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CHAPTER 9: INTRODUCTION TO ASSET PRICING MODELS Part 1: Capital Market Theory Brief Recap of Chapter 8 material Investors like more return with less risk. Investors are risk averse - More risk-averse investors have steeper indifference curves - Less risk-averse investors have flatter indifference curves The mean and variance of any asset can be plotted on the mean-standard deviation diagram. Investors would ideally like to be near the northwest corner of this diagram. But there are not enough opportunities for all investors to be there. One needs to look at how much we can reduce standard deviation (risk) while earning some return. Portfolios of two assets formed with different weights on the assets (e.g. 50%- 50%, 60%-40%, 90%-10% etc.) can be plotted on the mean-standard deviation diagram. At a correlation of +1, i.e. if the two assets are perfectly correlated, then, all portfolios formed from the two assets will lie on a straight line in the mean- standard deviation diagram. As correlation keeps decreasing from +1 to –1, this line will start curving towards the left. This can be seen below: -1.0 1.0 0.5 -0.5 0.0 1 The reason for this leftward shift is that by combining the assets into a portfolio, we are able to achieve lower standard deviation than either of the assets by themselves. This is diversification. In general, for all the assets in the economy, we can plot all the optimal risk- return combinations on the mean-standard deviation. This line is called the efficient frontier for risky assets. The efficient frontier looks like the following: Less risk averse invest or More risk averse invest or Investors locate along the efficient frontier according to their risk-aversion. This chapter introduces a risk-free asset along with the bunch of risky assets that we have so far considered. This will help us refine our understanding of the efficient frontier, and we can proceed to develop a theory. Risk-free asset: A risk-free asset is one whose returns are certain. To understand the characteristics of such an asset, let us consider the following example with two assets, one risky and one risk-free asset. Returns State Probability Risky Risk-free Recession 0.2 -20% 6% Normal 0.5 15% 6% Boom 0.3 30% 6% Exp. Return 13% 6% 2 - As you can see from the table, the return on a risk-free asset does not depend upon the state of the economy, which is uncertain; the risky asset’s return does depend upon the uncertain state. - Let’s calculate the variance and standard deviation of the risk-free asset: State Probability Return Deviation Sq. Dev Recession 0.2 6% 0% 0 Normal 0.5 6% 0% 0 Boom 0.3 6% 0% 0 All the squared deviations are zero, which means the variance and standard deviation of the risk-free asset are zero. 2(Rf) = 0 - Let’s calculate the covariance of the risk-free asset and the risky asset: Deviations Product State Probability Risky Risk-free of devns. Recession 0.2 -33% 0% 0% Normal 0.5 3% 0% 0% Boom 0.3 18% 0% 0% All the products of deviations are zero, which means the covariance of any asset with the risk-free asset is zero Cov(Ri,Rf)=0. Risk-free asset in a portfolio: - Let us understand a portfolio of one risk-free asset and one risky asset. Recall that for two assets, the expected return and variance of a portfolio are given by the following equations: E(Rp) = w1E(R1)+w2E(R2), and 2(Rp) = w122(R1)+w222(R2)+2w1w2Cov(R1,R2) If the weight on the risk-free asset is wRF, then the weight on the risky asset is (1-wRF). Substituting these weights into the above formula, E(Rp) = wRF(Rf)+(1-wRF)E(Ri), and 2(Rp) = wRF22(Rf)+(1-wRF)22(Ri)+2wRF(1-wRF)Cov(Ri,Rf) But we know that 2(Rf)=0 and Cov(Ri,Rf)=0, which means that: 2(Rp) = (1-wRF)22(Ri), and 3 (Rp) = (1-wRF).(Ri) - If we form all possible portfolios of the risk-free and the risky asset, where will they be on the mean-standard deviation diagram? Let us see with the risk-free and risky asset above. We had: E(Ri) = 13%, Rf=6%, (Ri)= 17.50% (verify this for yourself!) Let’s form some possible portfolios of the risky and risk-free assets using the above formulas: wRF (1-wRF) E(Rp) sigma(Rp) 0.0 1.0 12.5% 17.5% 0.2 0.8 11.2% 14.0% 0.4 0.6 9.9% 10.5% 0.6 0.4 8.6% 7.0% 0.8 0.2 7.3% 3.5% 1.0 0.0 6.0% 0.0% If we plot these portfolios as a mean-standard deviation diagram, we see the following: Portfolio of risky and risk-free asset 14.0% Portfolio expected return 12.0% 10.0% 8.0% 6.0% 4.0% 2.0% 0.0% 0.0% 5.0% 10.0% 15.0% 20.0% Portfolio standard deviation Note two points from this diagram: 1) The risk-free asset itself plots on the vertical axis (because it has zero standard deviation). This is the little circle in the diagram. 4 2) All portfolios formed with a risk-free asset and a risky asset (like the diamonds in the above figure) lie along a straight line in the mean-standard deviation diagram, extending from the risk-free asset to the risky asset. The risk-free asset changes the efficient frontier - In the last chapter, we chose the portfolios that are the best possible risk-return combinations, from among the infinite possible portfolios of all risky assets. - Now, we add a risk-free asset to all the risky assets. - There are many possible portfolios that can be formed by investing a proportion of the portfolio in the risk-free asset, and the remaining proportion in a risky portfolio that is on the efficient frontier. See below: Expected Return M Rf Portfolios like this were not feasible in the absence of the risk-free asset Standard Deviation Note the following from the diagram: - We can form more portfolios than before because we have a risk-free asset. - Since we want to be as much to the northwest as possible, we will find it optimal to combine the risk-free asset with the highest possible point on the risky-asset efficient frontier. - Geometrically, this means that we will draw a tangent from the risk-free asset to the efficient frontier, which is tangent at the risky portfolio M. - This tangent now becomes the new efficient frontier, because all points with optimal risk-return combinations (farthest to the northwest) lie along this tangent. - This tangent is called the Capital Market Line (CML). 5 - I took the GM, IBM, USX and Rf data from your first assignment, and came up with the risky-asset efficient frontier (for the three stocks) and the new efficient frontier in the presence of a risk-free asset. This diagram is shown in the next page. - The little dark circle on this diagram is the minimum variance portfolio among all risky assets. - The little diamond represents a portfolio that is like portfolio M in the above example. It is the tangency portfolio, the only portfolio that lies on the new efficient frontier and on the old efficient frontier. - The line joining the risk-free asset and the tangency portfolio is the Capital Market Line (CML) in a world consisting solely of GM, IBM and USX. Efficient Frontier for GM, IBM and USX 0.020 Portfolio expected return 0.018 0.016 0.014 0.012 0.010 0.008 0.006 0.004 0.002 0.000 0.00 0.05 0.10 0.15 0.20 0.25 Standard deviation - To give you a little flavor of the numbers, in this diagram, the tangency portfolio represents a portfolio of 85.1% in IBM, 28.9% in GM, and –14% in USX. (What does a negative proportion mean?) Investors preferences meet the Capital Market Line 6 - We know that investors want to be as far to the northwest as they can, based on their preferences. The best they can do is to be on the efficient frontier. - In the presence of a risk-free asset, investors locate on the Capital Market Line, which plots all optimal risk-return combinations. They locate at different points along the CML, according to their risk aversion. Expected Return Less risk-averse investor with a M flatter indifference curve Market Portfolio More risk-averse investor with a steeper indifference curve Standard Deviation - So, we are saying that all investors choose portfolios along the CML. That is every investor chooses to invest in a portfolio of the risk-free asset and the tangency portfolio. This tangency portfolio must then, include all risky assets. Why? Because, if there is a risky asset that does not belong to the tangency portfolio, then no investor would want to own that asset, and it would have no value at all. - This means that the tangency portfolio should be the market portfolio. This is a portfolio of all risky securities combined in market proportions. Market proportion for, say, Microsoft can be computed by dividing the total market value of Microsoft (no. of Microsoft shares outstanding multiplied by the price per share of Microsoft) by the total value of all risky assets. How do investors locate at different points along the CML? - We now know that every investor will invest some portion of his portfolio in the risk-free asset and the remaining in the market portfolio. How do they decide these proportions and what do the proportions mean? 7 - Consider a case where the expected return on the market is 12%, and the standard deviation of the market is 9%. The risk-free rate is 6%. There are two investors L and B. L wants an expected return on her portfolio of 8%, while B wants an expected return of 15%. How might they achieve these returns? Let us see. Case 1: Investor L L wants to combine the risk-free asset and the market portfolio to give her an expected return of 8%. She solves the following equation: E(RL) = wRF(Rf)+(1-wRF)E(RM) = 8% Plugging in the numbers for Rf and E(RM), we have: wRF(6%)+(1-wRF)12% = 8% 12% - wRF(6%) =8% wRF = (12%-8%)/6% = 4%/6% = 2/3 and (1- wRF) = 1/3 This means that out of every $100 of money invested, L will lend $66.67 to the risk-free borrower (like the U.S. govt.), and she will invest the remaining $33.33 in the market portfolio. The standard deviation on her portfolio is given by: (RL) = (1-wRF).(RM) =(1/3).9% = 3% Case 2: Investor B B wants to combine the risk-free asset and the market portfolio to give her an expected return of 15%. She solves the following equation: E(RB) = wRF(Rf)+(1-wRF)E(RM) = 15% Plugging in the numbers for Rf and E(RM), we have: wRF(6%)+(1-wRF)12% = 15% 12% - wRF(6%) =15% wRF = (12%-15%)/6% = -3%/6% = -1/2 and (1- wRF) = 1-(-1/2) = 3/2 This means that for every $100 of money invested, B will borrow $50 at the risk- free rate, and she will invest the entire $150 in the market portfolio. In our earlier 8 language (Chapter 4), she will do a margin transaction. (In reality, of course, you and I cannot borrow at the risk-free rate. But here we are assuming everyone can borrow and lend at the risk-free rate. A minor, but valid, concern. The standard deviation on her portfolio is given by: (RB) = (1-wRF).(RM) =(3/2).9% = 13.5% - Look at B and L on the following graph. Note that B is located between the risk- free asset and the market portfolio on the capital Market Line. L is located beyond the market portfolio on the CML. (Fill in the numbers yourself in class) - In fact, all portfolios beyond the market portfolio on the CML are obtained by borrowing at the risk free rate and investing in the market portfolio, while all portfolios between the risk-free asset and the market portfolio are obtained by lending some money at the risk-free rate, and investing the rest in the market portfolio. E(RP) B L (P) 9

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Chapter 7, standard deviation, Chapter 8, Portfolio management, Chapter 6, expected return, Chapter 13, systematic risk, Table of Contents, Object-Oriented Programming

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posted: | 2/21/2010 |

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