Case 035Forwards, futures and arbitrageCore
Kolahoi Pharma is at Rs 1,500 and pays a Rs 30 dividend in two months. The six-month future trades at Rs 1,510 with rates at 6.5% continuous. What is fair value, which way does the arbitrage run, and what does the trade need that is not always there?
1The situation
Kolahoi Pharma trades at Rs 1,500. The board has declared a dividend of Rs 30 a share, payable in two months. The six-month single-stock future is quoted at Rs 1,510, and the six-month rate is 6.5% continuously compounded. The lot size is 250 shares.
An arbitrage desk is asked whether the future is rich or cheap, how it would trade the difference, and what could stop it.
2Your task
Compute the fair futures price net of the dividend, say which side of the trade is cheap, set out the arbitrage leg by leg, and name the thing it needs that is not always available.
Quick check
Ignoring the dividend, the fair six-month future on a Rs 1,500 stock at 6.5% is about Rs 1,550. Including a Rs 30 dividend in two months, fair value moves to roughly:
Worked solution
Try it on paper, then open one step at a time.
30-second answerThe answer to give first
Fair value is about Rs 1518.89, so the future at Rs 1,510 is 8.89 a share cheap and the arbitrage is to sell the stock and buy the future. Spot less the present value of the dividend, Rs 1470.32, carried six months at 6.5% is Rs 1518.89. The trade needs the stock to be borrowed for six months, and the borrow fee and the risk of recall are what usually decide whether the Rs 8.89 is real.
Step 1What does a dividend do to a futures price?
Think of buying a flat that has a tenant paying rent for two of the next six months. If you agree today to take delivery in six months, the seller keeps that rent, so you pay less for the forward delivery than for the flat today plus the interest you save. A dividend is the rent: the stockholder gets Rs 30 in two months and the futures buyer does not, so fair value is spot less the present value of the dividend, all carried at the rate to expiry. The present value of Rs 30 in two months at 6.5% is Rs 29.68, which takes spot to Rs 1470.32; six months of carryThe interest cost of holding the stock until the future expires, less any income the stock pays in the meantime. at 6.5% adds Rs 48.57, so fair value is Rs 1518.89. Done the other way, Rs 1500 carried is Rs 1549.55, less the dividend grown four months to Rs 30.66, which is the same Rs 1518.89.
| S | spot, Rs 1,500 |
| D | dividend, Rs 30, paid at time t_d, two months |
| r | 6.5% continuous |
| T | six months to expiry |
Step 2Which way does the trade run, and what does each leg do?
The future is cheap, so buy it, and sell the thing that is expensive by comparison: the stock. The desk does not own Kolahoi, so it borrows the shares. Borrow 250 shares, sell them at Rs 1,500, invest the proceeds at 6.5%, buy one lot of futures at Rs 1,510; in two months pay the lender the Rs 30 dividend out of the invested cash; at expiry take delivery through the future at Rs 1,510 and return the shares. Per share the invested Rs 1,500 grows to Rs 1549.55, the dividend paid away costs Rs 30.66 by expiry, and the stock costs Rs 1,510 to buy back, leaving Rs 8.89, exactly the gap, Rs 2,223 a lot before costs. Had the future been rich, the trade would run the familiar way: buy stock, sell future, and no borrow needed.
| Step | When | Cash per share |
|---|---|---|
| Borrow the shares and sell them | Today | +1,500 |
| Buy the six-month future | Today | 0, margin aside |
| Pay the lender the dividend | Month 2 | -30, worth -30.66 at expiry |
| Invested proceeds at 6.5% | Month 6 | 1549.55 |
| Take delivery via the future, return the shares | Month 6 | -1,510 |
| Locked-in profit | Month 6 | +8.89, before borrow fee and costs |
Step 3What does the trade need that is not always there?
A lender. Selling stock you do not own means borrowing it for the full six months, through a securities lending facility, and that is the leg that can fail. The borrow fee comes straight out of the Rs 8.89: at 1% a year on Rs 1,500 for half a year it costs about Rs 7.5 a share, nearly all of the edge, and a lender who recalls the shares early forces the desk to buy them back at whatever the price is then. The Rs 30 dividend is also an estimate until it is paid; a bigger dividend makes the trade better for the short seller, a cancelled one worse. State the rest of the limit: margin on the future and on the borrow tie up cash for six months, and the Rs 8.89 is before brokerage and taxes on the sale. The future often looks cheap on paper precisely because the stock is hard to borrow, which is the market telling you the price of the missing leg.
Where candidates lose it
The common loss is subtracting the full Rs 30 from a no-dividend fair value, or forgetting the dividend altogether and calling the future Rs 40 cheap. Fair value nets out the dividend at its present value, then carries; the difference between Rs 30 and Rs 30.66 is small but an interviewer will hear it.
The second is setting up the arbitrage as buy stock, sell future by reflex. That is the trade for a rich future. Here the future is cheap, the stock must be sold, and the candidate who does not then say the word borrow has not seen where the trade actually breaks.
What the interviewer asks next
- The borrow fee on Kolahoi is 2% a year. Is there still a trade?
- The dividend is raised to Rs 45 after the trade is on. Who gains?
- How does the answer change if the future is on the Satpura 50 index rather than a single stock?
- Why might a stock that is hard to borrow show a persistently cheap future, and what does that tell you about the quoted price?
Company names and figures are illustrative.