# Cash-and-carry arbitrage

Hi, does anybody understand, why the profit from a cash-and-carry arbitrage is not just the difference between the Forward price quoted and the Fair Forward price calculated ? For example, in the 2008 exam, question 6, I would say that the profit is 315.12 - 313 = 2.21. But the correct answer is 2.19 (the difference is not due to approximation). Thanks in advance, Bern

Do the CFAI Reading 38 EOC. Three problems, well explained (except for tailed gold rationale). Covers all you need to know on this cash and carry topic. Shouldn’t take you more than 30 min.

I DID the EOC exercices. Still don’t see why the profit cannot be calculated directly from Fair Forward price - Quoted Price. Moreover, the table 8 page 182 shows that at time T, the CF (benefit) is the Fair Forward price - Quoted price.

313+4.78-319.97=2.19 313 = F(0,T) 4.78 = S0(e^delta*t-1)=316*(e^.18/12-1)=316*(0.01511)=4.77573~4.78 319.97=316*(e^rt)=316*e^(.05*3/12)=316*1.01258=319.9748

Actually I think you are correct. !!! The solution in 2008 is probably wrong !!! the difference comes from the fact that in your correct calculation you implicitly assume that you sell at spot lower amount of copper, whereas they in 2008 assume you sell the same amount of copper. this is the difference, I do not have enough time at the moment to go into more details, check page 181 of v5 section Forward prices and the Lease rate The Table 7 and 8 clearly say Buy/Sell lower amount at spot (PV using the lease rate if I may to use this terminology). This is how curriculum and exams evolve, I suspect…

Ahhhhhhh It’s good read you ! Thanks pfcfaataf ! You are the sunshine of my day (-7 by the way…)

Is there a difference between ‘cash and carry arbitrage’ and ‘reverse cash and carry arbitrage’? If so, what is it?

Yes, I the first case, you are long the spot, short the forward (because the quoted forward is to high). In the second it is the opposit.

pfcfaaraf and bern : why is this calculation wrong? It gives the 2.19 answer 313+4.78-319.97=2.19 313 = F(0,T) – Short Forward cash flow+ve at horizon 4.78 = S0(e^delta*t-1)=316*(e^.18/12-1)=316*(0.01511)=4.77573~4.78 – Buy spot +ve 319.97=316*(e^rt)=316*e^(.05*3/12)=316*1.01258=319.9748 — Borrow -ve

The calculation is not really “wrong”. But according to the material, you could earn a profit by being short a smaller amount at spot (look at table 7 and 8). Then the profit would have been the difference between the Fair Forward price and the Quoted Forward price => 2.21

7 days before the test and people are still saying CFAI is “wrong”…

saying the 2008 solution is wrong was just a shortcut, 2008 solution is not consistent with what 2011 curriculum teaches. the problem itself is a little bit more complicated and it is very dependent on the definition of all inputs/variables…

Yes you are correct bern and pfcfaataf. An integrated rate ( rf - lease rate ) is offered in the book, while 2008 exam has two separate transactions short the loan at rf , and long the commodity lease at delta . The difference is two cents , but if you can do it over a million copper contracts , the two cents could become real money. 2011 book: You’d have to line up a swap party that can give you a lease rate and finance your purchase at the same time . You lose two cents less to this party. 2008 exam: You can get by with two separate deals, one a short loan from a bank and another who leases the copper off you and gives you a lease rate. There is no integrated rate to one party. You lose two cents more on the net in the two deals

No error here. As I mentioned in a previous thread http://www.analystforum.com/phorums/read.php?13,1247401,1247437#msg-1247437 Don’t be confused by initial cash layout in cash and carry or reverse cash and carry situations. For the exam: Just focus on the resulting NET cash flow at the end. Table 7 and 8 and EOC 3 are consistent with the 2008 exam: Payoff = (futures - sum (FV(storage))) + sum (FV(lease)))- FV(spot) ) I wrote storage explicitly here and separate from (positive) lease for the sake of clarity. You can see storage as negative lease, thus not need to remember the sum (FV(storage))) component. Concerning your statement about buying less copper on spot, it is a misunderstanding from your part since you have not taken into account the PV of lease rate/storage which I don’t want to go into details here since it is on a few days to exam and I assume you guys don’t want lengthy theoretical explanation.

elcfa Wrote: ------------------------------------------------------- > No error here. > > As I mentioned in a previous thread > http://www.analystforum.com/phorums/read.php?13,12 > 47401,1247437#msg-1247437 > > Don’t be confused by initial cash layout in cash > and carry or reverse cash and carry situations. > > For the exam: Just focus on the resulting NET cash > flow at the end. > > Table 7 and 8 and EOC 3 are consistent with the > 2008 exam: > > Payoff = (futures - sum (FV(storage))) + sum > (FV(lease)))- FV(spot) ) > > I wrote storage explicitly here and separate from > (positive) lease for the sake of clarity. You can > see storage as negative lease, thus not need to > remember the sum (FV(storage))) component. > > Concerning your statement about buying less copper > on spot, it is a misunderstanding from your part > since you have not taken into account the PV of > lease rate/storage which I don’t want to go into > details here since it is on a few days to exam and > I assume you guys don’t want lengthy theoretical > explanation. i am more confused by you then the book… The question on the test follows the book 100% except for the tailed gold example. Seems fairly intuitive.

Paraguay What part confuses you?

elcfa, if you follow Table 7 , you do not get the answer as given in 2008 exam answers. You get a different answer. Please look at Table 7 again and use the rates given and see: Spot=316 Fwd=313 Rf=5% lease=6% time=0.25 Do the math. Table 7 would give you -2.21 . Exam 2008 gives -2.19. How is it same? And how is one or the other NOT wrong? Under what conditions would the wrong answer flip into the right answer?

janakisri You are right in your comment before concerning whether it is an integrated deal (as the table 7 and 8) or two separate deals. In the EOC’s, they always use the two separate deals: get/pay interest from spot borrowing/deposit AND storage cost/lease rate, thus my formula Payoff = (futures - sum (FV(storage))) + sum (FV(lease)))- FV(spot) ) Therefore, the 2008 question consistent with all the EOC’s and examples I have come across. In similar vein, you would use linear compound method if the question indicates so and not insisting on using table 7+8 since it uses continuously compound. Hope that it is clearer.

forward price is defined as F = S x e^(r - L) where L is lease rate (continously compounded) r and L are time adjusted (multiplied by t 0.25) the question in 2008 uses different calculation for calculating the arbitrage profit S x (e^r - e^L +1) - F here F is the quoted forward (you can input the numbers and will see the difference) any derivative is priced using some other instruments (no-arbitrage), whatever formula comes from this no-arbitrage condition. But if book uses F = S x e^(r - L) for calculating arbitrage-free forward price, the correct way to calculate the arbitrage profit is using the exactly same formula and this is what table 7 and 8 in 2011 book does. also storage costs are in the book specified typically in absolute value in USD per contract, the reason is simplicity, as this continously compounded rates and their adding or substracting them in exp. is not intuitive. and 2008 specified continously compounded, but I still think that it was differently presented in the 2008 books (table 7 and 8 I mean, if there were any at all) also in EOC gold 1.C is the logic of S x e^(r - L) formula (same as in table 7 and 8) always depends how these rates are specified. I am finished with this topic, good night (i am in europe)

Looks like a difference between add or geometrify . e^(L-F) is a geometric calculation while e^L-e^F is an arithmetic one