Case 029Factor investing and quantWarm up

A long-short fund is long Rs 100 crore of stocks with a beta of 1.3 and shorts a basket with a beta of 0.9. How large must the short be for beta neutrality, and what net money exposure results?

The situation

Samatol Long-Short Fund has built a long book of Rs 100 crore in stocks it expects to outperform. Measured against the market index, the long book has a beta of 1.3. The manager wants to remove market risk by shorting a basket of stocks she expects to lag, and that basket has a beta of 0.9.

The risk committee's rule is that the fund must be beta neutral: a move in the market, on its own, should leave the fund's value unchanged.

Your task

How big must the short be, what net money exposure does the fund then carry, and what does that net exposure mean for risk?

Quick check

How large a short makes the book beta neutral?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

Short about Rs 144.4 crore, which leaves the fund net short about Rs 44.4 crore in money. The long book carries Rs 130 crore of beta-weighted exposure; each rupee of a 0.9 beta basket offsets 0.9, so 130 over 0.9. The fund is market neutral only if both betas are right; the Rs 44 crore net short is a bet that low-beta stocks keep moving less than high-beta ones.

Step 1Why do you match beta-weighted rupees and not rupees?

Think of balancing two children on a see-saw. A heavier child sits nearer the middle; what balances is weight times distance, not weight alone. Beta is the distance: a rupee in a 1.3 beta stock moves 1.3 times as much as the market, so it takes more than a rupee of a 0.9 beta stock to cancel it. The long book's market exposure is 100 times 1.3, Rs 130 crore, and the short must also carry Rs 130 crore of beta-weighted exposure.

The relationship
S=L×βLβS=100×1.30.9=144.4Net=100−144.4=−44.4S = \frac{L \times \beta_L}{\beta_S} = \frac{100 \times 1.3}{0.9} = 144.4 \qquad \text{Net} = 100 - 144.4 = -44.4
Llong book, Rs 100 crore
\beta_L, \beta_Sbeta of the long book, 1.3, and of the short basket, 0.9
Sshort size, Rs crore
What it says in wordsThe short must be about Rs 144.4 crore, which makes the fund Rs 44.4 crore net short in money while flat to the market.
Beta neutral is not money neutral: the gap is a position of its ownRupees, Rs crore0+100.0Long-144.4Short-44.4Netbeta 1.3 long, beta 0.9 shortBeta-weighted rupees, Rs crore0+130.0Long-130.0Short0Net100 x 1.3 = 130; 144.4 x 0.9 = 130
In rupees Samatol is long Rs 100 crore and short Rs 144.4 crore, net short Rs 44.4 crore; in beta-weighted rupees both sides are Rs 130 crore and the net is zero, so the fund is beta neutral but not money neutral.
Step 2What does the Rs 44 crore net short actually bet on?

On the model, a 10% market rise makes the long book gain 13% of 100, Rs 13 crore, and the short lose 9% of 144.4, also Rs 13 crore: nothing. Now suppose the market rises 10% and every stock rises exactly 10%, as happens when betas collapse towards 1 in a broad rally: the long gains Rs 10 crore, the short loses Rs 14.4 crore, and the fund is down Rs 4.4 crore. The net short is a bet that high-beta stocks keep moving more than low-beta ones. The same happens if the long book's true beta is 1.1, not 1.3: a 10% rally then costs about Rs 2.0 crore.

Market up 10%Long P and LShort P and LFund
Betas hold at 1.3 and 0.9+13.0-13.00.0
Every stock moves 10%+10.0-14.4-4.4
Long beta really 1.1+11.0-13.0-2.0
The beta-neutral book is flat in a 10% rally only if the betas hold; if every stock moves 10%, or the long book's beta is really 1.1, the fund loses Rs 4.4 crore or Rs 2.0 crore, Rs crore throughout.
Step 3What else does the committee need to hear?

Three practical costs. Gross exposure is Rs 244.4 crore, 2.44 times the long book, so margin, stock borrow fees and the dividends owed on the short all scale with it. Betas are estimates from past data and drift, so a beta-neutral book has to be re-hedged as they move. And the alternative, a money-neutral short of Rs 100 crore, leaves Rs 40 crore of beta-weighted exposure: a 10% market rise earns Rs 4 crore that has nothing to do with stock picking. Neither is free of a bet; the manager has to choose which one she is willing to explain.

Where candidates lose it

The usual slip is shorting Rs 100 crore because the long is Rs 100 crore, which leaves Rs 40 crore of beta-weighted market exposure in a fund that calls itself neutral.

The second is stopping at 144.4 and calling the fund hedged. The net short in money is a real position: a bet on the gap between high-beta and low-beta stocks, and on the betas being right.

What the interviewer asks next

  • The short basket's beta rises to 1.0 after a sell-off. How much of the short do you cover?
  • How would you hedge the same long book with index futures instead of a basket, and what changes?
  • Why do low-beta shorts often cost more to hold than their beta suggests?

Company names and figures are illustrative.