1、RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS6-1Lecture 3Hafiz Hoque6-2Allocation to Risky AssetsInvestors will avoid risk unless there is a reward.The utility model gives the optimal allocation between a risky portfolio and a risk-free asset.Hafiz Hoque6-3Risk and Risk AversionSpeculationTak

2、ing considerable risk for a commensurate gainParties have heterogeneous expectationsHafiz Hoque6-4Risk and Risk AversionGamble Bet or wager on an uncertain outcome for enjoymentParties assign the same probabilities to the possible outcomesHafiz Hoque6-5Risk Aversion and Utility ValuesInvestors are w

3、illing to consider:risk-free assetsspeculative positions with positive risk premiumsPortfolio attractiveness increases with expected return and decreases with risk.What happens when return increases with risk?Hafiz Hoque6-6Table 6.1 Available Risky Portfolios (Risk-free Rate = 5%)Each portfolio rece

4、ives a utility score to assess the investors risk/return trade offHafiz Hoque6-7Utility FunctionU = utilityE ( r ) = expected return on the asset or portfolioA = coefficient of risk aversions2 = variance of returns = a scaling factor 21( )2UE rAsHafiz Hoque6-8Table 6.2 Utility Scores of Alternative

5、Portfolios for Investors with Varying Degree of Risk AversionHafiz Hoque6-9Mean-Variance (M-V) CriterionPortfolio A dominates portfolio B if:And BArErEBAssHafiz Hoque6-10Estimating Risk AversionUse questionnairesObserve individuals decisions when confronted with riskObserve how much people are willi

6、ng to pay to avoid riskHafiz Hoque6-11Capital 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 way: Manipulate the fraction of the portfolio invested in risk-f

7、ree assets versus the portion invested in the risky assetsHafiz Hoque6-12Basic Asset AllocationTotal Market Value$300,000Risk-free money market fund$90,000Equities$113,400Bonds (long-term)$96,600Total risk assets$210,00054. 0000,210$400,113$EW46. 000,210$600,96$BWHafiz Hoque6-13Basic Asset Allocatio

8、nLet y = weight of the risky portfolio, P, in the complete portfolio; (1-y) = weight of risk-free assets:7 . 0000,300$000,210$y3 . 0000,300$000,90$1 y378.000,300$400,113$:E322.000,300$600,96$:BHafiz Hoque6-14The Risk-Free AssetOnly the government can issue default-free bonds.Risk-free in real terms

9、only if price indexed and maturity equal to investors holding period.T-bills viewed as “the” risk-free assetMoney market funds also considered risk-free in practiceHafiz Hoque6-15Figure 6.3 Spread Between 3-Month CD and T-bill Rates Hafiz Hoque6-16Its possible to create a complete portfolio by split

10、ting investment funds between safe and risky assets.Let y=portion allocated to the risky portfolio, P(1-y)=portion to be invested in risk-free asset, F.Portfolios of One Risky Asset and a Risk-Free AssetHafiz Hoque6-17Example Using Chapter 6.4 Numbersrf = 7%s srf = 0%E(rp) = 15%s sp = 22%y = % in p(

11、1-y) = % in rfHafiz Hoque6-18Example (Ctd.)The expected return on the complete portfolio is the risk-free rate plus the weight of P times the risk premium of P( )()cfPfE rry E rr 7157yrEcHafiz Hoque6-19Example (Ctd.)The risk of the complete portfolio is the weight of P times the risk of P:yyPC22ssHa

12、fiz Hoque6-20Example (Ctd.)Rearrange and substitute y=sC/sP: CfPPCfCrrErrEsss2287 228PfPrrESlopesHafiz Hoque6-21Figure 6.4 The Investment Opportunity Set Hafiz Hoque6-22Lend at rf=7% and borrow at rf=9%Lending range slope = 8/22 = 0.36Borrowing range slope = 6/22 = 0.27CAL kinks at PCapital Allocati

13、on Line with LeverageHafiz Hoque6-23Figure 6.5 The Opportunity Set with Differential Borrowing and Lending RatesHafiz Hoque6-24Risk Tolerance and Asset AllocationThe investor must choose one optimal portfolio, C, from the set of feasible choicesExpected return of the complete portfolio:Variance:( )(

14、)cfPfE rry E rr222CPyssHafiz Hoque6-25Table 6.4 Utility Levels for Various Positions in Risky Assets (y) for an Investor with Risk Aversion A = 4Hafiz Hoque6-26Figure 6.6 Utility as a Function of Allocation to the Risky Asset, yHafiz Hoque6-27Table 6.5 Spreadsheet Calculations of Indifference Curves

15、Hafiz Hoque6-28Figure 6.7 Indifference Curves for U = .05 and U = .09 with A = 2 and A = 4 Hafiz Hoque6-29Figure 6.8 Finding the Optimal Complete Portfolio Using Indifference Curves Hafiz Hoque6-30Table 6.6 Expected Returns on Four Indifference Curves and the CALHafiz Hoque6-31Passive Strategies: Th

16、e Capital Market LineThe passive strategy avoids any direct or indirect security analysisSupply and demand forces may make such a strategy a reasonable choice for many investorsHafiz Hoque6-32Passive Strategies: The Capital Market LineA natural candidate for a passively held risky asset would be a w

17、ell-diversified portfolio of common 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).Hafiz Hoque6-33Passive Strategies: The Capital Market LineThe CML is given by a strategy that involves investment in two passive portfolios: 1.virtually risk-free short-term T-bills (or a money market fund) 2.a fund of common stock