10.3 Best Hedge

When considering finite or large changes in an asset holding, it is very useful to consider the change that would optimally hedge the rest of the portfolio. We can call this the best hedge—the position size that reduces the volatility as much as possible, or hedges the rest of the portfolio as effectively as possible.

To work out the best hedge position, consider that the marginal contribution attributable to a particular position may be either positive (adding to the portfolio risk) or negative (lowering the portfolio risk—acting as a hedge). At some point, the marginal contribution will be zero. This will be the position size that optimally hedges the rest of the portfolio.10 We can calculate the position size for asset k for which the marginal contribution is zero, given no changes in any other asset holdings. This means finding img that satisfies

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The point of zero marginal contribution is the point at which portfolio risk is minimized with respect to the size of asset k since the marginal contribution is the derivative of the volatility with respect to its position. This will be a best hedge in the sense of being the position in asset k that minimizes the portfolio volatility (all other positions unchanged).

Earlier, for a portfolio with two assets, we used the triangle ...

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