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.

Key Insight: RoR quantifies the probability of your bankroll falling below a critical threshold (e.g., 50% drawdown) within a specified number of bets. It is not just about whether you'll win in the long run — it is about whether you'll survive to reach the long run.

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:

WB = 1 + (F × (O - 1))
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:

EV = (WB × P) + (LB × (1-P)) - 1
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:

μ = P·ln(WB) + (1-P)·ln(LB)
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.

Technical Note: LP and μ are closely related but not identical. Since LP = exp(μ) − 1, they differ by higher-order terms. For small values (as in typical betting scenarios), LP ≈ μ to several decimal places, which is why the worked example shows them as equal when rounded to 0.000819.

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:

RoR ≈ Φ((-d - μ·n)/(σ·√n)) + exp((-2·μ·d)/σ²)·Φ((-d + μ·n)/(σ·√n))

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.

Note: This formula assumes continuous betting and uses the normal approximation of the underlying diffusion process. For small sample sizes or extreme probabilities, the approximation may be less accurate, but it provides an excellent practical guideline for typical betting scenarios.

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 favorable

This hierarchical decision tree ensures that we only accept bets that satisfy all three conditions:

1. Positive EV — The bet is statistically profitable
2. Positive LP — The bet exhibits sustainable geometric growth
3. 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

WB = 1 + (0.02 × (1.9091 - 1)) = 1.018182
LB = 1 - 0.02 = 0.98

Step 2: Calculate EV and LP

EV = (1.018182 × 0.55) + (0.98 × 0.45) - 1 = 0.001000
LP = (1.018182^0.55 × 0.98^0.45) - 1 = 0.000819

Step 3: Calculate μ and σ

μ = 0.55·ln(1.018182) + 0.45·ln(0.98) = 0.000819
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:

d = -ln(0.5) = 0.693
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:

RoR(F*) = tolerance

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.

Bottom Line: While EV tells you if a bet is profitable, and LP tells you if it is sustainable, RoR tells you if it is survivable. TMDA integrates all three perspectives to make more informed betting decisions.

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:

Win: +K units with probability p
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).

Notation: In Whelan's paper, μ denotes the expected arithmetic profit per round (i.e. E[X_i] = μ). In the TMDA framework above, μ denotes the expected log growth per bet. The two quantities are closely related but not identical — the log-drift is always smaller due to the variance penalty. When citing results from the paper below, μ refers to the paper's arithmetic convention unless otherwise noted.

At each step, wealth moves:

i → i + K    with probability p
i → i − 1    with probability (1 − p)

The probability of reaching each terminal state satisfies a difference equation whose characteristic equation is:

p·r^K+1 − r + (1 − p) = 0

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:

Win multiplier: M_W = 1 + F·a
Loss multiplier: M_L = 1 − F·b

The per-bet log-returns are:

X = ln(M_W) with probability p
X = ln(M_L) with probability (1 − p)

And TMDA's log-drift and log-volatility become:

μ = p·ln(M_W) + (1−p)·ln(M_L)
σ² = p·(ln(M_W) − μ)² + (1−p)·(ln(M_L) − 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

μ = 0.10·ln(1.09) + 0.90·ln(0.99) = −0.00043
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)

d = ln(100/50) = 0.693
RoR_TMDA ≈ 30.8%

Step 3: Exact Discrete RoR (Monte Carlo simulation)

RoR_exact ≈ 41%

Gap: The diffusion approximation underestimates ruin risk by approximately 10 percentage points. The many small losses push the bankroll toward the boundary faster than the continuous model predicts, because the discrete chain captures the true step-by-step dynamics of sequential losses between rare wins.

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:

K = a / b    (asymmetry ratio)
λ = 1 / (1 + K)    (blending weight on diffusion)

RoR_unified = λ · RoR_diffusion + (1 − λ) · RoR_exact

For our example with K = 9:

λ = 1/10 = 0.10
RoR_unified = 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:

1. 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.
2. 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.
3. 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.
4. 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.

Looking Ahead: In a forthcoming update, we may extend the interactive TMDA dashboard to support asymmetric payoff structures directly. The updated dashboard will compute diffusion-based, exact discrete, and unified RoR side by side — letting you see exactly how payoff asymmetry affects survivability for any strategy. It will also include a stake-grid visualisation showing where the three methods diverge.

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]

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