Forex for Beginner: November 2008

Mathematics in Trading: How to Estimate Trade Results
[Rashid Umarov, Rosh]

A certain level of mathematical background is required of any trader,
and this statement needs no proof. The matter is only: How can we
define this minimum required level? In growth of his or her trading
experience, trader often widens his or her outlook "single-handed",
reading posts on forums or various books. Some books require lower
level of mathematical background of readers, some, on the contrary,
inspire one to study or brush up one's knowledge in one field of pure
sciences or another. We will try to give some estimates and their
interpretations in this single article.
If I am going to be fooled by randomness; it better be of the
beautiful (and harmless) kind.
Nassim N. Taleb

Of Two Evils Choose the Least

There are more mathematicians in the world than successful traders.
This fact is often used as an argument by those opposing complex
calculations or methods in trading. We can say against it that trading
is not only ability to develop trading rules (analyzing skills), but
also ability to observe these rules (discipline). Besides, a theory
that would exactly describe pricing on financial markets have not been
yet created by now (I think it will never be created). The creation of
the theory (discovery of mathematical nature) of financial markets
itself would mean death of these markets which is an undecidable
paradox, in terms of philosophy. However, if we face the question of
whether to go to the market with not quite satisfactory mathematical
description of the market or without any description at all, we choose
the least evil: We choose methods of estimation of trading systems.
What is Abnormality of Normal Distribution?

One of basic notions in the theory of probability is the notion of
normal (Gaussian) distribution. Why is it named like this? Many
natural processes turned out to be normally distributed. To be more
exact, the most natural processes, at the limit, reduce to normal
distribution. Let us consider a simple example. Suppose we have a
uniform distribution on the interval of 0 to 100. Uniform distribution
means that probability of falling any value on the interval and
probability of that 3. 14 (Pi) will fall is the same as that of
falling 77 (my favorite number with two sevens). Modern computers help
to generate a rather good pseudorandom-number sequence.

How can we obtain normal distribution of this uniform distribution? It
turns out that, if we take every time several random numbers (for
example, 5 numbers) of a unique distribution and find the mean value
of these numbers (this is called 'to take a sample') and if the amount
of such samples is great, the newly obtained distribution will tend to
normal. The central limit theorem says that this relates to not only
samples taken from unique distributions, but also to a very large
class of other distributions. Since properties of normal distribution
have been studied very well, it will be much easier to analyze
processes if they are represented as a process with normal
distribution. However, seeing is believing, so we can see the
confirmation of this central limit theorem using a simple MQL4 indicator.

Let us launch this indicator on any chart with different N (amount of
samples) values and see that the empirical frequency distribution
becomes smoother and smoother.

[Indicator that creates a normal distribution of a uniform one]
Fig.1. Indicator that creates a normal distribution of a uniform one.

Here, N means how many times we took the average of pile=5 uniformly
distributed numbers on the interval of 0 to 100. We obtained four
charts, very similar in appearance. If we normalize them somehow at
the limit (adjunct to a single scale), we will obtain a several
realizations of the standard normal distribution. The only fly in this
ointment is that pricing on financial markets (to be more exact, price
increments and other derivatives of those increments), generally
speaking, does not fit into the normal distribution. The probability
of a rather rare event (for example, of price decreasing by 50%) on
financial markets is, whereas low, but still considerably higher than
at normal distribution. This is why one should remember this when
estimating risks on the basis of normal distribution.
Quantity Transforms into Quality

Even this simple example of modelling normal distribution shows that
the amount of data to be processed counts for much. The more initial
data there are, the more precise and valid the result is. The smallest
number in the sample is considered to have to exceed 30. It means
that, if we want to estimate results of trades (for example, an Expert
Advisor in the Tester), the amount of trades below 30 is insufficient
to make statistically reliable conclusions about some parameters of
the system. The more trades we analyze, the less the probability is
that these trades are just happily snatched elements of a not very
reliable trading system. Hence, the final profit in a series of 150
trades affords more grounds for putting the system into service than a
system estimated on only 15 trades.
Mathematical Expectation and Dispersion as Risk Estimate

The two most important characteristics of a distribution are
mathematical expectation (average) and dispersion. The standard normal
distribution has a mathematical expectation equal to zero. At that,
the distribution center is located at zero, as well. Flatness or
steepness of normal distribution is characterized by the measure of
spread of a random value within the mathematical expectation area. It
is dispersion that shows us how values are spread about the random
value's mathematical expectation.

Mathematical expectation can be found in a very simple way: For
countable sets, all distribution values are summed up, the obtained
sum being divided by the amount of values. For example, a set of
natural numbers is infinite, but countable, since each value can be
collated with its index (order number). For uncountable sets,
integration will be applied. To estimate mathematical expectation of a
series of trades, we will sum up all trade results and divide the
obtained amount by the amount of trades. The obtained value will show
the expected average result of each trade. If mathematical expectation
is positive, we profit in average. If it is negative, we lose in average.

[Chart of probability density of normal distribution]
Fig.2. Chart of probability density of normal distribution.

The measure of spread of the distribution is the sum of squared
deviations of the random value from its mathematical expectation. This
characteristic of the distribution is called dispersion. Normally,
mathematical expectation for a randomly distributed value is named
M(X). Then dispersion may be described as D(X) = M((X-M(X))^2 ). The
square root of dispersion is named standard deviation. It is also
defined as sigma (σ). It is a normal distribution having mathematical
expectation equal to zero and standard deviation equal to 1 that is
named normal, or Gaussian, distribution.

The higher the value of standard deviation is, the more changeable the
trading capital is, the higher its risk is. If the mathematical
expectation is positive (a profitable strategy) and equal to $100 and
if the standard deviation is equal to $500, we risk a sum, which is
several times larger, to earn each dollar. For example, we have the
results of 30 trades:
Trade Number X (Result)
1 -17.08
2 -41.00
3 147.80
4 -159.96
5 216.97
6 98.30
7 -87.74
8 -27.84
9 12.34
10 48.14
11 -60.91
12 10.63
13 -125.42
14 -27.81
15 88.03

Trade Number X (Result)
16 32.93
17 54.82
18 -160.10
19 -83.37
20 118.40
21 145.65
22 48.44
23 77.39
24 57.48
25 67.75
26 -127.10
27 -70.18
28 -127.61
29 31.31
30 -12.55

To find the mathematical expectation for this sequence of trades, let
us sum up all the results and divide this by 30. We will obtain mean
value M(X) equal to $4.26. To find the standard deviation, let us
subtract the average from each trade's result, square it, and find the
sum of squares. The obtained value will be divided by 29 (the amount
of trades minus one). So we will obtain dispersion D equal to 9
353.623. Having generated square root of the dispersion, we obtain
standard deviation, sigma, equal to $96.71.

The check data are given in the table below:
Trade
Number X
(Result) X-M(X)
(Difference) (X-M(X))^2
(Square of Difference)
1 -17.08 -21.34 455.3956
2 -41.00 -45.26 2 048.4676
3 147.80 143.54 20 603.7316
4 -159.96 -164.22 26 968.2084
5 216.97 212.71 45 245.5441
6 98.30 94.04 8 843.5216
7 -87.74 -92.00 8 464.00
8 -27.84 -32.10 1 030.41
9 12.34 8.08 65.2864
10 48.14 43.88 1 925.4544
11 -60.91 -65.17 4 247.1289
12 10.63 6.37 40.5769
13 -125.42 -129.68 16 816.9024
14 -27.81 -32.07 1 028.4849
15 88.03 83.77 7 017.4129
16 32.93 28.67 821.9689
17 54.82 50.56 2 556.3136
18 -160.10 -164.36 27 014.2096
19 -83.37 -87.63 7 679.0169
20 118.40 114.14 13 027.9396
21 145.65 141.39 19 991.1321
22 48.44 44.18 1 951.8724
23 77.39 73.13 5 347.9969
24 57.48 53.22 2 832.3684
25 67.75 63.49 4 030.9801
26 -127.10 -131.36 17 255.4496
27 -70.18 -74.44 5 541.3136
28 -127.61 -131.87 17 389.6969
29 31.31 27.05 731.7025
30 -12.55 -16.81 282.5761

What we have obtained is the mathematical expectation equal to $4.26
and standard deviation of $96.71. It is not the best ratio between the
risk and the average trade. Profit chart below confirms this:

[Balance graph for trades made]
Fig.3. Balance graph for trades made.
Do I Trade Randomly? Z-Score

The assumption itself that profit gained as a result of a series of
trades is random sounds sardonically for the most of traders. Having
spent a lot of time searching for a successful trading system and
observed that the system found has already resulted in some real
profits on a rather limited period of time, the trader supposes to
have found a proper approach to the market. How can he or she assume
that all this was just a randomness? That's a bit too thick,
especially for newbies. Nevertheless, it is essential to estimate the
results objectively. In this case, normal distribution, again, comes
to the rescue.

We don't know what there will be each trade's result. We can only say
that we either gain profit (+) or meet with losses (-). Profits and
losses alternate in different ways for different trading systems. For
example, if the expected profit is 5 times less than the expected loss
at triggering of Stop Loss, it would be reasonable to presume that
profitable trades (+ trades) will significantly prevail over the
losing ones (- trades). Z-Score allows us to estimate how often
profitable trades are alternated with losing ones.

Z for a trading system is calculated by the following formula:

Z=(N*(R-0.5)-P)/((P*(P-N))/(N-1))^(1/2)

where:
N - total amount of trades in a series;
R - total amount of series of profitable and losing trades;
P = 2*W*L;
W - total amount of profitable trades in the series;
L - total amount of losing trades in the series.

A series is a sequence of pluses followed by each other (for example,
+++) or minuses followed by each other (for example, --). R counts the
amount of such series.

[Comparison of two series of profits and losses]
Fig.4. Comparison of two series of profits and losses.

In Fig. 4, a part of the sequence of profits and losses of the Expert
Advisor that took the first place at the Automated Trading
Championship 2006 is shown in blue. Z-score of its competition account
has the value of -3.85, probability of 99.74% is given in brackets.
This means that, with a probability of 99.74%, trades on this account
had a positive dependence between them (Z-score is negative): a profit
was followed by a profit, a loss was followed by a loss. Is this the
case? Those who were watching the Championship would probably remember
that Roman Rich placed his version of Expert Advisor MACD that had
frequently opened three trades running in the same direction.

A typical sequence of positive and negative values of the random value
in normal distribution is shown in red. We can see that these
sequences differ. However, how can we measure this difference? Z-score
answer this question: Does your sequence of profits and losses contain
more or fewer strips (profitable or losing series) than you can expect
for a really random sequence without any dependence between trades? If
the Z-score is close to zero, we cannot say that trades distribution
differs from normal distribution. Z-score of a trading sequence may
inform us about possible dependence between consecutive trades.

At that, the values of Z are interpreted in the same way as the
probability of deviation from zero of a random value distributed
according to the standard normal distribution (average=0, sigma=1). If
the probability of falling a normally distributed random value within
the range of ±3σ is 99.74%, the falling of this value outside of this
interval with the same probability of 99.74% informs us that this
random value does not belong to this given normal distribution. This
is why the "3-sigma rule'' is read as follows: a normal random value
deviates from its average by no more than 3-sigma distance.

Sign of Z informs us about the type of dependence. Plus means that it
is most probably that the profitable trade will be followed by a
losing one. Minus says that the profit will be followed by a profit, a
loss will be followed by a loss again. A small table below illustrates
the type and the probability of dependence between trades as compared
to normal distribution.
Z-Score Probability of Dependence, % Type of Dependence
-3 99.73 Positive
-2.9 99.63 Positive
-2.8 99.49 Positive
-2.7 99.31 Positive
-2.6 99.07 Positive
-2.5 98.76 Positive
-2 95.45 Positive
-1.5 86.64 Indeterminate
-1.0 68.27 Indeterminate
0.0 0.00 Indeterminate
1.0 68.27 Indeterminate
1.5 86.64 Indeterminate
2.0 95.45 Negative
2.5 98.76 Negative
2.6 99.07 Negative
2.7 99.31 Negative
2.8 99.49 Negative
2.9 99.63 Negative
3.0 99.73 Negative

A positive dependence between trades means that a profit will cause a
new profit, whereas a loss will cause a new loss. A negative
dependence means that a profit will be followed by a loss, whereas the
loss will be followed by a profit. The dependence found allows us to
regulate sizes of positions to be opened (ideally) or even skip some
of them and open them only virtually in order to watch trade sequences.
Holding Period Returns (HPR)

In his book, The Mathematics of Money Management, Ralph Vince uses the
notion of HPR (holding period returns). A trade resulted in profit of
10% has the HPR=1+0.10=1.10. A trade resulted in a loss of 10% has the
HPR=1-0. 10=0.90. You can also obtain the value of HPR for a trade by
dividing the balance value after the trade has been closed
(BalanceClose) by the balance value at opening of the trade
(BalanceOpen). HPR=BalanceClose/BalanceOpen. Thus, every trade has
both a result in money terms and a result expressed as HPR. This will
allow us to compare systems independently on the size of traded
contracts. One of indexes used in such comparison is the arithmetic
average, AHPR (average holding period returns).

To find the AHPR, we should sum up all the HPRs and divide the result
by the amount of trades. Let's consider these calculations using the
above example of 30 trades. Suppose we started trading with $500 on
the account. Let's make a new table:
Trade Number Balance, $ Result, $ Balance at Close, $ HPR
1 500.00 -17.08 482.92 0.9658
2 482.92 -41.00 441.92 0.9151
3 441.92 147.8 589.72 1.3344
4 589.72 -159.96 429.76 0.7288
5 429.76 216.97 646.73 1.5049
6 646.73 98.30 745.03 1.1520
7 745.03 -87.74 657.29 0.8822
8 657.29 -27.84 629.45 0.9576
9 629.45 12.34 641.79 1.0196
10 641.79 48.14 689.93 1.0750
11 689.93 -60.91 629.02 0.9117
12 629.02 10.63 639.65 1.0169
13 639.65 -125.42 514.23 0.8039
14 514.23 -27.81 486.42 0.9459
15 486.42 88.03 574.45 1.1810
16 574.45 32.93 607.38 1.0573
17 607.38 54.82 662.20 1.0903
18 662.20 -160.10 502.10 0.7582
19 502.10 -83.37 418.73 0.8340
20 418.73 118.4 537.13 1.2828
21 537.13 145.65 682.78 1.2712
22 682.78 48.44 731.22 1.0709
23 731.22 77.39 808.61 1.1058
24 808.61 57.48 866.09 1.0711
25 866.09 67.75 933.84 1.0782
26 933.84 -127.10 806.74 0.8639
27 806.74 -70.18 736.56 0.9130
28 736.56 -127.61 608.95 0.8267
29 608.95 31.31 640.26 1.0514
30 640.26 -12.55 627.71 0.9804

AHPR will be found as the arithmetic average. It is equal to 1.0217.
In other words, we averagely earn (1.0217-1)*100%=2.17% on each trade.
Is this the case? If we multiply 2.17 by 30, we will see that the
income should make 65.1%. Let's multiply the initial amount of $500 by
65.1% and obtain $325.50. At the same time, the real profit makes
(627.71-500)/500*100%=25.54%. Thus, the arithmetic average of HPR does
not always allow us to estimate a system properly.

Along with arithmetic average, Ralph Vince introduces the notion of
geometric average that we shall call GHPR (geometric holding period
returns), which is practically always less than the AHPR. The
geometric average is the growth factor per game and is found by the
following formula:

GHPR=(BalanceClose/BalanceOpen)^(1/N)

where:
N - amount of trades;
BalanceOpen - initial state of the account;
BalanceClose - final state of the account.

The system having the largest GHPR will make the highest profits if we
trade on the basis of reinvestment. The GHPR below one means that the
system will lose money if we trade on the basis of reinvestment. A
good illustration of the difference between AHPR and GHPR can be
sashken's account history. He was the Championship's leader for a long
time. AHPR=9.98% impresses, but the final GHPR=-27.68% puts everything
into perspective.
Sharpe Ratio

Efficiency of investments is often estimated in terms of profits
dispersion. One of such indexes is Sharpe Ratio. This index shows how
AHPR decreased by the risk-free rate (RFR) relates to standard
deviation (SD) of the HPR sequence. The value of RFR is usually taken
as equal to interest rate on deposit in the bank or interest rate on
treasury obligations. In our example, AHPR=1.0217, SD(HPR)=0.17607, RFR=0.

Sharpe Ratio=(AHPR-(1+RFR))/SD

where:
AHPR - average holding period returns;
RFR - risk-free rate;
SD - standard deviation.

Sharpe Ratio=(1.0217-(1+0))/0.17607=0.0217/0.17607=0.1232. For normal
distribution, over 99% of random values are within the range of ±3σ
(sigma=SD) about the mean value M(X). It follows that the value of
Sharpe Ratio exceeding 3 is very good. In Fig. 5 below, we can see
that, if the trade results are distributed normally and Sharpe
Ratio=3, the probability of losing is below 1% per trade according to
3-sigma rule.

[Normal distribution of trade results with the losing probability of less]
Fig.5. Normal distribution of trade results with the losing
probability of less than 1%.

The account of Participant named RobinHood confirms this: his EA made
26 trades at the Automated Trading Championship 2006 without any
losing one among them. Sharpe Ratio=3.07!
Linear Regression (LR) and Coefficient of Linear Correlation (CLC)

There is also another way to estimate trade results stability. Sharpe
Ratio allows us to estimate the risk the capital is running, but we
can also try to estimate the balance curve smooth degree. If we impose
the values of balance at closing of each trade, we will be able to
draw a broken line. These points can be fitted with a certain straight
line that will show us the mean direction of capital changes. Let us
consider an example of this opportunity using the balance graph of
Expert Advisor Phoenix_4 developed by Hendrick.

[Balance graph of Hendrick, the Participant of the Automated Trading
Championship 2006]
Fig. 6. Balance graph of Hendrick, the Participant of the Automated
Trading Championship 2006.
We have to find such coefficients a and b that this line goes as close
as possible to the points being fitted. In our case, x is the trade
number, y is the balance value at closing the trade.
x (trades) y (balance)
1 11 069.50
2 12 213.90
3 13 533.20
4 14 991.90
5 16 598.10
6 18 372.80
7 14 867.50
8 16 416.80
9 18 108.30
10 19 873.60
11 16 321.80
12 17 980.40
13 19 744.50
14 16 199.00
15 17 943.20
16 19 681.00
17 21 471.00
18 23 254.90

x (trades) y (balance)
19 24 999.40
20 26 781.60
21 28 569.50
22 30 362.00
23 32 148.20
24 28 566.70
25 30 314.10
26 26 687.80
27 28 506.70
28 24 902.20
29 26 711.60
30 23 068.00
31 24 894.10
32 26 672.40
33 28 446.30
34 24 881.60
35 21 342.60

Coefficients of an approximating straight are usually found by least
squares method (LS method). Suppose we have this straight with known
coefficients а and b. For every x, we have two values: y(x)=a*x+b and
balance(x). Deviation of balance(x) from y(x) will be denoted as
d(x)=y(x)-balance(x). SSD (sum of squared deviations) can be
calculated as SD=Summ{d(n)^2}. Finding the straight by LS method means
searching for such a and b that SD is minimal. This straight is also
named linear regression (LR) for the given sequence.

[Balance value deviation from the straight of y=ax+b]
Fig. 7. Balance value deviation from the straight of y=ax+b

Having obtained coefficients of the straight of y=a*x+b using the LS
method, we can estimate the balance value deviation from the found
straight in money terms. If we calculate the arithmetic average for
sequence d(x), we will be certain that М(d(x)) is close to zero (to be
more exact, it is equal to zero to some calculation accuracy degree).
At the same time, the SSD of SD is not equal to zero and has a certain
limited value. The square root of SD/(N-2) shows the spread of values
in the Balance graph about the straight line and allows to estimate
trading systems at identical values of the initial state of the
account. We will call this parameter LR Standard Error.

Below are values of this parameter for the first 15 accounts in the
Automated Trading Championship 2006:
# Login LR Standard Error, $ Profit, $
1 Rich 6 582.66 25 175.60
2 ldamiani 5 796.32 15 628.40
3 GODZILLA 2 275.99 11 378.70
4 valvk 3 938.29 9 819.40
5 Hendrick 3 687.37 9 732.30
6 bvpbvp 9 208.08 8 236.00
7 Flame 2 532.58 7 676.20
8 Berserk 1 943.72 7 383.70
9 vgc 905.10 6 801.30
10 RobinHood 109.11 5 643.10
11 alexgomel 763.76 5 557.50
12 LorDen 1 229.40 5 247.90
13 systrad5 6 239.33 5 141.10
14 emil 2 667.76 4 658.20
15 payday 1 686.10 4 588.90

However, the degree of approximation of the balance graph to a
straight can be measured in both money terms and absolute terms. For
this, we can use correlation rate. Correlation rate, r, measures the
degree of correlation between two sequences of numbers. Its value may
lie within the range of -1 to +1. If r=+1, it means that two sequences
have identical behavior and the correlation is positive.

[Positive correlation example]
Fig. 8. Positive correlation example.

If r=-1, the two sequences change in opposition, the correlation is
negative.

[Negative correlation example]
Fig. 9. Negative correlation example.

If r=0, it means that there is no dependence found between the
sequences. It should be emphasized that r=0 does not mean that there
is no correlation between the sequences, it just says that such a
correlation has not been found. This must be remembered. In our case,
we have to compare two sequences of numbers: одна последовательность
из графика баланса, а вторая - соответствующие точки на прямой
линейной регрессии.

[Values of balance and points on linear regression]
Fig. 10. Values of balance and points on linear regression.

Below is the table representation of the same data:
Trade
Balance Regression Line
0 10 000.00 13 616.00
1 11 069.52 14 059.78
2 12 297.35 14 503.57
3 13 616.65 14 947.36
4 15 127.22 15 391.14
5 16 733.41 15 834.93
6 18 508.11 16 278.72
7 14 794.02 16 722.50
8 16 160.14 17 166.29
9 17 784.79 17 610.07
10 19 410.98 18 053.86
11 16 110.02 18 497.65
12 17 829.19 18 941.43
13 19 593.30 19 385.22
14 16 360.33 19 829.01
15 18 104.55 20 272.79
16 19 905.68 20 716.58
17 21 886.31 21 160.36

Trade Balance Regression Line
18 23 733.76 21 604.15
19 25 337.77 22 047.94
20 27 183.33 22 491.72
21 28 689.30 22 935.51
22 30 411.32 23 379.29
23 32 197.49 23 823.08
24 28 679.11 24 266.87
25 29 933.86 24 710.65
26 26 371.61 25 154.44
27 28 118.95 25 598.23
28 24 157.69 26 042.01
29 25 967.10 26 485.80
30 22 387.85 26 929.58
31 24 070.10 27 373.37
32 25 913.20 27 817.16
33 27 751.84 28 260.94
34 23 833.08 28 704.73
35 19 732.31 29 148.51

Let's denote balance values as X and the sequence of points on the
straight regression line as Y. To calculate the coefficient of linear
correlation between sequences X and Y, it is necessary to find mean
values M(X) and M(Y) first. Then we will create a new sequence
T=(X-M(X))*(Y-M(Y)) and calculate its mean value as M(T)=cov(X,
Y)=M((X-M(X))*(Y-M(Y))). The found value of cov(X,Y) is named
covariance of X and Y and means mathematical expectation of product
(X-M(X))*(Y-M(Y)). For our example, covariance value is 21 253 775.08.
Please note that M(X) and M(Y) are equal and have the value of 21
382.26 each. It means that the Balance mean value and the average of
the fitting straight are equal.

T=(X-M(X))*(Y-M(Y)) M(T)=cov(X,Y)=M((X-M(X))*(Y-M(Y)))

where:
X - Balance;
Y - linear regression;
M(X) - Balance mean value;
M(Y) - LR mean value.

The only thing that remains to be done is calculation of Sx and Sy. To
calculate Sx, we will find the sum of values of (X-M(X))^2, i.e., find
the SSD of X from its mean value. Remember how we calculated
dispersion and the algorithm of LS method. As you can see they are all
related. The found SSD will be divided by the amount of numbers in the
sequence - in our case, 36 (from zero to 35) - and extract the square
root of the resulting value. So we have obtained the value of Sx. The
value of Sy will be calculated in the same way. In our example,
Sx=5839. 098245 and Sy=4610. 181675.

Sx=Summ{(X-M(X))^2}/N Sy=Summ{(Y-M(Y))^2}/N r=cov(X,Y)/(Sx* Sy)

where:
N - amount of trades;
X - Balance;
Y - linear regression;
M(X) - Balance mean value;
M(Y) - LR mean value.

Now we can find the value of correlation coefficient as r=21 253
775.08/(5839. 098245*4610. 181675)=0.789536583. This is below one, but
far from zero. Thus, we can say that the balance graph correlates with
the trend line valued as 0.79. By comparison to other systems, we will
gradually learn how to interpret the values of correlation
coefficient. At page "Reports" of the Championship, this parameter is
named LR correlation. The only difference made to calculate this
parameter within the framework of the Championship is that the sign of
LR correlation indicates the trade profitability.

The matter is that we could calculate the coefficient of correlation
between the balance graph and any straight. For purposes of the
Championship, it was calculated for ascending trend line, hence, if LR
correlation is above zero, the trading is profitable. If it is below
zero, it is losing. Sometimes an interesting effect occurs where the
account shoes profit, but LR correlation is negative. This can mean
that trading is losing, anyway. An example of such situation can be
seen at Aver's. The Total Net Profit makes $2 642, whereas LR
сorrelation is -0.11. There is likely no correlation, in this case. It
means we just could not judge about the future of the account.
MAE and MFE Will Tell Us Much

We are often warned: "Cut the losses and let profit grow". Looking at
final trade results, we cannot draw any conclusions about whether
protective stops (Stop Loss) are available or whether the profit
fixation is effective. We only see the position opening date, the
closing date and the final result - a profit or a loss. This is like
judging about a person by his or her birth and death dates. Without
knowing about floating profits during every trade's life and about all
positions as a total, we cannot judge about the nature of the trading
system. How risky is it? How was the profit reached? Was the paper
profit lost? Answers to these questions can be rather well provided by
parameters MAE (Maximum Adverse Excursion) and MFE (Maximum Favorable
Excursion).

Every open position (until it is closed) continuously experiences
profit fluctuations. Every trade reached its maximal profit and its
maximal loss during the period between its opening and closing. MFE
shows the maximal price movement in a favorable direction.
Respectively, MAE shows the maximal price movement in an adverse
direction. It would be logical to measure both indexes in points.
However, if different currency pairs were traded,we will have to
express it in money terms.

Every closed trade corresponds to its result (return) and two indexes
- MFE and MAE. If the trade resulted in profit of $100, MAE reaching
-$1000, this does not speak for this trade's best. Availability of
many trades resulted in profits, but having large negative values of
MAE per trade, informs us that the system just "sits out" losing
positions. Such trading is fated to failure sooner or later.

Similarly, values of MFE can provide some useful information. If a
position was opened in a right direction, MFE per trade reached $3000,
but the trade was then closed resulting in the profit of $500, we can
say that it would be good to elaborate the system of unfixed profit
protection. This may be Trailing Stop that we can move after the price
if the latter one moves in a favorable direction. If short profits are
systematic, the system can be significantly improved. MFE will tell us
about this.

For visual analysis to be more convenient, it would be better to use
graphical representation of distribution of values of MAE and MFE. If
we impose each trade into a chart, we will see how the result has been
obtained. For example, if we have another look into "Reports" of
RobinHood who didn't have any losing trades at all, we will see that
each trade had a drawdown (MAE) from -$120 to -$2500.

[Trades distribution on the plane of MAE x Returns]
Fig. 11. Trades distribution on the plane of MAE x Returns

Besides, we can draw a straight line to fit the Returns x MAE
distribution using the LS method. In Fig. 11, it is shown in red and
has a negative slope (the straight values decrease when moving from
left to right). Parameter Correlation(Profits, MAE)=-0.59 allows us to
estimate how close to the straight the points are distributed in the
chart. Negative value shows negative slope of the fitting line.

If you look through other Participants' accounts, you will see that
correlation coefficient is usually positive. In the above example, the
descending slope of the line says us that it tends to get more and
more drawdowns in order not to allow losing trades. Now we can
understand what price has been paid for the ideal value of parameter
LR Correlation=1!

Similarly, we can build a graph of distribution of Returns and MFE, as
well as find the values of Correlation(Profits, MFE)=0.77 and
Correlation(MFE, MAE)=-0.59. Correlation(Profits, MFE) is positive and
tends to one (0.77). This informs us that the strategy tries not to
allow long "sittings out" floating profits. It is more likely that the
profit is not allowed to grow enough and trades are closed by Take
Profit. As you can see, distributions of MAE and MFE дgive us a visual
estimate and values of Correlation(Profits, MFE) and
Correlation(Profits, MAE) can inform us about the nature of trading,
even without charts.

Values of Correlation(MFE, MAE), Correlation(NormalizedProfits, MAE)
and Correlation(NormalizedProfits, MFE) in the Championship
Participants' "Reports" are given as additional information.
Trade Result Normalization

In development of trading systems, they usually use fixed sizes for
positions. This allows easier optimization of system parameters in
order to find those more optimal on certain criteria. However, after
the inputs have been found, the logical question occurs: What sizing
management system (Money Management, MM) should be applied. The size
of positions opened relates directly to the amount of money on the
account, so it would not be reasonable to trade on the account with $5
000 in the same way as on that with $50 000. Besides, an ММ system can
open positions, which are not directly proportional. I mean a position
opened on the account with $50 000 should not necessarily be 10 times
more than that opened on a $5 000 deposit.

Position sizes may also vary according to the current market phase, to
the results of the latest several trades analysis, and so on. So the
money-management system applied can essentially change the initial
appearance of a trading system. How can we then estimate the impact of
the applied money-management system? Was it useful or did it just
worsen the negative sides of our trading approach? How can we compare
the trade results on several accounts having the same deposit size at
the beginning? A possible solution would be normalization of trade
results.

NP=TradeProfit/TradeLots*MinimumLots

where:
TradeProfit - profit per trade in money terms;
TradeLots - position size (lots);
MinimumLots - minimum allowable position size.

Normalization will be realized as follows: We will divide each trade's
result (profit or loss) by the position volume and then multiply by
the minimum allowable position size. For example, order #4399142 BUY
2.3 lots USDJPY was closed with the profit of $4 056. 20 + $118.51
(swaps) = $4 174.71. This example was taken from the account of
GODZILLA (Nikolay Kositsin). Let's divide the result by 2.3 and
multiply by 0.1 (the minimum allowable position size), and obtain a
profit of $4 056.20/2.3 * 0.1 = $176.36 and swaps = $5.15. these would
be results for the order of 0.1-lot size. Let us do the same with
results of all trades and we will then obtain Normalized Profits (NP).

the first thing we think about is finding values of
Correlation(NormalizedProfits, MAE) and Correlation(NormalizedProfits,
MFE) and comparing them to the initial Correlation(Profits, MAE) and
Correlation(Profits, MFE). If the difference between parameters is
significant, the applied method has likely changed the initial system
essentially. They say that applying of ММ can "kill" a profitable
system, but it cannot turn a losing system into a profitable one. in
the Championship, the account of TMR is a rare exception where
changing Correlation(NormalizedProfits, MFE) value from 0.23 to 0.63
allowed the trader to "close in black".
How Can We Estimate the Strategy's Aggression?

We can benefit even more from normalized trades in measuring of how
the MM method applied influences the strategy. It is obvious that
increasing sizes of positions 10 times will cause that the final
result will differ from the initial one 10 times. And what if we
change the trade sizes not by a given number of times, but depending
on the current developments? Results obtained by trust-managing
companies are usually compared to a certain model, usually - to a
stock index. Beta Coefficient shows by how many times the account
deposit changes as compared to the index. If we take normalized trades
as an index, we will be able to know how much more volatile the
results became as compared to the initial system (0.1-lot trades).

Thus, first of all, we calculate covariance - cov(Profits,
NormalizedProfits). then we calculate the dispersion of normalized
trades naming the sequence of normalized trades as NP. For this, we
will calculate the mathematical expectation of normalized trades named
M(NP). M(NP) shows the average trade result for normalized trades.
Then we will find the SSD of normalized trades from M(NP), i.e., we
will sum up (NP-M(NP))^2. The obtained result will be then divided by
the amount of trades and name D(NP). This is the dispersion of
normalized trades. Let's divide covariance between the system under
measuring, Profits, and the ideal index, NormalizedProfits
cov(Profits, NormalizedProfits), by the index dispersion D(NP). The
result will be the parameter value that will allow us to estimate by
how many times more volatile the capital is than the results of
original trades (trades in the Championship) as compared to normalized
trades. This parameter is named Money Compounding in the "Reports". It
shows the trading aggression level to some extent.

MoneyCompounding=cov(Profits, NP)/D(NP)=
M((Profits-M(Profits))*(NP-M(NP)))/M((NP-M(NP))^2)

where:
Profits - trade results;
NP - normalized trade results;
M(NP) - mean value of normalized trades.

Now we can revise the way we read the table of Participants of the
Automated Trading Championship 2006:
# Login LR Standard error, $ LR Correlation Sharpe GHPR
Z-score (%) Money Compounding Profit, $
1 Rich 6 582.66 0.81 0.41 2.55 -3.85(99.74) 17.27 25 175.60
2 ldamiani 5 796.32 0.64 0.21 2.89 -2.47 (98.65) 28.79 15 628.40
3 GODZILLA 2 275.99 0.9 0.19 1.97 0.7(51.61) 16.54 11 378.70
4 valvk 3 938.29 0.89 0.22 1.68 0.26(20.51) 40.17 9 819.40
5 Hendrick 3 687.37 0.79 0.24 1.96 0.97(66.8) 49.02 9 732.30
6 bvpbvp 9 208.08 0.58 0.43 12.77 1.2(76.99) 50.00 8 236.00
7 Flame 2 532.58 0.75 0.36 3.87 -2.07(96.06) 6.75 7 676.20
8 Berserk 1 943.72 0.68 0.20 1.59 0.69(50.98) 17.49 7 383.70
9 vgc 905.10 0.95 0.29 1.63 0.58(43.13) 8.06 6 801.30
10 RobinHood 109.11 1.00 3.07 1.74 N/A (N/A) 41.87 5 643.10
11 alexgomel 763.76 0.95 0.43 2.63 1.52(87.15) 10.00 5 557.50
12 LorDen 1229.40 0.8 0.33 3.06 1.34(81.98) 49.65 5 247.90
13 systrad5 6 239.33 0.66 0.27 2.47 -0.9(63.19) 42.25 5 141.10
14 emil 2 667.76 0.77 0.21 1.93 -1.97(95.12) 12.75 4 658.20
15 payday 1686.10 0.75 0.16 0.88 0.46(35.45) 10.00 4 588.90

The LR Standard error in Winners' accounts was not the smallest. At
the same time, the balance graphs of the most profitable Expert
Advisors were rather smooth since the LR Correlation values are not
far from 1.0. The Sharpe Ratio lied basically within the range of 0.20
to 0.40. The only EA with extremal Sharpe Ratio=3.07 turned not to
have very good values of MAE and MFE.

The GHPR per trade is basically located within the range from 1.5 to
3%. At that, the Winners did not have the largest values of GHPR,
though not the smallest ones. Extreme value GHPR=12.77% says us again
that there was an abnormality in trading, and we can see that this
account experienced the largest fluctuations with LR Standard error=$9
208.08.

Z-score does not give us any generalizations about the first 15
Championship Participants, but values of |Z|>2.0 may draw our
attention to the trading history in order to understand the nature of
dependence between trades on the account. Thus, we know that Z=-3.85
for Rich's account was practically reached due to simultaneous opening
of three positions. And how are things with ldamiani's account?

Finally, the last column in the above table, Money Compounding, also
has a large range of values from 8 to 50, 50 being the maximal value
for this Championship since the maximal allowable trade size made 5.0
lots, which is 50 times more than the minimal size of 0.1 lot.
However, curiously enough, this parameter is not the largest at
Winners. The Top Three's values are 17.27, 28.79 and 16.54. Did not
the Winners fully used the maximal allowable position size? Yes, they
did. the matter is, perhaps, that the MM methods did not considerably
influence trading risks at general increasing of contract sizes. This
is a visible evidence of that money management is very important for a
trading system.
[Rashid Umarov, Rosh]

The 15th place was taken by payday. The EA of this Participant could
not open trades with the size of more than 1. 0 lot due to a small
error in the code. What if this error did not occur and position sizes
were in creased 5 times, up to 5.0 lots? Would then the profit
increase proportionally, from $4 588.90 to $22 944.50? Would the
Participant then take the second place or would he experience an
irrecoverable DrawDown due to increased risks? Would alexgomel be on
the first place? His EA traded with only 1.0-лот trades, too. Or could
vgc win, whose Expert Advisor most frequently opened trades of the
size of less than 1.0 lot. All three have a good smooth balance graph.
As you can see, the Championship's plot continues whereas it was over!
Conclusion: Don't Throw the Baby Out with the Bathwater

Opinions differ. This article gives some very general approaches to
estimation of trading strategies. One can create many more criteria to
estimate trade results. Each characteristic taken separately will not
provide a full and objective estimate, but taken together they may
help us to avoid lopsided approach in this matter.

We can say that we can subject to a "cross-examination" any positive
result (a profit gained on a sufficient sequence of trades) in order
to detect negative points in trading. This means that all these
characteristics do not so much characterize the efficiency of the
given trading strategy as inform us about weak points in trading we
should pay attention at, without being satisfied with just a positive
final result - the net profit gained on the account.

Well, we cannot create an ideal trading system, every system has its
benefits and implications. Estimation test is used in order not to
reject a trading approach dogmatically, but to know how to perform
further development of trading systems and Expert Advisors. In this
regard, statistical data accumulated during the Automated Trading
Championship 2006 would be a great support for every trader.
Created: 2007.08.15 Author: Rashid Umarov
Interview with Al Parsai

I would say while Money Management is a very important element, a
complete EA is the one that consists of a successful strategy, risk
management, and money management. While without money management the
odds of winning are very low, you also need to have a correct
understanding of risk and strategy. The bottom line is that they all
go hand in hand.

Interview with Andrey Vedikhin

The most common error of traders is that they try to write a universal
Expert Advisor, which would remain profitable regardless of the
current market state. In my opinion, the simplest and most effective
solution would be to diversify trading tactics.
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To add comments, please, log in or register

SmartPips wrote:
Will all these analysis be included in the MetaTrader 4 backtesting
reports?

Well, it's difficult to me to give you an exact answer.
Rosh
2007.08.29 13:13
Will all these analysis be included in the MetaTrader 4 backtesting
reports?
SmartPips
2007.08.24 15:30

Writing a MQL4 script would be good to perform the analysis. I was
wondering if anyone has already done it?
SmartPips
2007.08.24 14:57
Thank you, Trong.

1. I consider, is necessary to develop the trading strategy only with
fixed lot. And then to add methods of management of the capital for
maximization of profit and minimization of risk. Therefore, it is not
necessary to spend competitions with fixed lot.

2. I consider as more important smoothness of the balance's(equty's)
curve, than the total Net Profit to Maximal Drawdown ratio. Because
you never know, that current drawdown there will be a Stop Out.

3. You can easy download deep historical rates from History
Center(Press F2 into MetaTrader4 client terminal) . See video How
works downloading from History Center.
Rosh
2007.08.21 13:09

Hi Rosh,

Thank you very much for an informative article. I have a couple of
questions for you:

1) should the championship be with fix 1.0 lot trading? Pips counting
will provide us with the best EAs over the period of 3 months of the
competition. It does not guarantee us that the EAs will be good over term.

2) my risk to reward ratio is defined as maximum drawdown v.s. total
profit with fix 0.1 lot. Say over a period of 3 years, my total profit
is 10,000pips and my max drawdown is 1,000pips, I would have a ratio
of 10:1. All of my EAs are designed with at least 6:1 ratio. Note that
I only open 1 trade at any given time. Can you comment to this?

3) where can we obtain tickdata for backtesting? It seems not many are
aware about the limitation of bacttesting in MT4.

Trong

480
trohoang
2007.08.20 22:04

everlongh wrote:

Rashid, could you please tell us the tools you used for your analysis
and also post the mq4 indicator you used to generate the plots at the
beginning of your article?

See attach here - 'Mathematics in Trading: How to Estimate Trade
Results' (NormalDistribution.mq4)
Rosh
2007.08.20 20:03

Great article Rashid!! Also, appreciate the comment by Luis. The point
for me is that standard technical indicators and analysis only with
backtesting for profits is not enough. The approach of applying the
Scientific method and statistical inference is key to developing a
long term successful EA.

Rashid, could you please tell us the tools you used for your analysis
and also post the mq4 indicator you used to generate the plots at the
beginning of your article?
428
everlongh
2007.08.20 19:28
Mmmmm, it seems you rushed to answer without a proper consideration.
Anyway, thank you for the clever and stimulating article!
207
ldamiani
2007.08.16 14:44
Dear Luis Damiani.

Z allows us to judge about dependence between trades that are not made
at the same time. The interval between position openings is
insufficient for this. It is more important that there is a minimal
interval (at least 1 second) between closing of one position and
opening of another one. Your trades opened one by one without waiting
for close. So we cannot make any conclusions about the benefits from
the dependence you discovered.

See picture from your account:

I 'm agree with you that Sortino sometimes is better then Sharpe. MAE
and MFE of GUMASA trades is good.

I wish you good luck in the forthcoming Championship.
Rosh
2007.08.16 12:02

Interesting article, Rashid, but what kind of conclusions and
considerations would you make about GUMASA (my EA)? I know this
conclusions may not have a high degree of certanty (given the limited
history available), but I am really interested in them. I am bit
sketical about using Sharpe ratio. It may be useful to people who are
highly averse to risk. Developers should not use it as a parameter to
be optimized, my opinion. Perhaps a better risk/reward ratio to be
used as parameter of comparison and optimization is the Sortino's
ratio. What do you think?

Just to answer your question in:

we know that Z=-3.85 for Rich's account was practically reached due to
simultaneous opening of three positions. And how are things with
ldamiani's account?

All GUMASA's trades had at least 5 hours between them (built in
parameter), as you can see on the Account History (
http://championship.mql4.com/2006/users/ldamiani/ )

Luis Guilherme Damiani

May the odds be with you !!
207
ldamiani
2007.08.16 05:56

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