMENT PROPORTIONS BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

18 FIGURE 7.4 PORTFOLIO STANDARD DEVIATION AS A FUNCTION OF INVESTMENT PROPORTIONS BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

19 MINIMUM VARIANCE PORTFOLIO AS DEPICTED IN FIGURE 7.4 Standard deviation is smaller than that of either of the individual component assets Figure 7.3 and 7.4 combined demonstrate the relationship between portfolio risk BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

20 FIGURE 7.5 PORTFOLIO EXPECTED RETURN AS A FUNCTION OF STANDARD DEVIATION BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

21 CORRELATION EFFECTS The relationship depends on the correlation coefficient -1.0 < < +1.0 The smaller the correlation, the greater the risk reduction potential If = +1.0, no risk reduction is possible BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

22 FIGURE 7.6 THE OPPORTUNITY SET OF THE DEBT AND EQUITY FUNDS AND TWO FEASIBLE CALS BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

23 THE SHARPE RATIO Maximize the slope of the CAL for any possible portfolio, p The objective function is the slope: S P E( r ) P P r f BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

24 FIGURE 7.7 THE OPPORTUNITY SET OF THE DEBT AND EQUITY FUNDS WITH THE OPTIMAL CAL AND THE OPTIMAL RISKY PORTFOLIO BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

25 FIGURE 7.8 DETERMINATION OF THE OPTIMAL OVERALL PORTFOLIO BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

26 ASSET ALLOCATION Finding the optimal risky portfolio: II. Formally Intuitively BKM7 Figs. 7.6 and 7.7 improve the reward-to-variability ratio optimal risky portfolio tangency point (Fig. 7.8) Formally: 1 s.t. ] [ i i P f P w w r r E Max D E D E D D D D D E D D D E D w w w w R E w R E w Max w D, ) 2(1 ) (1 ] [ ] [ ) (1 1 /16/2010 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 26

27 ASSET ALLOCATION 18 formally (continued) 2 2 ) ] [ ( ) ] [ ( D f E E f D r r E r r E Den Den Num w D ), )cov( ] [ ( ) ] [ ( 2 E D f E E f D r r r r E r r E Num ), )cov( 2 ] [ ] [ ( E D f D E r r r r E r E 1 /16/2010 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 27

28 ASSET ALLOCATION 19 Example (BKM7 Fig. 7.8) 1. plot D, E, riskless 2. compute optimal risky portfolio weights w D = Num/Den = 0.4; w E = 1- w D = given investor risk aversion (A=4), compute w * w * 2 x E[ r P x bottom line: 25.61% in bills; ] r A x f 2 ( r P ) 2 x x x % in bonds (0.7439x0.4); rest in stocks BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

29 FIGURE 7.9 THE PROPORTIONS OF THE OPTIMAL OVERALL PORTFOLIO BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

30 MARKOWITZ PORTFOLIO SELECTION MODEL Security Selection First step is to determine the risk-return opportunities available All portfolios that lie on the minimumvariance frontier from the global minimumvariance portfolio and upward provide the best risk-return combinations BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

31 MARKOWITZ PORTFOLIO SELECTION MODEL Combining many risky assets & T-Bills basic idea remains unchanged 1. specify risk-return characteristics of securities find the efficient frontier (Markowitz) 2. find the optimal risk portfolio maximize reward-to-variability ratio 3. combine optimal risk portfolio & riskless asset capital allocation BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

32 MARKOWITZ PORTFOLIO SELECTION MODEL finding the efficient frontier definition set of portfolios with highest return for given risk minimum-variance frontier take as given the risk-return characteristics of securities estimated from historical data or forecasts n securities > n return + n(n-1) var. & cov. use an optimization program to compute the efficient frontier (Markowitz) subject to same constraints BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

33 MARKOWITZ PORTFOLIO SELECTION MODEL Finding the efficient frontier (cont d) optimization constraints portfolio weights sum up to 1 no short sales, dividend yield, asset restrictions, Individual assets vs. frontier portfolios BKM7 Fig short sales > not on the efficient frontier no short sales > may be on the frontier example: highest return asset BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

34 FIGURE 7.10 THE MINIMUM-VARIANCE FRONTIER OF RISKY ASSETS BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

35 MARKOWITZ PORTFOLIO SELECTION MODEL CONTINUED We now search for the CAL with the highest reward-to-variability ratio BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

36 FIGURE 7.11 THE EFFICIENT FRONTIER OF RISKY ASSETS WITH THE OPTIMAL CAL BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

37 MARKOWITZ PORTFOLIO SELECTION MODEL CONTINUED Now the individual chooses the appropriate mix between the optimal risky portfolio P and T-bills as in Figure 7.8 n n 2 P wiwjcov ri rj i1 j1 (, ) BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

38 FIGURE 7.12 THE EFFICIENT PORTFOLIO SET BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

39 CAPITAL ALLOCATION AND THE SEPARATION PROPERTY The separation property tells us that the portfolio choice problem may be separated into two independent tasks Determination of the optimal risky portfolio is purely technical Allocation of the complete portfolio to T- bills versus the risky portfolio depends on personal preference BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

40 FIGURE 7.13 CAPITAL ALLOCATION LINES WITH VARIOUS PORTFOLIOS FROM THE EFFICIENT SET BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

41 THE POWER OF DIVERSIFICATION Remember: n n 2 P wiwjcov ri rj i1 j1 If we define the average variance and average covariance of the securities n as: We can then express portfolio variance as: n i1 1 Cov nn ( 1) i n j1 i1 ji Cov( r, r ) 2 2 P n i j (, ) 1 n 1 Cov n n BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

42 TABLE 7.4 RISK REDUCTION OF EQUALLY WEIGHTED PORTFOLIOS IN CORRELATED AND UNCORRELATED UNIVERSES BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

43 RISK POOLING, RISK SHARING AND RISK IN THE LONG RUN Consider the following: p =.001 Loss: payout = $100,000 No Loss: payout = 0 1 p =.999 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

44 RISK POOLING AND THE INSURANCE PRINCIPLE Consider the variance of the portfolio: 1 n 2 2 P It seems that selling more policies causes risk to fall Flaw is similar to the idea that long-term stock investment is less risky BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

45 RISK POOLING AND THE INSURANCE PRINCIPLE CONTINUED When we combine n uncorrelated insurance policies each with an expected profit of $, both expected total profit and SD grow in direct proportion to n: E( n) ne( ) Var( n ) n Var( ) n SD( n) n BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

46 RISK SHARING What does explain the insurance business? Risk sharing or the distribution of a fixed amount of risk among many investors BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

47 AN ASSET ALLOCATION PROBLEM Question: Suppose that there are many stocks in the market and that the characteristics of Stocks A and B are given as follows: Stock Expected Return Standard Deviation A 10% 5% B 15% 10% Correlation = Suppose that it is possible to borrow at the risk-free rate, R f. What must be the value of the risk-free rate? BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

48 AN ASSET ALLOCATION PROBLEM 2 Perfect hedges (portfolio of 2 risky assets) perfectly positively correlated risky assets requires short sales perfectly negatively correlated risky assets BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 if 2 2 D D E E P w w 0 ) ( D D E D P w W 1 /16/

49 AN ASSET ALLOCATION PROBLEM 3 Answer: The trick is to construct a risk-free portfolio from Stocks A and B that rules out arbitrage. Since Stocks A and B are perfectly negatively correlated, a risk-free portfolio can be constructed and its rate of return in equilibrium will be the risk-free rate. To find the proportions of this portfolio (w A invested in Stock A and w B = 1 - w A in Stock B), set the standard deviation equal to zero. With perfect negative correlation, the portfolio standard deviation reduces to: p = w AA - w BB => 0 = 5w A - 10 (1-w A ) => w A = The expected rate of return on this risk-free portfolio is: E(R) = x 10% + ( x 15%) 11.67%. BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

50 Index Models CHAPTER 8

51 FACTOR MODEL Idea the same factor(s) drive all security returns Implementation (simplify the estimation problem) do not look for equilibrium relationship between a security s expected return and risk or expected market returns look for a statistical relationship between realized stock return and realized market return BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

52 FACTOR MODEL 2 Formally stock return = expected stock return + unexpected impact of common (market) factors + unexpected impact of firm-specific factors r i E[ r ] i m e i r i E[ r i ] Fe E[ m] E[ e ] i i 0 i BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

53 INDEX MODEL Factor model problem Index Model solution what is the factor? market portfolio proxy S&P 500, Value Line Index, etc. BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

54 ADVANTAGES OF THE SINGLE INDEX MODEL Reduces the number of inputs for diversification Easier for security analysts to specialize BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

55 SINGLE FACTOR MODEL r E( r ) m e i i i i ß i = index of a securities particular return to the factor m = Unanticipated movement related to security returns e i = Assumption: a broad market index like the S&P 500 is the common factor. BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

56 SINGLE-INDEX MODEL Regression Equation: R t R t e t ( ) ( ) ( ) t i t M i Expected return-beta relationship: E( R ) E( R ) i i i M BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

57 SINGLE-INDEX MODEL CONTINUED Risk and covariance: Total risk = Systematic risk + Firm-specific risk: Covariance = product of betas x market index risk: ( e ) i i M i Correlation = product of correlations with the market index Cov( r, r ) 2 i j i j M Corr( r, r ) Corr( r, r ) xcorr( r, r ) i j M i M j M i j i M j M i j i M j M BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

58 INDEX MODEL AND DIVERSIFICATION Portfolio s variance: Variance of the equally weighted portfolio of firm-specific components: When n gets large, ( ) e P P M P n P i1 n ( e ) ( ei) ( e) n 2 ( ) e P becomes negligible BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

59 FIGURE 8.1 THE VARIANCE OF AN EQUALLY WEIGHTED PORTFOLIO WITH RISK COEFFICIENT Β P IN THE SINGLE-FACTOR ECONOMY BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

60 FIGURE 8.2 EXCESS RETURNS ON HP AND S&P 500 APRIL 2001 MARCH 2006 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

61 FIGURE 8.3 SCATTER DIAGRAM OF HP, THE S&P 500, AND THE SECURITY CHARACTERISTIC LINE (SCL) FOR HP BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

62 TABLE 8.1 EXCEL OUTPUT: REGRESSION STATISTICS FOR THE SCL OF HEWLETT- PACKARD BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

63 FIGURE 8.4 EXCESS RETURNS ON PORTFOLIO ASSETS BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

64 ALPHA AND SECURITY ANALYSIS Macroeconomic analysis is used to estimate the risk premium and risk of the market index Statistical analysis is used to estimate the beta coefficients of all securities and their residual variances, σ 2 ( e i ) Developed from security analysis BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

65 ALPHA AND SECURITY ANALYSIS CONTINUED The market-driven expected return is conditional on information common to all securities Security-specific expected return forecasts are derived from various security-valuation models The alpha value distills the incremental risk premium attributable to private information Helps determine whether security is a good or bad buy BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

66 SINGLE-INDEX MODEL INPUT LIST Risk premium on the S&P 500 portfolio Estimate of the SD of the S&P 500 portfolio n sets of estimates of Beta coefficient Stock residual variances Alpha values BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

67 OPTIMAL RISKY PORTFOLIO OF THE SINGLE- INDEX MODEL Maximize the Sharpe ratio Expected return, SD, and Sharpe ratio: n1 n1 E( R ) E( R ) w E( R ) w P P M P i i M i i i1 i n1 n P P M ( ep ) M wi i wi ( ei ) i1 i1 ER ( P ) SP P BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

68 OPTIMAL RISKY PORTFOLIO OF THE SINGLE- INDEX MODEL CONTINUED Combination of: Active portfolio denoted by A Market-index portfolio, the (n+1)th asset which we call the passive portfolio and denote by M Modification of active portfolio position: 0 * wa wa 0 1 (1 ) w When A A * 0 1, w w A A A BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

69 THE INFORMATION RATIO The Sharpe ratio of an optimally constructed risky portfolio will exceed that of the index portfolio (the passive strategy): 2 s s A M ( e ) A 2 2 P BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

70 FIGURE 8.5 EFFICIENT FRONTIERS WITH THE INDEX MODEL AND FULL-COVARIANCE MATRIX BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

71 TABLE 8.2 COMPARISON OF PORTFOLIOS FROM THE SINGLE-INDEX AND FULL- COVARIANCE MODELS BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

72 INDEX MODEL: INDUSTRY PRACTICES Beta books Idea Merrill Lynch monthly, S&P 500 Value Line etc. regression analysis weekly, NYSE BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

73 INDEX MODEL: INDUSTRY PRACTICES 2 Example (Merrill Lynch differences, Table 8.3) total (not excess) returns slopes are identical smallness percentage price changes dividends? S&P 500 adjusted beta beta = (2/3) estimated beta + (1/3). 1 sampling errors, convergence of new firms exploiting alphas (Treynor-Black) BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

74 TABLE 8.3 MERRILL LYNCH, PIERCE, FENNER & SMITH, INC.: MARKET SENSITIVITY STATISTICS BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

75 TABLE 8.4 INDUSTRY BETAS AND ADJUSTMENT FACTORS BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

76 USING INDEX MODELS Question: Consider the two (excess return) index-model regression results for Stocks A and B: R A = 1% + 1.2R M R-square (A) = Residual standard deviation-n (A)= 10.3% R B = -2% + 0.8R M R-square (B) = Residual standard deviation-n (B) = 9.1% (a) Which stock has more firm-specific risk? (b) Which stock has greater market risk? (c) For which stock does the market explain a greater fraction of return variability? (d) Which stock had an average return in excess of that predicted by the CAPM? BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

77 USING INDEX MODELS 2 Answer: (a) Firm-specific risk is measured by the residual standard deviation. Thus, Stock A has more firm-specific risk, 10.3 > 9.1. (b) Market risk is measured by beta, the slope coefficient of the regression. Stock A has a larger beta coefficient, 1.2 > 0.8. (c) R-square measures the fraction of the total variation in asset returns explained by the market return. Stock A s R-square is larger than Stocks B s R-square, that is, > (d) The average rate of return in excess of that predicted by the CAPM is measured by alpha, the intercept of the security characteristic line (SCL). Alpha (A) = 1% is larger than alpha (B) = -2%. BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

78 USING INDEX MODELS 3 Question: Based on current dividend yields and expected growth rates, the expected rates of return on stocks A and B are 11% and 14%, respectively. The beta of stock A is 0.8, while that of stock B is 1.5. The T-bill rate is currently 6%. The expected rate of return on the S&P 500 index is 12%. The standard deviation of stock A is 10% annually, while that of stock B is 11%. (a) If you currently hold a well-diversified portfolio, would you choose to add either of these stocks to your holdings? (b) If instead you could invest only in bills plus only one of these stocks, which stock would you choose? Explain your answer using either a graph or a quantitative measure of the attractiveness of the stocks. BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/

79 USING INDEX MODELS 4 Answer: (a) The alpha of Stock A is: A = r A [r f + A (r M r f )] => A = 11 [ (12 6)] = 0.2%. For Stock B, B = r B [r f + B (r M r f )] => B = 14 [ (12 6)] = -1%. Thus, Stock A would be a good addition. Taking a short position in B may be desirable. (b) The reward to variability ratio of the stocks is: S A = (11 6)/10 = 0.50 S B = (14 6)/11 = 0.73 Stock B is superior when only one can be held. BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 1 /16/