Beta-Adjusted Hedge Against a Benchmark ETF
You are long a single stock and want to neutralize its exposure to a broad index that you can trade as an ETF. You estimate the stock's market beta by regressing the stock's returns on the ETF's returns. The slope of that regression is the stock's beta $\beta$.
For example, suppose you hold a 1,000,000 dollar long position in a stock with $\beta = 0.833$ against the FAANG ETF. How large an ETF position do you take to hedge, and in which direction? More generally, given a long position of notional $V$ in a stock with beta $\beta$, what is the beta-adjusted hedge?
Hints
- Beta is the slope from regressing the stock's returns on the ETF's returns -- it measures dollar exposure per unit of your notional.
- To hedge a long, take the opposite (short) position in the index instrument; the question is only how big.
- Match dollar betas: short ETF notional $H = \beta V$, since the ETF has beta $ to its own index.
Worked Solution
How to Think About It: Beta tells you how much the stock moves, on average, for a one-unit move in the index. If you are long the stock, you are implicitly long the index by an amount $\beta$ times your notional. To kill that exposure you short the index ETF by exactly that dollar amount. The hedge is not dollar-for-dollar -- it is scaled by beta. A low-beta stock needs a smaller hedge; a high-beta stock needs a larger one.
Approach: Match the dollar beta of the hedge to the dollar beta of the position.
Formal Solution: The position's exposure to the index, in dollar terms, is $\beta \cdot V$ (a one-unit index return moves the position by $\beta V$). The ETF has beta
$ to the index by construction, so a short ETF notional of $H$ contributes exposure $-H$. Setting the net index exposure to zero:$\beta V - H = 0 \implies H = \beta V.$
For the example: $H = 0.833 \times 1{,}000{,}000 = 833{,}000$ dollars. You sell (short) $833{,}000$ dollars of the FAANG ETF against the
{,}000{,}000$ dollar long stock position.Answer: Short the ETF in notional $H = \beta V$. In the example, sell $\$833{,}000$ of the ETF to hedge a $\
{,}000{,}000$ long with $\beta = 0.833$.Intuition
Beta hedging is the most basic risk-neutralization tool on any equity desk, and the single most common error is hedging dollar-for-dollar instead of beta-for-dollar. If your stock has beta $0.5$, a dollar-for-dollar short over-hedges and leaves you net short the market; if beta is
.5$, a dollar-for-dollar short under-hedges. The right size always scales the hedge notional by beta.The subtler point that good candidates raise: beta is estimated, so the hedge is only as good as your regression. Betas drift over time, are noisy on short windows, and a single-factor (one-ETF) hedge leaves you exposed to anything orthogonal to that index -- sector, size, idiosyncratic news. Real desks re-estimate beta regularly and often hedge with multiple factors. But the core arithmetic, hedge notional equals beta times position notional, is the foundation everything else builds on.