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I am calculating the historical portfolio variance of various long-short equity portfolios. For simplicity, assume the portfolio is long stock A with weight 1.0 and short stock B with weight -0.5. So cash/risk free is 0.5 for an overall portfolio weight of 1.0.

Since $\sigma^2_\text{risk free} = 0$ and $\sigma^2_{\text{risk free}, X} = 0$, I reduce the portfolio to a 2x2 covariance matrix for A and B with weights [1.0, -0.5]. However, the weights don't total 1 for this portfolio and I thought the weights have to total one?

Am I thinking about this correctly?

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We have weights $w_A$, $w_B$ and $w_C = 1 - w_A - w_B$ that sum to $1$.

With de-meaned returns $r_A$, $r_B$, and $r_C$, the portfolio variance is $$E\{[w_A r_A + w_B r_B + (1 - w_A - w_B)r_C]^2 \} = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2 w_A w_B\rho_{AB}\sigma_A \sigma_B,$$

assuming the cash volatility $\sigma_C$ is zero.

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  • $\begingroup$ And essentially $w_C$ is the weight in cash. In this case, the 0.5. $\endgroup$ – SRKX Feb 24 '17 at 2:28
  • $\begingroup$ @Stanford Wong: Yes -- that was the intent. Everything carries through as expected since with zero volatility the cash returns contribute nothing to portfolio variance. The stock returns are weighted appropriately even though they do not sum to $1$. $\endgroup$ – RRL Feb 24 '17 at 2:33

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