the spot price is approximately equal to the front contract price. Then Equation 2 becomes
EQUATION 3: Joined Price (Roll Date) ≈ Spot Price + Cumulative Roll Adjustment
If we use Equation 3 to calculate the profit of a long futures position that is put on and unwound on a roll date an instant before the roll, we get:
EQUATION 4: Futures Return (Roll Date to Roll Date) = Spot Return + Cumulative Roll Adjustment
And comparing this to Equation 1:
EQUATION 5: Roll Yield Between Roll Dates = Cumulative Roll Adjustment Between Roll Dates
Why is cumulative roll adjustment a useful concept? Since the roll adjustment does not represent the return of any instrument, there has been some debate about the usefulness of such a decomposition. The reason it is useful is that the two terms on the right side of Equation 2 can behave in dramatically different ways.
Fig.2 depicts what Lois would have experienced had she held a long position in the front contract since March 1998 (“Joined Price” curve). It also shows the spliced price (“Front Price” curve). The cumulative roll adjustment is simply the difference between them. It is non-zero on roll dates, and has extended periods where it is upwardly sloping, downwardly sloping, or flat. The long-term persistence across multiple roll days suggests that roll yield is more predictable than spot price. We have essentially decomposed the futures return into a part that looks
very noisy and a part that looks very smooth. To this point, for simplicity, our transformation has focused primarily on roll dates, when spot price approximates futures price. Let’s now introduce a more precise decomposition, which is valid on non- roll dates. Let’s rewrite Equation 2 as:
EQUATION 6: Joined Price = Spot Price + Front Price – Spot Price + Cumulative Roll Adjustment
These cancel out Spot/Futures basis
We have now introduced the concept of spot/ futures basis, which is the difference between the front futures price and the spot price. For markets in backwardation (/contango) the basis is generally negative (/positive). We introduce a fictitious futures market which is perpetually in backwardation, with the deferred contract always three points lower than the front contract. An investor decides to go long this market on the day the front month expires, when the spot price is 17.5 and the new front contract is 14.5. Fig.3 shows the spot price, basis, roll adjustments, and the joined price over time. Note that if we sum the spot, basis, and roll adjustment curves, we get the joined prices. The basis starts at -3 immediately after each roll, and slowly accretes to zero immediately prior to the next roll, as the front futures price converges to the spot price. Now, let’s go back to Equation 6:
EQUATION 7: Joined Price = Spot Price + Basis + Cumulative Roll Adjustment
Noisy, unbounded Bounded Strong momentum characteristics
This looks similar to Equation 3, which combined the spot and basis together. Conceptually, the return due to backwardation/contango is represented by the sum of the basis and roll adjustment. Furthermore, unless the spot price has a strong drift, the cumulative roll adjustment will be the dominant factor driving the futures return. Comparing Equation 7 to the defining equation for roll yield, we get:
EQUATION 8: Roll Yield = Basis Return + Cumulative Roll Adjustment
Note that this differs from Equation 5 in that we are no longer restricted to roll dates. Since the basis is bounded by the shape of the term structure, the basis return is also bounded. Thus over long periods the cumulative roll adjustment will dominate, and we have:
EQUATION 9: Roll Yield ≈ Cumulative Roll Adjustment where equality holds if the basis return is zero.
Implications for trend following The central idea behind trend following is that returns have some degree of persistence. Thus, if total returns have been positive (/negative), a trend- following strategy would go long (/short). Since for futures markets, the total return is reflected in the joined price series, we use joined prices in the calculation of the trend signal.
If we put “persistence” in front of the terms in Equation 4, we get:
EQUATION 10: Persistence in Futures Return = Persistence in Spot Return + Persistence in Cumulative Roll Adjustment
Fig.2 Joined futures price versus front price (i.e. “spliced price”) for S&P 500 index
The Cumulative Roll Adjustment is defined as the difference between the joined and front futures price. Unlike the spot return, the roll adjustment exhibits high autocorrelation. Front price (left axis)
Joined price (left axis) 2000 Dividend yield < funding cost 1600 1200 800 400 0 1998 2001 2004 2007 2010 2013
Dividend yield < funding cost Roll yield is negative; term structure is in contango
Cumulative roll adjustment (right axis)
Dividend yield > funding cost Roll yield is positive;
term structure is in backwardation Dividend yield ≈ funding cost
-50 -100 -150 -200 -250 -300 -350
Source: Bloomberg
0
35
Page 1 |
Page 2 |
Page 3 |
Page 4 |
Page 5 |
Page 6 |
Page 7 |
Page 8 |
Page 9 |
Page 10 |
Page 11 |
Page 12 |
Page 13 |
Page 14 |
Page 15 |
Page 16 |
Page 17 |
Page 18 |
Page 19 |
Page 20 |
Page 21 |
Page 22 |
Page 23 |
Page 24 |
Page 25 |
Page 26 |
Page 27 |
Page 28 |
Page 29 |
Page 30 |
Page 31 |
Page 32 |
Page 33 |
Page 34 |
Page 35 |
Page 36 |
Page 37 |
Page 38 |
Page 39 |
Page 40 |
Page 41 |
Page 42 |
Page 43 |
Page 44 |
Page 45 |
Page 46 |
Page 47 |
Page 48 |
Page 49 |
Page 50 |
Page 51 |
Page 52 |
Page 53 |
Page 54 |
Page 55 |
Page 56 |
Page 57 |
Page 58 |
Page 59 |
Page 60 |
Page 61 |
Page 62 |
Page 63 |
Page 64 |
Page 65 |
Page 66 |
Page 67 |
Page 68 |
Page 69 |
Page 70 |
Page 71 |
Page 72