Orthogonalizing Trading Signals and Returns for Performance Analysis

Summary

The document examines whether removing factor exposure from a trading signal or from forward returns produces the same performance estimate. With a projection matrix P and residual-maker M = I − P, it writes the strategy’s cross-sectional return as the inner product of the orthogonalized signal with raw returns, or of the raw signal with orthogonalized returns. These expressions are algebraically equal for a given observation when the same projection is applied.

The unresolved question is whether measuring variance over time from these equivalent expressions also gives the same strategy volatility. The author suspects the estimates should match but reports conflicting claims from AI assistants. No data, derivation beyond the stated matrix identities, or empirical comparison is included. The result may depend on implementation details such as how factor loadings and projections are estimated at each time, so the document poses a statistical question rather than presenting a tested conclusion.

Key ideas

• A signal’s performance can be evaluated by combining it with forward returns when the signal changes.
• The document uses a residual-maker matrix to remove factor exposures from signals or returns.
• For a fixed observation, the two stated inner-product expressions are algebraically equivalent under the same projection.
• The author asks whether their time-series variance estimates also agree.
• No empirical evidence or complete variance derivation is provided.

Tags

# Orthogonalizing returns or signal


# Orthogonalizing returns or signal












Say you want to test the performance of a signal, you can multiply it by forward returns. And do these each time the signal changes.

One can orthogonalize either the return or the signal with the factor loading hat matrix and obtain the same result. Let $M = I-P$

$$\mu_1 = s_\perp^\top r = (Ms)^\top r = s^\top M r$$ $$\mu_2 = s^\top r_\perp = s^\top Mr = \mu_2$$

Does this lead to the same variance estimate of the strategy? Mathematically, I believe I get the same answer, but for some reason chatgpt and claude argue that they do result in different variances and I have no idea why they would say that. If we take the variance of $s^\top r_\perp$ or $s^\top_\perp r$ overtime, it seems like the vol measure is the same?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.