In the Hull book, i saw the following exercice and its answer :

Suppose the one-year gold lease rate is 1,5% and the one-year risk-free rate is 5%. Both rates are compounded annually. Calculate the maximum one-year forward price Goldman Sachs should quote for gold when the spot price is 1200€.

Goldman Sachs can borrow 1 ounce of gold and sell it for 1200€. It invests the 1200€ at 5% so that it becomes 1260€ at the end of the year. It must pay the lease rate of 1,5% on 1200€. This is 18€ and leaves it with 1242€. It follows that if it agrees to buy the gold for less than 1242€ in one year it will make a profit.

In the point of view of the gold company, wouldn't it better to just sell the gold now, invests at 5% and get 1260€ instead of receiving 1242€ ? More generally, what kind of "arbitrage" the gold company could do ?


Arbitrage paths are commutative diagrams, and they usually start with no assets and end with no assets. Just like in the Goldman Sachs example.

If you insist on doing the paths for the gold company with a stock of 1: They would sell gold now for 1200, invest at 5%, get 1260 back, but to be commutative, they would need to buy a gold forward now, obviously for no more than 1260, so they can bid 1260 on the forward contract.

The "arbitrage" (it's really just a quality spread) is 1.5% because they don't have to borrow the gold.

  • $\begingroup$ what do you mean by "they can bid 1260 on the forward contract." ? That the max they can buy gold with a forward is 1260 ? $\endgroup$ – TmSmth Apr 26 '19 at 12:38
  • $\begingroup$ Yes, just like Goldman can pay at most 1242. $\endgroup$ – hroptatyr Apr 26 '19 at 13:01

the gold company could sell its gold stock 1200€ per ounce. But it could happen that gold prices rise by 10% and hence the gold company would make a (relative) loss. Instead of earning 10% on the gold it only earned 5% percent risk-free.

The idea of a forward is not (necessarily) to make profits, but to hedge against future movements in gold prices. If you know that you want to buy gold next year but not today (for whatever reason) there is the risk that gold prices might rise within this year. For example it could be that gold sells for 2000€ per ounce next year.

If you want to protect against that risk you can enter a forward contract and fix the price of the future transaction today: you agree to buy gold in one year for (e.g.) 1242€ per ounce.

From the perspective of the gold company it is the other way around. They are facing the risk of falling gold prices. Of course they can sell their whole stock now. But what about the gold stock they will mine within the next year?

So, in order not to be left with worthless gold after one year they can enter into a forward contract today and fix the price of one ounce of gold.

For either perspectives, the idea of a forward is to fix the price for future transaction, i.e. transaction which you cannot or do not want to do today.

  • $\begingroup$ But even if the price rises, the gold company will have to sell at 1242€, so it will also be a (bigger) relative lost. As Goldman Sachs won't make profit above 1242€ and the gold company can have more by selling now and invest, I don't understand the point to enter in this precise forward for the gold company. $\endgroup$ – TmSmth Apr 25 '19 at 15:15
  • $\begingroup$ I updated my answer $\endgroup$ – Cettt Apr 25 '19 at 15:47
  • $\begingroup$ @TmSmth you're addressing too many concerns at the same time. Hull is concerned with the bid side of a forward contract. You're involving a ficticious gold company, who says they have access to the 5%? There is something called a quality spread. $\endgroup$ – hroptatyr Apr 26 '19 at 7:14

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