Index future contract's relationship to underlying index

[Martinghoul]

Oh this sounds like fun....let me play.

So the return of the index over the long term usually has 3 components: risk free rate, div cash flows, risk premium. I like to add the long term inflation rate to the risk free rate since I need to be compensated for that long term as an opportunity cost assuming substituting bonds will compensate me for the inflation rate via long term rates.

The knowns: risk free rate (or as Marty says, our real internal borrowing costs), div cash flows (give or take), inflation rate. That leaves us with the risk premium. The risk premium is known long term. Long term, equities have historically provided about a 2% risk premium to bonds. This is actually pretty constant. So if all these factors are relatively known, we can use them as one large discount rate.

So while this is broadly correct, there are a few things I would question. Firstly, it's not clear whether inflation should be added or not. There's some evidence that, because firms have at least some pricing power, your equity returns have at least some "real" component to them. This is a subject of much debate and very little consensus, so it doesn't get included most of the time. As to equity risk premium, I would disagree that it's "pretty constant". In fact, there's a reason why there's an "equity risk premium puzzle". From that Wikipedia article, there's some sort of accepted wisdom that it's maybe arnd 3-7% in the really long run, but it's also wildly variable (from over 19% in the 1950s to 0.3% in the 1970s, according to the above). So it seems that such a nebulous and uncertain quantity would also be difficult to include. This seems to leave us with the two things that people commonly use, i.e. dividends and "risk-free" rate.

So the value of the S&P 500 today is equal to the present value of the future cash flows of ES (dividends). And this PV is derived from taking the FV of the dividends divided by the discount rate (risk free rate + inflation rate). The inflation rate incorporates the risk premium.

Am I getting warm?

Not at all sure about the inflation rate incorporating risk premium. Agree that S&P today can be calculated, in theory, using the dividend discount model. The question we're arguing about here is a little different, though.

 

[Martinghoul]

I'll start with an intuitive look. Right now the Dec ES futures are trading at about 7 points lower than the spot SPX. Assuming 0% interest, which is essentially the case right now, that 7 point delta will gradually decay as component stocks of the S&P 500 go ex-dividend until they match on the day the Dec ES futures expire. This is a monotonic decrease in the delta, I can be sure it will never go to an 8 point delta between now and then (keeping interest rates out of it). If you follow the ES/SPX delta you will see this happens without fail every time.
Now let's add interest rates. If you buy Dec futures you're paying less to get the Dec price than someone who bought all the stocks in the index at the right ratios (or SPY), because you just have to post margin, not the full purchase amount. This acts in the opposite direction of dividends, so if interest rates were 10%, for example, the Dec ES futures would trade at a premium to SPX. Because they're so low right now, they don't have much impact, but if interest rates went up dramatically tomorrow, for example, the delta between SPX and ES would decrease because of it, and if it were possible for them to go down dramatically the delta between the two would increase.
At no time will this delta move simply because the market feels that the S&P 500 will be higher or lower in Dec than today, it is purely a mechanical relationship based on dividends and interest. This is a required relationship based on the math. The only variables are those dividends and interest, if they don't pan out during the interim period like the market thought they would, i.e. dividends change appreciably from historic levels or interest rates change, then the delta between SPX and ES will change.
Now let's assume that somehow the Dec futures started to show a higher price than would be warranted by the equation, supposedly because the market thought it should be higher. At this point, I can short the Dec ES futures and buy SPY and I'd be sure to capture a risk free profit by at least the time the ES future expired in Dec. Since this particular market is highly efficient and arbitrageurs step in the microsecond it gets out of balance, the market's perception of what the future price should be will never be reflected in an ES price that varies from the no arbitrage formula.
One last intuitive point, the current price of any security in a freely trading market is always the current market consensus of its discounted net present value. If the market thought the S&P 500 would be up 200 point in Dec, it would push the S&P 500 up nearly 200 points now. It wouldn't show up in the futures and not the spot. The thought experiment is to think of a thing everyone thought would be worth $100 on Dec 16th, what would they collectively agree to pay for it Dec 15th? Probably just under $100 right, accounting for 1 day of interest to compensate for paying for it early? What about on Dec 14th? Again just under $100 right? And Dec 1st, Nov 15th.... As you can see adjusted for carrying costs if the market feels the value of something in the future is X, the value of that something will be X today.

I am not sure what you're arguing here, but it would appear that you're agreeing with me...

I can offer you a very simple thought experiment as well, if you like.

 

[Maverick74]

So while this is broadly correct, there are a few things I would question. Firstly, it's not clear whether inflation should be added or not. There's some evidence that, because firms have at least some pricing power, your equity returns have at least some "real" component to them. This is a subject of much debate and very little consensus, so it doesn't get included most of the time. As to equity risk premium, I would disagree that it's "pretty constant". In fact, there's a reason why there's an "equity risk premium puzzle". From that Wikipedia article, there's some sort of accepted wisdom that it's maybe arnd 3-7% in the really long run, but it's also wildly variable (from over 19% in the 1950s to 0.3% in the 1970s, according to the above). So it seems that such a nebulous and uncertain quantity would also be difficult to include. This seems to leave us with the two things that people commonly use, i.e. dividends and "risk-free" rate.

Not at all sure about the inflation rate incorporating risk premium. Agree that S&P today can be calculated, in theory, using the dividend discount model. The question we're arguing about here is a little different, though.

I think there are two things here. Short term pricing is based on the no arbitrage principle and to that I think we all agree. My post above is more of a long term expected return model that is very broad and obviously debatable. I'm not sure what the short term debate is either. I think all of us are saying the no arbritrage principle applies here.

 

[Martinghoul]

I think there are two things here. Short term pricing is based on the no arbitrage principle and to that I think we all agree. My post above is more of a long term expected return model that is very broad and obviously debatable. I'm not sure what the short term debate is either. I think all of us are saying the no arbritrage principle applies here.

I agree it's sorta semantics that we're arguing. I am stating that both the "no arbitrage" pricing logic is valid and also that all futures, including index futures, reflect mkt's expectations for where the underlying would be at maturity. In fact, the two statements are tautological.

 

[Sig]

I am not sure what you're arguing here, but it would appear that you're agreeing with me...

I can offer you a very simple thought experiment as well, if you like.

Based on your other posts I figured we were probably in violent agreement, as Maverick points out I probably didn't get the distinction between short term and longer expected return.

 

[JackRab]

@Martinghoul , by cashflow I mean the ones that flow into the company, not dividends. Dividends (which is a whole other discussion) is just a payout to shareholders... which is therefore a negative cashflow to the company. For example Apple, which has a ton of cash locked-up overseas because they would pay a huge tax when they payout as dividend. So even if a company doesn't pay any dividend they still have a positive CF when they make money.

That aside... I'm getting the feeling we all more or less agree.

Saying a future basically shows you the index/stock-price level at expiration is the same as saying the current stock price is the price at expiration... which to the inexperienced would mean he 'knows' exactly where the price is at t=x...
Which is completely false, because we will not know that, since there will be other factors in play which change the landscape. Any news etc.

So, I'm saying is that a the assumption that the future price points out the index price at t=x is too basic and therefore false.
Except maybe in the case of commodities, where a future has less to do with the current spot value of the underlying commodity and more to do with the supply/demand of the spot at t=x.

The current price of anything is always right according to the EMH but that doesn't mean it stays where it is, whatever is incorporated in the discounting.