What it answers
Every trader with a small account has two numbers in mind: the level that would change things (double the account, say)
and the level that would end them (losing most of it). This board gives the odds between the two. With your win rate,
reward-to-risk, size, stop and costs, it tells you the chance of reaching your goal before you blow up, the chance of
blowing up first, and when each tends to happen: within the horizon you choose, and if you keep trading the same way for
as long as it takes.
It answers in three ways that check each other: 20,000 simulated accounts traded day by day (the bars and dots), the
exact day-by-day odds of the same rules (the gold curves), and the classic smooth approximation that treats the account
like a drifting random walk (the violet dashed curves). When the gold and violet curves part, the smooth approximation is
wrong for your trading, and the board shows by how much.
How to use it
Move the sliders; every number and picture follows, and the address bar keeps your settings, so a link you copy opens
the board exactly as you left it. The settings below Your goal are the same as on the Day-Trading P&L Calculator, with
the same names, so a link from that tool opens this one with the same trader.
- Your goal: the goal as a gain on your starting capital (10% to 1,000%; 100% doubles it), the blow-up level as
the share of your starting capital left (20% means you have lost 80%), withdrawals in dollars per month (taken at
each close, a 21st of the month’s amount), the horizon in trading days (up to two years), and the day the first
panel shows.
- Strategy, Size, Account, Execution, Activity and Safety: your trading, as on the P&L tool. The
size is set at each day’s open from that morning’s equity and held all day; costs, scratches, gaps and the daily loss
limit work the same way.
An account stops at the first close at or above the goal, or at or below the blow-up level. An account too small to buy
one share counts as blown up.
The panels:
- Your account on day t: where the accounts still trading stand on that day, simulated (bars) and exact (gold), with
the share that has already reached the goal or blown up.
- The account over time: the exact chance of each level, day by day, as a colour map with labelled contour lines; the
simulated median and the 10% and 90% paths run over it.
- When it ends: on which days accounts reach the goal (above the axis) or blow up (below it).
- The odds by day: the chance of having reached the goal, or of having blown up, by each day; the dotted lines are
where those chances end up if you trade forever.
The readouts give, within your horizon, the chance of the goal first, of blowing up first and of neither (simulated,
exact, diffusion), the simulation’s own uncertainty (one standard error), and the median day each happens on, for the
accounts it happens to. If you trade this way forever gives the chance of the goal first and the expected number of
days until one or the other. The last group is the diffusion’s error against the exact answer, in percentage points.
The equations
The smooth approximation, the classic Fokker–Planck equation, treats the account $x$ as drifting by $\mu$ dollars a day
with a variance of $\sigma^2$, both taken from your one-day outcomes at that account size. Its density $p$ spreads and
drifts, and whatever touches the goal $G$ or the blow-up level $B$ is removed ($p$ is zero there):
The exact equation does not smooth anything. Write the account’s distribution today as $p_t$, and let $K$ be the one-day
matrix: $K(x,y)$ is the chance that one day of your trading takes the account from $x$ to $y$, with every count of wins,
losses, scratches and gaps, the daily limit included, and only levels strictly between $B$ and $G$ kept. With $f(x)$ the
chance that one day carries the account past the goal, tomorrow’s distribution is today’s times $K$, and $g_{t+1}$, the
share that reaches the goal on that day, is today’s times $f$:
Trading forever, the chance $u$ of reaching the goal first is the chance of getting there tomorrow, $f$, plus the chance of
getting there later from wherever tomorrow leaves you, $K u$. So $u = K u + f$: a Fredholm equation, one linear system.
The expected number of days $T$ until the goal or the blow-up comes from the same matrix:
Worked examples
A slow trader aiming to double. Open the board at these settings:
a 50% win rate at 1.5 to 1, 500 shares of a USD 10 stock with a 20-cent stop, two trades a day, USD 10,000 aiming for
USD 20,000 and stopping at USD 5,000. The edge is real: trading forever, 99.1% of such accounts double before they halve.
But it is slow. Within two years only 33.8% have doubled (the simulation says 33.3%, give or take 0.3), 0.7% have halved,
and 65.5% are still going; those that double take a median of 397 trading days, and the expected wait for one end or the
other is 665 trading days, about 2.6 years. Four trades a day instead of two
(open)
lifts the two-year chance to 86.5% and halves the expected wait to 334 days. For a trader like this the smooth
approximation is close: it says 34.6%, 0.9 points too high.
A hyper-scalper. Open the board at these settings:
300 trades a day at a 60% win rate and 1 to 1, 1,000 shares with a 5-cent stop, a USD 25,000 account with four times
buying power, a daily loss limit of 2%, and two trades in a thousand gapping through the stop. The goal is plus 20% and
the blow-up level minus 20%, within one month. Here everything happens fast: within 21 trading days 46.1% reach the goal
(simulated 45.7%), 2.8% blow up, and the median account that gets there does so on day 13. With 300 trades a day you might
expect the smooth approximation to be at its best. It is not: it says 52.8% and 6.2%, wrong by 6.7 and 3.4 points,
because the daily limit cuts the bad days short and a gap is a jump, and neither looks like a bell curve. Trading forever,
the exact chance of the goal first is 93.1%, the approximation’s 89.5%. Without the daily limit
(open)
the month’s chances are 65.4% and 4.0%: the limit protects the account and also slows it down.
A trader living off the account. Open the board at these settings:
the same slow edge, risking 1% of equity on each trade, four trades a day, USD 100,000 with twice the buying power, and
USD 4,000 a month taken out to live on. Within two years 52.7% of such accounts double and 15.4% halve first; trading
forever, 77.0% double first, and the expected wait is 452 trading days. Take USD 2,000 a month instead
(open)
and the chance of halving within two years falls to 2.8%; take nothing
(open)
and it is 0.3%. The withdrawal is the same every day, good or bad, so it weighs most when the account is down.
Assumptions and limits
- Everything the Day-Trading P&L Calculator assumes: trades are independent and the win rate is fixed (no streaks, no
good or bad weeks); every win is the reward-to-risk times the stop and every loss the stop plus the stop-out slippage;
the size is set at each day’s open and held all day; positions are intraday, with no taxes and no broker’s or
regulator’s rules beyond your own settings.
- The goal and the blow-up level are checked at each close, in the simulation and in the exact equation; the smooth
approximation checks them continuously, as if the account were watched every instant. An account too small to buy one
share counts as blown up.
- The daily loss limit is counted in trades at the first day’s size, as on the P&L tool.
- The exact equation holds the account on a grid of levels between the blow-up level and the goal (the status line says
how many). When every day moves the account by whole steps of the same few dollars, the grid is made of those steps and
the answer is exact; otherwise it is exact up to the grid’s spacing.
- The long run assumes you trade the same way, at the same settings, for as long as it takes.
- Withdrawals are taken every trading day as a 21st of the monthly amount, at the close, whatever the day’s result.