# CS代写 1 On the board Week 1 – cscodehelp代写

1 On the board Week 1
Gross returns: Excess returns:
Rt = Pt + Dt : Pt1
R te = R t R tf ;

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 =kE(!0Rte):V R~tp =V(k!0Rte)=k2V(!0Rte).So:

Posted in Uncategorized