Chris Luyckx93 wrote: 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?
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:
- Today you:
- Enter into the short position in the June contract, agreeing to deliver one ounce of gold for $1,488.20.
- Enter into the long position in the December contract, agreeing to purchase one ounce of gold for $1,497.
- Enter into the short position (pay floating, receive fixed) in a 5 × 11 FRA with a notional amount of $1,488.20 and a fixed rate of 5.6%.
- In June you:
- Borrow one ounce of gold to deliver against the June contract. (Note: the arbitrage transaction assumes that you can borrow the gold for free. In practice, you cannot, so the cost of borrowing the gold could wipe out your profit. This is where the 6-month risk-free rate comes into play; this is why I said that it mostly doesn’t matter.)
- Deliver one ounce of gold and receive $1,488.20.
- Receive the payoff on the FRA: {[(1.056^½ − 1.04^½] × $1,488.20} / 1.04^½ = $11.40.
- Invest $1,499.60 (= $1,488.20 + $11.40) at 4% for 6 months.
- In December you:
- Receive $1,529.30 (= $1,499.60 × 1.04^½) from your investment.
- Buy one ounce of gold for $1,497.
- Return the one ounce of gold that you borrowed in June.
- Enjoy a profit of $32.30.
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.