1 On the board Week 1
Gross returns: Excess returns:
Rt = Pt + Dt : Pt 1
R te = R t R tf ;
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where Rtf is the gross risk-free rate from time t 1 to time t, e.g., Treasury bill rate. Excess returns are returns to a ìzero-investmentîstrategy. E.g., borrow $1 at interest rate 1%, Rtf = 1:01. Put $1 in market and get, say, a 5% net return, Rt = 1:05. Excess return is 4%.
Can always write:
R t = R tf + R te : Risk premium and variance of returns:
Unconditional risk premium = Unconditional variance =
E (Rte) ; V (Re) :
Letís think about a portfolio choice problem. Let Rte be an N 1 vector: portfolio excess return = !0Rte;
where ! are an N 1 vector of exposures, ìportfolio wepights.îI want to invest in the portfolio with the highest Sharpe ratio (E (Rte) = V (Rte)):
min !0V (Rte) ! s.t. !0E (Rte) = m; !
where m is some scalar desired level of expected portfolio excess return. Solve for !
min!0V (Rte)!+(m !0E(Rte)); !
where is a Lagrange multiplier. FOC wrt !:
2V (Rte)! E(Rte)=0: Solve for optimal portfolio weights:
! = 2 V ( R te ) 1 E ( R te ) / V(Rte) 1E(Rte):
Recall: V (Rte) is N N and E(Rte) is N 1. I donít care so much about ,
as this is about the amount of leverage we take and it doesnít a§ect the ratio Rt = rV R~tp = pV (!0Rte): 1
ratio. Letís say we scale our portfolio returns with a constant k, R~p = k!0Re.
~ p E R~ tp E ( ! 0 R te )
tt E R~tp =kE(!0Rte):V R~tp =V(k!0Rte)=k2V(!0Rte).So:
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