Allocation to Risky Assets. Risk Aversion and Capital. Allocation to Risky Assets. Risk and Risk Aversion. Risk and Risk Aversion

Transcription

1 Allocation to Riky Aet 6- Rik Averion and Capital Allocation to Riky Aet Invetor will avoid rik unle there i a reward. The utility model give the optimal allocation between a riky portfolio and a rik-free aet. Rik and Rik Averion 6-3 Rik and Rik Averion 6-4 Speculation Taking coniderable rik for a commenurate gain artie have heterogeneou expectation Gamble Bet or wager on an uncertain outcome for enjoyment artie aign the ame probabilitie to the poible outcome

2 Rik Averion and Utility Value Invetor are willing to conider: 6-5 Table 6.1 Available Riky ortfolio (Rikfree Rate = 5%) 6-6 rik-free aet peculative poition with poitive rik premium ortfolio attractivene increae with expected return and decreae with rik. What happen when return increae with rik? ach portfolio receive a utility core to ae the invetor rik/return trade off Utility Function 6-7 Table 6. Utility Score of Alternative ortfolio for Invetor with Varying egree of Rik Averion 6-8 U = utility ( r ) = expected return on the aet or portfolio A = coefficient of rik averion = variance of return ½ = a caling factor 1 U () r A

3 Mean-Variance (M-V) Criterion timating Rik Averion ortfolio A dominate portfolio B if: And r A r B A B Ue quetionnaire Oberve individual deciion when confronted with rik Oberve how much people are willing to pay to avoid rik Capital Allocation Acro Riky and Rik- Free ortfolio Aet Allocation: I a very important part of portfolio contruction. Refer to the choice among broad aet clae. Controlling Rik: Simplet way: Manipulate the fraction of the portfolio inveted in rik-free aet veru the portion inveted in the riky aet 6-11 Baic Aet Allocation Total Market Value $300,000 Rik-free money market fund $90,000 quitie $113,400 Bond (long-term) $96,600 $113,400 $96,600 W 0.54 Total W $10,000 rik aet $10,000 B $10,00 6-1

4 Baic Aet Allocation The Rik-Free Aet Let y = weight of the riky portfolio,, in the complete portfolio; (1-y) = weight of rik-free aet: $10,000 $90,000 y y 0. 3 $300,000 $300,000 $113,400 $96,600 :.378 B :. 3 $300,000 $300,000 Only the government can iue default-free bond. Rik-free in real term only if price indexed and maturity equal to invetor holding period. T-bill viewed a the rik-free aet Money market fund alo conidered rik-free in practice Figure 6.3 Spread Between 3-Month C and T-bill Rate 6-15 ortfolio of One Riky Aet and a Rik-Free Aet 6-16 It poible to create a complete portfolio by plitting invetment fund between afe and riky aet. Let y=portion allocated to the riky portfolio, (1-y)=portion to be inveted in rik-free aet, F.

5 xample Uing Chapter 6.4 Number 6-17 xample (Ctd.) 6-18 r f = 7% rf = 0% (r p ) = 15% p = % y = % in p (1-y) = % in r f The expected return on the complete portfolio i the rik-free rate plu the weight of time the rik premium of ( rc ) rf y ( r ) r f r 7 y15 7 c xample (Ctd.) xample (Ctd.) The rik of the complete portfolio i the weight of time the rik of : Rearrange and ubtitute y= C / : C r r r C f 8 rf 7 C C y y r r f Slope 8

6 Figure 6.4 The Invetment Opportunity Set 6-1 Capital Allocation Line with Leverage 6- Lend at r f =7% and borrow at r f =9% Lending range lope = 8/ = 0.36 Borrowing range lope = 6/ = 0.7 CAL kink at Figure 6.5 The Opportunity Set with ifferential Borrowing and Lending Rate 6-3 Rik Tolerance and Aet Allocation 6-4 The invetor mut chooe one optimal portfolio, C, from the et of feaible choice xpected return of the complete portfolio: ( rc ) rf y ( r ) r f Variance: y C

7 Table 6.4 Utility Level for Variou oition in Riky Aet (y) for an Invetor with Rik Averion A = Figure 6.6 Utility a a Function of Allocation to the Riky Aet, y 6-6 Table 6.5 Spreadheet Calculation of Indifference Curve 6-7 Figure 6.7 Indifference Curve for U =.05 and U =.09 with A = and A = 4 6-8

8 Figure 6.8 Finding the Optimal Complete ortfolio Uing Indifference Curve 6-9 Table 6.6 xpected Return on Four Indifference Curve and the CAL 6-30 aive Strategie: The Capital Market Line 6-31 aive Strategie: The Capital Market Line 6-3 The paive trategy avoid any direct or indirect ecurity analyi Supply and demand force may make uch a trategy a reaonable choice for many invetor A natural candidate for a paively held riky aet would be a well-diverified portfolio of common tock uch a the S& 500. The capital market line (CML) i the capital allocation line formed from 1-month T-bill and a broad index of common tock (e.g. the S& 500).

9 aive Strategie: The Capital Market Line 6-33 aive Strategie: The Capital Market Line 6-34 The CML i given by a trategy that involve invetment in two paive portfolio: 1. virtually rik-free hort-term T-bill (or a money market fund). a fund of common tock that mimic a broad market index. From 196 to 009, the paive riky portfolio offered an average rik premium of 7.9% with a tandard deviation of 0.8%, reulting in a reward-to-volatility ratio of The Invetment eciion Top-down proce with 3 tep: Optimal Riky ortfolio 1.Capital allocation between the riky portfolio and rik-free aet.aet allocation acro broad aet clae 3.Security election of individual aet within each aet cla

10 iverification and ortfolio Rik 7-37 Figure 7.1 ortfolio Rik a a Function of the Number of Stock in the ortfolio 7-38 Market rik Sytematic or nondiverifiable Firm-pecific rik iverifiable or nonytematic Figure 7. ortfolio iverification Covariance and Correlation ortfolio rik depend on the correlation between the return of the aet in the portfolio Covariance and the correlation coefficient provide a meaure of the way return of two aet vary

11 Two-Security ortfolio: Return r p r w r w r w r w r ortfolio Return Bond Weight Bond Return quity Weight quity Return ( r ) w ( r ) w ( r ) p Two-Security ortfolio: Rik w w w w Cov r, r p Cov r, r = Variance of Security = Variance of Security = Covariance of return for Security and Security Two-Security ortfolio: Rik 7-43 Covariance 7-44 Another way to expre variance of the portfolio: w w Cov( r, r ) w w Cov( r, r ) w w Cov( r, r ) Cov(r, r ) =, = Correlation coefficient of return = Standard deviation of return for Security = Standard deviation of return for Security

12 Correlation Coefficient: oible Value Correlation Coefficient When ρ = 1, there i no diverification Range of value for 1, > > -1.0 w w If = 1.0, the ecuritie are perfectly poitively correlated If = - 1.0, the ecuritie are perfectly negatively correlated When ρ = -1, a perfect hedge i poible w 1 w Table 7. Computation of ortfolio Variance From the Covariance Matrix 7-47 Three-Aet ortfolio 7-48 ( r ) w ( r ) w ( r ) w ( r ) p w w w p w1 w 1, w1 w3 1,3 w w 3,3

13 Figure 7.3 ortfolio xpected Return a a Function of Invetment roportion 7-49 Figure 7.4 ortfolio Standard eviation a a Function of Invetment roportion 7-50 The Minimum Variance ortfolio 7-51 Figure 7.5 ortfolio xpected Return a a Function of Standard eviation 7-5 The minimum variance portfolio i the portfolio compoed of the riky aet that ha the mallet tandard deviation, the portfolio with leat rik. When correlation i le than +1, the portfolio tandard deviation may be maller than that of either of the individual component aet. When correlation i - 1, the tandard deviation of the minimum variance portfolio i zero.

14 Correlation ffect The amount of poible rik reduction through diverification depend on the correlation. The rik reduction potential increae a the correlation approache -1. If = +1.0, no rik reduction i poible. If = 0, σ may be le than the tandard deviation of either component aet. If = -1.0, a rikle hedge i poible Figure 7.6 The Opportunity Set of the ebt and quity Fund and Two Feaible CAL 7-54 The Sharpe Ratio Maximize the lope of the CAL for any poible portfolio,. The objective function i the lope: 7-55 Figure 7.7 The Opportunity Set of the ebt and quity Fund with the Optimal CAL and the Optimal Riky ortfolio 7-56 S ( r ) r f The lope i alo the Sharpe ratio.

15 Figure 7.8 etermination of the Optimal Overall ortfolio 7-57 Markowitz ortfolio Selection Model Security Selection The firt tep i to determine the rikreturn opportunitie available. All portfolio that lie on the minimumvariance frontier from the global minimum-variance portfolio and upward provide the bet rik-return combination 7-58 Figure 7.10 The Minimum-Variance Frontier of Riky Aet 7-59 Markowitz ortfolio Selection Model 7-60 We now earch for the CAL with the highet reward-to-variability ratio

16 Figure 7.11 The fficient Frontier of Riky Aet with the Optimal CAL 7-61 Markowitz ortfolio Selection Model 7-6 veryone invet in, regardle of their degree of rik averion. More rik avere invetor put more in the rik-free aet. Le rik avere invetor put more in. Capital Allocation and the Separation roperty The eparation property tell u that the portfolio choice problem may be eparated into two independent tak etermination of the optimal riky portfolio i purely technical. Allocation of the complete portfolio to T- bill veru the riky portfolio depend on peronal preference Figure 7.13 Capital Allocation Line with Variou ortfolio from the fficient Set 7-64

17 The ower of iverification The ower of iverification Remember: n n wiwjcov ri rj i1 j1 (, ) If we define the average variance and average covariance of the ecuritie a: 1 n i1 i 1 Cov nn ( 1) n n n j1 i1 ji Cov( r, r ) i j We can then expre portfolio variance a: 1 n 1 Cov n n Table 7.4 Rik Reduction of qually Weighted ortfolio in Correlated and Uncorrelated Univere 7-67 Optimal ortfolio and Nonnormal Return 7-68 Fat-tailed ditribution can reult in extreme value of VaR and S and encourage maller allocation to the riky portfolio. If other portfolio provide ufficiently better VaR and S value than the mean-variance efficient portfolio, we may prefer thee when faced with fat-tailed ditribution.

18 Rik ooling and the Inurance rinciple 7-69 Rik Sharing 7-70 Rik pooling: merging uncorrelated, riky project a a mean to reduce rik. increae the cale of the riky invetment by adding additional uncorrelated aet. The inurance principle: rik increae le than proportionally to the number of policie inured when the policie are uncorrelated Sharpe ratio increae A riky aet are added to the portfolio, a portion of the pool i old to maintain a riky portfolio of fixed ize. Rik haring combined with rik pooling i the key to the inurance indutry. True diverification mean preading a portfolio of fixed ize acro many aet, not merely adding more riky bet to an ever-growing riky portfolio. Invetment for the Long Run 7-71 Long Term Strategy Invet in the riky portfolio for year. Long-term trategy i rikier. Rik can be reduced by elling ome of the riky aet in year. Time diverification i not true diverification. Short Term Strategy Invet in the riky portfolio for 1 year and in the rik-free aet for the econd year.