It is January now. The current interest rate is 5.6%. The June futures price for gold is $1,488.20, while the December futures price is $1,497. Is there an arbitrage opportunity? How could you exploit if so?
I understand that arbitrage exists as the listed December futures price is $1,497 whereas according to parity it should be 1,488.20(1.056)^.5= $1,529.30.
but what do you do to exploit this?
Buy the cheap one, sell the dear one.
The key is locking in the interest rate between June and December.
So ask yourself: what mechanism is there to ensure a future interest rate?
So in this instance would you then long the December one and short the June one?
I added to my earlier comment. Ponder the addition.
An interest rate future?
Think of the derivatives we specifically study for Level I.
That answer says buy the December contract and sell the June contract. I understand why you would buy the December contract as it is cheap as the December spot is greater than the initial futures price.
But, I don’t understand why you sell the June contract? If it’s expensive what would that be relative to?
Maybe is it reverse your position when closing out?
Relative to the December contract.
The answer, as is common, is incomplete; it’s half an answer.
By selling (i.e., taking the short position in) the June contract and buying (i.e., taking the long position in) the December contract, you have effectively agreed to borrow $1,488.20 for 6 months, beginning in June, at a rate of 1.1861% (= $1,497 / $1,488.20)² − 1). To exploit the arbitrage opportunity, you have to agree to lend $1,488.20 for 6 months, beginning in June, at the market rate of 5.6%. Note that the 5.6% rate is the 6-month forward rate starting 5 months from today. (It would have been nice if the question had specified that.)
Therefore, in addition to the two positions in the gold futures contracts, you also need a short position in a 5 × 11 FRA, with a notional amount of $1,488.20, and a fixed rate of 5.6%. That guarantees that you’ll receive 5.6% on $1,488.20, for a net of 4.3889%, or $32.30 in December. (Note that the net isn’t simply 5.6% − 1.1861%; you have to uncompound each one to get the 6-month effective rate, then subtract, then compound to get the annual rate. Sigh.)
To have a specific example, let’s assume that in June the 6-month (spot) risk-free rate is 4%. (It turns out that it (mostly) doesn’t matter what it is, but having a specific number will make things clearer.) Here’s what happens:
You should work out the transactions if the 6-month spot rate in June is higher than 5.6%: say, 6%. The profit is the same either way: it doesn’t depend on the (floating) spot rate in June.
As I say, the rub here is that you cannot borrow that ounce of gold for free; the cost will reduce your profit and could, in fact, wipe it out.
In short, arbitrage ain’t easy.
That took a while to understand, but it makes sense now. Thank YOU SO MUCH. Clearly arbitrage is quite a process
I’m glad that it finally penetrated, Chris.
Some arbitrage transactions are simple; others are complicated. For example, the arbitrage transactions needed to set the correct price (i.e., fixed rate) for an FRA are (potentially) even more complicated than what we did here.
As I say, I find it irritating that the “standard” answer to questions like these is, at best, half an answer, and that they don’t explain what’s really going on.
Your answer was far more comprehensive which allowed me to understand both sides of the transaction. As you said, I think the answer they gave in the book was too simplistic.