Sizing the Overlay: Dollar-Beta Matching and the Limits of Replication
Match $2M of notional against a $2M micro-cap with a market beta of 1.35 and you are 35% under-hedged on day one. The position carries $2.7M of market-equivalent exposure. Your proxy covers $2M. The missing $700K rides naked through every drawdown — and your put spread, sized to the smaller basket, caps losses you no longer have while leaving the losses you do have uncapped.
Position sizing is where proxy hedges fail. Not in the regression. Not in the options chain. In the arithmetic between the illiquid name and the liquid basket.
Why Notional Matching Fails
Notional matching assumes a dollar in the proxy moves like a dollar in the micro-cap. It does not.
A micro-cap with β = 1.35 moves $1.35 for every $1.00 move in the market. A proxy constituent with β = 0.90 moves $0.90. Put $2M in each and a 1% market day splits them: ~$27,000 vs. ~$18,000. Across a 10% correction the gap is $90,000 — a shortfall no options overlay can fix, because the overlay was sized to the wrong base.
Dollar-beta matching corrects for this. It sizes the basket so that a 1% market move produces the same dollar P&L on both sides. That is the only definition of “matched” that matters for hedge sizing.
The rule is simple: high-beta names need less proxy notional than their face value suggests when the basket itself is high-beta, and more proxy notional when the basket is low-beta. Skip this step and every downstream number — put spreads struck, calls sold, premium collected — inherits the error.
The Dollar-Beta Formula
Dollar beta is notional multiplied by beta:
Dollar β = Notional × β
Matching means equating the two sides:
Basket Notional = (Micro-Cap Notional × Micro-Cap β) / Basket β
For the reference position:
- Micro-cap: $2,000,000 × 1.35 = $2,700,000 market-equivalent exposure
- Basket β (blended): 1.59
- Required basket notional: $2,700,000 / 1.59 = $1,698,113 ≈ $1.70M
That $1.70M figure is not a haircut. It is the notional amount of 1.59-beta instruments that generates the same systematic dollar swing as $2M of 1.35-beta stock. Size to $2M of basket instead and you over-hedge by ~18%. Size to $1.5M and you under-hedge by ~12%. Both errors compound through the options layer.
Recompute this monthly. Beta is an estimate, not a constant, and the quotient moves with it.
Blended Basket Beta Math
The basket never has a single beta. It has a weighted average of its components, and the weights are dollar weights.
Take the standard construction from the parent structure — 60% small-cap industrials ETF, 40% split across two liquid single names:
| Leg | Weight | Leg β | Contribution |
|---|---|---|---|
| Small-cap industrials ETF | 60% | 1.42 | 0.85 |
| Single name A | 20% | 1.75 | 0.35 |
| Single name B | 20% | 1.95 | 0.39 |
| Blended basket β | 1.59 |
Blended β = (0.60 × 1.42) + (0.20 × 1.75) + (0.20 × 1.95) = 1.59.
Three points operators miss:
- Weights drift. A 15% run in single name B pushes its weight from 20% to ~22%, lifting blended β to ~1.63. Basket notional required drops from $1.70M to $1.66M. Rebalance back to target weights or resize — pick one, but do not hold still.
- Substitution changes β. Swapping single name B (β 1.95) for a calmer constituent (β 1.40) drops blended β to 1.48. Required notional rises to $1.82M. Every instrument change forces a resizing pass.
- Estimate each leg separately. Do not regress the basket as a unit. Regress each leg against the same market index over the same 252-day window, then weight. A basket-level regression hides composition shifts.
Over-Hedging vs. Under-Hedging: What Each Costs You
Sizing errors are not symmetric. They fail in different directions.
Under-hedging — basket too small. A $1.50M basket at β 1.59 produces $2.39M of market-equivalent exposure against $2.70M of micro-cap exposure. The $310K gap (~12%) is unhedged systematic risk. The put spread pays on $1.50M of notional, not $1.70M. Premium undershoots too: fewer shares under the short calls, less 30Δ premium per month.
Over-hedging — basket too large. A $2.00M basket at β 1.59 produces $3.18M equivalent — $480K (~18%) more than the position. In a rally the short calls bleed against $3.18M of upside, capping gains on exposure you never owned. Hedge cost scales linearly: ~18% more put-spread premium per roll for protection on phantom notional.
The cost summary:
- Under-hedge: protection shortfall in selloffs, income shortfall in flat markets. Failure is quiet until it is sudden.
- Over-hedge: premium drag in rallies, excess hedge spend every roll. Failure is a steady bleed.
If forced to err, err 3–5% light rather than heavy. An under-hedge costs you in events; an over-hedge costs you every month, including the months that pay for the structure.
When Beta Drifts: The ±0.15 Rule
Betas move. The sizing that was exact in January is approximate by March. The operating rule: resize or rebalance when realized beta drifts more than 0.15 from target on either side.
Worked drift example on the $2M / β 1.35 position:
- Micro-cap β drifts to 1.50 (+0.15). Market-equivalent exposure rises to $3.00M. Against the unchanged $1.70M basket at β 1.59 ($2.70M equivalent), residual exposure is +$300K — an 11% under-hedge that appeared without a single trade. Fix: grow the basket to $3.00M / 1.59 = $1.89M, or rotate into higher-beta constituents.
- Micro-cap β drifts to 1.20 (−0.15). Exposure falls to $2.40M. The same $1.70M basket now over-hedges by $300K (~12%). Fix: shrink the basket to $2.40M / 1.59 = $1.51M, or rotate down-beta.
- Basket β drifts to 1.74 (+0.15, single names running hot). $1.70M of basket now throws $2.96M equivalent — a $260K over-hedge. Fix: trim to $2.70M / 1.74 = $1.55M.
- Basket β drifts to 1.44 (−0.15, ETF weight creeping up). $1.70M throws only $2.45M — a $250K under-hedge. Fix: add to $2.70M / 1.44 = $1.88M.
The ±0.15 band keeps residual exposure inside roughly ±10–12% without churning the book weekly. Tighter bands (±0.05) generate resize trades whose costs exceed the tracking improvement. Wider bands (±0.25) let residuals reach 20%+, at which point the put-spread strikes no longer match actual exposure.
Monitor on the same monthly regression cycle as the factor model. Log every drift reading — direction and persistence tell you whether the relationship is noisy or breaking.
Sizing Worksheet
Run this top to bottom every resize. Nothing here requires more than the latest regression output and current prices.
| Step | Input | Example Value |
|---|---|---|
| 1. Micro-cap notional | Current market value | $2,000,000 |
| 2. Micro-cap β | 252-day OLS vs. market | 1.35 |
| 3. Market-equivalent exposure | Step 1 × Step 2 | $2,700,000 |
| 4. Leg weights | Dollar weights, sum to 100% | 60 / 20 / 20 |
| 5. Leg betas | 252-day OLS per leg | 1.42 / 1.75 / 1.95 |
| 6. Blended basket β | Σ (weight × leg β) | 1.59 |
| 7. Required basket notional | Step 3 / Step 6 | $1,698,000 |
| 8. Allocate by weight | Step 7 × each weight | $1.02M / $340K / $340K |
| 9. Drift check | |Current β − Target β| > 0.15? | Resize if yes |
| 10. Residual exposure | Actual equivalent − Target equivalent | Must be < ±10% |
A completed worksheet is the sizing record. File it with the regression output. When the hedge underperforms, this sheet tells you within minutes whether sizing caused it or something else did.
What Sizing Cannot Fix
Dollar-beta matching aligns systematic dollar exposure. It does nothing for the idiosyncratic residual — the ~22% of micro-cap variance outside the factor model. No basket resize covers a single-name event with no market component.
Sizing also assumes the beta estimates are valid at the moment you need them. In acute stress, realized betas spike past trailing estimates and the ±0.15 band is breached intraday. The worksheet tells you the hedge is offsides. It does not put it back onside mid-crash.
Size correctly, monitor the drift, and accept the boundary: beta matching makes the hedgeable component hedgeable. The rest was never yours to hedge.