FRM Market and Products 3-c

[HC78]

Hello David,
difference between future price and expected future spot price is that for the latter we do not take not acount systematic risk of the asset ? In this case, is relation for Fo=E(ST) exp(k) with k=Risk free+Beta*(Excess risk premium relative to market) ?
If we talk about investment asset (not a consumption asset), there is no convenienve yield. So, could we use previous relation to infer "implied Beta" (which is price by the market)? Imagine we know risk free rate; storage cost; and eventually dividend yield. By observing Fo and reverse equation, we can determine Beta (a little bit like convenience yiel is used as a "plug in parameter" to respect cash and carry model) ? Put another way, is it only a ex post relation or can we use it as an ex ante relation ?

In this case, when we observe Fo > (or <) of So*exp(cost of carry), explanation is systematic risk. In this case, could we always talk about arbitrage and strategy that arbitrager set to gain b/c it is fully explain by systematic risk ?

Thank you for your help.
Hervé

 

[David Harper CFA FRM]

Hi Herve,

If we let T = 1 year to let that drop out and simplify, the relationship given by Hull is (see p 26 of 3.c):
F0 = E(St)*EXP(r-k), and per CAPM let k = rf + beta*MRP, where
The expected future spot price St = S0*EXP(k), such that:
F0 = S0*EXP(k) * EXP(r-k) = S0*EXP(r - k + k) = S0*EXP(r)

And we can see the implication of this theory (theory of normal backwardation): the future price (F0) does not take account systematic risk (i.e., if that means "is not influenced by it") but the expected future spot price does. The intuition here is, imagine you are long the forward contract (e.g., you promise to buy in the future at $X) and I take the short position (I promise to sell for $X). If there is systematic risk, F0 < E[S(t)] and you expect to realize a profit per this difference; I expect to lose the same. Viewed another way, as the long (investor) more systematic risk does not change the price you will pay because the increase in E(St) is offset by an increase in your discount rate.

Your inference idea is very interesting but I get stuck because, unlike F0 which is observed, E(St) is unobserved. So it's not an input. It's only an input, I think, if we use beta to produce it. I perceive we have the same issue as if we were trying to extract beta just from a regular DCF: how do we get the expected future value (without "boostrapping" with the beta)? (E(St) seems to me, by definition, an ex ante and theoretical idea. )
… specifically, it is not F0 <?> S0*EXP(cost of carry) because F0 = S0*EXP(cost of carry). Rather, we could solve for beta if we have F0 and E(St), or F0 and S(0)*EXP(discount rate). See my point? You need the discount rate, not the cost of carry, and under this model, you need beta to get the discount rate?! I may miss something but i just don't see it.

In case it's helpful, please note XLS http://www.bionicturtle.com/how-to/spreadsheet/3.e.2_mcdonalds_arbitrage_w_lease_rate/
This refers to assigned McDonald but the same concept is used (let lease rate = dividend). You'll note the implied forward price (10.345) uses only riskfree rate (and lease/dividend) while the E(future spot price, $10.5127) incorporates systematic risk b/c instead of 4% riskfree rate it uses a discount rate of 6% (as if 6% = 4% rf + 0.5 beta * 2% MRP, or something like that).

David

 

[HC78]

Hello David,
Thanks a lot for your reply. If I try to summarize with my own words (to check understanding again):
1/ Fo=So*exp(Rf) b/c of cost of carry model. Could we say that ST=Fo b/c there is no uncertainty in that sense that we know Rf, So and that we do not introduce expected parameter in the relation?
2/ E(ST)=So*exp(k) (with k=Rf+B*ERP) b/c we introduce an expected parameter to take into account systematic risk.
3/ In order to link 1/ and 2/ and respect cost of carry model, we set discount rate attached to E(ST) equal to exp*(r-k) such that Fo=E(ST)*exp(r-k)=So*exp(Rf). From that, it follows that (to paraphrase your own words) systematic risk does not change the price you will pay (Fo or the spot price in the future) because the increase in E(St) is offset by an increase in your discount rate. Indeed,
i) E(ST)=So*exp(k) (with k=Rf+B*ERP) so comparatively of Fo (cost of carry model without expectation), So growth more with a factor of Beta*ERP
ii) Fo=E(ST)*exp(Rf-k)= E(ST)*exp(Rf-Rf-B*ERP)= E(ST)*exp(-B*ERP), so you discount E(ST) with the same (marginal) factor. Finally, expectations have no impact on Fo. It is like in risk neutral world in which asset rate growth and discount rate are the same?

Another thing that I have some trouble to understand is what follows.
1/ In my mind, key pb in this matter is to compute value of the spot price in the future (ST for generic).
As an operator, I know cost of carry model, so I know relation that link Spot price and Future price. So, all operators would have the same Futures price computation (otherwise there are arbitrage opportunity and strategy that push future prices to respect cost of carry model). ST is Fo (=So*exp(Rf)).
In the same way, if each operator do is own expectation concerning influence of systematic risk on the spot price in the future, ST become E(ST). But that is an individual view. On market (aggregated) view, is it necessarily implies that all operators have same expectation ? Put another way if we want to compare Fo (which is unique) and E(ST), E(ST) should be unique.

Finally, concerning my inference idea. I do agree with you, I have made a mistake. Thanks to your reply it is now clear in my mind (i hope !). Idea consisted in comparing Fo and E(ST). Operator fist compute Fo with cost of carry model (So, Rf, Storage cost, dividend). Second, he observes E(St) on market. And third, b/c it is an investment asset there is no convenience yield and so there is only one unknown parameter in the equation which link Fo and E(ST) [F0 = E(St)*EXP(r-k)]. This unknown parameter is B*ERP. But as E(ST) is unobserved, it is not possible. Nevertheless, if I well understood article cited in this link (http://www.cxoadvisory.com/equity-premium/the-best-equity-risk-premium/ : that is web site that you already mentioned in News links. So, I think it is permits to mention it again): “Equity Risk Premiums (ERP): Determinants, Estimation and Implications – A Post-crisis Update”, it is possible to infer ERP from market price. So is it then possible to infer E(ST) by doing hypothesis that we deal with Market index, so that Beta =1 ?

Thank you for your big help.
Hervé

 

[David Harper CFA FRM]

Hi Herve,

1/ Fo=So*exp(Rf) b/c of cost of carry model. Could we say that ST=Fo b/c there is no uncertainty in that sense that we know Rf, So and that we do not introduce expected parameter in the relation?

Under this thy of normal backwardation, we cannot say St = F0 unless systematic risk (beta) = 0. Unlike F0, the future expected spot price, if it has systematic risk, is expected to grow above the Rf rate.

2/ E(ST)=So*exp(k) (with k=Rf+B*ERP) b/c we introduce an expected parameter to take into account systematic risk.
Yes, exactly!

3/ In order to link 1/ and 2/ and respect cost of carry model, we set discount rate attached to E(ST) equal to exp*(r-k) such that Fo=E(ST)*exp(r-k)=So*exp(Rf). From that, it follows that (to paraphrase your own words) systematic risk does not change the price you will pay (Fo or the spot price in the future) because the increase in E(St) is offset by an increase in your discount rate. Indeed,
i) E(ST)=So*exp(k) (with k=Rf+B*ERP) so comparatively of Fo (cost of carry model without expectation), So growth more with a factor of Beta*ERP
ii) Fo=E(ST)*exp(Rf-k)= E(ST)*exp(Rf-Rf-B*ERP)= E(ST)*exp(-B*ERP), so you discount E(ST) with the same (marginal) factor. Finally, expectations have no impact on Fo. It is like in risk neutral world in which asset rate growth and discount rate are the same?

Yes, I think your conclusion here follows, I agree because:
F0 = E(St) * EXP (r - (r + beta*ERP)) = E(St) * EXP(-beta*ERP)

In regard to "Finally, expectations have no impact on Fo. It is like in risk neutral world in which asset rate growth and discount rate are the same?" …. Well, I regret I may have misled in the original answer. I thought about this later. The comparison to risk-neutral and tempting and sort of works on a superficial level. However, it is not quite the same thing. In risk neutral valuation, the insight is that the riskless no-arbitrage portfolio produces a certain (riskless) future value which, therefore can be discounted at the riskless rate. It is not quite the same thing here. This is a mere theory (a mere viewpoint) which can be more easily challenged. Here the idea is more like: as the asset becomes more systematically riskly, the forward price is unchanged because the long position expects greater future profit (the long will profit = St - F) in the form of greater E(St) - F0; i.e., as systematic risk goes up, and F0 remains the same per cost of carry, the expected future profit (E(St) - F0) increases to compensate the long position. Rather than no-arbitrage, this is more like "greater expected return for more risk."

1/ In my mind, key pb in this matter is to compute value of the spot price in the future (ST for generic).
As an operator, I know cost of carry model, so I know relation that link Spot price and Future price. So, all operators would have the same Futures price computation (otherwise there are arbitrage opportunity and strategy that push future prices to respect cost of carry model). ST is Fo (=So*exp(Rf)).
In the same way, if each operator do is own expectation concerning influence of systematic risk on the spot price in the future, ST become E(ST). But that is an individual view. On market (aggregated) view, is it necessarily implies that all operators have same expectation ? Put another way if we want to compare Fo (which is unique) and E(ST), E(ST) should be unique.

I find this very interesting, I had not considered it. Clearly, you understand why I say that E(St) is unobserved and therefore, I like to say, contango is observed but normal contango/backwardation is not.
(and, yes, as the XLS above demonstrates, we can virtually prove that F0 has a no-arbitrage value!)
I am not sure I have a decisive view on your provocative viewpoint. My initial reaction is: as the CAPM assume equilibrium, so then does this model similarly, rigorously and (of course) unrealisitically (!) assume that all market participants share the same views and come to the same conclusions re E(St); i.e, IMO, the essence of "equilibrium theory" is that all participants have (i) the same information and (ii) importantly, the same view/perspective on the information. My initial hunch is: in theory E(St) has a single value per the rigorous equilibrium, but in practice I don't see why this precludes individual variations on E(St)….

Finally, concerning my inference idea. I do agree with you, I have made a mistake. Thanks to your reply it is now clear in my mind (i hope !). Idea consisted in comparing Fo and E(ST). Operator fist compute Fo with cost of carry model (So, Rf, Storage cost, dividend). Second, he observes E(St) on market. And third, b/c it is an investment asset there is no convenience yield and so there is only one unknown parameter in the equation which link Fo and E(ST) [F0 = E(St)*EXP(r-k)]. This unknown parameter is B*ERP. But as E(ST) is unobserved, it is not possible. Nevertheless, if I well understood article cited in this link (http://www.cxoadvisory.com/equity-premium/the-best-equity-risk-premium/ : that is web site that you already mentioned in News links. So, I think it is permits to mention it again): “Equity Risk Premiums (ERP): Determinants, Estimation and Implications – A Post-crisis Update”, it is possible to infer ERP from market price. So is it then possible to infer E(ST) by doing hypothesis that we deal with Market index, so that Beta =1 ?

Yes, that is terrific and I agree with your entire line of thinking…. Until I still perceive a similar "glitch" in the final step. We agree that F0 = E(St) * EXP(-beta*ERP) and I think the question is "Can E(St) be inferred from observable information?"
... We observe the F(0) as input
… We can assume an ERP as input
… alas, the beta is not 1.0 necessarily b/c we want the commodity beta. I do agree with you that, given a commodity beta, we can infer the E(St) but this seems like not much of an improvement as, given the commodity beta, we can just estimate E(St) straightaway.

Whether you agree or disagree, i hope this is further helpful and thanks for your deep dive...

David

 

[HC78]

Hi David,
thanks a lot again. That allow me to fix (and correct) lot of concepts. Another brick in my building block learning...
Sorry have been provocative...

Regards.
Hervé

 

[David Harper CFA FRM]

(Herve - I meant "provocative" as a compliment, for what it's worth. I find these discussions help me understand better, too. I intended not a whiff of negative only positive feedback)

 

[HC78]

Hi David,
Thanks for the compliment. Coming from you it is very motivating. Frankly, it is a pleasure to discuss about these kind of (sometimes theorical) ideas with you.
Regards
Hervé