Sizing a hedge by beta-weighted delta

Every hedge in this module has been sized against 100 shares of an index ETF. Real portfolios hold a dozen stocks with different prices and different sensitivities to the market, plus some option positions. To hedge that with index puts you need one number: how many ETF shares the whole portfolio behaves like. That is the beta-weighted delta, and computing it is the same addition as Module 1 with one extra factor per position.

Beta

A stock's beta is the historical sensitivity of its returns to the index's returns: beta 1.3 means the stock has tended to move 1.3% for every 1% the index moves. It is estimated from past data, usually a year or more of daily or weekly returns, and every platform publishes one. It is imprecise, it changes, and it is still the best single number for the job.

The formula

For each position: beta-weighted delta = position delta × price ÷ ETF price × beta, where position delta is in shares (100 per 100 shares, or the option position delta from Module 1). The result is expressed in ETF share-equivalents.

A portfolio, with the index ETF at $400:

The portfolio behaves like 121 shares of the ETF, or $48,400 of index exposure, on $40,000 of stock. Its effective beta is 1.21. A 1% fall in the index should cost about 121 × $4 = $484, not the $400 that "1% of $40,000" would suggest, because the holdings are on average more sensitive than the index.

What a 1% index drop costs, by position (beta-weighted)

  Stock A   ################################  $208
  Stock B   ##################               $120
  Stock C   ######################           $144
  XYZ sprd  ##                               $12
  -----------------------------------------------
  Total                                      $484

Sizing the hedge

Decide how much of the 121 to remove. Nobody hedges to zero; that would be selling the portfolio with extra steps. Suppose the target is to cut market exposure roughly in half over the next 90 days: remove about 60 ETF deltas.

The put's delta is the reason the arithmetic is only a starting point. At entry three $360 puts remove 66 deltas. If the index falls 10%, each put's delta moves toward −50 and the hedge is now removing about 150, more than the whole portfolio's exposure. That is convexity working for you: the hedge gets stronger as the decline deepens, which is what you want from insurance and would not get from short shares. It also means the position can be net short after a big drop; re-check the beta-weighted total after any large move and trim if that is not what you intend.

Note the XYZ spreads' contribution: 3 deltas. Options positions belong in the table too, and a book of short premium can carry a surprising amount of beta-weighted delta from its short puts (Module 1's book was +331 raw).

What beta-weighting does not do

• Beta is historical. A stock's beta over the last year says little about next month. Estimates on the same stock from different sources routinely differ by 0.2 or more. Round to the nearest 0.1 and do not pretend the third decimal means anything.
• Correlations rise in a crash. In a broad sell-off, most stocks fall together and low-beta names fall harder than their beta predicted. The hedge sized at beta 0.8 for Stock B will look too small on the day it is needed. Some hedgers size against a "stressed beta" of 1.0 minimum for every holding.
• Idiosyncratic risk is not hedged at all. If Stock A drops 30% on its own earnings, the index puts do nothing. Beta-weighting hedges the market factor only; single-name risk needs single-name hedges or, more simply, smaller positions.
• The number drifts. Prices change the notionals, options change their deltas, and betas get re-estimated. Recompute monthly, and after any move larger than 5%.

Key idea: Beta-weighted delta converts every holding into index-ETF share-equivalents: shares × price ÷ ETF price × beta, summed with the option positions. It tells you what a 1% index move costs and how many puts remove the fraction of that you want gone. Put deltas grow as the market falls, so the hedge strengthens on its own; beta is historical, correlations rise in crashes, and single-name risk is untouched.

Try it: Build the beta-weighted table for your own portfolio, including any option positions, using your platform's betas rounded to one decimal. Compute the cost of a 1% index drop. Then price the number of 90-day 10%-OTM index puts that would remove half of it and express their cost as a percentage of the portfolio. Repeat with every beta below 1.0 set to 1.0 and note how much the hedge size changes.

Recap

• Beta-weighted delta = shares (or option delta) × price ÷ ETF price × beta, summed across positions, in ETF share-equivalents.
• The example portfolio of $40,000 behaves like 121 ETF shares, so a 1% index drop costs about $484.
• Size the hedge to the fraction of exposure you want removed; three $360 puts remove about 66 deltas at entry.
• Put deltas grow as the market falls, so the hedge strengthens; re-check after large moves.
• Beta is historical, correlations rise in crashes, and index hedges do nothing for single-name risk.