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This is part 2 of the post.
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5. Continuously adjust the price series over time ("Perpetual")
This approach adds some of the prices from the current contract with some
from the next contract. The continuous futures contract value initially
would be composed of a large percentage of the current contract and a small
percentage of the next contract. As a contract expiration date becomes
closer, a progressively smaller percentage of the current contract and
larger percentage of the new contract is used. This results in a smooth
blend from one contract price level to the next.
The calculations are shown below:
Assume we are merging the prices of the 9/98 and the 12/98 contracts.
Assume the price in each contract can be represented by the following:
P1 = C + F * d1 + s1
P2 = C + F * d2 + s2
where:
P1 is the price of the 9/98 contract on any bar
P2 is the price of the 12/98 contract on any bar
d1 = days to expiration on the 9/98 contract
d2 = days to expiration on the 12/98 contract
s1 = a residual value indicating how the actual price differs
from "fair value" on the 9/98 contract
s2 = a residual value indicating how the actual price differs
from "fair value" on the 12/98 contract
F is the "Fair Value Factor" defined as follows:
F = C * (i - v) / 365 (about 13% then)
C = the value of the S&P cash index = about 1186 then
i = interest rate for fair value calculation = about 5% then
v = dividend rate of the S&P cash index = about 1% then
By manipulating the algebra, you can show that the merged contract price,
Px, equals:
Px = (91 * F) + (s1 * d1 / 91) + (s2 * (1 - d1 /91))
This consists of three terms:
(91 * F) = (91 * 0.13) = about 12 then. This is the premium at the
start of each contract due to the fair value calculation.
(s1 * d1 / 91) = the residual value of the 9/98 contract weighted
by the days remaining on the 9/98 contract.
(s2 * (1 - d1 /91)) = the residual value of the 12/98 contract
inversely weighted by the days remaining on the 9/98 contract.
So there is a fairly constant offset due to the fair value rates, plus the
weighted average of the residual price values of the two contracts. It will
be about 1% higher that the S&P cash index at all times with no
discontinuities and noisier because the volatility of the futures contract
is greater than that of the cash index. The residual value of the combined
price series will have volatility less than either of its two components
since uncorrelated random variations will partially cancel.
The characteristics of the resulting adjusted price will be quite different
than the price of either contract. The prices will not fall on multiples of
the minimum price movement such as 1/32 in the case of bonds. We could
round to those increments but this would introduce a new source of noise.
The volatility of the resulting price will be less than the real contracts,
particularly near expirations. Finally the price will always be about 1%
higher than the S&P cash index. This is unlike the futures price, which
decreases at about a 4% per year rate over time. All of these factors can
affect the performance of a trading system and could make the results of
backtesting differ from trading a single contract.
For a long-term system, one that stayed in the market for many months,
using such data would be very convenient since it would eliminate the need
to ever worry about expirations in backtesting.
For a medium-term system, say one that stayed in the market for many days
to weeks, such distortions would be insignificant.
For short-term trading, say under a few days, they become important.
For a day-trading system they could be serious sources of error. Systems
that trade on the volatility of the price data will not work as well on
such data since the volatility of the merged data is less than that of
either contract.
6. Use continuous adjustment for both backtesting and trading
The distortions mentioned above can be eliminated by actually trading a mix
of the two contracts. This is only practical if you are trading perhaps
over 20 contracts so that you can adjust the number of each in the proper
proportions over time. It also adds a level of complexity to the trading
system.
IV. Summary
There is no "best" method in an absolute sense. All the methods have
advantages and disadvantages. The best method in particular cases depends
on the type of market analysis being done or the type of trading strategy
being used.
For instance, trading systems that compare current prices to distant past
prices probably will back-test more realistically with Method 5, because
there is less long-term price distortion. However, Method 5 tends to
produce very unrealistic results with trading strategies that depend upon
price waveshape measurement, because there will be significant medium and
short-term waveshape distortion. Neural networks, exotic waveshape filters,
wavelet methods, Fourier methods, and many other advanced methods do not
back test realistically with that method. Method 3 or 4 will be much better
to use in those cases. Method 4, using backward price adjustment is the
most popular.
V. My Conclusions
I use Method 4 - back adjusted - because my objective is to simulate the
behavior of my trading system on actual contract data as accurately as
possible. My trading systems use the bar-to-bar price differences so are
unaffected by adding a constant to all prices. Since method 4 (and method
3) maintains the exact bar-to-bar price changes of each original contract,
this is what I prefer.
Needless to say, other people might consider other factors more important.
I have no interest in having my systems perform more poorly on backtesting
that they would have on actual contracts. (Maybe you could call it some
sort of "stress testing".)
If I were using a long term holding period, I would definitely use method 5
- the continuously adjusted contract but that is not what I am trading.
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I hope this has been useful. I would appreciate any feedback.
Bob Fulks
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