# What is the correct method to maximize information ratio ex ante of a 150/50 portfolio against the S&P500

I have no problem forecasting and minimize ex ante information ratio for a market neutral portfolio, because the benchmark is just zero and you are just minimizing portfolio variance while maximizing return, using some optimization software and wT x cov x w to calculate variance.

But how do I change the formulation when I am trying to maximize information ratio, aka the sharpe ratio of excess returns relative to tracking error? How do I calculate tracking error from the covariance matrix?

I have tried adding the index to the weights at weight -100 and then add index to the covariance matrix, but would wT x cov x w then output tracking error?

• This is just a general idea and not an answer. You can calc the tracking error ( some people view it as portfolio variance or portfolio standard deviation. the definition varies ). Assuming you have a factor model that provides residual variance, It will be $w Cov w + \sum_{i}^{n} w_{i} \sigma^2_i$ where sigma_{i} is the residual variance of each stock in the portfolio. This means you can construct expected return divided by tracking error. As far as maximizing that, I don't know.. Maybe a fractional programming approach ? it's complex because you have weights in the denom and numerator. Commented Jan 9 at 6:56
• Note that another view of tracking error is (portfolio return - benchmark return) but I don't think that's what you want here because that tracking error is kind of usefess after the fact. Commented Jan 9 at 6:57