Friday, October 02, 2026

Prix de l'Arc de Triomphe Preview: Hippos Handicapping - LLM Virtual Panel

WCMI Prix de l'Arc de Triomphe Preview: Hippos Handicapping - LLM Virtual Panel

The Hippos Handicapping Panel — where memory and mechanisms collide, but only the horses decide.

Our ongoing exploration of the role of Large Language Models (LLM) in sports trading.

Welcome to the Hippos Handicapping Panel — a virtual round‑table of racing minds brought to life with the help of an LLM. Each Hippo has a distinct voice:

  1. Mick – Aussie handicapper and professional punter
  2. Pearl – Canadian academic and causal analyst
  3. Philip – British host who keeps them honest and sneaks in his own Weekend Warrior longshots

Together they blend events and explanations into a lively debate that is equal parts analysis and paralysis.

Art vs Science of Picking Winners

Note: the panel discussion below blends verified racecard data with handicapping interpretation. Pace, track-bias, trainer-intent, and value judgments should be read as opinion rather than hard fact unless explicitly tied to the racecard.

🐴 Hippos Handicapping Panel — Qatar Prix de l'Arc de Triomphe (Gr.1)

Race Context & Likely Shape

Longchamp's sweeping right-handed mile-and-a-half is the great European leveller — a false straight, a stiff uphill pull to the line, and a camber off the Petit Bois turn that punishes anyone caught three-wide and travelling badly. Good To Soft is the operative description, and that matters enormously: it blunts pure speed, rewards a staying gallop, and quietly tips the scales toward horses with a cruising rhythm rather than a sudden sprint. Sixteen go to post, no geldings, three generations represented, and the eternal weight-for-age equation front and centre — the three-year-old colts in on 8st 13lb against the older brigade shouldering 9st 2lb to 9st 5lb.

The field splits cleanly. Graffard fires a two-pronged assault with the unbeaten-this-season Daryz (FR) at the head of the market and stablemate Varandir (FR). Ballydoyle, as ever, saturate the race — Diamond Necklace (IRE), Benvenuto Cellini (IRE) and the globe-trotting Minnie Hauk (IRE) carry the Aidan O'Brien colours behind three different pilots. Haggas sends Maltese Cross (FR), Balding the admirable mare Kalpana (GB), and there's a genuine Japanese dimension with Meisho Tabaru (JPN) and Admire Terra (JPN) flying the Rising Sun.

Market scaffolding: Daryz is a warm, almost protective 7/4 — the crowd has anointed him. Maltese Cross heads the three-year-olds at 5/1, Kalpana sits 6/1 as the top older performer, and there's a tight cluster of 10/1 shots that tells you the race is deep but the top of it is thin. Weight-of-money has quietly firmed Daryz and nibbled Thundering On at 8/1. Let's convene the panel.

🎙️ Philip (Host)

Good afternoon and welcome to the Panel's Arc preview — the race that stops a continent, run on ground with a bit of ease in it, which already starts rearranging my assumptions. Mick, you've seen more Arcs than you've admitted to in polite company. The market has crowned Daryz at 7/4 before the stalls have even been bolted together. Where does your memory take you first — and does the crowd have this right?

🗂️ Mick (Memory Lane)

Mate, I've been watching this race since the days you needed a short-wave radio and a prayer to catch it. And the first thing my gut tells me is that a 7/4 shot in an Arc is a rare beast — the market doesn't do that lightly, so you've got to respect what it's seeing.

Let's talk stable form, because that's where the money's buried. Graffard's yard is running at 100% RTF — that's not a stable in form, that's a stable on fire. And he's not here mob-handed by accident; Daryz at 7/4 has the profile of a plot that's gone exactly to plan. Form figures of 1-1131, RPR of 138 — the highest in the race — and a 28-day freshener into the biggest pot of the year. Seen it before, mate: Graffard doesn't bring them to Longchamp in October to make up the numbers. By Sea The Stars, who sires Arc-winning stamina in his sleep, out of a Zarak-family mare — he'll relish the ease in the ground.

Now, collateral form. Let me do the back-of-a-beer-mat sum. The three-year-olds are getting roughly half a stone off the older horses, and in a staying Arc on soft that allowance is worth — call it two, two-and-a-half lengths, give or take. That's not nothing. Maltese Cross at 5/1 is the one the whippersnappers point to — Haggas, Marquand, form of -11211, won his trial smoothly, and that WFA sweetener makes him dangerous. But Haggas's yard is only ticking at 50% RTF, which cools my coffee a touch.

The one that keeps nagging at me is Kalpana at 6/1. Top older horse on RPR bar Daryz, mare's allowance on top of it, Colin Keane in the plate, form 1-1211 and a TS of 133 that matches the favourite. She's hardy, she stays, and soft ground is no bother. For the place, she's rock solid.

So here's where I land. Daryz at 7/4 is the win pick — I don't love the price, but I love the plot more. For the safety each-way, Kalpana at 6/1 every day of the week. And the value swing — the one I reckon is overpriced — is Diamond Necklace at 10/1. Aidan's filly, 1-1112, beaten a whisker last time, Soumillon jocked up, and a double WFA-and-filly allowance. Ten-to-one about a horse with an RPR of 136 and that kind of trajectory? That's a drinks round waiting to be won.

Approximately right beats precisely wrong, Philip. I'll take the plot I can see over the model I can't.

🎙️ Philip to Pearl

Thank you, Mick — a yard "on fire," a beer-mat, half-a-stone, and a filly you've essentially accused the bookmakers of mispricing. Pearl, Mick leans on the plot and the pattern. But a plot is only a story until something causes a result. When you draw the arrows on this race, does the pathway lead to the same door — or does the ground do something to the structure Mick can't see from the pub?

🔗 Pearl (Meaningful Musings)

Thank you, Philip. Mick's instincts are, as usual, directionally sound — but let me draw the diagram, because the arrows matter more than the market here.

Start with the central node everyone treats as fixed: Daryz's 7/4. The crowd reads "unbeaten, highest-rated, hot stable" and collapses that into a short price. But notice the confounder hiding behind it — stable form. The yard's 100% RTF makes every Graffard runner look like the plot. That same signal attaches to Varandir at 10/1, which means the confidence we're reading into Daryz is partly a property of the barn, not solely the horse. That doesn't make Daryz wrong; it makes him slightly overdetermined in the market's mind. The price has eaten the stable narrative and the individual narrative at once.

Now the mediator — and on Good To Soft, running position becomes the whole race. The uphill Longchamp finish on an easy surface is a stamina sieve. The causal pathway to victory runs through travelling within yourself to the Petit Bois and still having a gallop left, not through acceleration. Model it as a three-way interaction: position × weight-for-age × going. The three-year-olds get the WFA allowance precisely where it's most valuable — a soft-ground staying test — which is a genuine, structural, non-narrative edge.

That brings me to Maltese Cross at 5/1. Here the causal pathway is clean: a progressive three-year-old by Sea The Stars, carrying 8st 13lb, on ground that amplifies the weight concession, with a trajectory (-11211) still climbing. His DSR of 46 is the one caution — a longer gap than ideal — but Haggas targets autumn with patience. The arrows point the right way.

Let me run the counterfactual Mick skipped. If the pace is truly strong — and with two Japanese raiders who like to roll along, it may be — then the hold-up, stamina-laden types are advantaged, and the collider slams shut on anything caught flat-footed. Under that scenario, Minnie Hauk at 16/1 becomes structurally interesting. An RPR of 137 — second only to Daryz — a four-year-old mare with the allowance, Buick aboard, and recent form (-15283) that looks patchy until you condition on the fact she's been running at the top level globally. Absence of a recent win is not absence of ability; it's a lagging indicator distorted by brutal company. She offers structural value because the market is pricing the form figures, not the pathway.

So, to selections. My win pick is Maltese Cross at 5/1 — the causal pathway through WFA-on-soft is the cleanest in the race, and I'll take him over the overdetermined favourite. For structural each-way value, Minnie Hauk at 16/1, who the pace scenario quietly favours. And for those seeking progressive upside, Diamond Necklace at 10/1 — Mick and I converge here, which is itself worth noting, because her filly-plus-WFA allowance is a double discount the market has only half-applied.

Prediction is not explanation, Philip. The favourite may well win — but if he does, it will be despite the structure giving the three-year-olds the edge, not because of it. And a price should reflect the pathway, not the plot.

🎙️ Philip challenges Mick

Mick, Pearl has just gently dismantled your headline act. She calls Daryz "overdetermined" — a horse whose 7/4 is inflated by his stablemate's glow as much as his own merit, and running against the structural grain because the three-year-olds hold the weight-for-age whip hand on this ground. You've backed the plot. She's telling you the plot is partly an optical illusion. Defend him.

🗂️ Mick (Rebuttal)

Fair shot, and I'll meet it head-on. "Overdetermined" — love the word, Pearl, I'll use it at the TAB and watch 'em scatter. But here's the practical punter's answer. Yes, the WFA allowance is real — I said as much, two-and-a-half lengths on the beer mat. But Daryz isn't a 4yo scraping to hold his form; he's improving at four, which is rarer and more valuable than a 3yo on the up, because he's already proven he trains on. The thing the three-year-olds haven't shown me is that they can do it on the first Sunday in October against hardened older horses — and the Arc has a long, long memory of three-year-old favourites who looked unstoppable in their trials and found the real thing a furlong too far.

And on the "stable glow" — mate, confounder or not, I'd rather my money sat in a yard running at 100% than one running at 50% like Maltese Cross's barn. The glow's real even if the cause is shared. I'm not married to the price — 7/4's skinny, I grant you — which is exactly why my value play is Diamond Necklace at 10/1, where we agree. Approximately right, Pearl. You've drawn a lovely map; I've just already walked the road.

🎙️ Philip challenges Pearl

Pearl, Mick lands a blow — Daryz is improving at four, which is the trait your WFA argument quietly ignores, and he'd sooner be in a 100% barn than a 50% one whatever the confounder. And your each-way structural play, Minnie Hauk at 16/1, rests on a strong-pace counterfactual that may simply not materialise. If they dawdle, your collider never closes, your hold-up angle evaporates, and she's a well-beaten mare off patchy form. Isn't your 16/1 just a hopeful bet on chaos?

🔗 Pearl (Rebuttal)

Both fair, and I'll take them in order. On Daryz improving at four — yes, and I weighted that; it's precisely why I didn't oppose him outright, only argued he's short. A horse can be improving and structurally disadvantaged by the weights simultaneously; those aren't contradictions, they're competing vectors, and at 7/4 the price has already paid for the improvement and left nothing for the WFA headwind. That's an expected-value statement, not a prediction he'll lose.

On Minnie Hauk and the pace — this is the sharper challenge, and Mick's right that I'm conditioning on a scenario. But look at the base rate: Arcs on Good To Soft with sixteen runners, two Japanese raiders bred to stay all day. The prior probability of a true gallop here is high, not speculative. I'm not betting on chaos; I'm betting on the modal pace scenario and taking 16/1 about a horse rated 137 who thrives in it. If the pace collapses, I lose — but a sound bet isn't one that wins every branch of the tree, it's one priced generously relative to its most likely branch. The map, Mick, is how you avoid walking the wrong road twice.

🎙️ Philip's Summary

So where does the Panel leave us? A genuine point of convergence first: both Mick and Pearl independently flag Diamond Necklace at 10/1 as underpriced — a filly carrying a double allowance off a trajectory of 1-1112. When a case-based reasoner and a causal analyst arrive at the same door by different roads, I pay attention; that's independent evidence, not groupthink.

The divergence is at the summit. Mick stands by the plot — Daryz at 7/4 — trusting the 100% barn and a four-year-old still improving. Pearl counters with structure, preferring Maltese Cross at 5/1 on the weight-for-age pathway that soft ground amplifies, and adds Minnie Hauk at 16/1 as a pace-dependent each-way play rated just a pound below the favourite.

My clarifications sharpened two things: Mick concedes the price is skinny but trusts the horse's proven durability over the three-year-olds' unproven October; Pearl concedes she's conditioning on a strong gallop but shows the base rate makes that the likely branch, not a wish.

Consolidating, I'll steer this way. The win pick is Maltese Cross at 5/1 — I side with Pearl's structural logic; the WFA edge on this ground is the cleanest arrow on the board and the price is fair against a cramped favourite. The each-way backup is Diamond Necklace at 10/1, the Panel's one true consensus value. And the risk add, for those who want the top-rated mare at a square price, is Minnie Hauk at 16/1, trusting the gallop to materialise.

As the old Longchamp hands say — the Arc is not won by the fastest horse, but by the one still answering at the top of the hill. Form is temporary, class is permanent, but stamina on soft is eternal.

🧢 Weekend Warrior — Philip's Live Longshot

Now the segment where I abandon reason and reach for romance. My outsider is Meisho Tabaru (JPN) at 33/1.

He's not really in Pearl's model — her arrows don't quite reach Japan — he's a faded memory at best in Mick's case base, and the market has shuffled him to the back of the queue. But consider the narrative: a hardy five-year-old by the staying sire Gold Ship, an RPR of 133 that puts him smack in the mix on raw figures, the great Yutaka Take in the plate chasing the one European prize that has eternally eluded the Japanese, and — here's my hidden angle — a DSR of 112 that reads as "undercooked" to everyone else but screams a trainer who has pointed his entire season at this one Sunday. On Good To Soft, a tough front-running stayer who likes to roll along is exactly the sort who steals an Arc when the favourites look at one another up the hill.

If he lands a place at 33/1, I shall be utterly, catastrophically insufferable — right up until our next preview, at the earliest. And if he trails in last, we shall never speak of him again.

📋 Quick Racecard Crib

  • Race: Qatar Prix de l'Arc de Triomphe (Gr.1), 3yo+, no geldings
  • Course/Trip: Longchamp, 1m 4f, right-handed, uphill false-straight finish
  • Going: Good To Soft — stamina-biased, blunts pure speed
  • Field: 16 runners · Winner approx £2,484,348
  • Top-rated: Daryz (RPR 138), Minnie Hauk (137), Kalpana & Diamond Necklace (136)
  • Key angle: Weight-for-age — 3yo colts in on 8st 13lb vs older on 9st 2lb–9st 5lb; allowance magnified on soft
  • Hot yard: Graffard 100% RTF (Daryz, Varandir)
  • Pace: Likely strong — Japanese raiders
  • WOM tell: Daryz firmed to 7/4; Thundering On nibbled at 8/1

💰 Guide Odds (Panel Selections)

Horse Price Panel Role
Maltese Cross (FR) 5/1 Pearl win / Philip consolidated win
Kalpana (GB) 6/1 Mick safety each-way
Daryz (FR) 7/4 Mick win pick
Diamond Necklace (IRE) 10/1 Panel consensus value each-way
Minnie Hauk (IRE) 16/1 Pearl structural E/W / Philip risk add
Meisho Tabaru (JPN) 33/1 Philip's Weekend Warrior longshot

🌐 Web Sites (Alphabetical)

  • Attheraces — attheraces.com
  • Betfair Exchange (WOM monitoring) — betfair.com
  • France Galop (official French racing) — france-galop.com
  • Racing Post (racecard source) — racingpost.com
  • Sporting Life — sportinglife.com
  • Timeform — timeform.com

Opinions are free; facts are sacred. Prices quoted are the validated current odds as supplied — always confirm live markets before committing.

Generated by Hippos Handicapping Preview - LLM Virtual Panel [ https://vendire-ludorum.blogspot.com/ ]

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Tuesday, September 29, 2026

Risk-of-Ruin Max Stake Calculator - Ceiling Not Target

WCMI Risk-of-Ruin Max Stake Calculator - Ceiling Not Target

From a Stylised Trade to a Real Session

In the TMDA post we asked whether one repeated wager is profitable, sustainable, and survivable, and found the largest stake fraction whose probability of a 50% drawdown inside 2,300 bets stayed under 5%. Clean inputs, clean answer.

A real exchange session is not like that. There is no single win probability and no single price. What there is is a history: a spreadsheet of every bet, with its stake and the market's net result. This post turns the survivability question loose on such a history and asks: what is the most I can put at risk in one market without a realistic chance, over time, of halving the starting bank?

Conservative ceiling
0.52%
€52.13 per market on a €10,000 bank
Basis
330 markets
5,880 bets · racing Win markets only
Observed return
+5.77%
€913.27 on €15,824.53 liability
Actual mean liability
€47.95
median €43.00 · max €252.98
Uncomfortable finding: the trader's average liability over the period (€47.95) sat within 8% of the ceiling the history itself implies (€52.13). The largest single-market liability was 4.9× over it. The session was profitable, but it was not being run comfortably under its own survivability limit - it was being run at it, with excursions well above.

The Question TMDA Left Open

So the question becomes: given this history, and a bank I refuse to see halved more than one time in twenty, what is the maximum liability I should carry into any one market?

Key Insight: the answer is a ceiling, not a target. It says where survivability breaks down, not where growth is fastest. Kelly asks the growth question and accepts a one-in-two chance of ever halving the bank; this asks for one-in-twenty, so the ceiling sits well below the Kelly fraction (roughly m / v, about 1.6% on this session).

The Model: Fixed-Stake Ruin

Treat the bank as a random walk with drift. Each market adds a result drawn from the same distribution: mean m and variance v per unit of liability. Stake a fixed fraction s of the starting bank in every market. The classical continuous-time result (Feller, Ross) gives the probability that the walk ever falls by D before drifting away:

r = exp( −2 · m · D / (s · v) )
s_max = 2 · m · D / ( v · ln(1/r) )
Symbol Meaning Value
s_max Maximum liability per market, as a fraction of the starting bank. Held fixed for a session; recalculated at session end. answer
m Mean market result per unit of liability history
v Variance of those results history
D Drawdown that counts as ruin, from the starting balance 50%
r Accepted probability of ever hitting that drawdown 5% → ln(1/r) = 2.9957

Two properties matter for what follows. First, s_max is linear in m and inversely proportional to v: how you measure the mean and variance from history decides the number. Second, if m ≤ 0 the ceiling is zero - no positive stake is safe without an edge, and the formula says so bluntly.

Attribution: this expression is often called the “Mason Malmuth formula” in poker circles. That attribution has been publicly retracted by Malmuth himself. It is the standard gambler's-ruin result for Brownian motion with drift and has no single eponym; cite Feller (1971, Vol. 2, Ch. XIV) or Ross (1996). Chen & Ankenman (2006, Ch. 22) give the poker-facing derivation.

The Data: An Exchange Report

The input is the exchange's own market P&L report, one row per bet: Date, Market, Ref No, Selection, Odds, Stake, Bet P&L, Market P&L, Net P&L. The file covers 2 March to 30 April 2026. A market is one betting event; its liability is the sum of stakes placed into it, and its result is the net P&L the exchange reports for it after commission. The calculator keeps racing Win markets only - other market types are other strategies and are sized separately - and drops any market whose recorded liability cannot bound its loss.

Quantity Win markets only
Bets (rows) 5,880
Markets 330
Total liability €15,824.53
Total net P&L €913.27
Overall return on liability 5.77%
Bets per market (mean) 17.8

The result per market is expressed as return on liability: net P&L divided by liability. A total loss is −1.00; scratching is 0.00; the best Win market in the period (€26 liability, €300.54 net) is +11.56. These ratios are the raw material for m and v.

Three Ways to Read the Same History

The formula wants a mean and a variance. There are at least three defensible ways to extract them from 330 market results, and they disagree by a factor of 3.3. Working through them is the point of the exercise.

Method 1 - Wins and Losses

The spreadsheet-trader's instinct: count profitable markets, compute a strike rate p, take the average odds of the winning selections O, and treat every market as a coin that pays O − 1 or loses 1.

p = 108 / 330 = 32.73%    O = 25.00 (mean odds of winning selections)
m = p · (O − 1) + (1 − p) · (−1) = 0.3273 × 24.00 − 0.6727 = 7.182
v = p · (24 − m)² + (1 − p) · (−1 − m)² = 137.60
s_max = 2 × 7.182 × 0.50 / (137.60 × 2.9957) = 1.74% = €174.22

Read m = 7.182 again. It claims the average market returns seven times its liability. The session's actual return was 5.77%. The method has silently assumed each win puts the whole liability on one selection at 25.0, when in fact the trader spreads eighteen bets across the field and a winning market returns about 2.6× liability, not 24×. Both m and v are inflated by orders of magnitude; the ratio happens to net out to a stake 2.5× too generous.

Why it flatters: Wins-and-Losses discards outcome size. It cannot distinguish a session of small wins and rare catastrophic losses from a session of large wins and modest losses, so long as the counts match. Prefer the size-weighted figures.

Method 2 - Profit and Loss (banded)

The spreadsheet approach: sort each market's return-on-liability into five bands, weight each band by how often it occurs and by its mean result. Sizes now survive, but each band is collapsed onto its mean.

Why these five bands, and not some other five: the cut points are not tuned to the data; they are the three values of return-on-liability that mean something to a trader. −1 is a total loss, the floor for anyone backing - you cannot lose more than you staked. 0 is a scratch: liability committed, nothing won or lost. +1 is the point where a market returns more than the liability that was risked in it. Those three points partition the number line into exactly five regions - total loss, partial loss (reds), scratch, modest win (greens), and outsized win - which is also how a trader describes a day. The regions do not overlap and, between them, cover every result a backer can record, so each market lands in exactly one band and the probabilities must sum to 100% by construction. When the checksum reads anything else, the bands are not what is wrong; the data is.
Band Meaning Count Probability Mean result Contribution to m
> 1 returned more than the liability 80 24.24% +258.72% +0.62720
0 .. 1 modest wins 28 8.48% +62.80% +0.05329
= 0 scratched 9 2.73% 0.00% 0.00000
−1 .. 0 partial losses 13 3.94% −66.86% −0.02634
= −1 total losses 200 60.61% −100.00% −0.60606
Total 330 100.00% m = +0.04809
m = Σ pi · xi = 0.04809    v = Σ pi · (xi − m)² = 2.2775
s_max = 2 × 0.04809 × 0.50 / (2.2775 × 2.9957) = 0.70% = €70.49

Now m is a believable 4.8% per unit of liability, in line with the observed 5.77% return. Three in five markets are total losses; one in four returns more than 2.5×. This is what a dutching session at long prices looks like, and the stake falls to 0.70%.

Method 3 - Raw Outcomes

Drop the bands. Weight all 330 market results equally at their own return on liability and compute the moments directly.

m = mean(xi) = 0.04809    v = var(xi) = 3.0796
s_max = 2 × 0.04809 × 0.50 / (3.0796 × 2.9957) = 0.52% = €52.13

The mean is identical to Method 2 - a probability-weighted mean does not care how finely you partition the outcomes. The variance is 35% higher. Banding replaced eighty individual wins ranging from +1.0 to +11.6 with a single value of +2.59, and two hundred losses with exactly −1; it threw away the spread within each band. The true per-market variance includes that spread, so the Raw stake is never higher than the banded one, and here it is a quarter lower.

Raw Outcomes is the honest method. It makes no distributional assumption beyond “the future draws from the same urn as the past”, it loses no information, and it is the one to act on.

Comparison and the Conservative Ceiling

Method m v s_max Cash Verdict
Wins and Losses 7.18179 137.602 1.74% €174.22 discards size
Profit and Loss (banded) 0.04809 2.278 0.70% €70.49 understates v
Raw Outcomes 0.04809 3.080 0.52% €52.13 ceiling
Wins and Losses
1.74% · €174.22
Profit and Loss
0.70% · €70.49
Raw Outcomes
0.52% · €52.13

Wins-and-Losses is 2.5× the banded figure; banded is 1.4× Raw. The calculator reports the minimum of the three as the conservative ceiling. In practice that is always Raw, and the other two are there so you can see how much the shortcut methods would have over-staked you.

Ceiling Versus What Was Actually Staked

The per-market detail the script writes alongside the report makes one more comparison possible: the ceiling against the liabilities the trader actually carried.

Ceiling (Raw, s_max)
€52.13
Median liability
€43.00
Mean liability
€47.95
Largest liability
€252.98

The median market was staked at 82% of the ceiling; the mean market at 92% of it; the largest at 485%. Caveat 1 says to stay comfortably underneath. This session was sitting close to the line and periodically jumping over it. It was profitable in the period - but with a 5% lifetime chance of halving at s_max, and materially worse odds on the €250 markets, “profitable in the period” is exactly what the 95% branch is supposed to look like.

Actionable: cap per-market liability at the ceiling and treat anything above two-thirds of it as needing a reason. On this bank that is roughly €35–€52 per market, recalculated at the end of each session (week or month) from the updated history.

Interactive Dashboard

To explore the ceiling across different parameters, we have created an interactive dashboard where you can:

  • Load the three measured (m, v) pairs from this session, or type your own from your report
  • Move the bank, the drawdown threshold and the risk tolerance
  • See s_max, the cash liability, and the round-trip RoR check update live
  • Read the ruin probability at multiples of the ceiling, with an Accept / Reduce / Avoid verdict


(Opens in a new window; allow popups if prompted)

How This Relates to TMDA

TMDA (Feb 2026) This calculator
Input One wager: P, O A history: 330 market results
Staking Fixed fraction of current bank (compounding) Fixed fraction of starting bank per market (fixed stakes)
Space Log-returns: μ, σ Arithmetic P&L per unit liability: m, v
Horizon Finite: n bets Infinite: “ever”
Question Is this stake survivable over n bets? What is the largest stake that is survivable at all?
Decision Accept / Reduce / Avoid A ceiling; m ≤ 0 ⇒ ceiling is zero (Avoid)

They are the same diffusion, asked two different questions. TMDA's finite horizon makes it less pessimistic than the “ever” version here; its compounding makes ruin in the strict sense impossible and so it measures drawdown instead, which is the same thing this calculator does with D.

The TMDA post's closing section on asymmetric payoffs is directly relevant. Whelan (2025) showed the diffusion approximation understates ruin when wins are rare and large. The Win session here has a 33% strike rate and winning selections at average odds of 25 - an asymmetry ratio of order K ≈ 24, far beyond the K = 9 example where diffusion missed by ten percentage points. So the 5% in “5% chance of halving at s_max” is best read as a lower bound; the discrete-chain figure will be materially higher. One more reason the number is a ceiling to stay under, not a target to reach.

Caveats

These travel with every result the script prints. Numbered so they can be cited.

  1. Treat s_max as a ceiling to stay comfortably underneath, not a target to reach.
  2. Asks whether you will ever hit the drawdown, not whether you will hit it in the next N markets. Pessimistic on that axis.
  3. Measures drawdown from the starting balance, not the running peak, which flatters the result - on balance the number errs generous.
  4. Assumes the reported history represents the session you are about to trade.
  5. Assumes fixed stakes: no resizing or moving down as the bank moves.
  6. Treats an observed edge as the true edge; optimistic on small samples. 330 markets is not many for a 33% strike rate at long prices.
  7. Continuous approximation of discrete outcomes. With K ≈ 24 the approximation understates ruin (Whelan 2025).
  8. Wins-and-Losses discards outcome size, so it flatters a session of small wins and rare large losses. Prefer the size-weighted figures.
  9. Profit-and-Loss collapses each band onto its mean, understating variance. Raw Outcomes is the honest one; prefer that.
  10. If m ≤ 0, s_max is 0%: no positive stake is safe without an edge.
Bottom Line: TMDA told you whether a bet is survivable. This tells you, from your own record, how big a bet could be before it stops being survivable. On this session the answer is about half a percent of the bank per market - and the session was already there. Using the observed market-return mean and variance, the fixed-cash-stake Brownian model gives a nominal ceiling of approximately 0.52% of the starting bank, or €52 on a €10,000 bank, for a modelled 5% probability of ever falling below half the starting balance. This is not a validated safe staking limit. The estimated edge is highly uncertain. The figure should therefore be treated as a conditional model output rather than an assurance of survivability.

References

  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, Ch. XIV. Wiley.
  • Ross, S. M. (1996). Stochastic Processes, 2nd ed. Wiley.
  • Chen, B. & Ankenman, J. (2006). The Mathematics of Poker, Ch. 22. ConJelCo.
  • Whelan, K. (2025). “Ruin Probabilities for Strategies with Asymmetric Risk.” University College Dublin. [PDF]
  • matekus (2026). TMDA - Profitable, Sustainable, Survivable.

Claude used for the final sanity check of the analysis and for the complete generation of the visual theme.

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