Hedging: How Do I Remove the Risk I Don't Want?

Chapter 12 removed an unwanted exposure by re-trading the portfolio. Often that is the wrong tool: the stock book may be the alpha engine you don’t want to disturb, trading it may be expensive or slow, or the unwanted exposure may be temporary. Hedging removes systematic exposure with overlay instruments, futures, ETFs, swaps, baskets, leaving the underlying portfolio intact. In factor language, it is arithmetic: add instrument exposures to the book’s exposures until the unwanted components cancel.

13.1 What hedging means in factor space

A hedge instrument is just another asset with a factor exposure vector. An index future’s exposures are those of the index it tracks. A sector ETF loads on its industry. A custom basket has whatever exposures its weights imply. Holding the portfolio plus notional positions $h = (h_1, \dots, h_M)$ in $M$ instruments with exposure vectors $x_{h_1}, \dots, x_{h_M}$ (per unit notional), the combined exposures are

$x_{\text{hedged}} = x_p + \sum_{m} h_m\, x_{h_m},$

and hedging is choosing $h$ to make selected components of $x_{\text{hedged}}$ zero (or to minimize the resulting variance). The goal is never “no risk”. It is keeping the intended exposures while neutralizing the unintended or unpaid ones.

13.2 Beta hedging: the single-factor case

The classic: a long equity book hedged with index futures. With one instrument, the variance-minimizing notional is the OLS answer:

$h^* = -\frac{\mathrm{Cov}(r_p, r_h)}{\mathrm{Var}(r_h)} \;=\; -\beta_{p,h},$

the portfolio’s beta to the hedge instrument, computed through the model as $\mathrm{Cov}(r_p, r_h) = x_p^\top F x_h$ (+ any specific-risk overlap, zero for a future on a broad index) and $\mathrm{Var}(r_h) = x_h^\top F x_h$ . Through the model, “beta” needs no time-series regression: it is implied by current exposures and the factor covariance, and therefore updates as fast as the exposures do (Chapter 4’s responsiveness argument, again).

The expression is more general than hedging. For any two portfolios, $\beta_{p,h} = (x_p^\top F x_h)/(x_h^\top F x_h)$ is the model’s predicted beta of $p$ to $h$ : set $h$ to a benchmark for benchmark beta, to a sector basket for sector beta, to another book to see how one tracks the other. It needs only the two exposure vectors and $F$ , reads forward where a regression beta reads backward.

Mini example: The manager’s portfolio against a future on the cap-weighted benchmark index: $\beta_{p,h} = x_p^\top F x_h / x_h^\top F x_h = 0.934$ , so $h^* = -\beta_{p,h} = -0.934$ : the minimum-variance hedge shorts 0.934x the portfolio’s value in index futures. Result: total volatility falls from 17.55% to 8.29%. (Why beta < 1 for a fully invested portfolio? The book tilts away from the index’s tech weight toward lower-vol financials/consumer and cheap stocks, so its model-implied co-movement with the index is slightly sub-unit.)

13.3 Multi-factor hedging

With several instruments and several target factors, choose $h$ to minimize hedged variance:

$\min_h \;\; (x_p + X_h^\top\, h)^\top F\, (x_p + X_h^\top h) \; + \; \text{specific terms},$

where $X_h$ is the $M \times K$ matrix of instrument exposures. Two regimes:

Exact neutralization of a chosen subset $S$ of factors: solve the $|S|$ linear equations $(x_p + X_h^\top h)_k = 0,\, k \in S$ . Requires $M \ge |S|$ instruments with linearly independent exposures in those factors. With $M = |S|$ the solution is unique. With $M > |S|$ use the freedom to also minimize variance over the rest.
Minimum-variance (unconstrained) hedging: the multi-instrument generalization of beta hedging, $h^* = -(X_h F X_h^\top + \Delta_h)^{-1} X_h F\, x_p$ , a multivariate regression of the portfolio on the instruments. It reduces every factor it can reach, including ones you wanted to keep: a min-variance hedge will happily sell off your value bet if the instruments span it. Choose exact neutralization when you have intended exposures to protect. Reserve min-variance hedging for pure de-risking.

Under- and over-determination in practice: more target factors than instruments ⇒ exact hedging impossible, minimize weighted residual exposure instead. More instruments than targets ⇒ pick the cheapest combination (costs enter the objective).

13.4 The instrument shelf

Instrument	Exposure profile	Basis risks
Broad index futures	the index’s exposures, no specific risk	roll cost, index ≠ your benchmark
Sector ETFs / futures	one industry + market	expense drag, sector definition mismatch with model industries
Factor ETFs / style indices	a (impure, Ch. 7) style + market + incidentals	tracking gap to the pure factor, capacity
Custom baskets / swaps	anything you can specify, including a pure factor portfolio	counterparty cost, the purest hedge at the highest setup cost
Single-name shorts	full row of $X$ + specific risk	borrow cost, adds specific risk while removing factor risk

The recurring trade-off: liquid instruments are cheap but approximate (their exposures only roughly match what you need to shed), precise instruments are expensive. The factor model is what lets you quantify the approximation. Compute the residual exposure left by the cheap hedge and decide if it’s tolerable.

13.5 Worked example: making the running portfolio market- and size-neutral

Mandate shift: keep the stock book (the value process and its stock selection), but remove market direction and the small-cap lean, e.g., the fund is being run as a hedged share class. Like Chapter 12’s repair, the example works from the month-1 book, Chapter 11’s buy-back set aside. Instruments: the benchmark index future and a small-cap index future with exposures (per unit notional):

	MKT	TECH	FIN	CONS	VALUE	MOM	SIZE
Index future	1.00	0.455	0.325	0.220	0	0	0
Small-cap future	1.05	0.35	0.30	0.35	0.10	−0.05	−1.20

The exposures here are total, not benchmark-relative: hedging removes market direction outright, so you work with the portfolio’s own $x_p$ (MKT component 1.0 for a fully invested book) rather than the active $x_a$ of Chapters 9–12 (whose MKT component was 0). The style components coincide, because the cap-weighted benchmark is style-neutral by construction, so SIZE reads −0.275 either way. Solve the 2x2 system for exact neutralization of MKT and SIZE ( $x_{p,\text{MKT}} = 1.0$ , $x_{p,\text{SIZE}} = -0.275$ ):

$h_{\text{index}} = -0.759, \qquad h_{\text{smallcap}} = -0.229$

Short 76% of NAV in index futures and 23% in small-cap futures, both shorts. Futures are what make an overlay like this practical: they are unfunded, margin posted against a notional rather than capital spent, so the shorts consume no capital and disturb no holdings, and the positions roll at each quarterly expiry. The small-cap leg does the size-hedging: shorting a future with a −1.20 SIZE loading adds positive SIZE, offsetting the book’s −0.275 (its long-small-cap, i.e. negative-SIZE, tilt). The index leg removes the remaining market exposure.

Post-hedge profile:

	MKT	TECH	FIN	CONS	VALUE	MOM	SIZE
Hedged exposures	0.000	−0.116	+0.104	+0.023	+0.362	−0.320	0.000

	Pre-hedge	Post-hedge
Total vol	17.55%	8.14%
factor vol	15.99%	3.76%
specific vol	7.22%	7.22% (unchanged)

Readings:

The value bet survived (+0.36, barely dented, the small-cap future’s incidental +0.10 VALUE loading shaved it slightly).
The momentum accident survived too (−0.32): hedging removes exactly what you aim at, nothing more. Could you have hedged momentum instead? Not sensibly. There is no momentum future. The tradable proxies (a shortable momentum ETF, a custom basket) sit in the expensive rows of the Section 13.4 shelf. And momentum is the fastest-decaying characteristic in the model, so a static overlay would be mis-sized within weeks, Section 13.7’s maintenance problem at its worst. The right tool for that accident was Chapter 12’s re-trade: overlays for what the book can’t trade away, re-trading for what it can.
Specific risk is now the dominant risk (7.22% of the 8.14%, factor risk shrank 4x, specific not at all). This is the structural limit of overlay hedging: index instruments carry no specific risk, so they cannot remove any. A hedged stock-picker is a specific-risk portfolio, which is the point of the exercise, but should be a conscious choice.
Compare the two hedging philosophies on the same book: min-variance single-instrument (Section 13.2) reached 8.29% with one trade but left a residual market stub and would shave intended bets if given richer instruments. Exact two-factor neutralization reached 8.14% while leaving every other exposure where it was. Similar risk, different composition. Selectivity is what you’re buying with the second instrument.

13.6 Neutrality: dollar, beta, factor

“Market neutral” is a claim with at least three inequivalent definitions, all factor-model-checkable:

Dollar neutral: long notional = short notional. Says nothing about risk. Long USD 100 of 2-beta tech vs. short USD 100 of 0.5-beta utilities is dollar neutral and wildly market-directional.
Beta neutral: zero model-implied covariance with the market, $x_p^\top F x_h = 0$ for a market-proxy instrument $h$ (the Section 13.2 condition). Removes the average market sensitivity. Industry and style exposures remain.
Factor neutral: $x_k = 0$ for all $k$ in a designated set (often: market + industries + size, keeping only the deliberate style or specific bets). The strictest, and what “neutral” should mean in a factor-aware mandate. Verify it with Chapter 9’s report, not with notionals.

The mini example post-hedge is market- and size-neutral but deliberately not value-neutral and accidentally not momentum-neutral, a precise statement no notional-based description could make.

13.7 Maintenance: hedges decay

Exposures drift, prices move, characteristics update, the book trades, so a hedge sized today is mis-sized in a month. Maintenance discipline: recompute $x_{\text{hedged}}$ at each model update. Rebalance the overlay when residual target-factor exposure exceeds a band: continuous rebalancing churns roll costs, so the band trades hedge slippage against trading. Measure hedge effectiveness ex post: realized variance reduction vs. predicted, a cousin of the bias statistic. A persistent shortfall is a model test in its own right (Chapter 15): it indicts the factor covariances the hedge ratios were built from.

13.8 One model to evaluate, another to hedge

Everything so far assumed that the model that flags an exposure in the risk report is also the model that sizes the overlay that removes it. Nothing forces that. A firm can run separate models for the two jobs, or one model calibrated in different ways, and there are reasons to do that.

Factor content is one: Momentum has to be in the evaluation model, or the −0.332 accident of Chapter 9 never appears in a report and never gets managed. As a hedging target it is close to useless (Section 13.5: the exposure decays faster than an overlay can track), so a hedging model may not offer it at all. Crowding is the reverse. A firm can leave a crowding factor (Chapter 16’s CROWD) out of PM-level evaluation on purpose, because crowding is often what being right early looks like. The PM sees a trade, enters it, and the rest of the market comes around and piles in behind them. Spotting the opportunity first is skill and should be rewarded, not penalized. The de-leveraging risk of the crowded book is real regardless of how the PM got there, so CROWD may still sit in the firm-level hedging model, where the central risk book can hedge it.

Speed is another: Chapter 8 split models by horizon per mandate. The same split works by function. A risk and performance evaluation model should run at the speed of the investment process it judges: a quarterly-rebalance value manager seen through a fast model gets a performance conversation about every two-week wobble. A hedging model should run at the speed of the hedging process, which for a weekly-reset overlay means covariances at least that fresh, or Section 13.2’s ratios are stale on arrival. One lens for risk and performance at the PM level, another for hedging at the firm level.

This is not free. Two models give two answers to “what is this book’s risk?” The PM’s report can be clean while the hedge desk’s model shows a crowding exposure it intends to hedge, and the two will not print the same tracking error for the same holdings. Attribution splits the same way: the firm’s CROWD hedge earns P&L that the PM’s model has no factor to assign, so it prints as specific return, Chapter 10’s reminder that “specific” means unspanned by this model. That gap is there by design, but it needs the same treatment as any other model difference: know which factors account for it, reconcile the numbers, and decide in advance which model arbitrates which decision. Otherwise the first the PM hears of the firm’s hedge is when it drags on a book their own report called clean.

13.9 Summary

Hedging is linear algebra on exposure vectors: instruments contribute exposures. Chosen notionals zero out targeted components ( $h$ from a small linear system) or minimize variance ( $h$ from a regression).
Min-variance hedging de-risks indiscriminately. Exact neutralization is selective. Protect intended bets by constraining what gets hedged.
Index-style instruments cannot touch specific risk. A fully hedged stock book becomes a pure specific-risk (plus residual-factor) portfolio, mini example: 17.55% -> 8.14%, of which 7.22% specific.
“Neutral” needs a definition. The factor report is the arbiter. Hedges drift and need maintenance bands and ex post effectiveness tracking.
The evaluation model and the hedging model can differ, in factor content (momentum is too fast to hedge; crowding is hedged at the firm level but left out of PM evaluation) and in speed (matched to the investment process vs. the hedging process). Two models means two risk views, a gap that must be tracked and explained.

Try it: in section 7 of mini_example.py, soften the small-cap future’s SIZE loading from −1.20 to −0.60 (the last entry of x_h2) and rerun. The small-cap short doubles to −0.458 and the index leg shrinks: a blunter instrument needs a bigger position for the same hedge, and a bigger position drags in more of its incidental exposures.