Bitcoin Forum
December 09, 2016, 03:42:51 PM *
News: To be able to use the next phase of the beta forum software, please ensure that your email address is correct/functional.
 
   Home   Help Search Donate Login Register  
Pages: [1] 2 »  All
  Print  
Author Topic: Goomboo's EMA10/21 Crossover Versus the rebuilder's Flipist Method  (Read 5235 times)
stochastic
Hero Member
*****
Offline Offline

Activity: 532


View Profile
January 28, 2012, 05:25:00 AM
 #1

Using modified code from here and here.  I ran some backtesting in R to compare the EMA 5 EMA 21 Crossover and a random buy/sell strategy based on a coin flip.

Why did I do this?  I was making my own trading strategies but got stuck on making one for bullish/bearish divergence and I thought some practice may help me find out what I am doing wrong.

The EMA 10 / 21 Crossover Strategy is:
1.  Buy when EMA 10 crosses over EMA 21.  So EMA 10 > EMA 21, BUY
2.  Sell when EMA 10 crosses below EMA 21.  So EMA 10 < EMA 21, SELL

The Flipist Strategy is:
1.  Buy when there is heads
2.  Sell when there is tails.

Trading dates will be between July 23, 2011 to Jan 22, 2012.  A total of 184 days.  Money management will be all in and all out.  Trades will be done according to the daily close.

Introducing constraints to the economy only serves to limit what can be economical.
Advertised sites are not endorsed by the Bitcoin Forum. They may be unsafe, untrustworthy, or illegal in your jurisdiction. Advertise here.
stochastic
Hero Member
*****
Offline Offline

Activity: 532


View Profile
January 28, 2012, 05:26:42 AM
 #2

Methods

Data aquisition

Data was acquired from bitcoincharts.com.  The entire data file was downloaded from the Bitcoincharts API ( http://bitcoincharts.com/t/trades.csv?symbol=bcmPPUSD&start=0 ~45mb)

The data was then converted from tick data to Open/High/Low/Close/Adjusted/Volume format.  The code for this conversion is:

Code:
#install.packages("xts") #install the xts package if you don't already have it.  Available on CRAN
require(xts)
ohlc <- function(ttime,tprice,tvolume,fmt)
{
    ttime.int <- format(ttime,fmt)
    data.frame(time = ttime[tapply(1:length(ttime),ttime.int,function(x) {head(x,1)})],
        .Open = tapply(tprice,ttime.int,function(x) {head(x,1)}),
        .High = tapply(tprice,ttime.int,max),
        .Low = tapply(tprice,ttime.int,min),
        .Close = tapply(tprice,ttime.int,function(x) {tail(x,1)}),
        .Volume = tapply(tvolume,ttime.int,function(x) {sum(x)}),
        .Adjusted = tapply(tprice,ttime.int,function(x) {tail(x,1)}))
}
data <- ohlc(data$time,data$price, data$volume,"%Y%m%d%H") #converts data in CSV to OHLC
data <- xts(data[,-1], order.by=data[,1]) #converts data frame to an XTS object

Introducing constraints to the economy only serves to limit what can be economical.
stochastic
Hero Member
*****
Offline Offline

Activity: 532


View Profile
January 28, 2012, 05:29:42 AM
 #3

Flipist Strategy Methods/code

This code will input the data, create the indicators, and then create the signals.  From the daily return and the buy or sell signals, equity curves are calculated.  Last charts are produced to show the performance and draw-down.

Code:
   # Load data
    # install.packages(c("quantmod","TTR"))
    library(quantmod)
    library(TTR)
    x = last(y,184) #gets the last 184 daily positions from the data (starts at July 23, 2011)

    # Calculate the random indicator
    set.seed(43) #include this to reproduce my results
    x$flip <- rbinom(length(x$.Open),1,0.5) #50% probability that 1 (heads/buy) will show and 50% probability that 0 (tails/sell) will show

    # Create the buy (1) and sell (-1) signals
    sigbuy <- ifelse(x$flip == 1, 1, 0)
    sigsell <- ifelse(x$flip == 0, -1, 0)

    # Lag signals to align with days in market,
    # not days signals were generated
    sigbuy <- lag(sigbuy,1) # Note k=1 implies a move *forward*
    sigsell <- lag(sigsell,1) # Note k=1 implies a move *forward*

    # Replace missing signals with no position
    # (generally just at beginning of series)
    sigbuy[is.na(sigbuy)] <- 0
    sigsell[is.na(sigsell)] <- 0

    # Combine both signals into one vector
    sig <- sigbuy + sigsell
    # Calculate Close-to-Close returns
    ret <- ROC(Cl(x))
    ret[1] <- 0

    # Calculate equity curves
    eq_up <- exp(cumsum(ret*sigbuy))
    eq_dn <- exp(cumsum(ret*sigsell*-1))
    eq_all <- exp(cumsum(ret*sig))

    # Equity Chart
    png(filename="flipist.png",width=720,height=720)
    plot.zoo( cbind(eq_up, eq_dn), plot.type="single", ylab=c("Long","Short"), col=c("green","red"), main="Flipist Strategy:\n 2011-07-23 through 2012-01-22" )
    dev.off()

    # Create a chart showing mtgoxUSD
    png("flipistchart.png",width=720,height=720)
    chartSeries(x, subset="last 184 days", type="line")

    # Add the total equity line
    addTA(eq_all)
    dev.off()

    # Evaluate the Strategy

    # install.packages("PerformanceAnalytics")
    require(PerformanceAnalytics)
    
    # chart equity curve, daily performance, and drawdowns
    png("performance-flipist.png",height=720,width=720)
    charts.PerformanceSummary(ret)
    dev.off()

Introducing constraints to the economy only serves to limit what can be economical.
stochastic
Hero Member
*****
Offline Offline

Activity: 532


View Profile
January 28, 2012, 05:32:18 AM
 #4

EMA5 / EMA21 Crossover Strategy

This code will also input the data, create the indicators, and then create the signals.  From the daily return and the buy or sell signals, equity curves are calculated.  Last charts are produced to show the performance and draw-down.

Since the EMA21 requires the past 21 days of history before trading can start, the data file used for this contains the last 205 days but trading cannot start until July 23, 2011.

  
Code:
# install.packages(c("quantmod","TTR"))
    library(quantmod)
    library(TTR)
    # Load presorted data
    x = last(y,205) #gets the last 205 daily positions from the data

    # Calculate the EMA indicators
    ema10 <- EMA(Cl(x),10)
    ema21 <- EMA(Cl(x),21)

    # Create the long (up) and short (dn) signals
    sigbuy <- ifelse(ema10 > ema21, 1, 0)
    sigsell <- ifelse(ema10 < ema21, -1, 0)

    # Lag signals to align with days in market,
    # not days signals were generated
    sigbuy <- lag(sigbuy,1) # Note k=1 implies a move *forward*
    sigsell <- lag(sigsell,1) # Note k=1 implies a move *forward*

    # Replace missing signals with no position
    # (generally just at beginning of series)
    sigbuy[is.na(sigbuy)] <- 0
    sigsell[is.na(sigsell)] <- 0

    # Combine both signals into one vector
    sig <- sigbuy + sigsell

    # Calculate Close-to-Close returns
    ret <- ROC(Cl(x))
    ret[1] <- 0

    # Calculate equity curves
    eq_up <- exp(cumsum(ret*sigbuy))
    eq_dn <- exp(cumsum(ret*sigsell*-1))
    eq_all <- exp(cumsum(ret*sig))

    # Equity Chart
    png("EMAcross.png",width=720,height=720)
    plot.zoo( cbind(eq_up, eq_dn),
    ylab=c("Long","Short"), col=c("green","red"),
    main="EMA5-EMA21 Crossover Strategy: 07-23-2011 to 01-22-2012" )
    dev.off()

    png(filename="emacross.png",width=720,height=720)
    plot.zoo( cbind(eq_up, eq_dn), plot.type="single", ylab=c("Long","Short"), col=c("green","red"), main="EMA Crossover Strategy:\n 2011-07-23 through 2012-01-22" )
dev.off()

    # Create a chart showing mtgoxUSD
    png("EMAcrosschart.png",width=720,height=720)
    chartSeries(x, subset="last 184 days", type="line")
    
    # Add the total equity line
    addTA(eq_all)
    dev.off()

    # Evaluate the Strategy

    # install.packages("PerformanceAnalytics")
    require(PerformanceAnalytics)
    
    # chart equity curve, daily performance, and drawdowns
    png("performance-EMAcross.png",height=720,width=720)
    charts.PerformanceSummary(ret)
    dev.off()

Introducing constraints to the economy only serves to limit what can be economical.
adamstgBit
Legendary
*
Offline Offline

Activity: 1904


Trusted Bitcoiner


View Profile WWW
January 28, 2012, 05:33:40 AM
 #5

Build and run!  Cheesy

bb113
Hero Member
*****
Offline Offline

Activity: 728


View Profile
January 28, 2012, 05:47:13 AM
 #6

Awesome.
organofcorti
Donator
Legendary
*
Offline Offline

Activity: 1960


Poor impulse control.


View Profile WWW
January 28, 2012, 05:59:26 AM
 #7

Using modified code from here and here.  I ran some backtesting in R to compare the EMA 5 EMA 21 Crossover and a random buy/sell strategy based on a coin flip.

Why did I do this?  I was making my own trading strategies but got stuck on making one for bullish/bearish divergence and I thought some practice may help me find out what I am doing wrong.

Why not generate your own? The R package rgp lets you evolve GAs. I've got some that consistently give me better than 'buy and hold'.

Bitcoin network and pool analysis 12QxPHEuxDrs7mCyGSx1iVSozTwtquDB3r
follow @oocBlog for new post notifications
stochastic
Hero Member
*****
Offline Offline

Activity: 532


View Profile
January 28, 2012, 06:00:19 AM
 #8

Flipist Results

**EDIT** Added this results table:
Code:
 
Signal # Trades    % Win Mean Win Mean Loss Median Win Median Loss Mean W/L Median W/L
1     -1       93 52.68817 7.132123 -6.507944   3.636423   -4.839239 1.095910  0.7514452
2      0        1  0.00000      NaN       NaN         NA          NA      NaN         NA
3      1       90 46.66667 4.551028 -4.431499   2.366660   -2.242254 1.026973  1.0554826

The first graph is the equity change based on the long (green) and short (red) positions.




Main graph.  This graph shows the exchange rate at Mtgox at the close of each day, along with the volume and the short and long positions.



Finally, we see a chart giving the cumulative return.  


Introducing constraints to the economy only serves to limit what can be economical.
stochastic
Hero Member
*****
Offline Offline

Activity: 532


View Profile
January 28, 2012, 06:03:26 AM
 #9

Using modified code from here and here.  I ran some backtesting in R to compare the EMA 5 EMA 21 Crossover and a random buy/sell strategy based on a coin flip.

Why did I do this?  I was making my own trading strategies but got stuck on making one for bullish/bearish divergence and I thought some practice may help me find out what I am doing wrong.

Why not generate your own? The R package rgp lets you evolve GAs. I've got some that consistently give me better than 'buy and hold'.

I just used a simple strategy as that is what these two strategies were doing.  Blotter along with quantstrat allows for better trading strategies.

I have not heard of the rgp.  I will check it out.  This was all done with the TTR and quantmod packages.

Introducing constraints to the economy only serves to limit what can be economical.
stochastic
Hero Member
*****
Offline Offline

Activity: 532


View Profile
January 28, 2012, 06:06:31 AM
 #10

Now for the EMA10-EMA21 crossover.

***EDIT*** Added the results table

Code:
 
    Signal  # Trades    % Win     Mean Win    Mean Loss     Median Win     Median Loss   Mean W/L    Median W/L
1      -1      132       56.81818   6.341372   -5.713935     2.840616   -4.183966      1.109808   0.678929
2        0       21         0.00000      NaN         NaN          NA             NA           NaN         NA
3        1       52        57.69231   5.068756   -3.911978     3.199519   -1.747152       1.295702   1.831277


****EDIT**** The following graphs title should say EMA10/21, not EMA5/12








Introducing constraints to the economy only serves to limit what can be economical.
organofcorti
Donator
Legendary
*
Offline Offline

Activity: 1960


Poor impulse control.


View Profile WWW
January 28, 2012, 06:14:42 AM
 #11

The EMA 10 / 21 Crossover Strategy is:
1.  Buy when EMA 10 crosses over EMA 21.  So EMA 10 > EMA 21, BUY
2.  Sell when EMA 10 crosses below EMA 21.  So EMA 10 < EMA 21, SELL

I think most systems based on SMA and EMA crossovers (eg GMMA) give 'buy' or 'sell' signal at the actual point of crossover - ie when (in this case) EMA10==EMA21. The rest of the time they indicate a hold. You shouldn't be changing positions that often.

Edit: Nice chartage though. What package are you using for the bottom one?


Bitcoin network and pool analysis 12QxPHEuxDrs7mCyGSx1iVSozTwtquDB3r
follow @oocBlog for new post notifications
stochastic
Hero Member
*****
Offline Offline

Activity: 532


View Profile
January 28, 2012, 06:18:08 AM
 #12

So what do others think of these graphs?

If I am reading the graphs right.  The flipist method quickly starts out badly, losing equity in both the short and long trades.  It never recovers its equity back.

In the EMA10/21 crossover strategy.  It starts with no change in equity, but it starts to lose money because it is bad at predicting when to sell.  Its long predictions seem pretty good but even the flipist method recovered pretty well in the December bull market.  I would  predict anyone that bought during that time did well.

If you look at the cumulative return.  The flipist method actually did slightly better.  Flipist -0.78 versus EMA10/21 -0.80.

Of course this is assuming I did this correctly.

Comments? Questions? Screams of fury?

Introducing constraints to the economy only serves to limit what can be economical.
stochastic
Hero Member
*****
Offline Offline

Activity: 532


View Profile
January 28, 2012, 06:25:30 AM
 #13

The EMA 10 / 21 Crossover Strategy is:
1.  Buy when EMA 10 crosses over EMA 21.  So EMA 10 > EMA 21, BUY
2.  Sell when EMA 10 crosses below EMA 21.  So EMA 10 < EMA 21, SELL

I think most systems based on SMA and EMA crossovers (eg GMMA) give 'buy' or 'sell' signal at the actual point of crossover - ie when (in this case) EMA10==EMA21. The rest of the time they indicate a hold. You shouldn't be changing positions that often.



That is true.  I will update the code.  There is a way to see how many trades was done but I can't remember right now.  I will search for that.

Introducing constraints to the economy only serves to limit what can be economical.
bb113
Hero Member
*****
Offline Offline

Activity: 728


View Profile
January 28, 2012, 06:26:57 AM
 #14

You need emotional investor and EMA10/21 contrarian controls. I'm not sure how to implement the emotional investor one...
stochastic
Hero Member
*****
Offline Offline

Activity: 532


View Profile
January 28, 2012, 06:34:59 AM
 #15

The EMA 10 / 21 Crossover Strategy is:
1.  Buy when EMA 10 crosses over EMA 21.  So EMA 10 > EMA 21, BUY
2.  Sell when EMA 10 crosses below EMA 21.  So EMA 10 < EMA 21, SELL

I think most systems based on SMA and EMA crossovers (eg GMMA) give 'buy' or 'sell' signal at the actual point of crossover - ie when (in this case) EMA10==EMA21. The rest of the time they indicate a hold. You shouldn't be changing positions that often.

Edit: Nice chartage though. What package are you using for the bottom one?



The bottom chart comes from PerformanceAnalytics, the middle is from quantmod, and the first is from from the zoo package.

I will try this for the signals.

Code:
    sigbuy = ifelse ((ema10 >= ema 21) & (ema10.old < ema21.old), 1, 0)  #buy signal.  yesterday's EMA10 was below yesterday's EMA21 and today it crossed over
    sigsell = ifelse ( (ema10 =< ema 21 & (ema10.0ld > ema21.old), -1, 0) #sell signal.  yesterday's EMA10 was above yesterday's EMA21 and today it crossed under

Introducing constraints to the economy only serves to limit what can be economical.
stochastic
Hero Member
*****
Offline Offline

Activity: 532


View Profile
January 28, 2012, 07:02:23 AM
 #16

I was going to modify the buy sell signals of the EMA10/EMA21 crossover strategy to:

Code:
   sigbuy = ifelse ((x$ema10 > x$ema21) & (x$ema10.old < x$ema21.old), 1, 0)  #buy signal.  yesterday's EMA10 was below yesterday's EMA21 and today it crossed over
    sigsell = ifelse ( (x$ema10 <= x$ema21) & (x$ema10.old > x$ema21.old), -1, 0) #sell signal.  yesterday's EMA10 was above yesterday's EMA21 and today it crossed under

Which means, if yesterday's EMA10 is below yesterday's EMA21 AND EMA10 is greater than or equal to EMA21 today then buy

Also, if yesterday's EMA10 is above yesterday's EMA21 AND EMA10 is less than or equal to EMA21 today then sell

But when I run the data there is only 1 trade at the beginning of December.  So what to do?  I can change the data to hourly.


Introducing constraints to the economy only serves to limit what can be economical.
mav
Full Member
***
Offline Offline

Activity: 168


View Profile
January 28, 2012, 07:32:33 AM
 #17


... when I run the data there is only 1 trade at the beginning of December.  So what to do?  I can change the data to hourly.


interesting 'predicament' but I guess there's no arguing with the notion of 'don't buy' during that big fall, and 'do buy' when confidence came back... I guess we're all action junkies at heart huh?

I find it very interesting that no matter the strategy, essentially you end up following the general market trend no matter what. Of course there are some differences between strategies, so the end result is 'one strategy is relatively better than the other', although when both loose it really reflects the market more than the strategy I reckon.

Also it's interesting that this only works for 'trivial' amounts, as anything more than that would have affected the market and the historical data becomes moot.

Great analysis, it's nice to see a bit of coding to pass the time.

edit:

do you suppose if you ran the flippest method many times and averaged the result at each point in time, you'd end up with a 'returns' chart that was pretty-much the same as the market price?
stochastic
Hero Member
*****
Offline Offline

Activity: 532


View Profile
January 28, 2012, 07:44:19 AM
 #18

To convert the data to hourly is pretty easy.  There is some function in one of the quantitative financial packages in R.  But I forget which one.  But that one function I had before can do the same thing with only a slight modification.

Code:
       
ohlc <- function(ttime,tprice,tvolume,fmt) {
             ttime.int <- format(ttime,fmt)
             data.frame(time = ttime[tapply(1:length(ttime),ttime.int,function(x) {head(x,1)})],
             mtgoxUSD.Open = tapply(tprice,ttime.int,function(x) {head(x,1)}),
             mtgoxUSD.High = tapply(tprice,ttime.int,max),
             mtgoxUSD.Low = tapply(tprice,ttime.int,min),
             mtgoxUSD.Close = tapply(tprice,ttime.int,function(x) {tail(x,1)}),
             mtgoxUSD.Volume = tapply(tvolume,ttime.int,function(x) {sum(x)}),
             mtgoxUSD.Adjusted = tapply(tprice,ttime.int,function(x) {tail(x,1)}))
             }

data <- ohlc(data$time,data$price, data$volume,"%Y%m%d%H") #converts data in CSV to OHLC,[b] notice the %H[/b]

The original data is taken from bitcoincharts and then put through this function.  The output is hourly data.

I also modified the code so that only crossovers indicate buy and sell signals.  Here is the new code for the EMA10/EMA21 crossover.

***EDIT*** Named the graph wrong.

Code:
   library(quantmod)
    library(TTR)
    # Load presorted data
    x = last(y,4436) #gets the last 4436 hourly positions from the data

    # Calculate the EMA indicators
    x$ema10 <- EMA(Cl(x),10)
    x$ema21 <- EMA(Cl(x),21)
    x$ema10.old = as.double(lag(x$ema10))
    x$ema21.old = as.double(lag(x$ema21))

    #omit NA's
    x = na.omit(x)

    # Create the long and short signals
    sigbuy = ifelse ((x$ema10 > x$ema21) & (x$ema10.old < x$ema21.old), 1, 0)  #buy signal.  yesterday's EMA10 was below yesterday's EMA21 and today it crossed over
    sigsell = ifelse ( (x$ema10 <= x$ema21) & (x$ema10.old > x$ema21.old), -1, 0) #sell signal.  yesterday's EMA10 was above yesterday's EMA21 and today it crossed under

    # Lag signals to align with days in market,
    # not days signals were generated
    sigbuy <- lag(sigbuy,1) # Note k=1 implies a move *forward*
    sigsell <- lag(sigsell,1) # Note k=1 implies a move *forward*

    # Replace missing signals with no position
    # (generally just at beginning of series)
    sigbuy[is.na(sigbuy)] <- 0
    sigsell[is.na(sigsell)] <- 0    

    # Combine both signals into one vector
    sig <- sigbuy + sigsell

    # Calculate Close-to-Close returns
    ret <- ROC(Cl(x))
    ret[1] <- 0

    # Calculate equity curves
    eq_up <- exp(cumsum(ret*sigbuy))
    eq_dn <- exp(cumsum(ret*sigsell*-1))
    eq_all <- exp(cumsum(ret*sig))

    # Equity Chart
    png(filename="emacrossnew.png",width=720,height=720)
    plot.zoo( cbind(eq_up, eq_dn), plot.type="single", ylab=c("Long","Short"), col=c("green","red"), main="EMA10/21 Crossover Strategy:\n 2011-07-23 to 2012-01-22" )
    dev.off()

    # Create a chart showing mtgoxUSD
    png("EMAcrosschart-new.png",width=720,height=720)
    chartSeries(x, subset="last 4415 hours", type="line")
    
    # Add the total equity line and EMA lines
    addEMA(n=c(10,21),col=c("orange","blue") )
    addTA(eq_all)
    dev.off()

    # Evaluate the Strategy
    # install.packages("PerformanceAnalytics")
    require(PerformanceAnalytics)
    
    # chart equity curve, daily performance, and drawdowns
    png("performance-EMAcrossnew.png",height=720,width=720)
    charts.PerformanceSummary(ret)
    dev.off()

Introducing constraints to the economy only serves to limit what can be economical.
stochastic
Hero Member
*****
Offline Offline

Activity: 532


View Profile
January 28, 2012, 07:50:07 AM
 #19


... when I run the data there is only 1 trade at the beginning of December.  So what to do?  I can change the data to hourly.


do you suppose if you ran the flippest method many times and averaged the result at each point in time, you'd end up with a 'returns' chart that was pretty-much the same as the market price?

I would expect in a trending market the ratio would be about breaking even, but I never looked before so I can only guess.

Introducing constraints to the economy only serves to limit what can be economical.
stochastic
Hero Member
*****
Offline Offline

Activity: 532


View Profile
January 28, 2012, 08:09:14 AM
 #20

Here is the EMA10/21 crossover using hourly data instead of daily.  This is still all buy or all sell.

I also found  how to get the number of trades.  

Code:
# This function gives us some standard summary
# statistics for our trades.
tradeStats <- function(signals, returns) {
# Inputs:
# signals : trading signals
# returns : returns corresponding to signals

# Combine data and convert to data.frame
sysRet <- signals * returns * 100
posRet <- sysRet > 0 # Positive rule returns
negRet <- sysRet < 0 # Negative rule returns
dat <- cbind(signals,posRet*100,sysRet[posRet],sysRet[negRet],1)
dat <- as.data.frame(dat)

# Aggreate data for summary statistics
means <- aggregate(dat[,2:4], by=list(dat[,1]), mean, na.rm=TRUE)
medians <- aggregate(dat[,3:4], by=list(dat[,1]), median, na.rm=TRUE)
sums <- aggregate(dat[,5], by=list(dat[,1]), sum)

colnames(means) <- c("Signal","% Win","Mean Win","Mean Loss")
colnames(medians) <- c("Signal","Median Win","Median Loss")
colnames(sums) <- c("Signal","# Trades")

all <- merge(sums,means)
all <- merge(all,medians)

wl <- cbind( abs(all[,"Mean Win"]/all[,"Mean Loss"]),
abs(all[,"Median Win"]/all[,"Median Loss"]) )
colnames(wl) <- c("Mean W/L","Median W/L")

all <- cbind(all,wl)
return(all)
}

print(tradeStats(sig,ret))

Code:
 Signal # Trades    % Win  Mean Win  Mean Loss Median Win Median Loss  Mean W/L Median W/L
1     -1       83 51.80723 0.8337976 -0.8799018  0.4516124  -0.5958119 0.9476030  0.7579782
2      0     4249  0.00000       NaN        NaN         NA          NA       NaN         NA
3      1       83 46.98795 0.6240971 -1.4239146  0.4776158  -0.5503347 0.4382968  0.8678642

Here are the graphs.

The equity curve graph with long (green) and short (red).  ***EDIT*** This graph's title should say EMA10/21, not EMA5/21



I tried to plot the EMA10 and 21 on here but the time span is so large it is hard to see.



Here is the cumulative return along with hourly return and draw-down.  Surprisingly or not it is the same as the other daily data set.


Introducing constraints to the economy only serves to limit what can be economical.
Pages: [1] 2 »  All
  Print  
 
Jump to:  

Sponsored by , a Bitcoin-accepting VPN.
Powered by MySQL Powered by PHP Powered by SMF 1.1.19 | SMF © 2006-2009, Simple Machines Valid XHTML 1.0! Valid CSS!