Chapter 2 of 5 · 12 min

Beta-hedged pair trades

Long one stock, short another, and size the short so the market washes out. What is left is the view.

By the end of this chapter you can

Size a short so a pair is beta-neutral
Split a pair's P&L into market and stock selection
Back out a stock's alpha from the P&L
Show what a dollar-neutral pair leaves behind

The intuition

You think one soft-drinks company will beat its rival, but you have no idea whether the stock market will rise or fall. Buy only the one you like and a crash could wipe out the gain even if you were right. Buy it and short the rival, and a crash hits both legs: the long loses, the short gains. What is left is the thing you had a view on: one company against the other.

That is a pair trade. It only removes the market if the two legs carry the same market exposure. A stock with a beta of 1.5 moves about 1.5% for every 1% the market moves, so a dollar of it counts for more than a dollar of a beta-1.0 stock. Size the legs by beta-dollars (dollars × beta), not by dollars.

The key idea

Each stock's return = alpha + beta × market return. Beta-neutral short = long × beta of the long ÷ beta of the short. In a beta-neutral pair the market P&L is zero, so the P&L = long × alpha of the long − short × alpha of the short.

Why it works

The conventions here: one market factor. Each stock's return is its alpha (its own, stock-specific return over the period) plus beta × the market return. No borrow, financing or dividend costs; the next chapter covers those.
Beta-neutral means long beta-dollars equal short beta-dollars, so any market move cancels exactly.
The short is bigger than the long when the short stock has the lower beta, and smaller when it has the higher beta.
Dollar-neutral means equal dollars. It only cancels the market when the betas are equal; otherwise long × (beta of long − beta of short) of market exposure stays in the book.
Attribution asks how much of the P&L was the market, which you did not mean to own, and how much was alpha, the view.
A pair can make money and still be wrong, or lose money and be right, because the legs are different sizes. Judge the view by the gap between the two alphas, not by the P&L.

Long $10M of a beta-1.5 stock, short a beta-1.0 stock; the market rises 4%; the long returns 9%, the short 3%

Beta-neutral short: 10 × 1.5 ÷ 1.0	$15M
Long P&L: 10 × 9%	+$0.90M
Short P&L: −15 × 3%	−$0.45M
Total	+$0.45M
Alphas: 9% − 1.5 × 4%, and 3% − 1.0 × 4%	3% and −1%
Alpha P&L: 10 × 3% − 15 × (−1%)	+$0.45M, all of it

Dollar-neutral instead ($10M short): P&L 0.90 − 0.30 = $0.60M, of which 10 × (1.5 − 1.0) × 4% = $0.20M is market the hedge missed.

The formulas

Stock return = alpha + beta × market return

Its own move plus its share of the market's.

Beta-neutral short = long × beta(long) ÷ beta(short)

Match beta-dollars, not dollars.

P&L = long × return(long) − short × return(short)

A short gains when its stock falls.

Market P&L = (long × beta(long) − short × beta(short)) × market return

Zero when the pair is beta-neutral.

Alpha P&L = long × alpha(long) − short × alpha(short)

What the view earned.

Long alpha = (P&L + short × return(short)) ÷ long − beta(long) × market

Run it backwards.

Worked example

Each leg's P&L first. Then strip beta × the market out of each stock's return to find its alpha.

Drawing the numbers…

See it move

Same pair of companies. Change the size of the long, both betas, the market's return and each stock's own return (its alpha).

Drawing the numbers…

Try this

Move the market. The beta-neutral P&L does not change at all; the dollar-neutral market bar moves, unless the two betas are equal.
Set the two betas equal. The dollar-neutral market bar disappears: with equal betas, equal dollars are beta-neutral.
Raise the beta of the short. The beta-neutral short gets smaller.
Raise the alpha of the short. Stock-selection P&L falls: the short loses when its own stock does well.

Run it backwards

Same pair, reversed: you know the P&L, the market's return and the short's return. What alpha did the long deliver?

Drawing the numbers…

Take the short leg out of the P&L to leave the long leg. Divide by the position for the long's total return, then remove beta × the market.

That alpha is what the analyst is paid for: the part of the return that came from being right about the company, not from the market.

Traps

Hedging with equal dollars.

Equal dollars only cancel the market when the betas match. Size the short by beta-dollars.

Getting the short size upside down.

Short = long × beta(long) ÷ beta(short). A lower-beta short needs more dollars, not fewer.

Forgetting the short leg's sign.

A short makes money when its stock falls: short P&L = −short × return.

Judging the trade by its total P&L.

Judge the view by the alpha gap. Different leg sizes can turn a right call into a loss.

Calling a beta-neutral pair fully hedged.

It still carries sector, size and style exposure wherever the two names differ.

Say it in the interview

The interviewer asks

“How would you hedge a long position, and how would you tell whether the trade worked?”

Say yours out loud first, then compare.

Check yourself

4 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.

Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.

0 of 4

Drawing your questions…

Remember

Short = long × beta(long) ÷ beta(short).
Beta-neutral: market P&L is zero, so P&L is all alpha.
Dollar-neutral leaves long × the beta gap of market.
Judge the call by the alpha gap, not the P&L.