Hi, I refer you to page 24 of chapter 16 for CM2 of the acted combined notes package 2021. The value of the portfolio at time t-1 = phi,t*St-1 + psi,t*Bt-1 = exp(r*[t-1])*(phi,t*Dt-1+psi,t) = exp(r[t-1])*Et-1 = Vt-1 (Could someone please explain the rational/substitution, in particular why the portfolio value, at time t-1, can be written as exp(r*[t-1])*(phi,t*Dt-1+psi,t)?) The value of the portfolio at time t = phi,t*St + psi,t*Bt = exp(rt)*(phi,t*Dt+psi,t) Could someone please explain the rational/substitution, in particular why the portfolio value, at time t, can be written as exp(rt)*(phi,t*Dt+psi,t)?) Many thanks, Darragh
At time t-1 the text says that phi,t units of the risky asset are held (valued at St-1), and that psi,t units of the cash account are held (valued at Bt-1). Therefore the portfolio value is: phi,t*St-1 + psi,t*Bt-1. But St-1 = Dt-1 * exp(r(t-1)) from Step 1, and Bt-1 = exp(r(t-1)) since its the risk-free cash account. Therefore the value of the portfolio can be written as: phi,t*Dt-1 * exp(r(t-1)) + psi,t * exp(r(t-1)) =(phi,t*Dt-1 + psi,t) * exp(r(t-1))
At time t the number of units of the risky and the units of the cash account remain unchanged, ie phi,t and psi,t respectively. The risky asset is now worth St and the cash account is worth Bt, so the value of the portfolio is now: phi,t*St + psi,t*Bt But St = Dt * exp(rt) from Step 1, and Bt = exp(rt) since its the risk-free cash account. Therefore the value of the portfolio can be written as: phi,t*Dt * exp(rt) + psi,t * exp(rt) =(phi,t*Dt + psi,t) * exp(rt)
