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Unformatted text preview: 1 The CAPM (Capital Asset Pricing Model) Berndt Chapter 2 1.1 Introduction assume that when investors act in the securities market, their behavior is perfectly rational in the sense that the only concern is assessing returns from their own investments. de&ne the rate of return on an investment as ( p 1 + d & p ) =p , where p 1 = price of security at the end of the time period d = dividends paid during the time period p = price of security at the beginning of the time period the return can be calculated ex post once the investment has been made, but the return is uncertain ex ante before the investment has been made. hereafter, write the ex ante return by R and interpret r as the expected return. investors are not just interested in the expected return r they are interested in the distribution of R . the risk of investment is typically characterized by the distribution of such possible returns. returns are often assumed to be distributed normally. in such cases the distribution can be completely described by two measures: the expected value r and the variance & 2 consider an investor has a portfolio consisting of a mix of two securities. let w j be the proportion of total funds invested in security j , j = 1 ; 2 the mix of these two securities generates an expected portfolio return r p and a variance of & 2 p . r p = E [ w 1 R 1 + w 2 R 2 ] = w 1 r 1 + w 2 r 2 & 2 p = V ar [ w 1 R 1 + w 2 R 2 ] = w 2 1 & 2 1 + w 2 2 & 2 2 + 2 w 1 w 2 & 12 , where & 12 = Cov ( R 1 ;R 2 ) 1.2 Derivation of a Linear Relationship between Risk and Return there is an asset with zero risk. the return of the asset is called the risk-free rate of return. denote it by r f . a real world example of r f : the 30-day U.S. Treasury bill rate held until maturity risk-free implies & 2 f = 0 . let the investor be able to borrow or lend inde&nitely at the risk-free rate of return r f . suppose an investor has a portfolio, called a , consisting of a mix of multiple assets. combine portfolio a with the risk-free asset into a new portfolio. 1 in such case, the expected return on the new portfolio becomes r p = w a r a + (1 & w a ) r f , where w a is the proportion of total funds invested in portfolio a . the variance of this portfolio becomes & 2 p = w 2 a & 2 a + (1 & w a ) 2 & 2 f + 2 w a (1 & w a ) & af = w 2 a & 2 a . rewrite & 2 p = w 2 a & 2 a to & p = w a & a . substitute it in r p = w a r a + (1 & w a ) r f = r f + [( r a & r f ) =& a ] & p . describe it in risk-return diagram: the slope is ( r a & r f ) =& a why do investors diversify a portfolio? suppose we have two securities, asset 1 and 2. expected return: r p = w 1 r 1 + (1 & w 1 ) r 2 variance of this portfolio: & 2 p = w 2 1 & 2 1 + (1 & w 1 ) 2 & 2 2 + 2 w 1 (1 & w 1 ) & 12 notice that & 2 p w 1 & 2 1 + (1 & w 1 ) & 2 2 , because & 12 & 1 & 2 ....
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- Spring '08