Risk Aversion and Capital Allocation to Risky Assets.ppt

1、CHAPTER 6,Risk Aversion and Capital Allocation to Risky Assets,6-2,Allocation to Risky Assets,Investors will avoid risk unless there is a reward. The utility model gives the optimal allocation between a risky portfolio and a risk-free asset.,6-3,Risk and Risk Aversion,Speculation Taking considerable

2、 risk for a commensurate gain Parties have heterogeneous expectations,6-4,Risk and Risk Aversion,Gamble Bet or wager on an uncertain outcome for enjoyment Parties assign the same probabilities to the possible outcomes,6-5,Risk Aversion and Utility Values,Investors are willing to consider: risk-free

3、assets speculative positions with positive risk premiums Portfolio attractiveness increases with expected return and decreases with risk. What happens when return increases with risk?,6-6,Table 6.1 Available Risky Portfolios (Risk-free Rate = 5%),Each portfolio receives a utility score to assess the

4、 investors risk/return trade off,6-7,Utility Function,U = utility E ( r ) = expected return on the asset or portfolio A = coefficient of risk aversion s2 = variance of returns = a scaling factor,6-8,Table 6.2 Utility Scores of Alternative Portfolios for Investors with Varying Degree of Risk Aversion

5、,6-9,Mean-Variance (M-V) Criterion,Portfolio A dominates portfolio B if: And,6-10,Estimating Risk Aversion,Use questionnaires Observe individuals decisions when confronted with risk Observe how much people are willing to pay to avoid risk,6-11,Capital Allocation Across Risky and Risk-Free Portfolios

6、,Asset Allocation:,Is a very important part of portfolio construction. Refers to the choice among broad asset classes.,Controlling Risk:,Simplest way: Manipulate the fraction of the portfolio invested in risk-free assets versus the portion invested in the risky assets,6-12,Basic Asset Allocation,6-1

7、3,Basic Asset Allocation,Let y = weight of the risky portfolio, P, in the complete portfolio; (1-y) = weight of risk-free assets:,6-14,The Risk-Free Asset,Only the government can issue default-free bonds. Risk-free in real terms only if price indexed and maturity equal to investors holding period. T

8、-bills viewed as “the” risk-free asset Money market funds also considered risk-free in practice,6-15,Figure 6.3 Spread Between 3-Month CD and T-bill Rates,6-16,Its possible to create a complete portfolio by splitting investment funds between safe and risky assets. Let y=portion allocated to the risk

9、y portfolio, P (1-y)=portion to be invested in risk-free asset, F.,Portfolios of One Risky Asset and a Risk-Free Asset,6-17,Example Using Chapter 6.4 Numbers,6-18,Example (Ctd.),The expected return on the complete portfolio is the risk-free rate plus the weight of P times the risk premium of P,6-19,

10、Example (Ctd.),The risk of the complete portfolio is the weight of P times the risk of P:,6-20,Example (Ctd.),Rearrange and substitute y=sC/sP:,6-21,Figure 6.4 The Investment Opportunity Set,6-22,Lend at rf=7% and borrow at rf=9% Lending range slope = 8/22 = 0.36 Borrowing range slope = 6/22 = 0.27

11、CAL kinks at P,Capital Allocation Line with Leverage,6-23,Figure 6.5 The Opportunity Set with Differential Borrowing and Lending Rates,6-24,Risk Tolerance and Asset Allocation,The investor must choose one optimal portfolio, C, from the set of feasible choices Expected return of the complete portfoli

12、o: Variance:,6-25,Table 6.4 Utility Levels for Various Positions in Risky Assets (y) for an Investor with Risk Aversion A = 4,6-26,Figure 6.6 Utility as a Function of Allocation to the Risky Asset, y,6-27,Table 6.5 Spreadsheet Calculations of Indifference Curves,6-28,Figure 6.7 Indifference Curves f

13、or U = .05 and U = .09 with A = 2 and A = 4,6-29,Figure 6.8 Finding the Optimal Complete Portfolio Using Indifference Curves,6-30,Table 6.6 Expected Returns on Four Indifference Curves and the CAL,6-31,Passive Strategies: The Capital Market Line,The passive strategy avoids any direct or indirect sec

14、urity analysis Supply and demand forces may make such a strategy a reasonable choice for many investors,6-32,Passive Strategies: The Capital Market Line,A natural candidate for a passively held risky asset would be a well-diversified portfolio of common stocks such as the S&P 500. The capital market

15、 line (CML) is the capital allocation line formed from 1-month T-bills and a broad index of common stocks (e.g. the S&P 500).,6-33,Passive Strategies: The Capital Market Line,The CML is given by a strategy that involves investment in two passive portfolios: virtually risk-free short-term T-bills (or

16、 a money market fund) a fund of common stocks that mimics a broad market index.,6-34,Passive Strategies: The Capital Market Line,From 1926 to 2009, the passive risky portfolio offered an average risk premium of 7.9% with a standard deviation of 20.8%, resulting in a reward-to-volatility ratio of .38

17、.,6-35,CHAPTER 7,Optimal Risky Portfolios,6-36,The Investment Decision,Top-down process with 3 steps: Capital allocation between the risky portfolio and risk-free asset Asset allocation across broad asset classes Security selection of individual assets within each asset class,6-37,Diversification an

18、d Portfolio Risk,Market risk Systematic or nondiversifiable Firm-specific risk Diversifiable or nonsystematic,6-38,Figure 7.1 Portfolio Risk as a Function of the Number of Stocks in the Portfolio,6-39,Figure 7.2 Portfolio Diversification,6-40,Covariance and Correlation,Portfolio risk depends on the

19、correlation between the returns of the assets in the portfolio Covariance and the correlation coefficient provide a measure of the way returns of two assets vary,6-41,Two-Security Portfolio: Return,6-42,Two-Security Portfolio: Risk,6-43,Two-Security Portfolio: Risk,Another way to express variance of

20、 the portfolio:,6-44,D,E = Correlation coefficient of returns,Cov(rD,rE) = DEDE,D = Standard deviation of returns for Security D E = Standard deviation of returns for Security E,Covariance,6-45,Range of values for 1,2,+ 1.0 r -1.0,If r = 1.0, the securities are perfectly positively correlated If r =

21、 - 1.0, the securities are perfectly negatively correlated,Correlation Coefficients: Possible Values,6-46,Correlation Coefficients,When DE = 1, there is no diversification When DE = -1, a perfect hedge is possible,6-47,Table 7.2 Computation of Portfolio Variance From the Covariance Matrix,6-48,Three

22、-Asset Portfolio,6-49,Figure 7.3 Portfolio Expected Return as a Function of Investment Proportions,6-50,Figure 7.4 Portfolio Standard Deviation as a Function of Investment Proportions,6-51,The Minimum Variance Portfolio,The minimum variance portfolio is the portfolio composed of the risky assets tha

23、t has the smallest standard deviation, the portfolio with least risk.,When correlation is less than +1, the portfolio standard deviation may be smaller than that of either of the individual component assets. When correlation is -1, the standard deviation of the minimum variance portfolio is zero.,6-

24、52,Figure 7.5 Portfolio Expected Return as a Function of Standard Deviation,6-53,The amount of possible risk reduction through diversification depends on the correlation. The risk reduction potential increases as the correlation approaches -1. If r = +1.0, no risk reduction is possible. If r = 0, P

25、may be less than the standard deviation of either component asset. If r = -1.0, a riskless hedge is possible.,Correlation Effects,6-54,Figure 7.6 The Opportunity Set of the Debt and Equity Funds and Two Feasible CALs,6-55,The Sharpe Ratio,Maximize the slope of the CAL for any possible portfolio, P.

26、The objective function is the slope: The slope is also the Sharpe ratio.,6-56,Figure 7.7 The Opportunity Set of the Debt and Equity Funds with the Optimal CAL and the Optimal Risky Portfolio,6-57,Figure 7.8 Determination of the Optimal Overall Portfolio,6-58,Figure 7.9 The Proportions of the Optimal

27、 Overall Portfolio,6-59,Markowitz Portfolio Selection Model,Security Selection The first step is to determine the risk-return opportunities available. All portfolios that lie on the minimum-variance frontier from the global minimum-variance portfolio and upward provide the best risk-return combinati

28、ons,6-60,Figure 7.10 The Minimum-Variance Frontier of Risky Assets,6-61,Markowitz Portfolio Selection Model,We now search for the CAL with the highest reward-to-variability ratio,6-62,Figure 7.11 The Efficient Frontier of Risky Assets with the Optimal CAL,6-63,Markowitz Portfolio Selection Model,Eve

29、ryone invests in P, regardless of their degree of risk aversion. More risk averse investors put more in the risk-free asset. Less risk averse investors put more in P.,6-64,Capital Allocation and the Separation Property,The separation property tells us that the portfolio choice problem may be separat

30、ed 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.,6-65,Figure 7.13 Capital Allocation Lines with Various Portfolios from the Efficient Set,6-66,The

31、 Power of Diversification,Remember: If we define the average variance and average covariance of the securities as:,6-67,The Power of Diversification,We can then express portfolio variance as:,6-68,Table 7.4 Risk Reduction of Equally Weighted Portfolios in Correlated and Uncorrelated Universes,6-69,O

32、ptimal Portfolios and Nonnormal Returns,Fat-tailed distributions can result in extreme values of VaR and ES and encourage smaller allocations to the risky portfolio. If other portfolios provide sufficiently better VaR and ES values than the mean-variance efficient portfolio, we may prefer these when faced with fat-tailed distributions.,6-70,Risk Pooling