Nonnegative Sector Weights Can Hide Negative Attribution Effects

Summary

The document formulates sector performance attribution as a constrained fit of S&P 500 returns using sector return time series. The proposed coefficients sum to one and are each restricted to the interval from zero to one. The author interprets larger coefficients as indicating greater sector impact, but points out that a sector may be negatively correlated with the market. Under the nonnegative constraint, its fitted coefficient can then be set to zero, which appears to erase a negative relationship.

The text poses this as an unresolved modeling question and gives no answer, alternative objective, or empirical example. It identifies a real interpretive tension between long-only attribution weights and coefficients intended to capture signed effects. Readers should not treat the proposed betas as a complete measure of sector impact without clarifying the attribution objective and constraints; the document itself does not determine which formulation is appropriate.

Key ideas

• The proposed attribution model fits index returns as a weighted combination of sector returns.
• Its coefficients are constrained to be nonnegative and to sum to one.
• A negatively correlated sector can receive a zero coefficient under these constraints.
• The document raises but does not answer how to represent negative sector effects appropriately.

Tags

# Constrained Optimization for performance attribution


# Constrained Optimization for performance attribution












I am trying to perform constrained opmitization for portfolio performance attribution analysis. Specifically, I am trying to determine the impact of sectors performance on the S&P 500 index.

Min y-(b1x1+b2x2+...+bnxn+bn+1xn+1+....bpxp)

subject to b1+b2+...+bp = 1,

0 <= bi <= 1 for i=1,2,...,p

`y`is the time-series return of S&P500 and xi is the time-series return of a sector for all `i = 1, 2, ..., p`.

The idea of the betas is that the higher the beta, the greater the impact of the variable (sector) is on the portfolio.

The problem with this optimization is that, sometimes, the sector and market are negatively correlated so beta is negative. But the constraint forces it to be between 0 and 1, so beta becomes 0, which suggests that the sector had no impact on the S&P 500 index (even though it DID have an impact, negatively).

What's the better way of solving this?

Thank you very much in advance!

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