FRM Fun 9 (Thur)

[Suzanne Evans]

FRM Fun 9 (S&P 500 Index in strange backwardation)

Yesterday, while writing the daily technical question, I looked up the prices for S&P 500 Index futures at http://www.cmegroup.com/. I was surprised to see the future curve in backwardation, I can't recall observing an investment commodity in backwardation (consumptions commodities, on the other hand, often are). Here is a snaphot:

Question: What can explain this? Is not there an obvious arbitrage?

As this is FRM prep, the best answer should address the cost of carry model contained in Hull and McDonald, which in theory should govern commodity forward curves (e.g., is consistent with, violates, doesn't apply because).

 

[ShaktiRathore]

Note:Block letters are Main parts of the answer
The Futures price of a commodity is given by:
F=S*exp(r+c-y)
where r: risk free rate, c:cost of carry, y: convenience yield
The Bond/stock futures pays a fixed dividend yield and thus this yield over a time can outweigh the investment commodity interest cost also that storage cost for investment commodity is 0.This suggest c=0,y>r , hence F<S the pattern which can be seen above.
From above equation it can also be seen that F<S when y>r+c thereby implying that the commodity has very large convenience yield that is the availability of commodity ensures that there is no shortage of supply and is readily available even at times of low supply.Also there can be higher yield that can be earned from the commodity. For e.g. the yield on gold may be large compared to the interest and storage cost or that convenience of holding Oil may far outweigh the the storage and interest cost.

Second part:
Arbitrage might not be possible. As by getting long on futures and short selling the commodity implies that we are foregoing the fixed yield on commodity at the same time which exactly offsets the return on Futures contract.Hence no arbitrage!!

 

[David Harper CFA FRM]

Hi ShaktiRathore,

Thank you, nicely done! I happen to agree and offer additionally as opinion:

• We don't have to claim that a fundamental model explains the forward can, we can "retreat" to technicals (liquidity; supply/demand)
• Even within fundamental explanatory models, we have some choices (see Kolb); e.g., we can attribute the observed forward to largely a function of consensus future expected prices ... here a pessimistic view about the expected future
• But the FRM assigns Hull and so basically follows a cost of carry model, as ShaktiRathore illustrates. In cost of carry model, we should conclude this INVESTMENT commodity has no storage cost, and we might conclude it has no convenience yield. Please note this is one way Hull distinguishes between an investment commodity and a consumption commodity: the latter has a convenience yield and consequently "weakens" or reduced the no-arbitrage implied forward price to an interval
• In this way, F(0) = S(0)*exp(interest rate - dividend yield)
• And backwardation implies, under this particular model with caveats applied, that dividend yield > interest rate; e.g., if S&P 500 dividend is ~2.0%, backwardation would be the natural implication if the rate < 2.0%.

Thanks I hope you found this interesting like i do!

 

[caramel]

To put it simply the dividend yield is synonymous with the convenience yield

 

[David Harper CFA FRM]

Hi caramel, I agree they are synonymous directionally: both reduce the forward price as they represent the (opportunity) cost of not owning.

But i would add: in Hull & McDonald the convenience yield is different due to its difficult-to-observe (or measure; or even subjective) quality: in McDonald, it causes the COC to imply a "region" for the forward; in Hull it is the plug variable, he would use the convenience yield to balance the equation. Unlike dividends which are somewhat objective, he would say the forward curve is information about a consensus convenience yield. Dividends don't unsettle the basic no arbitrage COC like the convenience yield.

(and i find it the subjective/objective difference interesting: dividends are the same to all owners, convenience varies subjectively i would think?), thanks!

 

[caramel]

Hi David and all

This is in connection with the above and hull
as per hull, Gold owners such as the central banks charge interest in the form of what is known as a gold lease rate when they lend gold. the same is true of silver. Gold and Silver can therefore provide income to the holder . Like other commodities they also have storage cost.

my interpretation here is that the lease rate is basically like a dividend or convenience yield

a question from the 2009 practice exam

1)if the gold lease rate is higher than the risk free rate, what is the market structure of the forward market for gold ?
a) contango
b) backwadation
c) inversion
d) need more information

 

[caramel]

I also want to add that gold is an investment commodity whereas silver can be both . ( maybe considering in India, gold is a huge consumption commodity)

 

[Aleksander Hansen]

I also want to add that gold is an investment commodity whereas silver can be both . ( maybe considering in India, gold is a huge consumption commodity)

@caramel: gold is both as well.

The Futures price of a commodity is given by:
From above equation it can also be seen that F<S when y>r+c thereby implying that the commodity has very large convenience yield that is the availability of commodity ensures that there is no shortage of supply and is readily available even at times of low supply.]

Arguably, the more fundamental forces of the price system guarantees an efficient allocation, so that no such thing as a shortage of supply exist [nor a shortage of demand] barring Government intervention.