Tuesday, December 14, 2010

So You Want To Understand S&P Futures Basis Trades (aka, Index Arb)?

I swear - there's nothing I'd rather do less than spend my time being the re-educator for all the incorrect crap that ZeroHedge spews into the Interwebs, but when a blog as widely read as they are publishes such utter and complete nonsense, threatening to miseducate masses of readers, well, I can't help myself.  So let's turn it into another educational lesson, right in my wheelhouse: today's topic is the "backwardation" in the S&P 500 futures.

Backwardation, although a term I've never heard applied to S&P futures before (that's a commodities futures markets term), means that the longer dated futures contracts are trading at a discount to the near contracts.  In other words, the price to buy the June 2011 S&P 500 future (herein referred to as the ESM1)  is lower than the price to buy the December 2010 S&P 500 future (herein referred to as the ESZ0).  This is a function of two things:  interest rates, and dividends, but not at all an indication of future supply and demand for stocks.

In the index arbitrage world, we want to know how the futures are trading versus their "fair value." The fair value of the futures vs. the cash index (underlying stock basket) is the difference in cash flows between holding one or the other.  The inputs are the "carry effect," derived from interest rates, the index level,  and time to maturity, and the "dividend effect," derived from the dividends that the companies in the index will pay between now and the expiration date on the futures.

When you buy the ESM1 - the June 2011 S&P 500 future, you don't actually have to pay for the full notional value of the future. You post margin.   Since you don't actually own the underlying stocks, you don't receive the dividends on them.  On the other hand, you earn "carry," because you can invest the money that you would have otherwise spent buying all the underlying stocks.  We can write a very simple equation to tell us what the equality would be if everything were trading at "fair value" and there was no arbitrage opportunity:

(ESM1) + carry = (SPX) + dividends

In other words, buying futures (that's why it's in parenthesis - it's a negative cash flow - we're spending money) and receiving interest on the money you don't have to spend buying the index itself is equal to buying the index itself and receiving dividends. Let's do some 6th grade algebra, rearrange those terms, and come up with the value of the basis: the difference between the futures and the cash index:

Carry - Dividends = ESM1 - SPX          aka, Futures Price - Cash Price, or "Fair Value"

Now, for my entire career, the "fair value" of S&P futures was always positive. The further out the curve you went (ie, if you looked at September instead of June futures,) the larger the fair value number got.  Why is this?  Simple:  the yield on the S&P was lower than the interest rate.  In other words, you'd rather own the futures than the entire basket of underlying stocks, because you can earn more money on the money that you don't have to spend buying the cash index than you can receive in dividends.  It's basically a simple interest rate equivalency.

Now, however, interest rates are zero, or near to it, so the fair value basis has turned negative - the futures are "less desirable" than the cash index because you'll earn dividends by holding the underlying stocks, but you don't have any opportunity cost that you're saving by buying the futures - because the alternative reinvestment rate on that cash is zero.  Thus, if you go out further on the futures curve, the basis becomes even more negative - because even more dividends are paid (yet the return on cash is still piddly).  

Which brings us to ZeroHedge, who writes:

"The ES futures curve is now at inverted term levels that have been unseen for months. For all who claim that by next summer the economy will be coasting well on its way to 3.5% growth or whatever imaginary number the crowd of lemming sell-side analysts pulls out of their pocket in their imitation of Goldman's upgrade, there sure is no actual conviction in this call. The differential between the Dec and the June ES contracts is a notable 10 points: December is at 1,246 while June is at 1,236. This is reminiscent of the curve last December, when those who bet that the market would be substantially lower half a year forward ended up being right on the money. For those who still believe in logic, a compression trade where one sells the Dec and buys the Jun contract may make sense, although with the only variable these days being what side of the bed Brian Sack wakes up on, we would be very cautious. As a reminder, the last time the VIX curve had a normal contango curve structure, was back in 2008, when the Bernanke Put was still being digested."

Of course, anyone with a rudimentary understanding of equity index arbitrage, even on the level I have just explained it above, will understand the reason that the June 2011 contract is trading at a 10  point "discount" to the December 2010 contract:  lotsa dividends will be paid between now and June, yet the interest rate curve out until June still provides for meager returns on riskless cash.   In other words, the 10 point discount isn't a discount at all - it's the difference between the June fair value and the Dec fair value. Of course, back in 2008, interest rates had not yet tanked, which is why the futures curve was upward sloping: the interest rate carry side of the equation was dominating the dividend side of the equation.  The equation has nothing to do with supply and demand for stocks. 

Class dismissed.  Ask questions in the comments if I've been unclear about anything here.


Now, in the interest of completeness, I should note a few slightly more advanced caveats.  First, the interest rate component is really based on the rate at which one can "fund" themselves.  In other words, for a bank, it's their cost of funds.  If I have a lower funding rate than you do, I can afford to perform the index arbitrage of selling futures and buying stocks before you can, because I require less of a futures premium to do the trade, since I have a lower funding hurdle.  Second, futures do of course sometimes trade rich or cheap to fair value.  In fact, most of my career, they constantly traded slightly rich, and many people on the street made a good living by selling futures at a slight premium to the calculated fair value, buying stocks, and holding those positions.  How do you do that?  You need a big balance sheet and cheap funding.  In other words, the futures richness is an indication of demand for FUNDS, not demand for stocks.  But I need another caveat here - the ZeroHedge post is talking about the roll trade - the difference between not futures and cash, but different futures months. I don't have the dividend and interest numbers handy, so I don't know if the -10 basis between June and December that they refer to is "rich" or "cheap" to fair value - but in either case, it's no indication of implied demand for stocks - rather, implied demand for balance sheet - for cash.   I would guess that it's trading pretty close to fair value.

Quick example:  suppose we calculate that the June S&P 500 future is trading 2 points higher than its "Fair value" (that would be a huge premium, by the way) - what do we do?  We sell the June S&P 500 future, we buy the basket of S&P 500 stocks, and we lock in our funding rate until June (so that if the Fed comes in and raises rates on us, we don't get crushed).  There's still a risk - that our dividend estimates were incorrect:  if we estimated that we'd get $12 in dividends, but only get $10, well then, there goes the $2 premium that we thought we had.  During my career, which was basically a bull market, this "risk" was actually more of a freeroll to the upside - because more companies increased dividend payouts on a surprise basis.  During 2008 and 2009, it must have been a risk, as companies slashed dividends - this would have been murder for index arb desks, who are generally long stock, short futures in the S&P 500.    However, a colleague tells me that as banks ran into trouble with their mortgage positions, they reduced balance sheet exposure elsewhere, and frequently lucked out of this risk by reducing large index arb positions.

Finally, I should mention the effect changes in various inputs have on the fair value basis, although readers can calculate this on their own easily from the equation.  If we hold all other inputs equal, higher index levels --> higher fair values, higher interest rates --> higher fair values, and higher dividends --> lower fair values.   We can easily understand why, because higher fair values mean holding futures is more desirable than holding the cash basket.   Higher index levels or higher interest rates each mean higher carry costs, which means futures are more desirable.  Of course, higher dividends means the cash basket is more desirable. 


Anonymous said...

yo kd,
i no longer believe zerohedge believes what they publish. the stuff is just too dumb. I think they have found that the goldbug/birther/tea-party/mish element on the internet are a really easy crowd for authoritative sounding financial speak. these people probably click thru and buy lots of the stuff on the ads on the site at much higher rates than the old set of ZH readers.... gold coins, books on trading, etc...

i think they are just using expert power to feed the santelli's of the world what they already believe, but with stuff that is general only in the purview of bd's. they are just playing the game of giving people new info that confirms a hypothesis they already have.

i would be 50 bid its purposefully disingenuous.

scharfy said...

Heres a good website I used to use "back in the day"

Gives updates on dividends, rates, fairvalues, etc... all in one spot. Kind of handy

Good post. You don't even need to work on a desk to know this stuff. Its college level finance.

EconomicDisconnect said...

Great stuff.

Caught the running dialogue over at your Seeking Alpha print on the silver story, all I can say is both wow and sorry. Keep brining it, I know it is thankless and hard at times.

Anonymous said...

The last time backwardation was busted out ad nauseum was when crude was hitting $149.

And purported to be on it's way to $300 and beyond.

Buy the dippin'. Not when most people are trippin'.

-HR Dobbs

Anonymous said...

Hey Kiddo,

calm down, will ya?

BTW, ZH is not talking about futures basis. They are talking about futures calendar spread.

Anyway, isn't the forward curve an unbiased estimate of the future spot price?

Therefore, if the fwd curve inverts, it means fwd mkt is bearish about the future spot price.

Kid Dynamite said...

anon - yes, I am well aware that ZH is talking about the calendar spread, and no, a negative calendar spread in the S&P futures has nothing to do with bearish market assumptions. (the Dec futures expire in 3 days, so their fair value is basically zero anyway) I think I explained that thoroughly in this post, didn't I? carry and dividends...

as for being fired up - it's frustrating, because for every million people ZeroHedge miseducates, I only get to re-educate a few thousand... that's a losing formula for increasing the general intelligence of the population

Onlooker said...

Thanks Kid. Count me as one who's been educated here, as I don't have a finance background and haven't spent any time learning about this realm. But it does jibe with my gut feeling about this, as I've heard this discussed before in this way.

Makes much more sense than the notion that the futures market is so much "smarter" than the cash market.

wcw said...

> isn't the forward curve an unbiased estimate of the future spot price?

No. It's just no-arbitrage prices.

This has been the latest in our public-service series of simple answers to stupid questions.

Marshall Jung said...

Thanks KD. This is why your website lives on my bookmarks bar.

Sometimes I feel like I'm living in a doomed world. It's about time to queue up the opening scene of the movie "Idiocracy".

Kid Dynamite said...

wcw - i don't want to discourage people from asking questions about this though - there are no stupid questions here on this one... at least not yet

Greycap said...

"there are no stupid questions here on this one... at least not yet"

The anon was cutting it pretty close, though, since the whole point of your post was to explain why forward equity prices are determined by arbitrage, not "unbiased estimates."

Although I enjoyed your post, I think you muddied the waters a bit by dragging in index arbitrage. The restrictions on calendar basis - a function of IR and expected dividends - hold for any individual equity, not just the index. If you can short the equity, that puts a floor on the forward price. If you can buy it in the spot market, that puts a cap on it. If you had concentrated on this, the anon might not have been confused.

Anonymous said...

"Anon said...i no longer believe zerohedge believes what they publish. the stuff is just too dumb. I think they have found that the goldbug/birther/tea-party/mish element on the internet are a really easy crowd for authoritative sounding financial speak"

Agreed. The day I checked out of ZH for good is the day they published some quote of impending doom from a source who also believed 30-40 foot tall people once roamed the earth.

ZH has a winning formula though. Fact of the matter is, doom, doom and more doom sells, and the sheeple will line up to drink it down with wild abandon.

BigShow said...

Sold at 1000s educated.

Anonymous said...


How can one properly assess "fair value" - channeling LTCM here - when there is an artificial cost of capital and Divs are by no stretch sticky when considering the net debt position of non financial corps. No contest on the technical "math/index arb" but this market stopped being about mechanics long ago. If one could halt time and remove all the artificial capital supports - ruling out a deflationary collapse mad max - and assume some form of normalized cost of capital (admitedly at this point bygone) perhaps ZH is right for the wrong reason. A quick look at historucal div ylds would tell you as much.

Kid Dynamite said...

anon - I'm not sure what you're saying. the capital (funding) part is pure arbitrage - it can be and does get locked in. There is risk in the dividend side of the estimate, but I'd bet that on a 3 month forward basis or even 6 month forward basis that this ends up being a miniscule risk. (although I don't have data handy for it)

In any case, if Mr. Market was worried about dividends, then the June future would be trading higher - less negative compared to Dec - remember, more dividends --> lower fair value (because futures don't get dividends and become less desirable).

similarly, if you assumed a "normalized" cost of capital, by which I assume you mean "higher" - then the june future would be trading higher, not lower.

but that's a moot point, because the futures don't trade of some historical or normalized cost of capital, they trade off actual cost of capital (and dividend estimates, which, on a short term basis, are pretty darn accurate)

EconomicDisconnect said...

I almost made a KD centered "LEAVE BRITTANY ALONE" video tonight, but thought better of it.

Daniel said...

You are confusing knowledge and intelligence. You can do something about knowledge, nothing about intelligence.

Anonymous said...

KD, thanks for post.

My point is that your discussion of the technical relationship between front month and forward has little to do with intrinsic underlying "value" per se (unless you still believe markets are always right, especially these markets). Was the forward month in May of 2007 a good predictor of bank stock dividends or rates or underlying intrinsic value?

Moreover, the vix curve showed pre meltdown how utterly useless futures are as a predictor of anything (divs, rates included) - different point i know

"similarly, if you assumed a "normalized" cost of capital, by which I assume you mean "higher" - then the June future would be trading higher, not lower."

Are you suggesting in the statement that if interest rates went to 6% that futures would be higher. Lets assume rates are rising for reasons other than sell side cheer leading - for if the US balance were actually healthy and growth was embedded there wouldn't need to be buying hundreds of billions on top of all their other guarantees etc.

Interestingly I looked at the chart of the front month vs. ylds and you are correct on the emini which interestingly was launched on in 1997 (1983 being the first launch which is another interesting year) - right when Rubin was getting warmed up with the strong dollar policy - which incidentally was about the time the gold started moving.

Anyway back to futures. As I read your post, there are 4 variables in that formula but you pay homage to only 2. The mean di yield is 4 percent or so. LEts say mean reversion occurs but the 10 yr goes to 6 plus percent. IF math is correct and you assume rates not moving up becasue of growth (or shall we say sustainable growth or terminal growth) then rates are moving up faser than di yield and underlying appreciation is capped by the secular challenges. Therefore Shouldn't futures in theory fall in this instance to normalize? I looked at the chart from 1997 onward and you are exactly correct that as rates rise so too did the futures - but isn't that at some level counter intuitive?

In other words perhaps you could expound on the index futures and rate relationship and why such a relationship makes sense as prescribed by the very real correlation

Apologies if this comes off as a stupid question - don't trade futures - I am particularly interested in the launch dates as they correspond to critical inflections in the evolution of US bond market

Kid Dynamite said...

anon - let's take this step by step

1) you wrote : "My point is that your discussion of the technical relationship between front month and forward has little to do with intrinsic underlying "value" per se (unless you still believe markets are always right, especially these markets). Was the forward month in May of 2007 a good predictor of bank stock dividends or rates or underlying intrinsic value? "

yes - I think that's my point - that the relationship between futures and cash or further futures is not based on predictions or "value" in the value stock sense - just a simple equation based on interest rates and dividends - the cash flow differences from holding stock or futures - that's it. Mathematical value, not balance sheet value. In May 2007 the futures were trading where they were based solely on expectations of dividends and actual interest rates.

2) you wrote "Moreover, the vix curve showed pre meltdown how utterly useless futures are as a predictor of anything (divs, rates included) - different point i know"

the VIX futures are a different animal. The S&P 500 futures are not utterly useless as a predictor of dividends and rates - they are a pure prediction of exactly that - rates which can be perfectly hedged, and dividends which are, I'm guessing, incredibly accurate on a 6 month forward basis (although I don't have data on that)

3) you wrote: "Are you suggesting in the statement that if interest rates went to 6% that futures would be higher."

I'm not just suggesting it - I'm saying it's a certainty, all other inputs equal. Higher rates make the futures more "desirable" than holding the basket of underlying stocks. It doesn't matter why rates are rising, it just matters what rates are.

I'll answer your final question in my next comment.

Kid Dynamite said...

Anon, to your final question:

you said that there are 4 variables but I paid homage to only two. SPX and ESM aren't variables - they are observable inputs. They just ARE. I think that illustrates a flaw in the final paragraph of your comment. When you looked at past charts of interest rates and "futures," you were probably looking at gross performance of each. That's not what we're talking about here.

I'm talking about the relative relationship between the futures and the cash (aka, spot). I don't think you would have seen this in any charts you could find.

In other words, I'm guessing you observed periods of rising interest rates and rising stock prices. Of course, with rising stock prices, you're also going to see rising futures prices.

Maybe try this: Imagine a board with two nails in it. One nail is the cash (S&P 500 index) and the other is the future. Put a rubber band around the two nails. Now, when the board moves around (up and down, price movements), both nails will move together of course. But what I'm talking about here is the LENGTH OF THE RUBBER BAND - the difference between the stocks and the futures. This changes when one of two things changes:
1) dividends
2) carry costs. Now, carry has other components other than just interest rates - like absolute index level. For example, on your board with the two nails, if the SPX went up 100 points and all other factors were constant (ie, rates unchanged), then the rubber band would actually stretch a little bit - fair value INCREASES because the carry costs go up due to the higher cost of buying all the underlying stocks.

If i didn't answer your question, please let me know. My potentially confusing analogy aside, just remember that we're talking about a mathematical arbitrage (derived from cash flow differences between cash and futures) - not predictions or "value" expectations.

Anonymous said...

I still have a hard time with the premise that SPX is an observable input - isn't the 10 or 2yr an observable input on that basis also and the div yield? It (spx) is the derived value of a series of expected cash flows assuming some discount rate and grow the rate (as derived from a PE for example)? If the cost of carry impacts the futures value then it certainly impacts the underlying value of the basket which is the index. If dividends and rates creep higher marginally but in disproportion to dimmer growth prospects would not the spx rise and in turn pull up the futures in view of the rubber band example?

Anonymous said...

sorry spx fall and pull futures lower despite observable inputs

Marylander said...

Excellent post. Thanks KD.

Kid Dynamite said...

Anon - the SPX price is an observable input because i know the price I have to pay to buy the index. The cost of capital is also an observable input. The dividend yield is an estimate, but again, I think it's going to be pretty accurate on the time period we're talking about. That's why this is an aribitrage - all the inputs in the equation are either 1) known 2) hedgeable or 3) estimatable with high accuracy.

the cost of carry impacts the futures value only in the sense that it determines the difference in value between futures and cash. not in the discounted cash flow valuation of the index.

I think you're confused because you're thinking that if interest rates go from 0% to 10% then it would result in a repricing of stocks, and similarly, if interest rates went from 10% to 0% it would result in a repricing of stocks. Both of those things are true, but we're not talking about stock valuations here - we're talking about futures vs stock fair value difference. maybe you should start from the top of the post again and go through it one more time.

specifically: this sentence: "We can write a very simple equation to tell us what the equality would be if everything were trading at "fair value" and there was no arbitrage opportunity:
(ESM1) + carry = (SPX) + dividends"

if you buy stocks, you get dividends. If you buy futures instead, you get carry. that's it - that's all. simple. those two cash flows must equate. (well, if they don't, then someone can make money by MAKING them equate)