## Wednesday, 8 May 2013

### Random Entries

Over on the Tradersplace forum there is a thread about random entries, early on in which I replied

"...I don't think the random entry tests as outlined above prove anything. First off, I think there is a subtle bias in the results. I could say that a 50/50 win/lose random entry by rolling dice with payoffs of -1, -2, -3, +4, +5 and +6 is good because there is an expected +9 every 6 rolls so random entries are great, whereas payoffs of +1, +2, +3, -4, -5 and -6 is bad with expectation -9 every 6 rolls, so random entries suck. These two "tests" prove nothing about random entries because the payoffs aren't taken into account.

In a similar fashion a 50/50 choice to be long or short means nothing if there is an extended trend in one direction or another such that the final price of the tradeable is much higher or lower in February 2013 than it was in January 1990. Of course a 50% chance to get on the right side of such a market will on average show a profit, just as with the first dice example above, but this profit results from the bias (trend) in the data and not the efficacy of the entry or exit. At a bare minimum, the returns of the tradeable should be detrended to remove this bias before any tests are conducted."

Later on in the thread the OP asked me to expand on my views and this post is my exposition on the matter. First off, I am going to use simulated markets because I think it will make it easier to see the points I am trying to make. If real data were used, as in the OP's tests, along with the other assumptions such as 1% risk, choice of markets etc. the fundamental truth of the random entries would be obfuscated by the "noise" these choices introduce. To start with I have created the following interactive Octave script
clear all

fprintf('\nMarket oscillations are represented by a sine wave.\n')

period = input('Choose a period for this sine wave: ')

% create sideways market - a sine wave
sideways_market = sind(0:(360.0/period):3600).+2 ;
clf ;
plot(sideways_market) ;
title('Sideways Market') ;

fprintf('\nDisplayed is a plot of the chosen period sine wave, representing a sideways market.\nPress enter to continue.\n')
pause ;

sideways_returns = diff(log(sideways_market)) ;

fprintf('\nThe total log return (sum of log differences) of this market is %f\n', sum(sideways_returns) ) ;

fprintf('\nPress enter to see next market type.\n') ;
pause ;

% and create a trending market
trending_market = ((0:1:size(sideways_market,2)-1).*1.333/period).+sideways_market ;
clf ;
plot(trending_market) ;
title('Trending Market') ;

fprintf('\nNow displayed is the same sine wave with a trend component to represent\nan uptrending market with 50%% retracements.\nPress enter to see returns.\n')
pause ;

trending_returns = diff(log(trending_market)) ;

fprintf('\nThe total log return of this trending market is %f\n', sum(trending_returns) ) ;

fprintf('\nNow we will do the Monte Carlo testing.\n')
iter = input('Choose number of iterations for loop: ')

sideways_iter = zeros(iter,1) ;
trend_iter = zeros(iter,1) ;

for ii = 1:iter

% create the random position vector
ix = rand( size(sideways_returns,2) , 1 ) ;
ix( ix >= 0.5 ) = 1 ;
ix( ix &lt; 0.5 ) = -1 ;

sideways_iter( ii , 1 ) = sideways_returns * ix ;
trend_iter( ii , 1 ) = trending_returns * ix ;

end

clf ;
hist(sideways_iter)
title('Returns Histogram for Sideways Market')

fprintf('\nNow displayed is a histogram of sideways market returns.\nPress enter to see trending market returns.\n')
pause ;

clf ;
hist(trend_iter)
title('Returns Histogram for Trending Market')

fprintf('\nFinally, now displayed is a histogram of the trending market returns.\n')


the output of which is shown in the desktop recording video below.

The purpose of this basic coding introduction is to show that random entries combined together with random exits have no expected value, evidenced by the histograms for Monte Carlo returns for both the sideways market and the trending market being centred about zero. The video might not clearly show this, so readers are invited to run the code for themselves. I would suggest values for the period in the range of 10 to 50.

However, in the linked thread the OP did not use random exits but instead used a "trailing 5 ATR stop based on a 20 day simple average ATR," i.e. a random entry with a heuristically pre-defined exit criteria. To emulate this, in the next code box I have coded a trailing stop exit based on the standard deviation of the bar to bar differences. I deemed this approach to be necessary for the simulated markets being used as there are no OHLC bars from which to calculate ATR. As in the earlier code box the code creates the simulated markets with user input prompted from the terminal, displays them, then performs the random entry Monte Carlo routine with a user chosen "standard deviation stop multiplier," produces histograms of the MC routine results, displays simple summary statistics and then finally displays plots of the markets along with the relevant trailing stops, with the stop levels being in red.
clear all

fprintf('\nMarket oscillations are represented by a sine wave.\n')

period = input('Choose a period for this sine wave: ')

% create sideways market - a sine wave
sideways_market = sind(0:(360.0/period):3600).+2 ;
clf ;
plot(sideways_market) ;
title('Sideways Market') ;

fprintf('\nDisplayed is a plot of the chosen period sine wave, representing a sideways market.\nPress enter to continue.\n')
pause ;

sideways_returns = diff(log(sideways_market)) ;

fprintf('\nThe total log return (sum of log differences) of this market is %f\n', sum(sideways_returns) ) ;

fprintf('\nPress enter to see next market type.\n') ;
pause ;

% and create a trending market
trending_market = ((0:1:size(sideways_market,2)-1).*1.333/period).+sideways_market ;
clf ;
plot(trending_market) ;
title('Trending Market') ;

fprintf('\nNow displayed is the same sine wave with a trend component to represent\nan uptrending market with 50%% retracements.\nPress enter to see returns.\n')
pause ;

trending_returns = diff(log(trending_market)) ;

fprintf('\nThe total log return of this trending market is %f\n', sum(trending_returns) ) ;

fprintf('\nNow we will do the Monte Carlo testing.\n')
iter = input('Choose number of iterations for loop: ')
stop_mult = input('Choose a standard deviation stop multiplier: ')

sideways_iter = zeros(iter,1) ;
trend_iter = zeros(iter,1) ;

sideways_position_vector = zeros( size(sideways_returns,2) , 1 ) ;
sideways_std_stop = stop_mult * std( diff(sideways_market) ) ;

trending_position_vector = zeros( size(trending_returns,2) , 1 ) ;
trending_std_stop = stop_mult * std( diff(trending_market) ) ;

for ii = 1:iter

position = rand(1) ;
position( position >= 0.5 ) = 1 ;
position( position &lt; 0.5 ) = -1 ;

sideways_position_vector(1,1) = position ;
trending_position_vector(1,1) = position ;

sideways_long_stop = sideways_market(1,1) - sideways_std_stop ;
sideways_short_stop = sideways_market(1,1) + sideways_std_stop ;

trending_long_stop = trending_market(1,1) - trending_std_stop ;
trending_short_stop = trending_market(1,1) + trending_std_stop ;

for jj = 2:size(sideways_returns,2)

if sideways_position_vector(jj-1,1)==1 && sideways_market(1,jj)&lt;=sideways_long_stop
position = rand(1) ;
position( position >= 0.5 ) = 1 ;
position( position &lt; 0.5 ) = -1 ;
sideways_position_vector(jj,1) = position ;
sideways_long_stop = sideways_market(1,jj) - sideways_std_stop ;
sideways_short_stop = sideways_market(1,jj) + sideways_std_stop ;
elseif sideways_position_vector(jj-1,1)==-1 && sideways_market(1,jj)>=sideways_short_stop
position = rand(1) ;
position( position >= 0.5 ) = 1 ;
position( position &lt; 0.5 ) = -1 ;
sideways_position_vector(jj,1) = position ;
sideways_long_stop = sideways_market(1,jj) - sideways_std_stop ;
sideways_short_stop = sideways_market(1,jj) + sideways_std_stop ;
else
sideways_position_vector(jj,1) = sideways_position_vector(jj-1,1) ;
sideways_long_stop = max( sideways_long_stop , sideways_market(1,jj) - sideways_std_stop ) ;
sideways_short_stop = min( sideways_short_stop , sideways_market(1,jj) + sideways_std_stop ) ;
end

if trending_position_vector(jj-1,1)==1 && trending_market(1,jj)&lt;=trending_long_stop
position = rand(1) ;
position( position >= 0.5 ) = 1 ;
position( position &lt; 0.5 ) = -1 ;
trending_position_vector(jj,1) = position ;
trending_long_stop = trending_market(1,jj) - trending_std_stop ;
trending_short_stop = trending_market(1,jj) + trending_std_stop ;
elseif trending_position_vector(jj-1,1)==-1 && trending_market(1,jj)>=trending_short_stop
position = rand(1) ;
position( position >= 0.5 ) = 1 ;
position( position &lt; 0.5 ) = -1 ;
trending_position_vector(jj,1) = position ;
trending_long_stop = trending_market(1,jj) - trending_std_stop ;
trending_short_stop = trending_market(1,jj) + trending_std_stop ;
else
trending_position_vector(jj,1) = trending_position_vector(jj-1,1) ;
trending_long_stop = max( trending_long_stop , trending_market(1,jj) - trending_std_stop ) ;
trending_short_stop = min( trending_short_stop , trending_market(1,jj) + trending_std_stop ) ;
end

end

sideways_iter( ii , 1 ) = sideways_returns * sideways_position_vector ;
trend_iter( ii , 1 ) = trending_returns * trending_position_vector ;

end

mean_sideways_iter = mean( sideways_iter ) ;
std_sideways_iter = std( sideways_iter ) ;
sideways_dist = ( mean_sideways_iter - sum(sideways_returns) ) / std_sideways_iter

mean_trend_iter = mean( trend_iter ) ;
std_trend_iter = std( trend_iter ) ;
trend_dist = ( mean_trend_iter - sum(trending_returns) ) / std_trend_iter

clf ;
hist(sideways_iter)
title('Returns Histogram for Sideways Market')
xlabel('Log Returns')

fprintf('\nNow displayed is a histogram of random sideways market returns.\nPress enter to see trending market returns.\n')
pause ;

clf ;
hist(trend_iter)
title('Returns Histogram for Trending Market')
xlabel('Log Returns')

fprintf('\nFinally, now displayed is a histogram of the random trending market returns.\n')

sideways_trailing_stop_1 = sideways_market .+ sideways_std_stop ;
sideways_trailing_stop_2 = sideways_market .- sideways_std_stop ;

trending_trailing_stop_1 = trending_market .+ trending_std_stop ;
trending_trailing_stop_2 = trending_market .- trending_std_stop ;

fprintf('\nNow let us look at the stops for the sideways market\nPress enter\n')
pause ;

clf ;
plot(sideways_market(1,1:75),'b',sideways_trailing_stop_1(1,1:75),'r',sideways_trailing_stop_2(1,1:75),'r')

fprintf('\nTo look at the stops for the trending market\nPress enter\n')
pause ;

clf ;
plot(trending_market(1,1:75),'b',trending_trailing_stop_1(1,1:75),'r',trending_trailing_stop_2(1,1:75),'r')


Discussion
As my above reply was concerned with the OP's application of random entries on trending data I will start with the trending market output of the code. For the purposes of a "trending market" the code creates an upwardly trending market that has retracements to an imagined 50% Fibonacci retracement level of the immediately preceding upmove. The "buy and hold" log return of this trending market for a selected period of 20 is 2.036665. Below can be seen four histograms representing 5000 MC runs with the stop multiplier being set at 1,2, 3 and 4 standard deviations, running horizontally from the top left to the bottom right.

From the summary statistics output (readers are invited to run the code themselves to see) not one shows an average log return that is greater, to a statistically significant degree, than the "buy and hold" return. In fact, for standard deviations 2, 3, and 4 the returns are less than "buy and hold" to a statistically significant degree. What this perhaps shows is that there are many more opportunities to get stop choices wrong than there are to get them right! And even if you do get it right (standard deviation 1), well, what's the point if there is no significant improvement over buy and hold? This is the crux of my original assertion that the OP's random entry tests don't prove anything.

However, the above relates only to random entries in a trend following context, with a trailing stop being set wide enough to "allow the trend to breathe" or "avoid the noise" or other such sayings common in the trend following world. What happens if we tighten the stop? Several 5000 MC runs with a standard deviation stop multiplier of 0.6 shows significant improvement, with the "buy and hold" returns being in the left tail or more than 2 standard deviations away from the average random entry return. To achieve this a fairly tight stop is required, as can be seen below.

Now rather than being a "trend following stop" this could be characterised as a "swing trading stop" where one is trying to get in and out of the market at swing highs and lows. But what if the market is not making swings but is in fact strongly trending as in
which can be achieved by altering a code line thus:

% and create a trending market
trending_market = ((0:1:size(sideways_market,2)-1).*4/period).+sideways_market ;

The same stops as above on this market all show average positive returns, but none so great as to be significantly better than this market's "buy and hold" return of 3.04452. A typical histogram for these tests is
which is quite interesting. The large bar at the right represents those MC runs where the first random position is a long which is not stopped out by the wide stops, hence resulting in a log return for the run of the "buy and hold" log return. The other bars show those MC runs which start with one or more short trades and obviously incur some initial loses prior to the first long trade. Drastically tightening the stop for this market doesn't really help things; the result is still an extreme right hand bar at the "buy and hold" level with a varying left tail.

What all this shows is that the best random entries can do is capture what I called the "bias" in my above quoted thread response, and even doing this well is highly dependent on using a suitable trailing stop; as also mentioned above there are many more opportunities to choose an unsuitable trailing stop. I also suggested that "At a bare minimum, the returns of the tradeable should be detrended to remove this bias before any tests are conducted." The point of this is that by removing the bias it immediately becomes obvious that there is no value in random entries - it will effectively produce the results that the above code gives for the sideways market, shown next.

Standard deviations 2 and 3 are what one might expect - histograms centred around a net log return of zero, standard deviation 4 provides a great opportunity to shoot yourself in the foot by choosing a terrible stop for this market, and standard deviation 1 shows what is possible by choosing a good stop. In fact, by playing around with various values for the stop multiplier I have been able to get values as high as 14 for the net average log return on this market, but such tight stops as these cannot be really be considered trailing stops as they more or less act as take profit stops at the upper and lower levels of this sideways market.

So, what use are random entries? In live trading I believe there is no use whatsoever, but as a "research tool" similar to the approach outlined here there may be some uses. They could be used to set up null hypotheses for statistical testing and benchmarking, or they could be used to see what random looks like in the context of some trading idea. However, the big caveat to this is that if one uses real data how can the randomness in the data, plus any possible non-linear effects of parameter choices, be distinguished from the effects of the "injected randomness" supplied by the random entries? All the forgoing discussion has been based on the clean, simple and predictable data of a fully-known, simulated market, with the intent of illustrating my belief about random entries. When applied portfolio wide on real data, with portfolio heat restrictions, position sizing choices etc. the whole test routine may simply have too many moving parts to draw any useful conclusions about the efficacy of random entries given that statistically speaking, even on the idealised market data used in this post, random entry trend following returns are indistinguishable from buy and hold returns.