Case 015Portfolio constructionWarm up

A fund holds Rs 50 crore of stocks with a portfolio beta of 1.3 and wants to be market neutral with index futures. How much notional should it short, and what risk remains?

The situation

Aranyam Fund Managers runs a Rs 50 crore stock-picking book. Measured against the broad index, the book's beta is 1.3, with a standard error on that estimate of about 0.15. The book's total volatility is 22% a year and the index's is 15%.

The PM believes in the stock selection but has no view on the market, and wants to hedge the market out with index futures. For the arithmetic, assume one futures contract carries about Rs 15 lakh of notional; the exchange sets the actual lot size, so confirm the current figure before trading.

Your task

How much index futures notional should the fund short, roughly how many contracts, and what risk is left after the hedge?

Quick check

How much index futures notional makes the book market neutral?

Worked solution

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

30-second answerThe answer to give first

Short about Rs 65 crore of index futures, roughly 433 contracts, because the book behaves like 1.3 times its size in the index. In a 10% fall the stocks lose about Rs 6.5 crore and the futures gain Rs 6.5 crore. What remains is stock-specific risk, about 10.2% a year or Rs 5.1 crore, plus the error in beta itself: a 0.15 miss leaves Rs 7.5 crore of market exposure.

Step 1Why is the hedge Rs 65 crore and not Rs 50 crore?

Because the futures must offset how much the book moves, not what it costs. A betaThe sensitivity of a portfolio to the market: the percentage move in the portfolio expected for a 1% move in the index. of 1.3 means a 1% index fall takes about 1.3% off the book. Rs 50 crore moving 1.3 times as much as the index is the same market exposure as Rs 65 crore of the index itself, so Rs 65 crore of short futures is the neutral hedge. Two people on a seesaw balance by weight times distance, not by weight alone; beta is the distance from the pivot.

The relationship
Nfutures=β×V=1.3×50=65 crorecontracts≈650.15≈433N_{\text{futures}} = \beta \times V = 1.3 \times 50 = 65 \text{ crore} \qquad \text{contracts} \approx \frac{65}{0.15} \approx 433
\betathe book's beta against the index, 1.3
Vvalue of the stock book, Rs 50 crore
0.15assumed notional per futures contract, Rs crore; confirm the current lot size
What it says in wordsThe futures notional to short is beta times the value of the book, and the contract count is that notional divided by one contract's notional.
Balance the beta-weighted exposure, not the rupeesLong stocks Rs 50 crorex beta 1.3 = Rs 65 croreShort futures Rs 65 crorex beta 1.0 = Rs 65 croreAbout 433 contracts at an assumed Rs 15 lakh notional eachIf the market falls 10%:Stocks, beta 1.3-6.5Futures, beta-weighted Rs 65 crore+6.5Rupee hedge instead: net-1.5
Rs 50 crore of stocks at beta 1.3 balances Rs 65 crore of short index futures at beta 1, so a 10% market fall costs the stocks about Rs 6.5 crore and pays the futures Rs 6.5 crore, while a rupee-for-rupee Rs 50 crore hedge would leave Rs 1.5 crore of loss.
Step 2What risk is left once the market is hedged?

Split the book's variance. With 22% total volatility and an index at 15%, the market part is 1.3 times 15%, which is 19.5%, and the rest is the stocks' own risk. Hedging removes the 19.5% and leaves idiosyncratic riskRisk specific to the individual holdings, such as a company missing its earnings, that does not move with the market. of about 10.2% a year, roughly Rs 5.1 crore of one-standard-deviation P&L on a Rs 50 crore book. That is the risk the PM wants: it is where the stock selection either pays or does not. Hedged, the book's volatility falls from about Rs 11 crore to about Rs 5.1 crore.

Rs crore a year, one standard deviationBefore hedgeAfter beta hedge
Market risk, 1.3 x 15% on Rs 50 crore9.750.00
Stock-specific risk5.095.09
Market risk from a 0.15 beta error1.12
Total volatility11.005.22
The beta hedge removes Rs 9.75 crore of market risk and leaves Rs 5.09 crore of stock-specific risk; a one-standard-error miss in beta adds back only about Rs 1.12 crore, so total risk falls from Rs 11 crore to about Rs 5.2 crore.
Step 3What can make the hedge drift away from neutral?

Four things, each worth a sentence in the interview. Beta is an estimate: a standard error of 0.15 means the true exposure could be Rs 7.5 crore either side of zero after the hedge. Beta also changes as the book trades and as markets move, so the hedge needs re-sizing. Futures carry basis riskThe risk that the futures price and the index, or the hedged portfolio, do not move exactly together over the life of the hedge., and they must be rolled at expiry. Finally, the futures need margin, and a sharp rally can demand cash at the very moment the stocks have not yet paid for it.

Close with the sector point. A book can be beta neutral and still carry a large bet on banks or on small caps, which the broad index hedge does not touch. If the PM truly wants only stock selection, a sector or factor hedge on top is the next conversation.

Where candidates lose it

The common loss is hedging rupee for rupee, shorting Rs 50 crore of futures. That leaves 0.3 of beta, Rs 15 crore of market exposure, and in a 10% fall the book still loses about Rs 1.5 crore.

The second is saying the hedged book is riskless. Hedging the market leaves every rupee of stock-specific risk, which here is about half the original volatility, and that is exactly the risk the PM is paid to take.

What the interviewer asks next

  • The index rallies 8% over a month and the hedge loses money. How do you explain that to the PM?
  • How would you estimate beta, and over what window?
  • The book is also heavy in banks. How would you hedge that sector exposure?

Company names and figures are illustrative.