If exposures are the units of risk, you eventually want to trade them directly. Two constructions do that.
What is a factor portfolio, and how do you hedge with one?
A factor portfolio, sometimes called a pure factor portfolio, has unit exposure to one factor and zero exposure to every other, and its return is therefore an estimate of that factor's return. Building one is a linear solve. With three assets, two style factors and a budget constraint that the weights sum to one, you have three equations and three unknowns.
On the three Indian index products in Worked Example 5, the pure value portfolio is 78.24% in the first, -7.87% in the second and 29.63% in the third, and it carries value exposure of exactly 1.000 and size exposure of exactly 0.000.
A tracking portfolio is the same solve with a different right-hand side: match a specified set of exposures rather than isolate one. Every hedge is a tracking portfolio, and so is every attempt to replicate a benchmark's factor profile with a small number of instruments.
The hedge itself needs one more step. A pure factor portfolio with weights summing to one still carries the market. Subtract a portfolio with zero style exposures and weights summing to one, and the difference has weights summing to zero, unit exposure to the target factor and nothing else. That zero-investment overlay is the hedging instrument. On the same three products the size overlay is (+0.2315, +0.5093, -0.7407), which sums to zero and carries a size exposure of 1.000.
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Is hedging an unwanted exposure to zero the best hedge?
Then comes the result that surprises people, and it is worth stating carefully because the instinct it corrects is very strong. Hedging an unwanted exposure to zero is usually not the tracking-error-minimising choice.
A fund with a value exposure of +0.62 and an unwanted size exposure of +0.35 has a tracking error of 5.1682%. Hedge the size exposure away entirely and it falls to 4.9070%, a reduction of only 5.05% of the tracking error, because value and size are correlated at -0.25 in this covariance matrix and the size bet was partly offsetting the value bet. Solve for the exposure that actually minimises tracking error, x_size = -Cov(V,S) x_value / Var(S) = +0.1163, and the tracking error falls to 4.8180%.
The optimal trade removes about two-thirds of the exposure and stops.
When does hedging to zero raise tracking error?
Push the correlation far enough and the instinct inverts completely. With a value exposure of 0.80, a size exposure of 0.30 and a correlation of -0.40 between the two, hedging the size exposure to zero raises tracking error from 4.7424% to 4.8826%, because the size bet was working as a partial hedge on the value bet all along. The break-even is at an exposure of 0.3556, below which hedging to zero always makes things worse. Any exposure worth hedging deserves the quadratic form rather than the instinct.