Lease rate for commodities

[wrongsaidfred]

Hi David,

I have a couple of questions about the lease rate and cost of carry.

The first is: the lease rate seems to be defined in two very seperate ways. One is growth rate minus discount rate (lease markets) and the ther is convenience yield minus storage costs (carry markets). Maybe I missed some of the finer points in the reading, but is this accurate? Carry markets obviously exist in the real world. Do lease markets or is this just a theoretical example to set up the cost of carry model? If so, what types of commodities are in each? Are any of these really observable besides storage costs?

Next, the terminology seems to be a little mixed up between Hull and McDonald. Hull considers carry to be everything except the convenience yield (r-q+u, r-r(foreign), etc). From your video and the reading, it seems like carry just refers to storage. Is this accurate? Is there one set definition for carry that we should go by?

Thanks,
Mike

 

[David Harper CFA FRM]

Hi Mike,

I don't mean to flatter you, but you have a knack for identifying the sore spots in the curriculum (of course, careful preparation helps!)

1. You are correct that, due to one of McDonald's definition of lease rate [not in Hull], there are two definitions. However, they are not inconsistent. The first tab of this XLS (http://www.bionicturtle.com/how-to/spreadsheet/3.d.1-arbitrage-mcdonald/ ) illustrates their compatibility. The reason is that the growth rate (g) is not used in the pricing of the forward, F0, which uses the riskfree rate; rather, the growth rate informs the expected future spot, E(St). To me, the easiest analogy is to a stock, if you think about the total return on a stock, we have something like:

Total return (TSR) = capital appreciation + dividend yield; analogously,
Discount rate (alpha) = growth (g) + lease rate (delta); i.e., McDonald's [lease = discount - expected growth]

For the exam, all that really matters is that the lease rate acts like a dividend, but is NOT directly observed and therefore is equivalent to Hull's convenience yield. (McDonald illustrates "lease markets" and there assumes, in the cost of carry, that it is the only factor ... but that is just the application of the carry model assuming only lease, zero storage, zero convenience since lease is used instead). In this way, although the FRM to my knowledge has not precisely clarified the Hulll/McDonald slight theoretical differences, the thing to do is: treat lease just like we treat convenience, which is to say as the intangible (not directly observed) benefit to ownership, so that a "universal" cost of carry gives:

F(0) = S(0)* exp(Rf + storage - dividend - [convenience | lease] )
... "universal" in the sense no commodity will use all five factors!

but, consumption commodities tend to have storage cost, maybe convenience but no dividend; while investment commodities have no storage, no convenience but maybe dividend.

"Carry markets" connote storage, but "cost of carry" employs the above for investment too.
Gold clearly has a lease rate (I marked this dense piece from early in the week, http://ftalphaville.ft.com/blog/2011/09/14/677021/why-gold-forward-rate-inversion-is-important/ ). And silver. Beyond that, I'm not knowledgeable sorry ... so apparently, at least with respect to gold, the lease rate is very real
... But they are not, unlike dividend yield, directly observable. So, what do we know from the carry model? That if lease is the only factor, we can infer the lease rate exactly from the spot and forward. However, if there are other inexact factors, like the convenience yield, it becomes dependent on other variables.

2. Agreed there is a semantic difference between Hull & McDonald with respect to carry. IMO, Hull's just being a bit more theoretical to encompass investments into carry as his (useful!) point is that both can fit under the same/similar model.

Hull distinguishes between consumption vs. investment commodities, such that the key difference is consumption commodities tend to require storage and confer convenience, and due to this difference, bound the forward price ("<=") instead of solve for it ("=').
McDonald, I think more in tune with practice, restricts "carry" to the storable consumption commodities.
... exam-wise, i don't think the subtle difference in "carry" matters. What we care about is: the universal carry model is same for both; but they use different factors

Hope that helps, sorry for length (oops), David

 

[wrongsaidfred]

Hi David,

Thank you for the explanation and absoluetly no appologies necessary for the detail. One of the things I told the man who interviewed me about BT was that this is one of the main reasons I remain active on this sight. Since the writings are just flat out inconsistent in some sesnses, I need someone to explain if I am misinterpreting something or if they are just theoretical and play no real part in problem solving.

So in a lease market, lease rate is equivalent to a convenience yield (both non-observable), but in a carry market the lease rate is the convenience yield minus storage costs. Does this sound accurate?

Forgive me for saying this, but it seems like there is still an inconsistency. I could be reading something incorrectly, but on p 179 of McDonald he actually uses the formula F=So*exp[(r-delta)t] to describe a forward price in a lease market, where delta = commodity discount rate - growth rate. In this sense isn't he using the lease rate to find to the forward price?

Thanks,
Mike

 

[David Harper CFA FRM]

Thanks Mike, I appreciate that, some people've told me i spend too much time in the forum but I'm not sure they realize that the forum conversations have greatly benefited the overall program. It's amazing how even innocent seeming questions spark a better understanding. (I've learned more from the forum, in trying to help with answers, than in any program i've taken)

1. Re: "So in a lease market, lease rate is equivalent to a convenience yield (both non-observable), but in a carry market the lease rate is the convenience yield minus storage costs. Does this sound accurate?"

I agree with "So in a lease market, lease rate is equivalent to a convenience yield (both non-observable)" but the next clause is problematic, I just don't think it's helpful. I think you are getting at the semantic confusion in the sense that, in theory, there can be a convenience component and a convenience component (and, with gold, I would actually make that argument, that the forward price includes both ... but then you have two variables to solve for...). For practical purposes, and at the risk of not exactly reconciling with McDonald, I prefer something like (but to your original point, I perceive this can be debated at the margin):

"in a lease market, lease rate is either equivalent to a convenience yield (both non-observable) or includes a convenience yield [as loaning gold with interest is a convenience!], but in a carry market the lease rate is probably just the convenience yield"
(but see how impossible it is to parse lease vs. convenience strictly? IMO, maybe i've not read the right expert)
... the more i think about it, i tend to prefer lease = convenience (unless persuaded otherwise!)

2. Yes, agreed, but i think i said the growth rate (g) is not used in the forward price? McDonald's 6.10 works greatly because lease rate is acting just like a convenience yield; ie, subtracting from the risk free rate. My point--which i could only grasp myself after i worked it out in the XLS above--is his 6.6: E(St) = S(0)*exp(gT), which employs a growth rate not a risk free rate. And is consistent with Hull's normal backwardation thy that, as g > r, F < E(St)

David

 

[wrongsaidfred]

Hi David,

Correct. You said that growth rate was NOT used in calculating the price of a forward, but in 6.10, the growth rate is a component of delta and delta, along with the risk free rate, is used directly in computing the forward price.

By your statement, did you just mean that delta, which happens to be the discount factor minus the growth rate, is used in pricing the forward but that neither the discount factor nor the growth rate are known or used directly? I know this is getting a bit philosophical, sorry for that.

I appologize for my other line of questions. Now that I looked over it in more detail, nowhere in McDonlad does he actually refer to delta (as defined by convenience yield minus storage) as the lease rate. Unfortunately he does say that the lease rate is the negativ of the storage costs. Makes my head hurt a bit.

My sincere thanks for helping me out with this. These readings are making this whole topic a lot more difficult than it actually has to be. Quite a shame, actually.

Mike

 

[David Harper CFA FRM]

Hi Mike,

McDonald 6.8 gives: delta = discount - growth, where delta = lease.
Lease rate as negative of storage is consistent with lease rate ~ convenience yield.

Re: the philosophical point, it's circular b/c the E(St) can be expressed as a function of F(0).
The best way i can handle it, mentally myself is:
(6.10) F = S*exp(r - lease); i.e., just like dividend
and just as the dividend = TSR - appreciation, so too: lease = discount rate - growth

... i think we start to get into thorny stuff when we treat the lease rate as something other than convenience yield (works for me is: lease ~= convenience)

FYI, I just wrote the following email to GARP:

"Hi Bill & Chris,

I hope you are well. Non-urgently I wanted to share this thread from today http://forum.bionicturtle.com/viewthread/5200/
… only because this arises every year: Hull gives the basic “cost of carry” model, then the McDonald assignment invariably confuses due to introduction of the “lease rate” into the model
(the specific theoretical issue, in my opinion, is: is there a difference between convenience yield and lease rate. My answer is, in theory probably, but any theoretical difference here is way too much trouble for the exam).

It is only a confusion when trying to reconcile Hull & McDonald. Of course, ironically and typically, it is only the careful and diligent candidate who ever notice this.

Thanks,
David Harper, CFA, FRM, CIPM"

 

[David Harper CFA FRM]

Bill just replied that he will "look at this issue next week." It's progress! Gotta love transparency David

 

[wrongsaidfred]

Nice! Please let me know if they give any good info.

There just seem to be so many circular definitions and unmentioned double negatives like charging negative rates. Borrowers and lenders who in the recontext of forward contracts are neither borrowing nor lending.

The more I talk about it the more I feel like I am losing grasp of what is actually happening.

The simple version is: the owner (a farmer, we will say) of a commodity will charge someone who wants to enter a long forward contract instead of buying the commodity now. The more it costs the farmer to store, the more he will charge the person getting long the contract (higher forward price). The more convenient it is for the farmer to hold onto, the less he will charge that same person (lower forward price). You said that lease rate=convenience yield but also lease=-storage. That does not seem plausible in this situation. If you could explain how the term "lease rate" applies in this situation and how it is quantified I will call it a day.

Thanks again,
Mike

 

[David Harper CFA FRM]

Hi Mike,

Your example looks good to me. There may not be a lease market for the farmer's commodity, but let's just assume there is: let's say he can lend his commodity out, while he owns it, awaiting to deliver in the future to his counterparty who is long (in a totally separate transaction). Let's say he can lend at lease rate of delta. In this case, we are avoiding the exact theoretical morass I refer to by assuming lease = convenience. Now, the farmer can either keep the commodity in his barn (convenience which may just be the "optionality" of owning; note: keeping it may entail both storage costs and offsetting convenience yield. Gold is that way) which confer delta convenience; or, he lends it out (not for free!) to be returned to him, plus delta interest, just in time to deliver to the forward. In either case, keep (convenience) or loan (lease), it is a benefit of ownership that offsets the cost of storage. I hope that helps, David

 

[wrongsaidfred]

Hi David,

Since you actually have the people from GARP looking into this I figured I might as well throw one more thing at you that adds even more confusion. The reading goes out of its way to talk about how if there are storage costs that the lender will pay the borrower, so that the "dividend" we get from lending can actually turn out to be more of a tax. This concept throws everything we have talked about upside down. So now convenience yield is negative, because even if you lease it out you have to pay the borrower and this convenience is now a burden. This would put the curve in permanent backwardation, which is not a problem but interesting to take note of

Here is (hopefully) an easy queqstion: you say that the shape of the forward curve depends on whether the lease rate is greater than or less than r. This seems true in the lease market, but in the carry market the price of the forward is defined as S*exp(r+storage-convenience). Does this this mean that in this market, the shape of the curve depends on whether or not (r+storage-convenience) is positive (contango) or negative (backwaration) or do we have to use the formula (6.11 or 6.12) to find the lease rate and then see whether it is greater than or less than r? Or does that formula even apply for carry markets?

At one point the video states that lease rate depends on convenience and storage and in the reading (and our discussion) it says it that convenience sometiemes just means the lease rate.

It is all semantics and that is what sucks about this. If these terms are more clearly defined and what (if any) mathematical relationship they have to each other is clearly expressed it could be a pretty simple topic.

Thanks for listening to me vent.

Mike

 

[FaisalAnjum]

Hi David,

I have a similar problem and wondering if you got any reply from GARP on this.

Thanks,
FA

 

[David Harper CFA FRM]

Hi FA,

No, I did not. And the 2012 AIMs (with McDonald repeating) do not even implicitly address the issue. Sorry, David

 

[FaisalAnjum]

Hi David,

Thanks for the prompt reply. I am preparing for ERP (May' 2012) by GARP . I am wondering if you cover that as well. Also, if you can guide me to the best resources on your site regarding the same.

P.S: I am not able to download any spreadsheets from the website, specifically "Cash & Carry Arbitrage"

Regards,
FA

 

[David Harper CFA FRM]

Hi FA,

We do not have a saleable ERP product yet; we are writing slowly fresh practice quesitons. We are looking for an ERP subject matter expert (SME) to lead it; hard to find. Questions are collecting in the lower paid boards.

Re spreadsheets: I assume you are a paid member? 90% of the time, it's a browser issue due to direct link such that it works if you: right-click and save as rather then try to open directly.

Thanks, David

 

[lRRAngle]

Hi David,

From your pdf readings on McDonald you state (pg. 11) "If you own physical gold directly: you forgo a “lease rate” but you also bear storage costs." Why do you say forgo? Isn't the lease rate considered a benefit of ownership?

Thanks

 

[David Harper CFA FRM]

Hi @lRRAngle Yes, agreed, for sure mathematically the lease rate is a benefit of ownership: its positive value decreases the theoretical forward price. My current interpretation actually is that the lease rate is a special case of the convenience yield (i.e., the convenience yield generalizes the lease rate per my model here at https://forum.bionicturtle.com/thre...rate-in-commodities-futures.10762/#post-54740). You will notice there is a long history of "debate" in the forum w.r.t. lease rate (!)

My use of the term "forgo" is simply to acknowledge (my interpretation of) McDonald's definition of the lease rate as as an explicit borrowing rate paid by the borrower to the lender of the commodity, see quote below; in the case gold, to realize the lease rate the owner would need to actually lend the gold. My interpretation of McDonald is that the lease rate is the explicit benefit of ownership (i.e., realized if the commodity is actually lent at a borrowing rate) as opposed to the implicit benefit (i.e., the convenience yield which remains but where the commodity is not lent). But I do think you can argue this implicit benefit is also a lease rate. But I reconcile by viewing convenience yield as a generalization of the lease rate (my model led me to this conclusion).

So I do not currently like the sentence to which you refer, I agree it is confusing. Because, in any case, clearly the gold futures curve is meant to allow for the lease rate (regardless of lent status). But I also think a previous sentence is confusing, too: "The forward price tends to be a gradually increasing function of maturity; this implies a lease rate."

Not to belabor but I would propose to replace the two paragraphs on page 11 with the edited (revised) copy below. Notice how I propose to change at least two aspects:

Gold is durable with low storage costs. The forward price tends to be a gradually increasing function of maturity. The upward slope reflects non-seasonal, predictable storage and financing cost; this may be tempered with downward pressure due to a lease rate that is typically but not always positive in practice. Exposure to gold can be achieved by ownership or (indirectly) by a long position in gold Futures.

If you own physical gold directly, you bear storage costs. To mitigate the storage costs, you can lend the gold and receive lease payments (called the lease rate). The lease rate is similar to dividends paid (to the owner) by short-seller of a stock who borrows the stock. Although even if you do not lend the gold, you still enjoy the convenience yield.

If instead you have a synthetically long position in gold: you have no storage costs, but you are exposed to credit risk. The text says that synthetic exposure is preferable, assuming you ignore credit (counterparty) risks.

McDonald on lease vs. convenience:

• "Lease rate. The short-seller of an item may have to compensate the owner of the item for lending. In the case of financial assets, short-sellers have to compensate lenders for missed dividends or other payments accruing to the asset. For commodities, a shortseller may have to make a payment, called a lease payment, to the commodity lender. The lease payment typically would not correspond to dividends in the usual sense of the word.
• Convenience yield. The owner of a commodity in a commodity-related business may receive nonmonetary benefits from physical possession of the commodity. Such benefits may be reflected in forward prices and are generically referred to as a convenience yield." -- McDonald, Robert L.. Derivatives Markets (3rd Edition) (Pearson Series in Finance) (Page 166). Pearson HE, Inc.. Kindle Edition.

Please let me know if you think that's an improvement? I hope that's helpful, thank you for your attention to detail!

 

[lRRAngle]

yup, I think it's an enhancement.