Covariance Optimization Versus Historical Portfolio Volatility

Summary

The document compares two proposed ways to choose weights for a long–short portfolio of S&P 500 stocks while minimizing estimated volatility. One approach estimates a covariance matrix from historical returns and minimizes the quadratic portfolio variance. The other feeds the historical return observations directly to an optimizer and minimizes portfolio volatility over the same lookback period. The central question is whether these procedures produce the same weights when they use identical data.

The document does not answer the question or specify practical details such as constraints, return frequency, or the precise volatility objective. Mathematically, when both procedures use the same observations, conventional sample covariance and the corresponding portfolio variance calculation encode the same quadratic objective, subject to consistent centering and scaling. Differences can arise if the data processing, objective, constraints, or volatility definition differs. Since the source is only a question, it offers no worked example or performance evidence, and its focus is the formulation of historical ex-ante risk minimization rather than realized future risk.

Key ideas

• The document frames two ways to minimize historical ex-ante volatility for a long–short equity portfolio.
• One approach optimizes portfolio variance from an estimated covariance matrix.
• The other optimizes volatility computed directly from historical portfolio returns.
• Equivalent data processing and objectives make the formulations correspond, while differing constraints or definitions can change the result.

Tags

# Two different ways to optimize ex-ante correlation of a long / short portfolio


# Two different ways to optimize ex-ante correlation of a long / short portfolio












Assuming I am making a long/short portfolio of S&P500 stocks and I would like to use the historical correlations to minimize ex-ante portfolio volatility. I can think of two ways of doing this:

First method

- Calculate covariance matrix based on some look-back period.

- Use optimizer to find weights that minimize wT @ cov @ w

Second method

- Gather return data over some look-back period.

- Use optimizer to find weights which minimize the volatility of the portfolio throughout the lookback period.

They're both using the same look-back period data so would they yield the same results?

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