Question on portfolio variance formula
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Hi Jack,
It's a *great* question that causes confusion. The way I approach this is to keep in mind that variance (and covariance) are in squared units; variance itself is not intuitive. All three terms in your equation, if we correct it with squared weights, are in squared units, unintuitively. If you take square root of the squared-weight term, you will be back to something more in line with your intuition; the units used by the standard deviation. I get stuck on this sometimes, but then i try and realize that the units of variance are not natural to me, rather their square root is. So, just conceptually, if you think about variance as (unintuitive) squared-units, perhaps the squared weights are acceptable? Perhaps this formula cannot be grasped intuitively. You'll note that even your third term is *really* in squared-units also, and hence, is not immediately sensible: (w1)(w2) is analogous to squared; and cov(1,2) is also a squared concept (that's why it needs to be divided by the squared volatility to render it into unitless correlation).
But mathematically, the explain starts with Gujarati 3.15: var(aX) = a^2(X). The constant is squared, and so the weight (w) being a constant, is squared in portfolio variance. Gujarati 3.17 is relevant:
var (aX + bY) = a^2var(X) + b^2var(Y), under independence
and ideally this is combined with Gujarati 3.32 to give the most general formula:
var (aX + bY) = a^2var(X) + b^2var(Y) + 2(a)(b)Cov(a,b)
Hope helps, David
It's a *great* question that causes confusion. The way I approach this is to keep in mind that variance (and covariance) are in squared units; variance itself is not intuitive. All three terms in your equation, if we correct it with squared weights, are in squared units, unintuitively. If you take square root of the squared-weight term, you will be back to something more in line with your intuition; the units used by the standard deviation. I get stuck on this sometimes, but then i try and realize that the units of variance are not natural to me, rather their square root is. So, just conceptually, if you think about variance as (unintuitive) squared-units, perhaps the squared weights are acceptable? Perhaps this formula cannot be grasped intuitively. You'll note that even your third term is *really* in squared-units also, and hence, is not immediately sensible: (w1)(w2) is analogous to squared; and cov(1,2) is also a squared concept (that's why it needs to be divided by the squared volatility to render it into unitless correlation).
But mathematically, the explain starts with Gujarati 3.15: var(aX) = a^2(X). The constant is squared, and so the weight (w) being a constant, is squared in portfolio variance. Gujarati 3.17 is relevant:
var (aX + bY) = a^2var(X) + b^2var(Y), under independence
and ideally this is combined with Gujarati 3.32 to give the most general formula:
var (aX + bY) = a^2var(X) + b^2var(Y) + 2(a)(b)Cov(a,b)
Hope helps, David
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