Forward and Futures Market (lease rate)

[WhizzKidd]

Hi David,

With regard to gold futures, why do they have an increasing futures curve (contango)? When we have a positive lease rate, because would the +delta not decrease the forward price... So*exp((r-delta)T)? I am struggling to understand why the curve is upward sloping (I am a bit confused on how the lease rate is supposed to work, if it acts like a convenience yield then it should drop the curve).

Trying to understand why holding the future is better than the gold.

 

[ShaktiRathore]

Hi,
The futures curve plots the futures price Vs Time to maturity(T). So that if r=5% and delta=4%, S0=100 then the future price=100*exp((5%-4%)T) hence if T increases the future price increases therefore the futures curve the plot of futures price Vs Time to maturity(T) is upward sloping.
thanks

 

[David Harper CFA FRM]

Hi @HTHNIK It's a great question. My short answer to the question (why would gold ever be in contango given the lease rate) is that (a) gold has a storage cost and (b) the lease rate might be zero or negative. The long answer is that it seems to be complicated, and in the FRM's case, complicated by the fact that Hull doesn't really address lease rate (i.e., does not really incorporate it into the cost of carry model, where it belongs) and McDonald (IMO) is both confusing and altered his stance slightly between versions.

We can start with the easy part, to agree with you: in the cost of carry model, the lease rate directionally behaves like a dividend/income or convenience yield. It is a quantifiable benefit of gold ownership and therefore lowers the futures price; i.e., ceteris paribus, tends away from contango and toward backwardation. The caveat is that, unlike stock dividends (which get paid in any scenario), the lease rate is collected only when the gold is loaned (still, this ought to be reflected in prices). In the actual market, they seem to compute the lease rate = GOFO - LIBOR lease rate = LIBOR - GOFO, where GOFO is a proxy for the forward return, that is ln(F1/S0), (see http://seekingalpha.com/article/1542652-what-gofo-is-and-why-its-now-very-bullish-for-gold), so you can see this is directionally consistent with Hull's COC: F = S*exp[(r -L)*T], where L is the lease rate and r is the riskless rate (e.g., LIBOR).

In terms of Hull (I need to stay in the FRM, to be honest only because I am not an expert on gold), I think it is safe to say that the COC remains valid such that F = S*exp(r + u -L)*T; but note that gold has a storage cost such that storage costs can explain the contango.

In terms of McDonald, his approach is very interesting. His COC implicitly seems to omit storage because he infers lease rate, L = r - 1/t*ln(F/S). Notice that is F = S*exp[(r-L)*T]; i.e., McDonald infers the lease rate as accounting for any offset to the funding (risk-free) rate! In a zero or near interest rate environment, we are back to your point: gold should be in backwardation not contango! Indeed, between the second and third versions of McDonald, he switched from asserting gold is naturally in contango to the finding of a negative lease rate (indeed, he is not the only one to have struggled to explain this phenom). Here is McDonald (emphasis mine), I hope this helps!

[after he illustrates that an actual recent gold forward curve exhibits contango above the riskfree rate; i.e., riskfree rate of 0.00342 is exceeded by forward rate of 0.00508]"The negative lease rate seems to imply that gold owners would pay to lend gold. With significant demand in recent years for gold storage, the negative lease rate could be measuring increased marginal storage costs. It is also possible that LIBOR is not the correct interest rate to use in computing the lease rate. Whatever the reason for negative lease rates, gold in recent years has been trading at close to full carry." -- McDonald, Robert L.. Derivatives Markets (3rd Edition) (Pearson Series in Finance) (Page 177). Pearson HE, Inc.. Kindle Edition.

 

[WhizzKidd]

Hi David,

Ok, thank you.

Given the form F = S*exp[(r-L)*T]; i.e., McDonald, we also know that the Lease Rate=Storage Cost-Convenience yield. So if it's negative. then should the storage cost not be lower than the convenience yield?

So why is this true: "the negative lease rate could be measuring increased marginal storage costs."

Perhaps, we are going to deep into this, and all that is needed is (a) gold has a storage cost and (b) the lease rate might be zero or negative, that is why contango exists.

 

[David Harper CFA FRM]

Hi @HTHNIK I think your point is excellent, because you are showing how McDonald actually does account for the convenience yield (which I forgot frankly in my note above). I am going to use Hull's notation (with McDonald's) formula to avoid my own confusion. But please note that, I think instead of "Lease Rate=Storage Cost-Convenience yield" you mean "Lease Rate=Convenience yield-Storage Cost;" i.e., the higher the convenience and/or the lower the storage cost, the more the commodity lender will charge the borrower. Or, illustrated another way, if the storage cost of gold is very high, the lender may even prefer a negative lease rate in order so that the borrower can store.

• So if we have lease rate = convenience yield - storage cost; i.e., L = y - u
  
• McDonald also gives the same COC as Hull, Cost of carry: S = F*exp[(r + u - y)*T]
• Therefore, McDonald implies: S = F*exp[(r + u - y)*T] = F*exp([(r - (y - u)]*T) = F*exp([(r - L]*T); in this way McDonald's lease rate, L, appears to impound (in gold's case anyway) the difference between storage and convenience!

Re: "So if it's negative. then should the storage cost not be lower than the convenience yield?" If L = y - u, then negative lease rate implies storage costs exceed convenience. Overall, we still have funding rate, r, so overall contango (i.e., F > S) is formula-wise explained by (y-u) < r ,or L < r, or y < r + u.

Re: "the negative lease rate could be measuring increased marginal storage costs." You can see this makes sense in the context of L = y - u, where higher (u) implies lower (L). And, intuitively, as storage price goes up, at some point the lender might well pay the borrower (with a negative lease rate) rather than charge a positive lease rate.

Thanks, this helped me too!

 

[FlorenceCC]

Hi @David Harper CFA FRM,

Allow me to follow-up on the thread above, as I have a question that is closely related to that topic yet slightly different,
- if a lease rate of L is associated with my ownership of gold, the futures price F(0,T) will be expressed as F(0,T) = So*exp[r-L]*T -> now, what a contango curve tells me is that r > L, that is, my lease rate, though it diminishes my CoC (because I lease the commodity I own) is not as high as my cost of financing, (r).

- So I don't understand why in your notes, you explain "if you own physical gold directly: you forgo a “lease rate”. Isn't the lease rate one of the possible advantages of holding a commodity, that you can lease it? Should I read it instead as meaning, "if I own, and store instead of leasing it, my gold, I forgo my lease rate and bear storage cost"? I apologize, I don't mean to be nitpicking or ask a silly question, I just wanted to make sure I understood.

- then I would express my futures price as below (under the assumption that L= y - u, convenience yield minus storage cost)
F(0,T)=S0*exp[r+u-y]. What a contango curve tells me is that the cost of financing and storing is higher than any convenience yield I get from holding on to my gold hence the interest in a synthetic position. +u-y represents -L, i.e. amount we do forgo by owning but not leasing our gold.

Am I articulating this correctly or am I misunderstanding something about the lease rate? I have used the formula above (L=y-u) based on GARP assumption in mock exam 2017, question73, so I figured it was "safe". Would you think it's ok to operate under this assumption for exam purposes?

Many thanks in advance for any feedback you may have.

Florence

 

[David Harper CFA FRM]

Hi @FlorenceCC Yes, I agree with everything you posted. Re: our notes reading "if you own physical gold directly: you forgo a 'lease rate'," I do agree that it's a fair criticism that this is imprecisely expressed. It should say something like, "If I hold the gold physically, then I incur the storage cost but enjoy the convenience yield; if I lend the gold, I forgo (save) the storage cost but forfeit the convenience. Therefore, my lending rate (aka, lease rate) should be the difference between the convenience yield and the storage cost. In this way, were L = lease rate, L = y - u. And the cost of carry can be expressed in terms of the lease rate: F(0) = S(0)*exp[(r-L)*T]. Substituting L = (Y-u), we have F(0) = S(0)*exp[(r-(y-u))*T] = F(0) = S(0)*exp[(r+ u - y)*T], which is a regular instance of the cost of carry. It frankly just took a few years for everybody, including GARP, to work this out primarily due to different authors (e.g., Hull doesn't strictly define, two versions of McDonald, other authors treat it synonymous with convenience yield).

It just happens that my latest YouTube video illustrates the lease rate. I've taken to referring to the lease rate, L = y - u, as the "net convenience yield" as in "net of storage cost." See below, thanks for your insights!

 

[FlorenceCC]

Thank you David. Yes, when looking at the forum for answers to my questions, I could see how the gaps in McDonald vs. Hull apporoaches created understandable confusion between lease and convenience yield, which is why I wanted to make sure I got this right.
That said, thank you for sharing your latest video, it cleared everything!!!