The fundamentals of the derivation of the CAPM can be outlined as follows:

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1 Summary & Review o the Capital Asset Pricing Model The undamentals o the derivation o the CAPM can be outlined as ollows: (1) Risky investment opportunities create a Bullet o portolio alternatives. That is, each investment opportunity stands by itsel. It has an expected return and some risk attached to it. However, i the investor/consumer chooses to invest in multiple projects, in their combined orm, the portolio return and risk are dierent than any one in isolation. Indeed, portolio investments generate lower risk or any level o expected return than can be achieved by investing in a single project. This is the process o diversiication. () The Bullet o risky portolio choices can be linked to the risk-ree return to create the Capital Market Line. The CML is the eicient rontier o investment opportunities because it employs not only the diversiication beneits o portolio investing but also the risk hedging potential o the risk-ree return to create the highest possible expected return or any given level o risk. An excellent illustration o the power o diversiication is ound by drawing a bullet o two assets such that the lower return asset actually has a return that is less than the risk ree return. The obvious question is posed: Why would an investor hold an asset which is risky which has a return that is less than the risk ree. The answer is because the risky asset has portolio diversiication power. Holding it reduces the portolio risk associated with other risky assets in a way that cannot be achieved with the risk-ree asset. Portolio diversiication is truly as close as we come in the social sciences to magic. (3) Given the Bullet and the CML, all assets are priced by the equality o the slope o the Bullet and the CML. That is, all assets have to oer an expected return given their risk that make them attractive to investors who are choosing among other risky assets and the risk-ree return. To see the pricing phenomenon o the CAPM, let s consider a comparative static experiment. 1 Let s consider the eect on the pricing o assets that occurs when the risk-ree return changes. Assume that there are only two risky assets, X and Y. Start with a value or the risk-ree return o r 0. There is an eicient portolio o {X,Y} designated as point a. That is, point a tells us the optimal portolio allocation o invested wealth between X and Y. {Graph to come.} The consumer/investor then chooses how much risk to bear by holding more or less wealth in the risk-ree asset. Next, let the risk-ree asset increase rom r 0 to r 1. A line drawn rom the new risk-ree rate to the old bullet o X and Y is tangent at point b. Point b is not an equilibrium, but rather depicts a tendency. Given the new risk-ree rate, investor/consumers are now interested in holding more o asset Y and less o asset X. This is the prediction o the CAPM. Because o the change in the risk-ree rate, investors will optimally sell asset X and buy asset Y. Selling X causes its price to all. Buying Y causes its price to rise. When the price o X alls, its expected return increases. When the price o Y rises, its expected return decreases. In risk-return space, point X shits up and point Y shits down. 1 That is, let s consider an experiment using CAPM akin to the experiments that we employ in Supply and Demand analysis. In S&D analysis we shit the demand curve and then describe how the equilibrium price changes. 5.doc; Revised: September 19, 006; M.T. Maloney 1

2 The Bullet rotates. There is a new equilibrium tangency between the CML and the Bullet. Label this point c. Point c is the eicient allocation o investment wealth between assets X and Y, given the risk-ree rate r 1. At point c, investors hold more Y and less X than they did at point a. But just like point a, at point c, assets X and Y are priced based on the equality o the slope o the Bullet and the slope o the CAPM. The CAPM explains how buying and selling pressure on assets will reach an equilibrium. (This is just the same way that S&D analysis tells us how buying and selling pressure in commodity markets reaches an equilibrium.) The CAPM tells us when utility maximizing investor/consumers will buy and sell assets. It tells us how that buying and selling activity will diminish as an equilibrium is approached. And, it gives us an equilibrium condition, i.e., it describes a situation in which there is no incentive to buy or sell assets. The price level o assets is based on the expected uture cash lows that those assets will enjoy relative to the cash lows o other assets. The CAPM assumes that the joint distribution o cash lows is independent o the current price o the assets. Hence, the current prices adjust relative to one another until the CAPM equilibrium is reached. This is shown in the picture by the vertical shits in X and Y as the risk-ree return changes. The exogenous change in the risk ree causes consumers to reallocate their portolio o risky assets. This portolio adjustment changes the level o the asset prices, but it does not change the distribution o the uture cash lows. Hence, an equilibrium is reached between (among in the n asset case) prices based on their expected returns and the unchanging distribution o cash lows. The distribution o cash lows is deined by the standard deviation o cash lows to the two assets separately, and the covariance between the cash lows o each pair. The CAPM prices assets looking orward. That is, the CAPM price o assets is based on the expectation o uture cash lows. We commonly use the past as a prediction o the uture, but it is not necessarily a perect predictor. In this sense the standard expression o the CAPM in the market model, r i = αi + βirm + ε i, is based on the idea that β is deterministically identiied by the joint distribution o uture cash lows. Our estimation o β using past observations on r i and r m is just an estimate which has its own sampling errors and potential errors o measurement. A common question that bright students ask is, How can one make money using the CAPM? The proper answer is the ollowing: Assuming that you have an accurate measure o β, i you ind assets priced such that [ Er ( ) r] > β [ Er ( ) r] i i m buy. I the asset is priced such that the inequality goes the other way, sell. I you can ind assets exhibiting these characteristics, you can make a proit. The problem is that you will not likely ind many because that is what all investors are trying to do. One inal point, and a point to which we will return, the E(r i ) given by the CAPM is the interest rate by which the cash lows o irm i are eiciently discounted in the DCF ormula. 5.doc; Revised: September 19, 006; M.T. Maloney

3 Capital Asset Pricing Y Expected Return X a c b Risk Bullet-1 Rebalance Line CML-1 Bullet- CML- 5.doc; Revised: September 19, 006; M.T. Maloney 3

4 Empirical Tests o the CAPM: The model is constructed as: ( r r ) = γ +γ ( r r ) +γ ( r r ) it 0 1 mt mt where r is either the return on Treasury bills or the return on a zero beta portolio. The model is estimated over various periods, and dierent speciications or i. Most researchers treat i as a portolio o assets. The CAPM prediction is that γ 0 and γ should be zero, and γ 1 is the value o β or asset group i. Generally, researchers ind that γ is zero but that γ 0 is not. Example o the analysis using Rodgers' data: {Dell, Google, Nike, Pizer, Sony}, the S&P 500, and the 0 yr constant maturity gov't bond yield. 33 data one; set one; 34 r=(1+ir/100)**(1/365)-1; 35 netr=sp-r; netp=portolio-r; 36 netr=netr**; The SAS System 10 13:58 Wednesday, September 14, 005 The MEANS Procedure Variable Label N Mean Std Dev Minimum Maximum ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Date Date Portolio Portolio SP SP ir VALUE netp netr netr E r E doc; Revised: September 19, 006; M.T. Maloney 4

5 The SAS System 9 13:58 Wednesday, September 14, 005 The REG Procedure Model: MODEL1 Dependent Variable: netp Number o Observations Read 56 Number o Observations Used 53 Number o Observations with Missing Values 3 Analysis o Variance Sum o Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coe Var Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept netr <.0001 netr Fama and French multi-actor models: E( r ) r = b[ E( r ) r ] + se( SmB) + he( HmL) P m CAPM is augmented by SmB, which is the dierence in the returns to small versus large capitalized stocks, and HmL, which is the dierence in the returns to high versus low book-tomarket stocks. These are so-called mimicking portolio returns built on irm size and the ratio o book to market equity, respectively. A mimicking portolio return is a return that is the dierence between the returns to two classes o irms. For instance, in the size mimicking portolio return, the returns o big irms are subtracted rom the returns to small irms. The result is a portolio return dierential that relects the commonality o returns to big irms as that is dierent rom returns to small irms, and both net o the return to all irms, which is held constant by the relation between the i th irm's return and the market. 5.doc; Revised: September 19, 006; M.T. Maloney 5

6 The coeicients on the mimicking portolios describe common actors in returns across irms i the coeicients consistently associate irms with their peers. That is, a given big irm should have negative return when SmB is positive; likewise, when SmB is positive, a small irm should have a positive return. The implication is that s should be positive or small irms and negative or big irms i size is a actor common in the rational risk pricing o assets. Similarly, HmL is a portolio return dierential between high book-to-market irms and low book-to-market irms. I book-to-market is a common risk actor, h should be positive or high book-to-market irms and negative or low book-to-market irms. On the other hand, i CAPM captured all attributes o asset pricing, then s and h should be zero and b would be the portolio beta. Fama and French show that size and book-to-market are indeed common risk actors. Roll's critique: There is no market portolio; hence, there can be no test o CAPM. Roll's critique is interesting because i there were a true market portolio, there would be no need to rebalance portolios. Everyone would hold everything and as everything was repriced individuals portolios would automatically rebalance. Because there is no market portolio, it is necessary to rebalance as expected cash lows and relative risk change. And we know there is a lot o rebalancing. In some ways the question about the accuracy o CAPM depends on the question. I argue that CAPM is a predictor o an equilibrium repricing process. While not necessarily perectly accurate, it indicates the direction that prices will move. Hence, here are a couple o "tests" o CAPM that I think are appropriate: 1) Estimate beta up to day t-1. Use beta to predict the direction o change in each stock based on the direction o change o the average o all stocks. Also estimate the magnitude. High beta stocks should change more in absolute value than low beta stocks. ) Compare changes in stock prices to changes in the risk ree rate. As the risk ree increases, low beta stock prices should all and high beta stock prices should rise. 5.doc; Revised: September 19, 006; M.T. Maloney 6

7 Revisiting CAPM Chris Kirby The purpose o this lecture is to revisit the capital asset pricing model by more explicitly examining (1) Expected Utility Maximizing and () Portolio Optimization. As we have shown beore, we will demonstrate that utility maximizing consumers will choose to hold a single portolio o risky assets, which then will result in equilibrium pricing o all assets. Let s start with a speciic utility unction, i.e., the negative exponential: UW ( ) = e aw This unction has several properties that are useul. Most importantly, it is convenient. Moreover, it has constant absolute risk aversion, which is identiied in the parameter, a. aw aw ARA = U / U = a e / ae = a The shape o the utility unction is upward sloping and concave. It starts in the negative quadrant at -1 and asymptotes to zero. Portolio Optimization: Consider the consumer/investor s choice between two assets. One is risky; the other is riskree. The risky asset has return R with a expected value o µ and a standard deviation o σ. The riskree has a known return o r. Let x be the proportion o the portolio wealth that is held in the risky asset. For simplicity, allow the initial value o the portolio to be 1. The portolio return can then be written as: R = xr + (1 x) r P or R = r + x( R r ) P where the expression in parentheses is called excess return. The expected return on the portolio is: and the variance o the portolio return is: ER ( P) = Er ( + xr [ r]) = r + x( µ r ) Var( RP) = x σ 5.doc; Revised: September 19, 006; M.T. Maloney 7

8 Maximizing Expected Utility: The consumer maximizes expected utility over the choice o x: ( ) max { x} EUW ( ( )) = E exp( ar [ + xr ( r )]) To operationalize this expression, we assert that the return on the risky asset ollows some distribution. Thus, we can write: 1. max { x} ( ( )) = exp( [ + ( )]) ( ; µ, σ, θ) EUW ar xr r R dr or some general density o R. It is extremely useul to assume that the density (.) is the normal. This is true because o two things: (a) The normal is ully described by its mean and standard deviation, so the extra parameter θ vanishes. We will see later that this becomes a crucial assumption. (b) The other reason that it is useul to assume that the return on the risky asset is distributed normal is because it allows or a very simple expression o expected utility. For any random variable that is normally distributed, i.e., y~n(m,s ), we know that: cm cy Ee ( ) = e + cs Thus, the expected utility o wealth given by (1) above can be written as:. axσ EUW ( ( ( x))) = exp ar [ + x( µ r)] + The consumer/investor maximizes () by the choice o x. The FOC looks like: which gives the optimal portolio choice: EUW ( ( ( x))) = µ σ () = x 1 ( µ r ) 3. x* = a σ [ a( r ) xa ]exp. 0 Equation (3) makes sense. The consumer adjusts her portolio toward more risky assets as her degree o risk aversion, a, declines. Riskiness is deined in terms o the excess return divided by the variance o the return on the risky asset. All consumer/investors are characterized by (3). In general, the density will be described by its moments where we use θ to stand or the higher moments. 5.doc; Revised: September 19, 006; M.T. Maloney 8

9 Notice that i we graph portolio choice in expected return and risk space, {E(R), Std Dev.}, we have simply a straight line starting at r on the vertical axis and going through the point {µ,σ}. The consumer s choice o x determines where she is on this line. For consumers that are very risk averse (high a) lie somewhere to the let o {µ,σ}. Consumers who are not very risk averse can borrow at the riskree rate so that they are to the right o {µ,σ}. Notice that they way that we have developed this analysis, {µ,σ} is just a point in risk-return space. Now consider what happens i the consumer has choices among risky assets. That is, assume that there are many {µ,σ} i ; these are described by the many risky assets in the world and by the investment rontier o possible portolio choices, that is, the bullet. Which risky portolio will the consumer choose. This is the same thing as asking which {µ,σ} maximizes: ax* σ EUW ( ( ( x*))) = exp ar [ + x*( µ r)] + Clearly the expected utility o wealth is maximized by choosing the tangent portolio, that is the portolio with the highest excess return relative to its risk: µ r σ. So we are back to the plank that connects the riskree asset to the risky investment rontier. All consumer/investors will choose the same risky asset in order to maximize their expected utility o wealth. This risky asset will be a portolio o all risky assets. All consumer/investors hold the same risky portolio and vary the proportion o their wealth held in this portolio and the proportion held in riskree asset based on their degree o risk aversion. Importantly, because all individuals hold the same portolio o risky assets, all assets are valued the same by all individuals. Thus, we have a pricing equilibrium: CAPM. Recognize the degree to which this equilibrium theorem is built on the assumption o the normal distribution. By assuming normality in the return to the risky asset we are able to describe the distribution o returns and the expected utility o wealth in terms o two parameters, {µ,σ}, and all individuals have the same valuation o these parameters. I we construct the theory based on a distribution other than the normal, the higher moments o distribution will enter the utility unction and these will be valued dierently based on each individual s degree o risk aversion. Hence, we will not have an equilibrium pricing model. 5.doc; Revised: September 19, 006; M.T. Maloney 9