TMDA - Profitable, Sustainable, Survivable
Beyond Expected Value and Likely Profit: Adding Risk of Ruin
In a previous post, we introduced the Dual-Metric Decision Algorithm (DMDA), which combined Expected Value (EV) and Likely Profit (LP) to provide a more comprehensive framework for evaluating betting decisions. While this approach was an improvement over solely relying on EV, it still lacked a critical component: explicit risk management.
Today, we are extending that framework with the Tri-Metric Decision Algorithm (TMDA), which adds a third dimension: Risk of Ruin (RoR). This addition addresses a fundamental question that every bettor must answer: "What is the probability that my bankroll will decline to an unacceptable level within a given time horizon?"
Missing Piece: Risk of Ruin
Expected Value tells you if a bet is statistically profitable. Likely Profit tells you if it is sustainable in terms of geometric growth. But neither metric explicitly addresses the volatility risk — the chance that short-term fluctuations will devastate your bankroll before the long-term edge materializes.
Consider this: you might have a positive EV bet with decent LP, but if your stake size is too aggressive relative to the volatility of outcomes, you could hit your drawdown threshold long before experiencing the expected growth. This is where Risk of Ruin (RoR) becomes essential.
Building the Foundation
Let's start with our familiar canonical example and build up to the full TMDA framework:
| Parameter | Value |
|---|---|
| Initial Bankroll (B) | $10,000 |
| Decimal Odds (O) | 1.9091 |
| Win Probability (P) | 55.00% |
| Stake Fraction (F) | Variable |
| Drawdown Threshold | 50% |
| Time Horizon | 2,300 bets |
Win-Balance and Loss-Balance Multipliers
As before, we calculate the bankroll state after wins and losses:
LB = (1 - F)
Where WB represents the bankroll multiplier after a win, and LB represents the bankroll multiplier after a loss.
Expected Value and Likely Profit
The first two metrics remain unchanged from DMDA:
LP = (WB^P × LB^(1-P)) - 1
EV represents the arithmetic mean return per bet, while LP represents the geometric mean return, which accounts for compounding effects.
Log-Drift and Log-Volatility
To calculate Risk of Ruin, we need two additional statistics derived from the log-space representation of bankroll growth:
σ² = P·(ln(WB)-μ)² + (1-P)·(ln(LB)-μ)²
Here, μ (mu) represents the expected log-growth per bet (drift), and σ (sigma) represents the standard deviation of log-returns (volatility). These metrics transform the problem into a continuous-time random walk, which allows us to apply diffusion approximations.
Risk of Ruin Formula
Using the reflection principle from probability theory, we can approximate the probability of hitting a drawdown threshold d within n bets:
Where Φ is the cumulative distribution function of the standard normal distribution, and d = -ln(drawdown_fraction). For a 50% drawdown threshold, d = 0.693.
Tri-Metric Decision Algorithm
With all three metrics in hand, we can now construct the TMDA decision framework:
if EV ≤ 0:
decision = 'Avoid' # Statistically unprofitable
elif LP ≤ 0:
decision = 'Reduce stake' # Geometric decay despite positive EV
elif RoR > tolerance:
decision = 'Reduce stake' # Risk exceeds acceptable threshold
else:
decision = 'Accept' # All metrics favorableThis hierarchical decision tree ensures that we only accept bets that satisfy all three conditions:
- Positive EV — The bet is statistically profitable
- Positive LP — The bet exhibits sustainable geometric growth
- Acceptable RoR — The risk of significant drawdown is within tolerance
Worked Example
Let's examine our canonical example with a stake fraction of F = 2%:
Step 1: Calculate WB and LB
LB = 1 - 0.02 = 0.98
Step 2: Calculate EV and LP
LP = (1.018182^0.55 × 0.98^0.45) - 1 = 0.000819
Step 3: Calculate μ and σ
σ² = 0.55·(ln(1.018182)-0.000819)² + 0.45·(ln(0.98)-0.000819)² = 0.000362
σ = 0.01901
Step 4: Calculate RoR
For a 50% drawdown over 2,300 bets:
RoR ≈ 4.15%
Step 5: Apply TMDA
- EV = 0.001000 ✓ Positive
- LP = 0.000819 ✓ Positive
- RoR = 4.15% ✓ Below 5% tolerance
Decision: Accept — The bet has positive EV, positive LP, and the risk of experiencing a 50% drawdown within 2,300 bets is within our 5% tolerance threshold.
Finding the Optimal Stake
One powerful application of TMDA is determining the maximum stake size that keeps RoR within acceptable bounds. Using binary search or numerical optimization, we can find the stake fraction F* that satisfies:
The dashboard computes this optimal stake numerically using the findStakeForTargetRoR() function.
For any given parameters, the "Optimal Stake for Target RoR" section in the dashboard will display the maximum
stake fraction that meets your risk tolerance, along with the corresponding dollar amount for your bankroll.
Interactive Dashboard
To explore TMDA across different parameters, we have created an interactive dashboard where you can:
- Adjust odds, probabilities, and bankroll amounts
- Compare multiple stake fractions simultaneously
- Visualize the relationship between EV, LP, and RoR
- Find the optimal stake for your risk tolerance
(Opens in a new window; allow popups if prompted)
Conclusion
The Tri-Metric Decision Algorithm represents a significant evolution beyond traditional EV-only approaches. By incorporating Likely Profit, we account for geometric compounding effects. By adding Risk of Ruin, we explicitly manage volatility risk and ensure that our stake sizing aligns with our risk tolerance.
However, TMDA is not a silver bullet. It assumes:
- Independent, identically distributed bets
- Accurate probability estimates
- Continuous betting (for the RoR approximation)
- Fixed odds and probabilities across all bets
Real-world betting involves correlated outcomes, model uncertainty, and dynamic market conditions. TMDA should be viewed as a framework for thinking rather than a mechanical system. It provides a structured approach to balancing profitability and risk, but successful implementation requires judgment, experience, and continuous refinement.
Asymmetric Payoffs and Ruin Theory
The TMDA framework developed above treats every bet as a two-outcome event with a single pair of win/loss multipliers derived from decimal odds. That model works well when the payoff structure is roughly symmetric — for example, a coin-toss-style wager where the amount you can win and the amount you can lose are of similar magnitude. But many real-world risk-taking environments feature asymmetric payoffs: losses are small and frequent, while gains are large but rare.
The Problem with Symmetric Assumptions
Consider venture capital, where most individual investments fail but a single outsized success can return the entire fund. Or options trading, where a strategy of buying out-of-the-money puts involves paying small, regular premiums in exchange for rare but massive payoffs during market crashes. Tail-risk hedging strategies follow the same pattern.
In all these cases, the payoff structure looks like:
Loss: −1 unit with probability (1 − p)
where K is the asymmetry ratio — the number of loss-units that a single win recovers. When K = 1 we are back to the symmetric case. When K = 9, a single win recovers nine consecutive losses. When K = 20, a single win recovers twenty.
The question is: does TMDA's diffusion-based Risk of Ruin remain accurate when payoffs are this skewed? Recent work by Whelan (2025) provides the analytical tools to answer this question precisely.
Discrete Markov-Chain Ruin Model
Whelan generalises the classical gambler's ruin problem to asymmetric payoffs. The wealth process is modelled as a discrete Markov chain on states 0, 1, 2, …, T, where the investor starts at wealth n, is ruined at 0, and succeeds at T (or T + 1, …, T + K − 1).
At each step, wealth moves:
i → i − 1 with probability (1 − p)
The probability of reaching each terminal state satisfies a difference equation whose characteristic equation is:
This polynomial has K + 1 roots. Using the matrix method of Harper & Ross (2005), the full set of absorption probabilities can be computed exactly via matrix inversion, giving exact ruin probabilities without any diffusion approximation.
Bridging Discrete and Continuous
TMDA works in log-return space with drift μ and volatility σ. The discrete asymmetric game can be mapped into this space by converting step sizes into fractional (multiplicative) returns.
Let the stake fraction be F, the fractional win be a, and the fractional loss be b. Then:
Loss multiplier: ML = 1 − F·b
The per-bet log-returns are:
X = ln(ML) with probability (1 − p)
And TMDA's log-drift and log-volatility become:
σ² = p·(ln(MW) − μ)² + (1−p)·(ln(ML) − μ)²
These (μ, σ) values are exactly the inputs TMDA already uses. In the small-step limit (large bankroll, small fractional stakes), the discrete chain converges to TMDA's continuous diffusion — so the two frameworks are mathematically consistent.
Where Diffusion Breaks Down
The critical insight from Whelan's analysis is that the diffusion approximation becomes increasingly inaccurate as payoff asymmetry grows. Specifically, for positive-EV games with high K, the diffusion-based RoR underestimates the true ruin probability — sometimes dramatically.
Consider a concrete example:
| Parameter | Value |
|---|---|
| Win probability (p) | 0.10 |
| Win payoff (a) | +9% |
| Loss payoff (b) | −1% |
| Asymmetry ratio (K = a/b) | 9 |
| Bankroll (W0) | 100 |
| Ruin threshold (Wmin) | 50 (50% drawdown) |
| Horizon (N) | 500 bets |
Step 1: Compute μ and σ
σ ≈ 0.0287
Note the negative log-drift despite a positive arithmetic EV — this is the classic "volatility drag" effect, amplified by asymmetry.
Step 2: Diffusion-based RoR (TMDA)
RoRTMDA ≈ 30.8%
Step 3: Exact Discrete RoR (Monte Carlo simulation)
Unified TMDA
To correct for this, we can blend the diffusion estimate with the exact discrete result using a weighting that reflects the degree of asymmetry:
λ = 1 / (1 + K) (blending weight on diffusion)
RoRunified = λ · RoRdiffusion + (1 − λ) · RoRexact
For our example with K = 9:
RoRunified = 0.10 × 0.308 + 0.90 × 0.41 ≈ 0.40
This unified estimate of ~40% matches the exact discrete result almost perfectly, while the standard TMDA diffusion would have reported only ~31%.
| Method | RoR | Notes |
|---|---|---|
| Exact discrete ruin | 41% | Ground truth (Monte Carlo) |
| Standard TMDA (diffusion) | 31% | Underestimates risk due to ignoring asymmetry |
| Unified TMDA | 40% | Matches exact result; asymmetry-aware |
Practical Implications
Whelan's analysis yields several findings that directly inform how TMDA should be applied to asymmetric strategies:
- Positive-EV strategies can be destroyed by variance. Even when expected return per play is held constant at μ = +0.01, increasing asymmetry from K = 1 to K = 20 raises the ruin probability from ~13% to ~64% (for an investor staking 1% with initial wealth n = 100, targeting a tripling of wealth). The rare big wins simply do not arrive often enough to prevent early ruin.
- Negative-EV strategies can be partially rescued by variance. In the symmetric case with μ = −0.01, ruin is near-certain (~98%). But at K = 20, ruin falls to ~71% and expected wealth recovers to ~90% of the initial amount. The occasional large win can rescue an otherwise losing game.
- Stake-size effects diminish as K grows. For near-symmetric games (K ≈ 1), stake size has a huge impact on outcomes — consistent with the Kelly criterion. But for highly asymmetric games (K ≥ 20), changing the stake fraction makes relatively little difference, because variance is dominated by the payoff structure itself rather than position sizing.
- The Kelly criterion connects naturally. For an asymmetric game with expected return μ and winning profit K, the Kelly-optimal stake fraction is approximately μ/K. When TMDA is applied with stakes above this level, the model correctly flags elevated ruin risk — but only if the asymmetry is accounted for via the unified RoR.
References
- Feller, W. (1950). An Introduction to Probability Theory and Its Applications, Wiley.
- Harper, J.D. and K.A. Ross (2005). "Stopping Strategies and The Gambler's Ruin," Mathematics Magazine, 78, 255–268.
- Kelly, J.L. (1956). "A New Interpretation of Information Rate," Bell System Technical Journal, 35, 917–926.
- Whelan, K. (2025). "Ruin Probabilities for Strategies with Asymmetric Risk," University College Dublin. [PDF]
Bet Parameters
Risk Parameters
About TMDA Decision Rules
Rule 2: If LP ≤ 0 → Reduce stake (geometric decay)
Rule 3: If RoR > tolerance → Reduce stake (risk too high)
Rule 4: Otherwise → Accept (all metrics favorable)
TMDA Results
| Stake (F) | EV | LP | μ | σ | RoR | Decision |
|---|---|---|---|---|---|---|
| Click "Calculate TMDA" to generate results | ||||||
Optimal Stake for Target RoR
Interpretation Guide
LP (Likely Profit): Geometric growth rate. LP is always ≤ EV. Negative LP indicates unsustainable compounding.
μ (mu): Log-drift, the expected log-growth per bet.
σ (sigma): Log-volatility, measures bankroll fluctuation risk.
RoR (Risk of Ruin): Probability of hitting the drawdown threshold within the time horizon.
Key Insight: A bet can have positive EV but negative LP if the stake is too aggressive for sustainable growth.
