CAPITAL ASSET MARKET (11) i R mx Rk (,k=1, 1), i.e., the risk margins are such that the ratio between the total risk compensation paid for an asset and the variance of the total stock of the asset is the same for all assets 4. COMPOSITION OF EQUILIBRIUM PORTFOLIOS We can now derive an important property of an individuals equilibrium port When(10)is substituted back in(9), the result is x2∑k2x (12)∑0y Now define for each individual z=xi/,(=l,,n-1),i.e, zj is the proportion of the outstanding stock of asset j held by individual i. Further, let that 2abja=l. Then (12)can be written (13)∑b2=∑bk2z (,k=1,…,n-1) It is easily proved that these equations imply that the z are the same for all j 4 What this means is that in equilibrium, prices must be such that each individual will hold the same percentage of the total outstanding stock of all risky assets. This percentage will of course be different for different individuals, but it means that if an individual holds, say, 2 per cent of all the units outstanding of one risky asset, he also holds 2 per cent of the units outstanding of all the other risky assets. note that we cannot conclude that he also holds the same percentage of the riskless asset; this proportion will depend upon his attitude towards risk, as expressed by his utility function. But the relation nevertheless permits us to summarize the description of an individuals portfolio by stating (a) his holding of the riskless asset, and(b) the percentage z held of the outstanding stock of the risky assets. We 5 Let the common value of the n-1 terms 2a=i bja zd be d, and let ca be the elements of the inverse of the matrix of the bia(assuming nonsingularity). It is well known that when sabia=1 then also 2acja-1. The solutions for the ff are then: zf= 2acaa'=a'2acja=a, which proves our proposition has content downl ued stube to sT oR ems aecondtp23013020-0 AMCAPITAL ASSET MARKET 775 (1) mjRi _ Mk Rk (j, k=1, ...,I n-1), Vi Vk i.e., the risk margins are such that the ratio between the total risk compensation paid for an asset and the variance of the total stock of the asset is the same for all assets. 4. COMPOSITION OF EQUILIBRIUM PORTFOLIOS We can now derive an important property of an individual's equilibrium port￾folio. When (10) is substituted back in (9), the result is: Z 7jaX~a UkaXXa (12) a - a Z 5fja x Z?ka Xa a a Now define for each individual zJ =X/X (j=1, ..., n- 1), i.e., zJ is the proportion of the outstanding stock of asset j held by individual i. Further, let bia = jaXa Z ijaXa a so that Zabja= 1. Then (12) can be written (13) Zbjaz =Z bka (j, k-1, ... 1) . a a It is easily proved5 that these equations imply that the zJ are the same for all j (equal to, say, zi), i.e., (14) zJ-Zk=4z (j, k=1, ..., n-1). What this means is that in equilibrium, prices must be such that each inidividual will hold the same percentage of the total outstanding stock of all risky assets. This percentage will of course be different for different individuals, but it means that if an individual holds, say, 2 per cent of all the units outstanding of one risky asset, he also holds 2 per cent of the units outstandlng of all the other risky assets. Note that we cannot conclude that he also holds the same percentage of the riskless asset; this proportion will depend upon his attitude towards risk, as expressed by his utility function. But the relation nevertheless permits us to summarize the description of an individual's portfolio by stating (a) his holding of the riskless asset, and (b) the percentage zi held of the outstanding stock of the risky assets. We 5 Let the common value of the n-I terms La= I bja z be a', and let cj, be the elements of the inverse of the matrix of the bc,, (assuming nonsingularity). It is well known that when Efbj= 1, then also 4.cja = 1. The solutions for the z, are then: z= Eaciaa'= aiaC;a = ai, which proves our proposition. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:20:50 AM All use subject to JSTOR Terms and Conditions