Bet Sizing

Expected Value tells you whether to invest. Bet Sizing tells you how much.

The tool is a growth rate function g(f) that takes a bet size (as a fraction of your available capital) and returns the expected growth rate of your capital over many sequential bets. Plot it, and you get a curve shaped like a hill - zero investment gives zero growth, small investments grow your capital slowly, there's a peak that maximizes growth, and past that peak, larger investments actually shrink your capital over time.

That peak is the optimal bet size. It's computed from two inputs:

1. 1)The probability of success p (from your Expected Value analysis)
2. 2)The payoff multiple b (net gain per dollar invested on success)

Risk Tolerance then acts as a hard cap: if the optimal amount exceeds what you can afford to lose, you invest less.

The derivation below produces a result studied in probability theory as the Kelly Criterion. But you don't need to take it on authority - the math falls out of First Principles once you ask: what bet size maximizes my capital growth rate over many sequential investments?

One important caveat: the derivation assumes binary outcomes - full success or full failure. Real Capital Investments often have partial success, staged funding, and a range of outcomes. The framework generalizes to continuous outcome distributions (the core insight holds: there is still a single peak, and oversizing still destroys growth), but the formula applies directly to the binary case. For most resource allocation decisions - fund this feature or don't, hire this person or don't - the binary approximation is a strong starting point.

Most Operators who understand Expected Value size bets by conviction: 'This feature will crush it' becomes 'put 80% of engineering behind it.' The growth rate function reveals why this fails: the penalty for oversizing is larger than the penalty for undersizing.

Undersizing a bet means slower growth - you capture less of the opportunity. But oversizing depletes the capital base that funds every future bet. The losses compound forward. Three years of overconcentrated bets followed by forced Cost Reduction, and you've fallen permanently behind an Operator who sized methodically and let Compounding do the work.

The growth rate function makes this asymmetry precise. Capital discipline - deploying the calculated fraction, not the emotionally satisfying one - stops being a vague instinct and becomes a quantitative practice.

Setting Up the Growth Rate Function

You have capital C (your Discretionary Cash for the period). You're evaluating an opportunity where you invest some fraction f of C. The opportunity is binary:

• •Succeeds with probability p: you gain b dollars per dollar invested (net of the investment). Your capital becomes C × (1 + b·f).
• •Fails with probability (1 - p): you lose the investment. Your capital becomes C × (1 - f).

If you only made one bet ever, you'd maximize Expected Value and go all-in on anything with positive Expected Return. But Operators make many sequential Capital Investment decisions - features, hires, Marketing Spend experiments - and the Compounding of these decisions is multiplicative. After N bets, your capital is C times the product of all growth factors.

To maximize a product, maximize the sum of its logarithms. The expected log-growth per bet - g(f) - is:

g(f) = p · ln(1 + b·f) + (1 - p) · ln(1 - f)

Finding the Optimal Fraction

Take the derivative with respect to f and set it to zero:

g'(f) = p·b / (1 + b·f) - (1 - p) / (1 - f) = 0

Solve for f:

p·b·(1 - f) = (1 - p)·(1 + b·f)

p·b - p·b·f = (1 - p) + (1 - p)·b·f

p·b - (1 - p) = b·f·[p + (1 - p)] = b·f

**f* = p - (1 - p) / b**

Your optimal bet size is your probability of success minus the probability of failure divided by the payoff multiple. Equivalently: **f* = Expected Return / b**, where Expected Return per dollar = p·b - (1 - p).

Three Properties That Matter

**1. f* is generally more conservative than raw Expected Return.** Since f* = Expected Return / b, the optimal fraction is strictly less than the Expected Return per dollar whenever b > 1. When b = 1, they are equal (Exercise 3 below demonstrates this). The growth function is more conservative because it accounts for the Compounding cost of losses across many bets - something single-period Expected Return ignores.

**2. f* shrinks as uncertainty increases.** If you hold Expected Value constant but increase Variance (by pushing *p toward 0.5 and adjusting b* to compensate), the optimal fraction drops. Higher uncertainty demands smaller bets.

**3. Oversizing past f* destroys capital rapidly.** Past the peak, g(f) drops steeply. At some fraction between f* and 1 - the exact point depends on the bet parameters - growth crosses zero. Beyond that crossing, positive-Expected Value bets actively shrink your capital through Compounding losses. The drop-off past the peak is steeper than the rise to it: oversizing by 20% of f* costs more growth than undersizing by the same amount. This asymmetry is the mathematical core of why overconcentration kills.

The Risk Tolerance Constraint

The growth function gives you the growth-maximizing fraction. But growth-maximizing doesn't mean survivable. Your actual bet is:

**Actual bet = min(f* × C, your loss capacity)**

Where loss capacity = liquid assets minus Emergency Fund (from Risk Tolerance). If f* says invest $100K but you can only absorb a $60K loss, you invest $60K.

Practical Adjustment: Half the Optimal

In practice, most experienced Operators invest about half of f. Why? Your probability estimate p has its own uncertainty - you're uncertain about how uncertain you are. Betting half the optimal fraction gives approximately 75% of the maximum growth rate in many common scenarios, with significantly less Volatility. The exact percentage varies by bet parameters, but the tradeoff is consistently favorable: you sacrifice a modest amount of growth for a large reduction in downside risk. This is a heuristic, not a law - but it's a good default. Treat full f as a ceiling, not a target.

Estimating p and b

The formula requires estimates of p (success probability) and b (payoff multiple). Getting these right is the hard part - the math amplifies whatever you feed it. Two practices make the inputs usable:

Use reference classes. What is the historical success rate of similar projects in your organization? If you've shipped 20 features and 11 succeeded, your base case for p is around 0.55 - not the 0.80 your team pitched in the planning meeting. For b, use actual Revenue outcomes from past launches, not the optimistic projections.

Track your estimates against outcomes. After 10-15 resolved bets, compare what you predicted to what happened. Most Operators discover they systematically overestimate p (optimism bias) or overestimate b (Revenue projections run hot). This calibration data is what makes the formula operational instead of theoretical.

Use Bet Sizing when:

• •You're doing Capital Budgeting and deciding how much Budget to commit to each initiative - not whether to commit, but how much
• •You're allocating engineering capacity across multiple features in a quarter
• •You're setting Marketing Spend levels for experiments with uncertain Returns
• •You're deciding how much Discretionary Cash to deploy vs. hold in reserve
• •Multiple positive-Expected Value opportunities are competing for the same Budget

Skip Bet Sizing when:

• •The investment is small relative to your capital (under 5% of Discretionary Cash). The growth function is flat near zero - just make the bet.
• •The outcome has very low Variance (near-Guaranteed Return). Just use Expected Value.
• •You're making a single, irreversible decision with no future bets to protect. Use a decision tree and Sensitivity Analysis instead.
• •You can't estimate p and b even roughly. Bet Sizing amplifies bad estimates - get better inputs first.

Decision rule for the threshold: If a single bet failing would force you into Cost Reduction, debt, or canceling other funded projects, it's big enough to size formally.

Bet Sizing is where Expected Value, Variance, and Risk Tolerance converge into a single operational tool: the probability and payoff go into the formula, Risk Tolerance caps the output, and uncertainty is why the formula exists at all (if there were no Variance, you'd just maximize Expected Value directly). Downstream, this feeds into two frameworks. Portfolio Construction addresses what Bet Sizing assumes away: correlation between bets. Sizing features independently is an approximation - if Feature A and Feature B both depend on the same customer segment, their outcomes are correlated, and the combined Allocation needs adjustment. Portfolio Construction handles those dependencies. The Efficient Frontier then maps the set of all possible Portfolio allocations to their Risk-Adjusted Returns, showing you which combinations of bets produce the highest growth rate for a given level of Volatility. Finally, Pipeline Velocity determines how fast the Compounding engine actually runs: more bets resolved per quarter means more iterations of g(f), which means faster capital growth - but only if each bet is properly sized.