R20.P1.T3.FIN_PRODS_McDonald_Ch6_Topic: Cash_Flows_Arbitrage Transaction in Commodity Forw

[gargi.adhikari]

In Reference to R20.P1.T3.FIN_PRODS_McDonald_Ch6_Topic: Cash_Flows_Arbitrage Transaction in Commodity Forwards:-
Hi,
I having some trouble wrapping my head around the Cash Flow depicted below:-
The Short Forward Payoff is ( K-St)= ( 11- 10.5127) So the money that we spent in buying the Spot is getting accounted for when we subtract St from K and get the effective Payoff= (K- St)
But to get the Net of .6955 = [ ( .48 + 10.5127 ) - 10.305 ] So we are adding back St(10.5127) to the Net. So effectively Net= [ (K- E[S(t) ] ) + E[S(t) ] - S0.e^-(r- lease rate).T ]

So, effectively, we are not accounting for the fact that we paid E[S(t) ] (= S0. e^ growthRate.T) to buy the Spot....?

 

[David Harper CFA FRM]

Hi @gargi.adhikari Kudos for thinking it through in the specific! You are correct that under the cash and carry (i.e., sell the trading-expensive forward and buy the trading-cheap commodity) in the future we do not buy the commodity (because we already did today). Below I copied this scenario below but with three different future (commodity) spot prices to show that--just as your formula shows--the future spot commodity price does not matter (this is an arbitrage: it must guarantee a profit regardless of the future spot price!). In the notes, I followed McDonald by illustrating a future spot price that just happens to be realized as the expected future spot price; but it cancels out in the future cash flow. (my xls is here @ https://www.dropbox.com/s/2pebxfzra6takic/0409-cash-and-carry.xlsx?dl=0) Because here is how I would describe the process:

• The initial (time zero) position is sort of simple: we have nothing and we spend nothing. Sweet ... and sort of zen. How so? First, we borrow everything we need to buy the (spot) commodity; hence the cash inflow (borrowing) to fund the purchase of the commodity. Second, simultaneously, we enter the short forward contract (with zero cost).
• Now fast forward to the future. What is the situation? Our position has three parts: 1. We continue to own the commodity which we have carried, 2. We have an outstanding loan to repay (because we bought the commodity to carry it), and 3. We have an open futures contract.
  • Say the future spot price is $9.00 (instead of $10.51271). We can deliver the commodity that we already own to fulfill the futures contract obligation; in exchange, we receive the $11.00 as promised. We use the $11.00 to repay the loan balance of $10.305 due, for a net profit of $0.6955. But that's not exactly what is illustrated, I suppose. What's shown is: We close out the futures contract for a profit of +$2.00; we continue to own an asset worth $9.00 and have a loan balance due. Our "balance sheet position" is the same +$0.6955, and we can sell the asset and use the ($2 + 9) to retire our debt, to the same conclusion! The key, shown in your formula, is the future commodity price is subtraction with respect to our forward contract but an addition with respect to our ownership (because we bought it in the cash and carry), so it cancels itself out. And we really have the forward price (as "revenue") minus the carried spot price, S(0)*exp(r-q), as the factors in our cash flow: but they are known at time zero.
  • If the future spot price is $13.00, we lose on the short forward: either if we close out for a -$2.00 loss; or if we deliver the spot. But it doesn't mater. If we close-out, then we have an expensive commodity we can sell for $13.00; if we delivered, we receive the full $11.00 that we were promised from the forward contract counterparty. I hope the clarifies this "missing" future spot!

 

[gargi.adhikari]

@David Harper CFA FRM Thanks a trillion Quantrillion (if there was such a thing ! I see the point I was overlooking ! Thank you ! Any discussion with you is such an AHA moment !

 

[gargi.adhikari]

@David Harper CFA FRM Hi I have a follow up question about the Cash Flows above :-

1) Invest/ lend at the RF Rate and receive $ 10.305
2) Use the cash earned to Buy the Long Forward -> Pay the Forward Input Price : - $ 11 and Receive Spot/Commodity worth E(St) : + $ 10.513
3)Short the Spot/ Commodity at E(St) = $10.513 -> Give away the Spot?Commodity/Asset worth $10.513=> So, - $10.513. But we also get paid the + $ 10.513 for selling the Commodity. So why don't we add that back...?
So Net position = ( - 11 + 10.513) [From the Long forward position] + (10.305) [ the Cash from the Investment] + ( - 10.513 + 10.513 ) [ From the Short Spot/Commodity]
instead of ( - 11 + 10.513) [From the Long forward position] + (10.305) [ the Cash from the Investment] + ( - 10.513 ) [ From the Short Spot/Commodity]

For Financial Assets, a Long and a Short position can cancel each other out completely if the premiums paid for both are the same and the they have the same strike price etc. But we do factor in the premiums paid. In this case, we are factoring the price paid for the Long position ( $ 11- the pre determined forward price-input) . So why are we not factoring in the amount received for Selling the Short...? :-( :-(

 

[David Harper CFA FRM]

Hi @gargi.adhikari But there is no premium paid/received for the futures contract: the initial cash flow associated with the futures contract is always zero; then at maturity, when the asset has a price of S(T), the cash flow associated with the futures contract is simply the difference between S(T) and the strike (ie, assumes cash settlement). I just quickly added a sheet for the reverse cash and carry to the same workbook ( https://www.dropbox.com/s/mee5g9zy71r1yg9/0503-reverse-cash-and-carry.xlsx?dl=0 ). Under this scenario the futures contract is initially under-priced (aka, trading cheap) which warrants the reverse cash and carry. Please note:

• As before, the initial net cash flow is zero because in the reverse C&C, we lend the same cash that we receive from shorting the commodity; in this case, we collect +9.90 from shorting the commodity and immediately lend it at the risk free rate. That's all that happens cash-wise! The futures is just a contract we enter into (I notice the above note has a typo: lending at risk-free rate should be -9.90 as it is an outflow. Like I have it below. FYI, we yesterday published the new Hull study notes and I can see that we did correct this in the new version).
• Then at maturity, we receive back the cash we lent (+10.3105) and most/all of that cash is used to purchase the commodity that we earlier shorted. Additionally, but separately, the forward pays or loses depending. I hope that clarifies!

 

[gargi.adhikari]

@David Harper CFA FRM Got it ! Thanks much much much