Case 050Forwards, futures and arbitrageWarm up

Saramati Cement trades at Rs 800 and its three-month futures at Rs 820, lot size 500. The rate is 7% continuous and no dividend is due. What is fair value, are the futures rich or cheap, and what does the cash-and-carry lock in per lot?

The situation

Saramati Cement trades at Rs 800 on the exchange. Its three-month single-stock futures trade at Rs 820, with a lot size of 500 shares. No dividend or other corporate action is due before expiry. The arbitrage desk of a broking house can fund itself at 7% a year, continuously compounded, for three months.

A new analyst on the desk is asked to check whether the futures price is in line with the stock, and if not, to set out the trade that captures the difference and what it earns on one lot.

Your task

Compute the fair futures price, say whether the quoted futures are rich or cheap, and lay out the cash-and-carry with its profit per lot before costs.

Quick check

Fair value of the three-month future is about:

Worked solution

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

30-second answerThe answer to give first

Fair value is Rs 814.12, so the futures at Rs 820 are Rs 5.88 rich, and the cash-and-carry locks in Rs 2,938 a lot before costs. Borrow Rs 4,00,000, buy 500 shares, sell one lot of futures at Rs 820; at expiry deliver the shares, receive Rs 4,10,000 and repay Rs 4,07,062. The result does not depend on where Saramati ends, but the edge disappears if the desk's funding costs more than about 9.9%.

Step 1What should the future be worth?

A future is an agreement to buy the stock later at a price fixed now. Someone who wants Saramati in three months can either buy the future or buy the stock today with borrowed money and hold it. Think of buying a scooter on a booking that delivers in three months against buying it today on a loan: the two should cost the same, once the loan interest is counted. So fair value is spot plus the cost of carrying it, Rs 800 times e to the 7% times a quarter of a year, Rs 814.12. No dividend is due, so there is no income to subtract.

The relationship
F∗=S erT=800×e0.07×0.25=814.12F^{*} = S\,e^{rT} = 800 \times e^{0.07 \times 0.25} = 814.12
Sspot, Rs 800
rfunding rate, 7% continuous
Ttime to expiry, a quarter of a year
F^{*}fair futures price
What it says in wordsThe future should cost the stock today plus the interest on the money tied up in it until expiry.
Step 2Which way does the trade run, and what does it make?

The future at Rs 820 is Rs 5.88 above fair value, so sell what is expensive and buy what is cheap. Borrow Rs 4,00,000 at 7%, buy 500 Saramati shares at Rs 800, and sell one lot of futures at Rs 820; at expiry deliver the shares against the futures for Rs 4,10,000 and repay the loan with interest, Rs 4,07,062, leaving Rs 2,938, Rs 5.88 a share. If Saramati ends at Rs 700 or Rs 900 the answer is the same: the shares are delivered at the Rs 820 fixed today, and the loan is fixed too. This is the cash-and-carryBuying the asset with borrowed money and selling the future against it, to earn the gap between the futures price and fair value., and the gap between the futures price and spot, Rs 20 here, is called the basis.

Cash-and-carry on one lot: locked in today, collected at expiryTodayExpiry, 3 monthscarry at 7%: Rs 800 grows to Rs 814.12Borrow at 7%+4,00,000Buy 500 shares at 800-4,00,000Sell 1 lot of futures at 8200, margin asideDeliver shares at 820+4,10,000Repay loan with interest-4,07,062Profit before costs+2,938Fair value 800 x e^(0.07 x 0.25) = 814.12; the futures at 820 are 5.88 rich.The trade earns exactly the gap, 5.88 x 500 = Rs 2,938, whatever Saramati does by expiry.It only pays if the desk can fund below about 9.9%, the rate the futures price implies.
Borrowing Rs 4,00,000 to buy 500 Saramati shares and selling one lot of futures at Rs 820 locks in Rs 2,938 at expiry, the Rs 5.88 a share by which the futures exceed the fair value of Rs 814.12, whatever the stock does.
StepWhenCash, one lot, Rs
Borrow at 7%Today+4,00,000
Buy 500 shares at 800Today-4,00,000
Sell one lot of futures at 820Today0, margin aside
Deliver shares against the futuresExpiry+4,10,000
Repay loan with interestExpiry-4,07,062
Profit before costsExpiry+2,938
The cash-and-carry leg by leg; nothing is paid today on balance, and the profit at expiry equals the futures' excess over fair value times the lot size.
Step 3What could stop the trade paying?

Three things, in order of size. Costs first: brokerage, exchange charges and taxes on buying and delivering the stock come straight out of Rs 5.88 a share, and at an assumed Rs 2 a share round trip the profit falls to about Rs 1,938 a lot. Second, funding: the futures price implies a carry rate of 9.88%, so a desk that funds at 7% earns the gap, while one that funds at 10% would lose on the same trade. Third, margin: the short futures position needs margin, and if Saramati rallies the desk must post more cash before expiry even though the trade is locked in; that cash also has to be funded. A desk can also close early if the basis narrows before expiry, selling the shares and buying back the future at a profit, which is how most of these trades actually end.

Where candidates lose it

The common loss is calling Rs 20 the profit, futures less spot. Rs 14.12 of that is the fair cost of carrying the stock for three months, so only Rs 5.88 is mispricing.

The second is using a full year of interest or discounting instead of compounding, which gives a wrong fair value and sometimes the wrong direction. Carry the spot forward for the life of the future, then compare; an interviewer listens for the quarter of a year.

What the interviewer asks next

  • The futures trade at Rs 808 instead. What trade would you put on, and what does it need that this one does not?
  • Saramati announces a Rs 10 dividend payable before expiry. What is fair value now?
  • Why does the basis shrink to zero at expiry, and what does that do to a desk that closes the trade early?
  • How do margin calls on the short futures affect the economics of the trade if the stock rallies 15%?

Company names and figures are illustrative.