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CHAPTER 6 Risk Aversion and Capital Allocation to Risky Assets Three Steps in Investment Decisions – Top-down Approach I. Capital Allocation Decision Allocate funds between risky and risk-free assets Made at higher organization levels II. Asset Allocation Decision Distribute risk investments across asset classes – smallcap stocks, large-cap stocks, bonds, & foreign assets III. Security Selection Decision Select particular securities within each asset class Made at lower organization levels 6-2 Risk and Risk Aversion • Speculation – Considerable risk • Sufficient to affect the decision – Commensurate gain • Gamble – Bet or wager on an uncertain outcome 6-3 Risk Aversion and Utility Values • Risk averse investors reject investment portfolios that are fair games or worse • These investors are willing to consider only risk-free or speculative prospects with positive risk premiums • Intuitively one would rank those portfolios as more attractive with higher expected returns 6-4 6-5 Table 6.1 Available Risky Portfolios (Risk-free Rate = 5%) 6-6 Utility Function 1 2 U E (r ) As 2 Where U = utility E ( r ) = expected return on the asset or portfolio A = coefficient of risk aversion s2 = variance of returns 6-7 Table 6.2 Utility Scores of Alternative Portfolios for Investors with Varying Degree of Risk Aversion 6-8 Estimating Risk Aversion • Observe individuals’ decisions when confronted with risk • Observe how much people are willing to pay to avoid risk – Insurance against large losses 6-9 Figure 6.2 The Indifference Curve 6-10 Table 6.3 Utility Values of Possible Portfolios for an Investor with Risk Aversion, A = 4 6-11 Capital Allocation Across Risky and Risk-Free Portfolios • Control risk – Asset allocation choice • Fraction of the portfolio invested in Treasury bills or other safe money market securities 6-12 The Risky Asset Example Total portfolio value = $300,000 Risk-free value = 90,000 Risky (Vanguard & Fidelity) = 210,000 Vanguard (V) = 54% Fidelity (F) = 46% 6-13 The Risky Asset Example Continued Vanguard 113,400/300,000 = 0.378 Fidelity 96,600/300,000 = 0.322 Portfolio P Risk-Free Assets F Portfolio C 210,000/300,000 = 0.700 90,000/300,000 = 0.300 300,000/300,000 = 1.000 6-14 The Risk-Free Asset • Only the government can issue default-free bonds – Guaranteed real rate only if the duration of the bond is identical to the investor’s desire holding period • T-bills viewed as the risk-free asset – Less sensitive to interest rate fluctuations 6-15 Figure 6.3 Spread Between 3-Month CD and T-bill Rates 6-16 Portfolios of One Risky Asset and a Risk-Free Asset • It’s possible to split investment funds between safe and risky assets. • Risk free asset: proxy; T-bills • Risky asset: stock (or a portfolio) 6-17 Example Using Chapter 6.4 Numbers rf = 7% srf = 0% E(rp) = 15% sp = 22% y = % in p (1-y) = % in rf 6-18 Expected Returns for Combinations E (rc ) yE (rp ) (1 y )rf rc = complete or combined portfolio For example, y = .75 E(rc) = .75(.15) + .25(.07) = .13 or 13% 6-19 Combinations Without Leverage If y = .75, then sc = .75(.22) = .165 or 16.5% If y = 1 s c = 1(.22) = .22 or 22% If y = 0 s c = (.22) = .00 or 0% 6-20 Capital Allocation Line (CAL) E(rc) = yE(rp) + (1 – y)rf = rf +[(E(rp) – rf)]y (1) σc = yσp → y = σc/σp (2) From (1) and (2) E(rc) = rf +[(E(rp) - rf)/σp]σc (CAL) 6-21 Figure 6.4 The Investment Opportunity Set with a Risky Asset and a Risk-free Asset in the Expected Return-Standard Deviation Plane 6-22 Capital Allocation Line with Leverage Borrow at the Risk-Free Rate and invest in stock. Using 50% Leverage, rc = (-.5) (.07) + (1.5) (.15) = .19 sc = (1.5) (.22) = .33 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 – Trade-off between risk and return – Expected return of the complete portfolio is given by: E (rc ) rf y E (rP ) rf – Variance is: s ys 2 C 2 2 P 6-25 Table 6.5 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 Analytical Solution where U = E(rc) – (1/2)Aσc2 (1) E(rc) = yE(rp) + (1-y)rf σc = yσp (2) (3) Substituting (2) and (3) into (1), we obtain U = yE(rp) + (1-y)rf – (1/2)A(yσp)2 From dU/dy = E(rp) – rf – Ayσp2 = 0, y* = (E(rp) – rf)/Aσp2 6-28 y* = (E(rp) – rf)/Aσp2 = (0.15 – 0.07)/4*(0.22)2 = 0.413 6-29 Indifference curve We can trace combinations of E(rc) and σc for given values of U and A. From U = E(rc) – (1/2)Aσc2 E(rc) = U + (1/2)Aσc2 Example: E(rc) = 0.05 + (1/2)(2)σc2 6-30 Table 6.6 Spreadsheet Calculations of Indifference Curves 6-31 Figure 6.7 Indifference Curves for U = .05 and U = .09 with A = 2 and A = 4 6-32 Figure 6.8 Finding the Optimal Complete Portfolio Using Indifference Curves 6-33 Passive Strategies: The Capital Market Line • E(rc) = rf +[(E(rM) - rf)/σM]σc • Passive strategy involves a decision that avoids any direct or indirect security analysis • Supply and demand forces may make such a strategy a reasonable choice for many investors 6-34 Passive Strategies: The Capital Market Line Continued • A natural candidate for a passively held risky asset would be a well-diversified portfolio of common stocks • Because a passive strategy requires devoting no resources to acquiring information on any individual stock or group we must follow a “neutral” diversification strategy 6-35