Suppose there are n risky assets in the economy and assume the data generating process for the generic risky asset return i over the risk-free rate rf to be rt=i+1,iff1+2,irf2+et i is the deterministic expected excess return of i,rfi is the random excess return on factor i{1,2}, which is a traded portfolio, e1 is an error term, which is independent from r~1,rf1, and rf2, has zero mean, E[et]=0, and positive variance, V[et]=t2>0. Let's derive a two-factor APT by (asymptotic) noarbitrage. 1. The first step is to construct an asymptotic arbitrage (i.e. a strategy that becomes an arbitrage as n goes to infinity) a. Take a generic portfolio P of the n assets, write down its data generating process (rp= ?) b. Note that every factor is a well-diversified portfolio constructed to have beta of 1 on its own beta and beta of 0 on any other factor. Write down the data generating process for the error terms are. c. Form an asymptotic arbitrage strategy investing in P, factor 1 , factor 2 and the risk-free asset with weights w=[wP,wf1,wf2,wf], Hint 1: an asymptotic arbitrage strategy is a set of weights w such that i. w1=wP+wf1+wf2+wf=0 iii. V[wr] goes to 0 as n goes to infinity Hint 2: select w such that wr=P+eP and impose iii 2. The second step is to impose the necessary structure such that the asymptotic arbitrage from step 1 is not achievable Hint: what must happen to the parameter(s) of the data generating process for the arbitrage strategy that you constructed in 1 not to be an arbitrage anymore? Suppose there are n risky assets in the economy and assume the data generating process for the generic risky asset return i over the risk-free rate rf to be rt=i+1,iff1+2,irf2+et i is the deterministic expected excess return of i,rfi is the random excess return on factor i{1,2}, which is a traded portfolio, e1 is an error term, which is independent from r~1,rf1, and rf2, has zero mean, E[et]=0, and positive variance, V[et]=t2>0. Let's derive a two-factor APT by (asymptotic) noarbitrage. 1. The first step is to construct an asymptotic arbitrage (i.e. a strategy that becomes an arbitrage as n goes to infinity) a. Take a generic portfolio P of the n assets, write down its data generating process (rp= ?) b. Note that every factor is a well-diversified portfolio constructed to have beta of 1 on its own beta and beta of 0 on any other factor. Write down the data generating process for the error terms are. c. Form an asymptotic arbitrage strategy investing in P, factor 1 , factor 2 and the risk-free asset with weights w=[wP,wf1,wf2,wf], Hint 1: an asymptotic arbitrage strategy is a set of weights w such that i. w1=wP+wf1+wf2+wf=0 iii. V[wr] goes to 0 as n goes to infinity Hint 2: select w such that wr=P+eP and impose iii 2. The second step is to impose the necessary structure such that the asymptotic arbitrage from step 1 is not achievable Hint: what must happen to the parameter(s) of the data generating process for the arbitrage strategy that you constructed in 1 not to be an arbitrage anymore
