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1 Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation Fina2802: Investments and Portfolio Analysis Spring, 2010 Dragon Tang Lecture 9 Capital Allocation February 9, 2010 Readings: Chapter 6 Practice Problem Sets: 1-5,12-14,26-28

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2 Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation Asset Allocation Objectives: Characterize the risk and return of portfolios containing risky and risk-free assets. 1.Evaluate the performance of a passive strategy.

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3 Portfolio Selection 1.Asset allocation 2.Security selection Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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4 Asset Allocation John Bogle: “Asset allocation accounts for 94% of the differences in pension fund performance” Identify investment opportunities (risk-return combinations) Choose the optimal combination according to investor’s risk attitude Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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5 Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation Portfolios: Basic Asset Allocation The complete portfolio is composed of: The risk-free asset: Risk can be reduced by allocating more to the risk-free asset The risky portfolio: Composition of risky portfolio does not change (market portfolio) This is called Two-Fund Separation Theorem. The proportions depend on your risk aversion.

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6 Risk-free Investment Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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7 Stock Returns Are Uncertain Example: Risky investment with ten possible rates of return Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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8 Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Risk and Risk Premium Risk-free rate: determined by demand/supply and intermediaries (such as Fed) Risk premium (=Risky return –Risk-free return) Example The expected return on the S&P500 is 9% The return on a 1-month T-bill is 3% The risk premium is 6% (9%-3%) Risk aversion E(r P ) – r f = ½ A σ p 2 Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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9 Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation Complete Portfolio Expected Return Example: Let the expected return on the risky portfolio, E(r P ), be 15%, the return on the risk-free asset, r f, be 7%. What is the return on the complete portfolio if all of the funds are invested in the risk-free asset? What is the risk premium? 7% 0 What is the return on the portfolio if all of the funds are invested in the risky portfolio? 15% 8%

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10 Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation Complete Portfolio Expected Return Example: Let the expected return on the risky portfolio, E(r P ), be 15%, the return on the risk-free asset, r f, be 7%. What is the return on the complete portfolio if 50% of the funds are invested in the risky portfolio and 50% in the risk-free asset? What is the risk premium? 0.5*15%+0.5*7%=11% 4%

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11 Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation Complete Portfolio Risk Premium In general: Equal to 0

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12 Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation Portfolio Standard Deviation where c - standard deviation of the complete portfolio P - standard deviation of the risky portfolio rf - standard deviation of the risk-free rate y - weight of the complete portfolio invested in the risky asset

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13 Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation Portfolio Standard Deviation Example: Let the standard deviation on the risky portfolio, P, be 22%. What is the standard deviation of the complete portfolio if 50% of the funds are invested in the risky portfolio and 50% in the risk-free asset? 22%*0.5=11%

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14 Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation Capital Allocation Line We know that given a risky asset (p) and a risk- free asset, the expected return and standard deviation of any complete portfolio (c) satisfy the following relationship: Where y is the fraction of the portfolio invested in the risky asset

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15 Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation Capital Allocation Line Risk Tolerance and Asset Allocation: More risk averse - closer to point F Less risk averse - closer to P

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16 Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation Slope of the CAL S is the increase in expected return per unit of additional standard deviation S is the reward-to-variability ratio or Sharpe Ratio

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17 Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation Example: Let the expected return on the risky portfolio, E(r P ), be 15%, the return on the risk-free asset, r f, be 7% and the standard deviation on the risky portfolio, P, be 22%. What is the slope of the CAL for the complete portfolio? S = (15%-7%)/22% = 8/22 Slope of the CAL

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18 Historical Risk-Return Trade-off RiskRealSharp Asset Class Prem.(%)Retn(%)Ratio Sm Stk13.9 14.6 0.35 Lg Stk9.3 8.9 0.45 LT Gov1.9 2.6 0.23 T-Bills--- 0.7 --- Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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19 Measuring Risk-Return Trade-off Mean-Variance Plot Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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20 Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation Borrowing where r B is the borrowing rate y > 1 Usually borrowing rate > lending rate

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21 Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation Borrowing Example Example: Let the expected return on the risky portfolio, E(r P ), be 15%, the return on the risk-free asset, r f, be 7%, the borrowing rate, r B, be 9% and the standard deviation on the risky portfolio, P, be 22%. What is the slope of the CAL for the complete portfolio for points where y > 1? S=(15%-9%)/22%=6/22 Note: For y 1, the slope is as indicated above if the lending rate is r f.

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22 Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation Investment Opportunity Set with Differential Borrowing and Lending Rates

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23 Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation Passive Strategies Assumes securities are fairly priced Avoids cost of security analysis Indexing - value-weighted portfolio Assume that the search for mispriced securities (performed by active strategies) keeps prices fair

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24 Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation Capital Market Line (CML) SPECIAL CASE OF CAL (I.e., P=MKT) The line provided by one-month T-bills and a broad index of common stocks (e.g. S&P500) Consequence of a passive investment strategy based on stocks and T-bills

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25 Which Portfolio to Choose? E(r) E(R m ) = 12% r f = 3% 20% 0 M F S=0.45 P1? P3? P2? Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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26 Key Determinant of Asset Allocation: Attitude towards Risk Risk Preference –Risk averse »Require compensation for taking risk –Risk neutral »No requirement of risk premium –Risk loving »Pay to take risk Utility Values: A is risk aversion parameter Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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27 Utility Function U = E ( r ) – 1/2 A 2 Where U = utility E ( r ) = expected return on the asset or portfolio A = coefficient of risk aversion = variance of returns Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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28 Utility Scores of Alternative Portfolios for Investors with Varying Risk Aversion Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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29 The Trade-off Between Risk and Returns of a Potential Investment Portfolio Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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30 Utility Values of Possible Portfolios for an Investor with Risk Aversion, A = 4 Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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31 Figure 6.2 The Indifference Curve Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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32 Utility Indifference Curves A=5 A=2 U1 U2 Increasing utility Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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33 Risk Aversion and Asset Allocation Greater levels of risk aversion lead to larger proportions of the risk free rate. Lower levels of risk aversion lead to larger proportions of the portfolio of risky assets. Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations. Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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34 Investor ’ s Willingness to Pay for Catastrophe Insurance Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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35 Spread Between 3-Month CD and T-bill Rates Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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36 Find the Optimal Allocation Solve the maximization problem: Two approaches: 1.Try different w1 2.Use calculus: Solution: Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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37 Asset Allocation Rules: When rm-rf increases, w1 increases When A increases, w1 decreases When m increases, w1 decreases W1 is constant when all three are fixed Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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38 Illustration of Solution E(r) E(R m ) = 12% r f = 3% 20% 0 M F S=0.45 P1! P3 P2 Utility indifference curves (A=4) If A=4 then w1=0.56 11.2% 8% Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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39 Which Portfolio to Choose? For Jack, risk aversion A = 2, the optimal choice is 112.5% (of total capital) in the market, financed by selling short 12.5% (of total capital) in T-bills For Jill, risk aversion A = 5, the optimal choice is 45% in the market and 55% in T-bills. The weight in the market: Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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40 Utility Comparison 45% 113% Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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41 Utility Levels for Positions in Risky Assets for an Investor with Risk Aversion A = 4 Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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42 Utility as a Function of Allocation to the Risky Asset, y Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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43 Finding the Optimal Complete Portfolio Using Indifference Curves Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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44 Expected Returns on Four Indifference Curves and the CAL Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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45 Average Annual Return on Stocks and 1-Month T-bills; S. Dev. and Reward to Variability of Stocks Over Time Fin 2802, Spring 08 - Tang Chapter 6: Asset Allocation Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation

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46 Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation Summary Capital allocation line (CAL) All combinations of the risky and risk-free asset Slope is the reward-to-variability ratio Capital market line (CML) Passive strategy Market index portfolio as the risky asset Risk aversion determines position on the capital allocation line Next: Market Efficiency

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