diego domenzain September 2020 @ Colorado School of Mines

You want to invest money, and you want to make sure you do not lose it. So, you make the trick of betting for and against.

How do you find how much betting for and against?

You minimize f(w) using Lagrange multipliers,

f(w) = w.' S w - l1*(r.'w - rho) - l2*(e.'w - mu)

where w is a vector of weights, S is the covariance matrix of the time-series stock values, r is a vector of the expected returns of the stock values, rho is the target expected return, e is a vector of just ones, and mu is the available quantity for investment.

w.' S w is the variance of the portfolio return.

This transforms to a matrix problem of the form,

[ 2*S -r -e ]   [w ]   [0  ]
[ r.'  0  0 ] * [l1] = [rho]
[ e.'  0  0 ]   [l2]   [mu ]

where the task is to find w, l1 and l2. However, we are really only interested in w.

This is assuming rho = 50 and mu = 1. Note the short position on asset No. 2.