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Risk Aversion and Capital Allocation to Risky Assets
CHAPTER 6 Risk Aversion and Capital Allocation to Risky Assets With Shakil Al Mamun
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Risk and Risk Aversion Speculation Gamble Considerable risk
Sufficient to affect the decision Commensurate gain Gamble Bet or wager on an uncertain outcome Speculation: the assumption of considerable business risk in obtaining commensurate gain. Considerable risk: the risk is sufficient to affect the decision. Individual might reject a prospect that has a positive risk premium cz the added gain is insufficient to make up for the risk involved. Commensurate gain: we mean a positive risk premium, that is an expected profit greater than the risk-free alternative. To turn a gamble into a speculative prospect requires an adequate risk premium to compensate risk-averse investors for the risks they bear.
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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 Fair games: a prospect that has a zero premium.
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Table 6.1 Available Risky Portfolios (Risk-free Rate = 5%)
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Utility Function Where U = utility
E ( r ) = expected return on the asset or portfolio A = coefficient of risk aversion s2 = variance of returns
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Table 6.2 Utility Scores of Alternative Portfolios for Investors with Varying Degree of Risk Aversion
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Figure 6.1 The Trade-off Between Risk and Returns of a Potential Investment Portfolio, P
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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
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Figure 6.2 The Indifference Curve
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Table 6.3 Utility Values of Possible Portfolios for an Investor with Risk Aversion, A = 4
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Table 6.4 Investor’s Willingness to Pay for Catastrophe Insurance
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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
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The Risky Asset Example
Total portfolio value = $300,000 Risk-free value = ,000 Risky (Vanguard & Fidelity) = 210,000 Vanguard (V) = 54% Fidelity (F) = 46%
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The Risky Asset Example Continued
Vanguard 113,400/300,000 = Fidelity 96,600/300,000 = Portfolio P 210,000/300,000 = Risk-Free Assets F 90,000/300,000 = Portfolio C 300,000/300,000 = 1.000
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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
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Figure 6.3 Spread Between 3-Month CD and T-bill Rates
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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)
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Example Using Chapter 6.4 Numbers
rf = 7% rf = 0% E(rp) = 15% p = 22% y = % in p (1-y) = % in rf
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Expected Returns for Combinations
rc = complete or combined portfolio For example, y = .75 E(rc) = .75(.15) + .25(.07) = .13 or 13%
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Combinations Without Leverage
= .75(.22) = .165 or 16.5% If y = .75, then = 1(.22) = .22 or 22% If y = 1 = (.22) = .00 or 0% If y = 0
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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 c = (1.5) (.22) = .33
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Figure 6.4 The Investment Opportunity Set with a Risky Asset and a Risk-free Asset in the Expected Return-Standard Deviation Plane
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Figure 6.5 The Opportunity Set with Differential Borrowing and Lending Rates
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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: Variance is:
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Table 6.5 Utility Levels for Various Positions in Risky Assets (y) for an Investor with Risk Aversion A = 4
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Figure 6.6 Utility as a Function of Allocation to the Risky Asset, y
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Table 6.6 Spreadsheet Calculations of Indifference Curves
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Figure 6. 7 Indifference Curves for U =. 05 and U =
Figure 6.7 Indifference Curves for U = .05 and U = .09 with A = 2 and A = 4
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Figure 6.8 Finding the Optimal Complete Portfolio Using Indifference Curves
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Table 6.7 Expected Returns on Four Indifference Curves and the CAL
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Passive Strategies: The Capital Market Line
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
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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
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Table 6.8 Average Annual Return on Stocks and 1-Month T-bills; Standard Deviation and Reward-to-Variability Ratio of Stocks Over Time
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