Expectancy Management

Quoting FXEZ

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I suppose you're talking in terms of a strategy that has an entry / stop / target, or simply entry and some sort of exit / flat period.

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Quote

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Are you defining correlation in some other manner?

Quoting PipMeUp

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This is really counter-intuitive. It means that when you accumulated a huge amount of money, you should stop trading it. That is favouring under-capitalization... This is the fractal dimension of the market. Because the price doesn't move in straight line, doubling the target more than double the time to reach it. This reduces the frequency of trading. {image}

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Quoting PipMeUp

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Because I do my homework even, and especially, when I'm told it is useless... I ran two Monte-Carlo simulations of two Bernoulli trials. In these two games either you lose 1 or you win R. Probability of winning is P. The first one simulates a system with 70% win rate and a R:R of 1. Its expectancy is 0.4. The second one simulates a system with only 28% win rate but a R:R of 4. Its expectancy is 0.4 as well. Apple to apple! In the first 2 pictures below, 100 trades are simulated 1000 times each. Because drawing 1000 curves doesn't look good, the...

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expectancy <- function(p, reward=1, risk=1, numTrades=100) {
  outcomes <- runif(numTrades, min=0, max=1)
  winners <- sum(outcomes <= p)
  losers <- sum(outcomes > p)
  structure(
    (winners*reward - losers*risk)/numTrades,
    data=outcomes,
    p=p,
    class='expectancyAnalysis'
  ;)
}

expectancyMC <- function(p, reward=1, risk=1, numTrades=100, numSimulations=10000) {
  sapply(
    1:numSimulations, 
    function(x) expectancy(p, reward, risk, numTrades), 
    simplify=FALSE
  ;)
}


expectationAnalysis <- function(
  experimentName, prWin, numTrades, reward, 
  risk=1, openingBalance=100, simulations=10000, outputPath="~/tmp",
  saveEquityCurves=F
) {
  result <- expectancyMC(prWin, reward, risk, numTrades, simulations)
  
  # Mean expectation from the simulations
  theExps <- sapply(result, function(x) as.numeric(x))
  meanExp <- mean(theExps)
  cat("Mean expectancy is: ", meanExp, "\n")
  
  # Calculate equity curves
  # Assumption that risk express a percentage of capital e.g. risk = 1 implies each trade risk 1% etc.
  equityCurves <- sapply(
    result,
    function(x) {
      outcomes <- attr(x, 'data') <= prWin
      balances <- c(openingBalance)
      rp <- risk/100
      for (idx in 2:length(outcomes)) {
        if ( balances[idx-1] <= 0 ) {
          newBalance <- 0
        } else {
          newBalance <- ifelse(outcomes[idx], balances[idx-1]*(1+rp), balances[idx-1]*(1-rp))
        }
        balances <- c(balances, newBalance)
      }
      balances
    }
  ;)
  
  
  # Plot expectation results
  png(
    paste(outputPath, paste(experimentName, "_expectancy_hist.png", sep=''), sep='/'), 
    width=1920, height=1080
  ;)
  hist(theExps, prob=T, xlab=sprintf("Expectancy (p=%3.2f)",prWin),  
       main=sprintf("Experiment %s expectancy distribution (E_{0}=%3.3f, N=%d)", experimentName, meanExp, simulations))
  abline(v=meanExp, lty=1, lwd=5, lend=1, col=rgb(1,0,0,0.5))
  lines(density(theExps), lty=2, lwd=2, col='blue')
  dev.off()
  
  # Plot equity curves
  col.best <- rgb(0, 204/255, 51/255)
  col.average <- rgb(51/255, 153/255, 1)
  col.worst <-rgb(154/255, 0, 51/255)
  png(
    paste(outputPath, paste(experimentName, "_equitycurves.png", sep=''), sep='/'), 
    width=1920, height=1080
  ;)  
  best <- which(equityCurves[100,] == max(equityCurves[100,]))
  worst <- which(equityCurves[100,] == min(equityCurves[100,]))
  plot(
    x=c(1, numTrades), y=range(c(0, pretty(equityCurves))), type='n', 
    main=paste("Equity curves (Experiment: ", experimentName, ")", sep=""), 
    xlab='Trade', ylab='Equity ($)'
  ;)
  lower <- apply(equityCurves, 1, min)
  upper <- apply(equityCurves, 1, max)
  polygon(c(1:numTrades, numTrades:1), c(lower, rev(upper)), col=rgb(0.8, 0.8, 0.8, .5), border=NA)
  lines(x=1:numTrades, y=equityCurves[, best], lwd=3, col=col.best)
  lines(x=1:numTrades, y=apply(equityCurves, 1, mean), lwd=2, lty=2, col=col.average)
  lines(x=1:numTrades, y=equityCurves[, worst], lwd=3, col=col.worst)
  legend("topleft", c("Best", "Worst", "Average"), col=c(col.best, col.worst, col.average), bty='n', lty=c(1,1,2), lwd=2)
  dev.off()
  
  # Save equity curves?
  if (saveEquityCurves) {
    write.csv(equityCurves, paste(outputPath, paste(experimentName, "_equitycurves.csv", sep=''), sep='/'))
  }
  
  theExps
}


expecations <- expectationAnalysis(
  experimentName="70perc_1t1", prWin=0.7, numTrades=100, reward=1, 
  risk=1, openingBalance=100, simulations=10000, outputPath="~/dropbox/personal/Dropbox/Public/ff/expectancy",
  saveEquityCurves=T
)
expecations <- expectationAnalysis(
  experimentName="28perc_4t1", prWin=0.28, numTrades=100, reward=4, 
  risk=1, openingBalance=100, simulations=10000, outputPath="~/dropbox/personal/Dropbox/Public/ff/expectancy",
  saveEquityCurves=T
)

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It is weird that the second curve points down in average as the expectancy is positive... I'm the one who is going to change my mind the most in this thread.

  equityCurves <- sapply(
    result,
    function(x) {
      outcomes <- attr(x, 'data') <= prWin
      balances <- c(openingBalance)
      rp <- risk/100
     [b][highlight] rw <- reward/100[/highlight][/b]
      for (idx in 2:length(outcomes)) {
        if ( balances[idx-1] <= 0 ) {
          newBalance <- 0
        } else {
          newBalance <- ifelse(outcomes[idx], [b][highlight]balances[idx-1]*(1+rw)[/highlight][/b], balances[idx-1]*(1-rp))
        }
        balances <- c(balances, newBalance)
      }
      balances
    }
  ;)

numsims <- 100000
expecations <- expectationAnalysis(
  experimentName="70perc_1t1", prWin=0.7, numTrades=100, reward=1, 
  risk=1, openingBalance=100, simulations=numsims, outputPath="~/dropbox/personal/Dropbox/Public/ff/expectancy",
  saveEquityCurves=T, doEquityCurveAnalysis=F
)
exp.sd <- sd(expecations)
exp.mu <- mean(expecations)
prWin <- 0.7
hist(
  expecations, prob=T, xlab=sprintf("Expectancy (p=%3.2f)",prWin),
  main=sprintf("Experiment %s expectancy distribution (E=%3.3f, SD=%3.3f, N=%d)", "70perc_1t1", exp.mu, exp.sd, numsims),
  cex.main=0.75
)
lines(density(expecations, adj=1.5), col='blue', lty=2)
abline(v=exp.mu, lwd=5, lend=1, col=rgb(1,0,0,0.5))
abline(v=exp.mu-exp.sd, lwd=5, lend=1, col=rgb(0,1,0,0.5))
abline(v=exp.mu-2*exp.sd, lwd=5, lend=1, col=rgb(0,1,0,0.5))
text(
  exp.mu-exp.sd*c(1:2), par()$usr[4] *.8, 
  c(expression(-sigma), expression(-2*sigma)), adj=c(0,-.6)
)

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So if you develop a system with a win rate and pay off (R) so that the (mean expectancy) - (1 standard deviation) is still positive then you have a higher chance of being ahead.

Quoting PipMeUp

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{quote}That is 1.77 times more. But four entries are needed which multiplies the risk by 4 as the SL will be more or less the same. The increase of profit is much less than the increase of risk. The \"let the profits run\" still holds. I was puzzled by these conclusions. Use a small R:R and let the profits run sounds like a contradiction. I realized that you can do both by using a bigger SL. This also usually increases the win rate. {image}

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Quoting charvo

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I am not sure if you can determine the 'increase of risk' by multiply SL by 4...... that's only the maximum of risk, but not the expectancy......

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Quoting PipMeUp

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{quote} My point was more about the time and the frequency of trading.

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Quoting PipMeUp

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{quote} Again this favors a small R:R (once the expectancy is fixed) because a smaller winrate will always have a wider std deviation...

But R:R=1 isn't perfect. With a fixed size MM, a winner and a loser balance each other: 1 + -1 = 0. Using a compounding MM this is no longer the case. Using 40% risk for instance a winner multiplies the account by 1.4 and a loser multiplies it by 0.6. So a pair winner/loser multiplies the account by 1.4 x 0.6 = 0.84 < 1. This introduces a bias towards zero. A R:R of 1.67 restores the balance: (1 + 0.4 x 1.67)...

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x <- seq(0.1, 10.0, 0.1)

expectancy <- function(p, reward=1, risk=1, numTrades=100) {
  outcomes <- runif(numTrades, min=0, max=1)
  winners <- outcomes <= p
  losers <- outcomes > p
  structure(
    (sum(winners*reward) - sum(losers*risk))/numTrades,
    data=data.frame(
      pr.win= p,
      outcomes.pr=outcomes,
      outcome=outcomes <= p,
      pay.off = reward,
      risk = risk
    ;),    
    class='expectancyAnalysis'
  ;) 
}

expectancyMC <- function(p, payoff.range=c(0.1, 10), payoff.mean=2, payoff.probs=NA, risk=1, numTrades=100, numSimulations=10000) {
  payoffs <- seq(payoff.range[1], payoff.range[2], 0.1)  
  if (is.na(payoff.probs)) {
    payoff.probs <- c(
      dnorm(payoffs[payoffs < payoff.mean],payoff.mean,payoff.mean/4.0),
      dnorm(payoffs[payoffs >= payoff.mean],payoff.mean,payoff.mean/2.0)
    ;)
  }
  sapply(
    1:numSimulations, 
    function(x) expectancy(p, sample(payoffs, numTrades, T, payoff.probs), risk, numTrades), 
    simplify=FALSE
  ;)
}

expectationAnalysis <- function(
  experimentName, prWin, numTrades, 
  payoff.range,  payoff.mean,
  risk=1, openingBalance=100, simulations=100000, outputPath="~/tmp",
  saveEquityCurves=F, doEquityCurveAnalysis=T
) {
  result <- expectancyMC(p=prWin, payoff.range=payoff.range,  payoff.mean=payoff.mean, risk=risk, numTrades=numTrades, numSimulations=simulations)
  
  # Mean expectation from the simulations
  theExps <- as.numeric(result)
  meanExp <- mean(theExps)
  cat("Mean expectancy is: ", meanExp, "\n")
  if (doEquityCurveAnalysis) {
    # Calculate equity curves
    # Assumption that risk express a percentage of capital e.g. risk = 1 implies each trade risk 1% etc.
    equityCurves <- sapply(
      result,
      function(x) {        
        data <- attr(x, 'data')
        balances <- c(openingBalance)            
        for (idx in 2:nrow(data)) {
          if ( balances[idx-1] <= 0 ) {
            newBalance <- 0
          } else {
            newBalance <- ifelse(data$outcome[idx-1], 
                                 balances[idx-1]*(1 + data$pay.off[idx-1]/100), 
                                 balances[idx-1]*(1 - data$risk[idx-1]/100))
          }
          balances <- c(balances, newBalance)
        }
        balances
      }
    ;)
        
    # Plot equity curves
    col.best <- rgb(0, 204/255, 51/255)
    col.average <- rgb(51/255, 153/255, 1)
    col.worst <-rgb(154/255, 0, 51/255)
    col.plume <- rgb(0.8, 0.8, 0.8, .5)
    png(
      paste(outputPath, paste(experimentName, "_equitycurves.png", sep=''), sep='/'), 
      width=1920, height=1080
    ;) 
    best <- which(equityCurves[100,] == max(equityCurves[100,]))[1]
    worst <- which(equityCurves[100,] == min(equityCurves[100,]))[1]
    plot(
      x=c(1, numTrades), y=range(c(0, pretty(equityCurves))), type='n', 
      main=paste("Equity curves (Experiment: ", experimentName, ")", sep=""), 
      xlab='Trade', ylab='Equity ($)'
    ;)
    abline(h=openingBalance, lwd=1, col='black')
    lower <- apply(equityCurves, 1, min)
    upper <- apply(equityCurves, 1, max)
    polygon(c(1:numTrades, numTrades:1), c(lower, rev(upper)), col=col.plume, border=NA)
    lines(x=1:numTrades, y=equityCurves[, best], lwd=3, col=col.best)
    lines(x=1:numTrades, y=apply(equityCurves, 1, mean), lwd=2, lty=2, col=col.average)  
    lines(x=1:numTrades, y=equityCurves[, worst], lwd=3, col=col.worst)
    leg1 <- legend(
      "bottomleft", 
      c("Best", "Worst", "Average", "Data range"), 
      col=c(col.best, col.worst, col.average, col.plume), bty='n', lty=c(1,1,2,1), lwd=c(2,2,2,0), 
      pch=c(NA,NA,NA, 15), cex=2, pt.cex=2
    ;)    
    dev.off()
    
    
    # Save equity curves?
    if (saveEquityCurves) {
      write.csv(equityCurves, paste(outputPath, paste(experimentName, "_equitycurves.csv", sep=''), sep='/'))
    }    
  }  
  
  # Plot expectation results
  sdExp <- sd(theExps)
  
  png(
    paste(outputPath, paste(experimentName, "_expectancy_hist.png", sep=''), sep='/'), 
    width=1920, height=1080
  ;)
  hist(theExps, prob=T, xlab=sprintf("Expectancy (p=%3.2f)",prWin),  
       main=sprintf("Experiment %s expectancy distribution (E=%3.3f, N=%d)", experimentName, meanExp, simulations))
  abline(v=meanExp, lty=1, lwd=5, lend=1, col=rgb(1,0,0,0.5))
  abline(v=meanExp-c(1:2)*sdExp, lwd=5, lend=1, col=rgb(0,1,0,0.5))
  lines(density(theExps), lty=2, lwd=2, col='blue')
  text(
    meanExp-c(1:2)*sdExp, par()$usr[4] *.8, 
    c(expression(-sigma), expression(-2*sigma)), adj=c(0,-.6)
  ;)  
  dev.off()
  
  write.csv(theExps, paste(outputPath, paste(experimentName, "_expectancies.csv", sep=''), sep='/'))
  theExps
}

for (prWin in seq(0.05, 0.95, 0.05)) {
  cat("Pr(win) =", prWin, "\n")
  result <- expectationAnalysis(
    experimentName=sprintf("%dperc_variaible_payoff", as.integer(prWin*100)),
    prWin=prWin, numTrades=100, 
    payoff.range=c(0.1,10),  payoff.mean=2,
    risk=1, openingBalance=100, simulations=10000, outputPath="~/dropbox/personal/Dropbox/Public/ff/expectancy/variable_R/",
    saveEquityCurves=T, doEquityCurveAnalysis=T    
  ;)
}

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PipMeUp I respect that this is your thread so please tell me if you would like me to post the research and findings else where.

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It took only about 9 minutes to run on my machine (an i7 3770k or something like that). It only needed about 400 meg of ram.

Pr_0(equity=1000)=1

Pr_1(equity=1001)=0.6
Pr_1(equity=999)=0.4
Pr_1(equity=X)=0 for any other values of X

Pr_2(equity=1002)=0.6*0.6=0.36          // two winners
Pr_2(equity=999)=0.4*0.4=0.16           // two losers
Pr_2(equity=1000)=0.4*0.6+0.6*0.4=0.48  // win+loss or loss+win
Pr_2(equity=X)=0 for any other values of X

Pr_N(equity=X)=delta(1000) (*) S^(*)N

# reward to risk ratio R
minR = 1.0
maxR = 5.0
stepR = 0.1
# percent P
minP = 0.01
maxP = 0.99
stepP = 0.01

borderP = 0
borderR = 0
borderCount = 0
for (R in seq(minR, maxR, stepR)) {
    lastE = -1.0
    for (P in seq(minP,maxP, stepP)) {
        E = R * P - (1-P)
        if (E >=0 && lastE < 0) {
            # record the values of the "Border" R and P
            borderCount = borderCount + 1
            borderP[borderCount] = P
            borderR[borderCount] = R
            break
        }
        lastE = E
    }
}
plot(borderR, borderP, t='l', col='yellow', lwd=3)
# intersection of red and cyan lines is at 2:1 reward:risk and 35% win%
abline(v=2, col='red')
abline(h=0.35, col='cyan')

Quoting FXEZ

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You may have heard of the efficient frontier.

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