1、INVESTMENTS | BODIE, KANE, MARCUS Copyright 2011 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin CHAPTER 6 Risk Aversion and Capital Allocation to Risky Assets INVESTMENTS | BODIE, KANE, MARCUS Allocation to Risky Assets Investors will avoid risk unless there is a reward. Th
2、e utility model gives the optimal allocation between a risky portfolio and a risk-free asset. INVESTMENTS | BODIE, KANE, MARCUS Risk and Risk Aversion Speculation Taking considerable risk for a commensurate gain Parties have heterogeneous expectations INVESTMENTS | BODIE, KANE, MARCUS Risk and Risk
3、Aversion Gamble Bet or wager on an uncertain outcome for enjoyment Parties assign the same probabilities to the possible outcomes INVESTMENTS | BODIE, KANE, MARCUS Risk Aversion and Utility Values Investors are willing to consider: risk-free assets speculative positions with positive risk premiums P
4、ortfolio attractiveness increases with expected return and decreases with risk. What happens when return increases with risk? INVESTMENTS | BODIE, KANE, MARCUS Table 6.1 Available Risky Portfolios (Risk- free Rate = 5%) Each portfolio receives a utility score to assess the investors risk/return trad
5、e off INVESTMENTS | BODIE, KANE, MARCUS 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 2 1 ( ) 2 UE rAs INVESTMENTS | BODIE, KANE, MARCUS Table 6.2 Utility Scores of Alternative Portfolios
6、for Investors with Varying Degree of Risk Aversion INVESTMENTS | BODIE, KANE, MARCUS Mean-Variance (M-V) Criterion Portfolio A dominates portfolio B if: And BA rErE BA ss INVESTMENTS | BODIE, KANE, MARCUS Estimating Risk Aversion Use questionnaires Observe individuals decisions when confronted with
7、risk Observe how much people are willing to pay to avoid risk INVESTMENTS | BODIE, KANE, MARCUS Capital Allocation Across Risky and Risk- Free Portfolios Asset Allocation: Is a very important part of portfolio construction. Refers to the choice among broad asset classes. Controlling Risk: Simplest w
8、ay: Manipulate the fraction of the portfolio invested in risk-free assets versus the portion invested in the risky assets INVESTMENTS | BODIE, KANE, MARCUS Basic Asset Allocation Total Market Value$300,000 Risk-free money market fund $90,000 Equities$113,400 Bonds (long-term)$96,600 Total risk asset
9、s$210,000 54. 0 000,210$ 400,113$ E W46. 0 00,210$ 600,96$ B W INVESTMENTS | BODIE, KANE, MARCUS Basic Asset Allocation Let y = weight of the risky portfolio, P, in the complete portfolio; (1-y) = weight of risk-free assets: 7 . 0 000,300$ 000,210$ y 3 . 0 000,300$ 000,90$ 1 y 378. 000,300$ 400,113$
10、 :E 322. 000,300$ 600,96$ :B INVESTMENTS | BODIE, KANE, MARCUS 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-bills viewed as “the” risk-free asset Money market funds also considere
11、d risk-free in practice INVESTMENTS | BODIE, KANE, MARCUS Figure 6.3 Spread Between 3-Month CD and T-bill Rates INVESTMENTS | BODIE, KANE, MARCUS Its possible to create a complete portfolio by splitting investment funds between safe and risky assets. Let y=portion allocated to the risky portfolio, P
12、 (1-y)=portion to be invested in risk-free asset, F. Portfolios of One Risky Asset and a Risk-Free Asset INVESTMENTS | BODIE, KANE, MARCUS rf = 7% s srf = 0% E(rp) = 15% s sp = 22% y = % in p(1-y) = % in rf Example Using Chapter 6.4 Numbers INVESTMENTS | BODIE, KANE, MARCUS Example (Ctd.) The expect
13、ed return on the complete portfolio is the risk-free rate plus the weight of P times the risk premium of P ( )() cfPf E rry E rr 7157yrE c INVESTMENTS | BODIE, KANE, MARCUS Example (Ctd.) The risk of the complete portfolio is the weight of P times the risk of P: yy PC 22ss INVESTMENTS | BODIE, KANE,
14、 MARCUS Example (Ctd.) Rearrange and substitute y=sC/sP: CfP P C fC rrErrEs s s 22 8 7 22 8 P fP rrE Slope s INVESTMENTS | BODIE, KANE, MARCUS Figure 6.4 The Investment Opportunity Set INVESTMENTS | BODIE, KANE, MARCUS Lend at rf=7% and borrow at rf=9% Lending range slope = 8/22 = 0.36 Borrowing ran
15、ge slope = 6/22 = 0.27 CAL kinks at P Capital Allocation Line with Leverage INVESTMENTS | BODIE, KANE, MARCUS Figure 6.5 The Opportunity Set with Differential Borrowing and Lending Rates INVESTMENTS | BODIE, KANE, MARCUS Risk Tolerance and Asset Allocation The investor must choose one optimal portfo
16、lio, C, from the set of feasible choices Expected return of the complete portfolio: Variance: ( )() cfPf E rry E rr 222 CP yss INVESTMENTS | BODIE, KANE, MARCUS Table 6.4 Utility Levels for Various Positions in Risky Assets (y) for an Investor with Risk Aversion A = 4 INVESTMENTS | BODIE, KANE, MARC
17、US Figure 6.6 Utility as a Function of Allocation to the Risky Asset, y INVESTMENTS | BODIE, KANE, MARCUS Table 6.5 Spreadsheet Calculations of Indifference Curves INVESTMENTS | BODIE, KANE, MARCUS Figure 6.7 Indifference Curves for U = .05 and U = .09 with A = 2 and A = 4 INVESTMENTS | BODIE, KANE,
18、 MARCUS Figure 6.8 Finding the Optimal Complete Portfolio Using Indifference Curves INVESTMENTS | BODIE, KANE, MARCUS Table 6.6 Expected Returns on Four Indifference Curves and the CAL INVESTMENTS | BODIE, KANE, MARCUS Passive Strategies: The Capital Market Line The passive strategy avoids any direc
19、t or indirect security analysis Supply and demand forces may make such a strategy a reasonable choice for many investors INVESTMENTS | BODIE, KANE, MARCUS Passive Strategies: The Capital Market Line A natural candidate for a passively held risky asset would be a well-diversified portfolio of common
20、stocks such as the S&P 500. The capital market 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). INVESTMENTS | BODIE, KANE, MARCUS Passive Strategies: The Capital Market Line The CML is given by a strategy that involves inves
21、tment in two passive portfolios: 1. virtually risk-free short-term T-bills (or a money market fund) 2. a fund of common stocks that mimics a broad market index. INVESTMENTS | BODIE, KANE, MARCUS 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.
