Somewhere, back in the first 25 or 30 pages of this thread, there were a lot of questions about the exact risk involved in this portfolio methodology. Clearly, the posters who were asking those questions were hoping for hard numbers — for example, [I]“if you open n positions, you will be risking x% of your account.”[/I] Answers that specific were never given, because such answers are not possible. The risk associated with opening multiple positions in a forex portfolio is dependent on a number of variables, some of which must be estimated.
The final step in forex portfolio risk analysis involves some hairy mathematics called binomial probability distribution, involving laborious calculations which you don’t even want to attempt to do by hand. But, those calculations are a snap for computers. And fortunately there are free online Binomial Calculators which will spit out binomial probabilities to 15 decimal places in just a few seconds, with just a few inputs.
And, therein lies the problem: those 15-decimal outputs are only as good as the inputs we provide; and, as it turns out, we have to make a lot of assumptions, in order to come up with those inputs. So, before we resort to a Binomial Calculator, let’s talk about trade risk in general terms, and let’s see whether we can make the necessary assumptions with an adequate level of confidence.
It’s common for forex traders to refer to risk as the [I]loss[/I] (in dollars or in percentage of account balance) which will result, if things go wrong — without regard to the [I]probability[/I] of things going wrong.
For example, two traders open [I]different[/I] positions in GBP/USD. Each trader has a $20,000 account. Each trader’s position size is one standard lot. And each trader protects his position with a 40-pip stop-loss.
Almost everyone would agree that these two traders are taking the same amount of risk; specifically, each trader is risking 40 pips, which is $400 (in a 1-lot position), and this $400 risk represents 2% of each trader’s account.
But, look what happens when we take into account just one additional metric — [I]win-ratio[/I] — involved in these trades.
Let’s say that one trader has an average win-ratio (in trades similar to his GPB/USD trade) of 55%. And let’s say that the other trader has an average win-ratio of 80%. The trader whose strategy has a 55% win-ratio is much more likely to lose $400 on [I]his[/I] GBP/USD trade, than the trader whose win-ratio is 80%.
Specifically, one trader has a 45% probability of a $400 loss, and the other trader has a 20% probability of a $400 loss.
Furthermore, over the course of many trades, the trader with the 55% win-ratio is much more vulnerable than the other trader to devastating [I]strings of losses.[/I]* (see the note at the bottom of this post)
Shouldn’t these realities be included in our concept of “risk”?
The idea that [I]“risk is simply the loss that will result if price hits your stop-loss”[/I] is so commonly held that we might as well just accept it and use it, when we’re discussing individual trades. But, for the more complex situation where several trades (or many trades) are running simultaneously in a portfolio, the winners and losers offset one another — so, we know intuitively that there has to be more to this idea of risk than just stop-losses.
If we know the number of positions in a portfolio, and if we know the [I]average win-ratio[/I] and the [I]average reward/risk ratio[/I] for the trades in the portfolio, we can calculate the number of losers required to tilt the overall portfolio from profit into loss.
If most, or all, of the positions in a portfolio were to result in losses, the overall portfolio loss would be substantially larger than most traders are accustomed to. So, it would be a good idea for us to determine the probability of such an event. Which brings us back to the binomial probability distribution mentioned above.
Let’s start with the metrics which Mastergunner has posted in this thread for his portfolio trading so far this year. Then, let’s plug his numbers into a Binomial Calculator to determine the probability of various numbers of losses.
Mastergunner has posted the following numbers:
• number of trades = 56
• net pips = 3434.4 pips
• average winner = 239.7 pips
• average loser = 93.26 pips
• 100 pips = approximately 1% of account
• broker leverage = 50:1 (maximum allowable leverage in the U.S.)
• typical portfolio size = 20 positions
Using his numbers, we can calculate the following:
• number of winners = 26
• number of losers = 30
• win-ratio = 46.4%
• reward/risk ratio = 2.57/1
• approximate average risk per trade = 0.9326% of account
• [I]implied[/I] average risk per 20 trades = 0.9326% risk per trade x 20 trades = 18.65% of account
• required margin per trade = 2% of account (approximate average)
• required margin per 20 trades = 40% of account (approximate average)
From these numbers, we can generalize that, in a typical portfolio of 20 positions, 40% of the account balance is “escrowed” as margin, and 18.65% of the account is nominally “at risk”. That is, if all 20 positions were to result in losses, averaging 93.26 pips each, then 18.65% of the account would be lost. (Even in this worst-case scenario, more than 40% of the account would be untouched by margins or losses; so, margin calls would not be an issue.)
If [I]x[/I] trades result in average profits (of 239.7 pips each), and [I]20-x[/I] trades result in average losses (of 93.26 pips each), then the “break-even” ratio of winners/losers is 6 winners/14 losers. That is, if the number of winners is 6 or more, then the 20 trades in the portfolio represent an [I]aggregate[/I] profit; if the number of winners is 5 or fewer, then the 20 trades in the portfolio represent an [I]aggregate[/I] loss.
We want to use a Binomial Calculator to determine the probability of 5 or fewer winners, in a portfolio of 20 positions, given Mastergunner’s win-ratio of 46.4%. This calculator works well.
We plug in the numbers, and get this result: Cumulative probability: P(X =/< 5) = 0.0427… (to 15 decimal places) — which means that there is a 4.27% probability that the number of winners (X) will be equal to, or less than, 5.
So, given the numbers which Mastergunner has posted, his portfolio has (on average) at any given time a 4.27% probability of being in aggregate loss, and a 95.73% probability of being in aggregate profit. This is an “idealized” result, which does not take into account the frequent removal and replacement of positions within the portfolio.
However, this result demonstrates that there is [I]a low probability of realizing overall portfolio loss,[/I] when compared to the way we normally evaluate risk.
And this result [I][U]implies[/U] that there must be an extremely low probability of realizing a total loss[/I] of all 20 positions in the portfolio (which would result in the 18.65% account loss previously calculated).
We can quickly calculate the probability of [I]total loss,[/I] using the Binomial Calculator. And we find that, given a 46.4% win-ratio, the probability of [I]zero winners and 20 losers[/I] in a 20-position portfolio is 3.83…E-06 (note the exponential notation) — which equals 0.00000383%. In other words, [I]the probability of total loss is less than 4 ten-thousandths of 1%.[/I]
As long as Mastergunner’s portfolio maintains a reward/risk ratio of 2.57/1, the “break-even” ratio of winners/losers for a 20-position portfolio will remain at 6 winners/14 losers.
However, his portfolio’s [I]win-ratio[/I] could improve. And relatively small improvements in the win-ratio of his portfolio would significantly reduce the probability of overall portfolio loss.
• Mastergunner’s current 46.4% win-ratio corresponds to a 4.27% cumulative probability of
achieving no more than 5 winners out of 20 positions, resulting in overall portfolio loss.
• Increasing his win-ratio to 50% would reduce the probability of overall portfolio loss to 2.07%.
• Increasing his win-ratio to 53% would reduce the probability of overall loss to 1.05%.
• Increasing his win-ratio to 56% would reduce the probability of overall loss to 0.5%.
• And increasing his win-ratio to 59% would reduce the probability of overall loss to 0.2%.
You can verify these results, by plugging the relevant numbers into the Binomial Calculator.
- Note concerning strings of losses for the two traders in the GBP/USD example —
For the trader with the 55% win-ratio, a string of 5 losses in a row will occur, on average, once in every 54 trades; whereas for the trader with the 80% win-ratio, a string of 5 losses in a row will occur, on average, once in every 3,125 trades. In general, for any number of consecutive losses specified, the average frequency of occurrence will be much higher for the trader with the 55% win-ratio, than for the trader with the 80% win-ratio.
The equation for calculating strings of losses can be found here.
Strings of losses should not be confused with binomial probability distributions. Strings of losses are, by definition, consecutive losses; whereas binomial probability distributions refer to numbers of losses (or wins) occurring in any order, whether consecutive or not.