Deriving a Stock’s Correlation with an Index Excluding That Stock

Summary

The document considers how to calculate the correlation between a stock and an index reconstructed without that stock. It defines the reduced index return by removing the stock’s weighted return from the full index return and rescaling by the remaining index weight. Given the stock and full-index volatilities, their correlation, and the stock’s index weight, it asks how to obtain the new correlation.

The proposed derivation uses covariance linearity: covariance distributes over sums and scales with constants. Applying these properties expresses the covariance of the stock with the reduced index in terms of its covariance with the original index and its own variance. The correlation then follows by dividing that covariance by the stock volatility and the reduced index volatility. The document gives the covariance step and a volatility expression, but does not show the final correlation formula or discuss estimation error, changing weights, or assumptions about return measurement.

Key ideas

• The index return excluding a stock is obtained by subtracting its weighted return and renormalizing the remaining weight.
• Covariance with the reduced index can be expanded using linearity.
• The stock’s covariance with the full index is determined by their correlation and volatilities.
• The resulting covariance must be standardized by both volatilities to obtain correlation.

Tags

# Correlation between a stock and the index without the stock


# Correlation between a stock and the index without the stock












The weight of stock i in index m is w. The returns are, respectively, r(i) and r(m).

I know the volatility s(i) of i, the volatility s(m) of m and the correlation rho(i, m) between i and m.

The index x is m without i:

r(x) = (r(m) - w*r(i)) / (1 - w)

s(x) = sqrt[(w/(1-w)*s(i))^2 + (1/(1-w)s(m))^2 - 2rho(i, m)s(i)s(m)(w/(1-w))(1/(1-w))]

Can I get the correlation rho(i, x) between i and x?

(I am sorry for not using LaTeX)

--- edit ---

I may have found it.

The relevant properties should be:

- COV(cX, Y) = cCOV(X, Y)

- COV(X + Y, Z) = COV(X, Z) + COV(Y, Z)

In my problem:

COV(i, x) = COV(i, (1/(1-w))*m - (w/(1-w))*i) = (1/(1-w))*COV(i, m) - (w/(1-w))*VAR(i)

and I have all the inputs needed.

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