Exercise 2 : We consider a stock exchange with n stocks j = 1,..., n. The number of shares of stock j is denoted a; assumed time independent and its price is Pj,t. We assume in the exercise zero dividends. The value of the market portfolio at date t is we = ajPjt j=1 n Pj,t - Pj.t-1 2.1 The return on asset j is : Yj,t = and the return on the market Pj,t-1 W - Wt-1 portfolio is yt = W-1 Show that yz = Cj,19jt . Give the expression of C;1. Is it possible j=1 to get the coefficients Cjt-1 independent of time ? n 2.2 We assume that the basic asset returns Yj,t, j = 1, ..., n, t= 1,...,T are independent, with a distribution which can depend on the asset. Are the market portfolio returns independent ? with a same distribution ? Can we assume yjt~ N(14j, oz), say? 2.3 Show that : Pi,t (1+Yj,t)(1 + Yj,t-1)... (1 + Yj,1)P.;0 1 = Pjo exp{t(log(1 + Y3jt) + ... + log(1 + 43,1)]}. 2.4 Apply the Law of Large Number to get the equivalence : llog(1 + Yje) + ... + log(1 +93,1)] ~ Elog(1+Y;!)]. 2.5 Deduce the behaviour of the prices of the basic assets in the long run (t + 0). What type of constraints on the return distribution will you introduce to avoid asymptotic (t large) arbitrage opportunities? 2.6 What is the behaviour of the value of the market portfolio for large t ? Exercise 2 : We consider a stock exchange with n stocks j = 1,..., n. The number of shares of stock j is denoted a; assumed time independent and its price is Pj,t. We assume in the exercise zero dividends. The value of the market portfolio at date t is we = ajPjt j=1 n Pj,t - Pj.t-1 2.1 The return on asset j is : Yj,t = and the return on the market Pj,t-1 W - Wt-1 portfolio is yt = W-1 Show that yz = Cj,19jt . Give the expression of C;1. Is it possible j=1 to get the coefficients Cjt-1 independent of time ? n 2.2 We assume that the basic asset returns Yj,t, j = 1, ..., n, t= 1,...,T are independent, with a distribution which can depend on the asset. Are the market portfolio returns independent ? with a same distribution ? Can we assume yjt~ N(14j, oz), say? 2.3 Show that : Pi,t (1+Yj,t)(1 + Yj,t-1)... (1 + Yj,1)P.;0 1 = Pjo exp{t(log(1 + Y3jt) + ... + log(1 + 43,1)]}. 2.4 Apply the Law of Large Number to get the equivalence : llog(1 + Yje) + ... + log(1 +93,1)] ~ Elog(1+Y;!)]. 2.5 Deduce the behaviour of the prices of the basic assets in the long run (t + 0). What type of constraints on the return distribution will you introduce to avoid asymptotic (t large) arbitrage opportunities? 2.6 What is the behaviour of the value of the market portfolio for large t