Cash & Carry Model

[RiskRat]

Reference AIM: Calculate an arbitrage payoff and describe how arbitrage opportunities are ephemeral.




1. Cash and carry: Short forward +Borrow cash to Buy spot commodity


2. Reverse cash and carry: Long forward +Short spot commodity and Lend/invest cash proceeds


For the point number 1 above i.e for Cash and Carry Model I understand the following parts:
i. Short forward
ii. Buy Spot

Above two are true by the principle of Buy Cheap and Sell costly .
From where the point number three i.e iii. Borrow Cash is deduced?

Similarly in Reverse cash and carry how the "Lend/invest cash proceeds" is derived" from the equation ?

 

[David Harper CFA FRM]

Hi @RiskRat

Good question! The cash is harder to infer directly from the carry formula because it's really "in" the risk free rate, r. There are a few ways to approach the thinking here (btw, Kolb goes into great detail on the arbitrage dynamics implied by the carry model in his http://www.amazon.com/Futures-Options-Swaps-Robert-Kolb/dp/1405150491), but to me this is the simplest:

First, realize this arbitrage is costless: there is no initial outlay. The relationship is between:

• observed forward price, F(0) and
• cost of carrying the commodity asset, S(0)*exp(rT), or in annual compounding terms, S(0)*(1+r)^T

For example, say the spot price of gold is $1,000 while the riskfree rate is 6.0%. For a one year time horizon, the choice is between two positions:

• Forward, F(0), versus
• Commodity, S(0)*(1+r), which in the case equals $1,000*1.06 = $1,060; ie., the carried commodity price is $1,060

If the forward price, F(0), is $1,100 (ie, a flat forward curve), then because 1,100 < 1,060, the forward is "expensive" and the carried commodity is "cheap" (relative to each other). Our arbitrage is:

• Short the futures contract because we sell the expensive thing
• Buy gold the commodity because it's cheap
• But our arbitrage is costless, so we need to borrow cash to pay for the commodity on the spot market, so we need to borrow exactly $1,000. We need to "go short" cash in order to fund the commodity, which implies a net outlay of zero.

Then, in one year:

• Recover our gold from storage, deliver the gold against the futures contract and collect $1,100 per our short forward position
• Repay the loan with 1000*1.06 = $1,060.
• Profit $40 per the cash and carry.

So, the cash enters simply because we need to borrow in order to fund the purchase and ensure our outlay is zero

If the F(0) is instead only $1,000, then the futures contract is cheap so initially we go long (buy the cheap thing). But that's a zero outlay. We also short the commodity but that creates an inflow of $1,000. What do we do with the cash inflow in order to ensure net zero outlay? In this case, we must lend the cash (i.e., go long cash) in order to ensure net zero inflow/outflow at time zero. So, you can see, the way that I look at it, the decision with cash is just a function of the decision with the commodity. I hope that's helpful!

 

[RiskRat]

Thank You So much @David Harper CFA FRM