Futures and Normal backwardation/contango

[Ened]

I have a question which is a bit difficult to write down, but let's try:

On the slides on futures one of the first ones says:
F=E(S)

Later when discussing normal backwardation/contango the conditions are
F<E(S) and F>E(S)

This does not rhyme with the starting condition that F=E(S) in my view. If the two are equal to each other, how/why would they deviate? Got my point? Please explain what the conditions are in mathematical terms and in financial terms (words).

Thanks!

 

[David Harper CFA FRM]

Ened,

Excellent! Distinguish between a pricing framework and actual (observed) prices. For example, elsewhere is talk about the AIM that refers to C-STRIPS/P-STRIPS that "trade rich" or "trade cheap." This refers to actual prices that differ from the price given by a model (pricing framework). Much of our study is pricing frameworks that we do not actually, realistically expect to hold in the real world.

In John Hull, his COST OF CARRY MODELS are price frameworks:

F = Expected (Future Spot Price) = Today's Spot Price * EXP[costs of carry - benefits of carry].

so the cost of carry says the best (unbiased) estimate of the future spot price is the futures price:
F0= E(ST) = S0*EXP(r)
(Note S0 versus ST).

This sort of contains (ends) the practical (testable) aspect. I mean, test question are likely to be cost of carry: given riskless rate, convenience yield, spot price, now compute the forward price.

Now transition to the real world and the less testable aspect:

Notice John Hull uses the RISKLESS RATE (r) in cost of carry. Further evidence he is not in the real world
But he knows that...that's why at the very end he "gets real" by adding the discount rate (k):

F = E(ST)EXP[r - k]
where k = discount rate (investor's expected return).

All thru Hull's cost of carry, until at the end, he is assuming r = k, so that F = E(ST).
(Why? Because he assumes no systemic risk; e.g., if you already have a well diversified portfolio and add a corn futures contract, he is saying, that is ALL IDIOSYNCRATIC RISK/NO SYSTEMIC RISK. Therefore, you can only expect riskless rate as your discount rate!)

But that is a simplifying assumption per a simplistic (cost of carry) pricing framework. By adding the discount rate (k), which conveniently is hard to observe, we can now say:

"Our cost of carry model predicts F0 = E(ST). The unreal assumption."
"Yet we observe F0 < E(ST). The real observed."
"The difference must be the investor discount rate (k), which must be different than r."
This part we see all the time: the asset price go down, and we say "investors are pricing in a higher risk premium."

and we resolve by declaring: investor's must expect more than the riskless rate
as k > r
this is NORMAL BACKWARDATION
maybe in words "my cost of carry predicts F0 but i observe <F0, so i am in normal backwardation and this can be explained by SUPPY/DEMAND. More specifically, investors are risk averse in their demand."
(and this is why it's called "normal" backwardation - it fits our expectation that investor want to make money)

I got an email yesterday from somebody who challenged my publishing on contango/backwardation b/c, he said, "the [actual] oil forward curve is not following your cost of carry model." I haven't replied but i mean to say: of course not, I never said the cost of carry is useful (ha ha). Especially energy commodities respond to supply/demand which is not built into the simple cost of carry. The cost of carry framework will not produce the typical oil forward or corn futures curve - it is too simple for that. The particulars of supply/demand give them shape.

This will re-appear in the cram session, for sure!

David

append:
I just wanted to connect that back to "trade rich, trade cheap" concept we see in the STRIPS AIM. And this may or may not help.
I think we can say that *normal* backwardation is when the futures contract "trades cheap" (i.e., the F0 is less than predicted by the cost of carry framework) and *normal* contango is when the futures contract "trades rich" (F0 > predicted by cost of carry).

 

[Ened]

Thanks David,
Your answer is exactly what I was looking for: the missing link. Much clearer now!