Risk of Ruin Explained: The Formula and Tables

Risk of ruin is the chance that a run of losses takes an account down to a limit you chose, at some point. The limit can be a loss amount, a daily or total drawdown rule, or the point where you would stop. The number depends on three things: how often you win, how big a win is next to a loss, and how far the limit is, counted in trades risked. For equal wins and losses at a win rate p above one half, it is ((1 − p) ÷ p)^N, where N is the limit divided by the amount risked per trade.

The model, and what it assumes

Everything below is for one simple, made-up model. Read the list before the numbers.

• Units. The limit is counted in losing trades. If you risk 0.5% of the starting balance on every trade and the limit is a 10% fall, the limit is 10 ÷ 0.5 = 20 units. The same sum works in money.
• Every trade is the same. It wins w units with probability p, otherwise it loses exactly 1 unit. The amount risked never changes, so there is no compounding.
• Trades are independent and there are no costs. Spread, commission and swap would lower p, or w.
• Ruin means touching the limit. The first two tables allow any number of trades. The third also counts ruin within a fixed number.
• Wins are whole units. The root below is exact for w = 1, 2, 3. A 1.5R win needs a different sum.

This is arithmetic on invented inputs. It is not a statement about your results and not a suggestion of how much to risk.

Even payoff: wins and losses the same size

When a win is worth 1 unit and a loss costs 1 unit, the chance of ever touching a limit N units away is ((1 − p) ÷ p)^N, for p above one half. At p = 55%: (0.45 ÷ 0.55) = 9 ÷ 11 = 0.8182. Ten units away, that is 0.8182^10 = 13.4%. Twenty units away, 0.8182^20 = 1.81%.

• At exactly 50% the answer is 100%. With no edge and no limit on the number of trades, a long enough run of losses reaches any limit. That is a property of the model with no end date, which the third table corrects.
• Small changes in win rate move it a lot. At 20 units, moving from 52% to 55% takes the figure from 20.2% to 1.81%.
• Distance matters more still. At 55%, going from 10 units to 50 takes it from 13.4% to 0.004%, because each extra unit multiplies by 0.82.

When a win is bigger than a loss

With a win of w units the sum changes: the chance is r^N, where r is the root below 1 of p × r^(w+1) − r + (1 − p) = 0. For w = 1 this is the ratio above. For p = 40% and w = 2 it is 0.4 r³ − r + 0.6 = 0, which factors as (r − 1)(0.4 r² + 0.4 r − 0.6) = 0, so r² + r − 1.5 = 0 and r = (√7 − 1) ÷ 2 = 0.8229. At 20 units: 0.8229^20 = 2.03%.

The grid shows 20 units from the limit. A cell reads 100% when the average trade is not positive (win rate × w − loss rate × 1 is zero or less), because then there is no root below 1. See expectancy for that sum, or type your own figures into the expectancy calculator.

Read across a row: the same win rate, with a bigger win next to the loss, lowers the figure. Read down a column: at a fixed payoff, a higher win rate lowers it. A win rate on its own does not set the figure: 40% with a win worth 3 units is 0.062%, while 50% with a win worth 1 unit is 100%. It is the same point win rate against risk-reward makes about break-even.

The same model with a time limit

The figures above allow endless trading. Real accounts trade for a while and stop, so the no-limit figures overstate the risk over a short run. This table counts, exactly and with no random numbers, the chance of touching the limit within a set number of trades, at an even payoff.

• No edge is not certain ruin over 100 trades. At 50% and 10 units it is 32.0% within 100 trades, and still climbing toward 100% as the number grows.
• With an edge, most of the risk comes early. At 55% and 10 units, 9.04% within 100 trades and 13.3% within 500, against 13.4% with no limit.

What it cannot tell you

• Your win rate is an estimate. The table treats p as known. A win rate measured on a few dozen trades could easily be several points off, and the figures above move fast with p. See how many trades before your stats mean anything.
• Trades are not identical. Real wins and losses vary in size, and results can come in streaks. The model uses one size and treats each trade as unrelated to the last.
• Size usually changes. If you risk a share of the current balance, the amount falls after a loss and the balance never reaches exactly zero; the answer then depends on where you set the limit.
• It is a probability, not a date. A 2% chance of touching the limit means two runs in a hundred, in the model. It says nothing about when, or about the run you are in.

Using it with your own numbers

1. Pick the limit, in money or as a percentage of the starting balance.
2. Divide it by the amount you risk on one trade. That is N, the number of units.
3. Take your win rate and your average win and loss in R. If wins vary a lot, treat the result as rough.
4. For an even payoff, work out ((1 − p) ÷ p)^N. For another payoff, find the root of the equation above, or use the grid as a guide.
5. Recompute when the win rate or risk changes. Compare the figure with a Monte Carlo run of your own trades, which needs no formula and shows the spread of falls.

What the same trades do to a balance after a fall is on the gain needed after a loss, and position sizing explains what sets the amount risked. To see what is left of a limit of your own after a fall, type it into the drawdown calculator.

Where Trepeat is

We're working on a free journal and a backtester, with no date yet. Today Trepeat is a trade copier for MT4, MT5 and cTrader, currently on demo accounts, and you can add notes, a grade and screenshots to any copied trade once it closes, with a calendar of your results.

We do not sell a risk-of-ruin calculator, and the journal does not calculate it or expectancy today. The journal documentation says what it holds, and the lot size calculator works out the size that makes the amount risked what you meant.

Written by the Trepeat team.