CAPITAL ASSET PRICING MODEL (CAPM)

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1 Finnce -- 5 y 202 CAPIAL ASSE PRICING ODEL (CAP)

2 Finnce -2-5 y 202 Portfolio of one riskless nd one risky sset Consider portfolio consisting of the riskless sset with men return per dollr 0 = + r nd mutul fund mde up of risky ssets with men, stndrd devition nd price P. One dollr buyers frction / P of the risky sset nd thus men return per dollr of / P nd stndrd devition per dollr of /. P Purchsing portfolio of shres in the two ssets mps out the line segment joining the (, ) combintions. By selling sset 0 short (borrowing) the investor cn P move to points further long the mrket opportunity line. he slope of this line is + r P ( + r) ( + rp ) = (.) P P

3 Finnce -3-5 y 202 Portfolios of risky ssets P + r P he blue curve represents the minimum vrince of dollr portfolio of risky ssets for every different men. We cn think of ech such portfolio s mutul fund. hen the line joining ech mutul fund mrker nd the riskless sset mrker is set of fesible opportunities. he best such line is the one tht just touches the curve. Let be the mutul fund t the tngency. he line is clled the mrket opportunity line nd the slope is the Shrpe Rtio.

4 Finnce -4-5 y 202 Vrince of portfolio ( q, q 2) of 2 ssets (review) x = qz + qz 2 2, ( x) = q+ q22, x ( x) = q( z ) + q2( z2 2). hen ( x ( x)) = ( q ( z ) + q ( z )) = ( q ( z )) + 2( q ( z ))( q ( z )) + ( q ( z )) = q ( z ) + 2 qq ( z )( z ) + q ( z ) he vrince is the expecttion of hen ( x ( )) 2 x which we write s 2 vr( x) E{( x ( x)) } Vr( x) = q E{( z ) } + 2 q q E{( z )( z )} + q E{( z ) } = q + 2qq + q

5 Finnce -5-5 y 202 Equilibrium Pricing (CAP) Everyone wnts to hold the sme mutul fund. But the supply of sset is Q= ( Q,..., Q A ) = (,,...,). In equilibrium supply = demnd so the mutul fund must be frction of the (totl) mrket portfolio. Let z be the totl return to the mrket portfolio z m A = z. = his hs price P m A = = P hus we cn think of ech investor choosing between the riskless sset nd single risky sset with return z..

6 Finnce -6-5 y 202 Now consider the cse of risky sset with return z, nd price P, the mrket portfolio z nd price P nd the riskless bond. he opportunity set for these two risky ssets is depicted below. P + r P Remember tht blue curve mps out the minimum for ech men cross ll portfolios while the red curve is the minimum vrince using only two ssets. hus the red curve cnnot cross the blue curve. his men tht the red curve cnnot cross the line. hus the slope of the red curve must be equl to the Shrpe rtio when ( q, q ) = (0, q ).

7 Finnce -7-5 y 202 rginl nlysis If the investor spends Pq + Pq per dollr on risky ssets he hs ( Pq Pq) left to spend on the riskless bond. His return per dollr is then x = ( Pq P q )( + r) + q z + q z. he men nd vrince per dollr re therefore nd ( q, q ) = + r+ q ( ( + r) P ) + q ( ( + r) P ) (.2) ( q, q ) = q + 2q q + q (.3) Note tht in equilibrium q = 0. herefore ( q, q ) = (0, q ) = q = q nd so (0, q ) = q (.4)

8 Finnce -8-5 y From (.3) ( q, q ) = q + 2q q + q. herefore nd = ( ( + rp ) ) q 2 ( q, q ) = 2q + 2q q Substituting from (.4) nd noting gin tht in equilibrium q = 0, 2q = 2q q hen the trdeoff is hence = q d d q ( + r) P = = q

9 Finnce -9-5 y 202 From (.) the slope of the mrket opportunity line is herefore nd so Hence ( ) P + r. ( + rp ) P( + r) = ( ) + rp = 2 ( P( + r)). P = 2 ( P) + r + r hus ech sset cn be priced in terms of its men return nd its covrince with the mrket portfolio.

10 Finnce -0-5 y 202 Review: Covrince of n sset with portfolio Asset returns Z. Devitions from men z = Z Portfolio X = qz + q Z + q Z Devitions from men x= qz + q z + q z Smple covrince cov( Z, X) = cov( z, x) = (...) z x = t t z qz + q z + q z + t t 2 2t 3 3t t= t= herefore = zqz +... t t zqz + t 2 2t zqz + t 3 3t t= t= t= = q z z + q... t t 2 z z + q t 2t 3 z z + t 3t t= t= t= = q cov( z z ) + q cov( z z ) + q cov( z z ) cov( Z, X ) = sumproduct of sset holdings nd row of the covrince rry

11 Finnce -- 5 y 202

Econ 401A Version 3 John Riley. Homework 3 Due Tuesday, Nov 28. Answers. (a) Double both sides of the second equation and subtract the second equation

Econ 401A Version 3 John Riley. Homework 3 Due Tuesday, Nov 28. Answers. (a) Double both sides of the second equation and subtract the second equation Econ 40 Version John Riley Homeork Due uesdy, Nov 8 nsers nser to question () Double both sides of the second eqution nd subtrct the second eqution 60q 0q 0 60q 0q 0 b b 00q 0 hen q 0 (b) he vlue of the