This article introduces the kelly fraction, the practical variant of the Kelly criterion used to size repeated bets or predictions. It focuses on the simple definition, the core formulas you need, and how to pick a scaling factor c that fits your tolerance for volatility and the constraints of challenge-style environments. You will learn the binary-bet formula, see a step-by-step calculator walkthrough, and get actionable guidance for multi-bet portfolios and sports prediction challenges. The goal is to provide a clear, testable approach rather than a promise of results.

Fractional Kelly scales the full-Kelly stake by c to reduce volatility while keeping the growth-oriented logic.

Half-Kelly is a common default because it often captures much of the long-run growth with lower drawdowns.

In portfolios, fractional Kelly is equivalent to de-levering the growth-optimal allocation.

What kelly fraction means, a plain definition and why it matters

The kelly fraction is the short name many traders and bettors use for a scaled version of the Kelly criterion where you bet a constant share of the full-Kelly stake. In simple terms, you compute the growth-optimal stake fstar, then multiply by a safety factor c between 0 and 1 to get f, so f = c times fstar. This scaling reduces the volatility and deep drawdowns that can come from betting the full-Kelly amount while preserving the core log-growth logic of the original rule Kelly 1956 paper

Simple spreadsheet checklist to compute fstar and fractional stake

p (win probability)
b (net odds)
c (scaling factor)
bankroll

Use conservative defaults when testing

Think of the kelly fraction like setting cruise control at a lower speed. Full Kelly aims for the fastest long-run growth, but that speed comes with bigger bumps. Scaling by c is a deliberate reduction in speed to smooth the ride and lower the chance of ruin when your probabilities or odds are estimated imperfectly Edward Thorp paper

Why does this matter in practice? If you are making repeated, favorable predictions or bets, the Kelly criterion prescribes the stake that maximizes expected logarithmic growth. But real forecasts come with estimation error and occasional surprises, so many practitioners use a fractional approach to balance long-term growth against short-term stability Kelly theory and practice

Growth-optimal betting in one sentence

The original Kelly goal (see Kelly criterion) is to maximize the expected log of capital across repeated bets, which translates into a specific percentage stake called fstar for any given edge and odds Kelly 1956 paper

Why you might not bet the full Kelly

Full Kelly is mathematically optimal under perfect knowledge, but it is sensitive to errors in your estimated probability and to rare extreme losses, so reducing exposure by using c less than 1 is a common and sensible hedge against model risk Long-term capital growth analysis

The math behind it, core formulas and a simple derivation

$Full frame minimalist spreadsheet mockup showing columns p b c fstar and fractional stake with one example row highlighted in Funded Plays colors illustrating kelly fraction$

Start with the binary-bet full-Kelly formula for a single repeated wager. For a bet that pays net odds b when you win and has win probability p, the growth-optimal fraction fstar is fstar = (b*p - (1 - p)) / b. In that expression, b is the net odds you receive on a winning stake and p is your probability of winning. This formula is the textbook binary case for the Kelly criterion and shows how both odds and edge determine the recommended fraction Edward Thorp paper

When bets are even-money, which means net odds b = 1, the formula simplifies cleanly to fstar = 2*p - 1. That makes the algebra and intuition easy: if your forecasted probability p is greater than 0.5 you have a positive edge, and the recommended full-Kelly stake is twice the edge above break-even. Fractional Kelly simply sets f = c times fstar, with 0 < c < 1, so for even-money bets f = c*(2*p - 1) Kelly notes by Sharpe

Run the worked example in a spreadsheet

Try the worked example below in a spreadsheet or calculator to see how different c values change stake size and volatility

Open a calculator template

It is important to note the allowed range for c and the interpretation of special cases. By definition c lies strictly between zero and one for fractional Kelly; c = 1 is full Kelly and c approaching zero approaches a no-bet policy. If the computed full-Kelly fstar is zero or negative because your edge is zero or adverse, the formula implies you should not place a positive stake or you should consider a short exposure in contexts where shorting is permitted. These sign rules follow directly from the binary formula and guard against misapplied sizing Classic Kelly derivation

Binary-bet formula for fstar and intuition

The binary formula shows two forces at work: the size of your edge, which pushes fstar up, and the offered odds, which scale how much you can profit relative to stake. When b is large, a given edge supports a smaller fraction of bankroll because the potential payoff amplifies returns; when b is small, you must stake more to capture the same expected log growth Edward Thorp paper

Even-money simplification fstar = 2p minus 1

For quick mental math on even bets, remember fstar = 2p - 1. If your probability p is 0.6, full Kelly says stake 20 percent of your bankroll; with c = 0.5, half-Kelly would stake 10 percent. That simple relation is why many bettors and modelers prefer working in even-money terms when teaching the method Kelly notes by Sharpe

How fractional Kelly is applied, f equals c times fstar

Applying fractional Kelly is mechanically straightforward: compute fstar from the formula for your bet, choose a scaling factor c consistent with your tolerance for volatility, and set f = c times fstar. The rest is operational: convert f into a stake size given your bankroll, and enforce caps or position limits as needed by your rules or platform Kelly theory and practice

A short calculator walkthrough, compute fractional Kelly from a forecast

This section gives a step-by-step path you can copy into a spreadsheet or a small script. Minimal inputs are your forecasted win probability p, the net odds b for the market or bet, and the scaling factor c you choose. From those three numbers compute fstar with the binary formula, then multiply by c to get f. Finally, multiply f by your current bankroll to obtain a stake in currency units Edward Thorp paper

$Minimalist 2D vector showing portfolio allocation bars shrinking left to right as c moves from one to zero point two five with subtle correlation arrows in Funded Plays palette kelly fraction$

Step 1, enter p as a decimal probability between 0 and 1. Step 2, enter net odds b, which for even-money is 1. Step 3, compute fstar = (b*p - (1 - p)) / b. Step 4, pick c and compute f = c times fstar. Step 5, convert to stake = f times bankroll. Keep the spreadsheet cells labeled so you can run sensitivity checks quickly Kelly notes by Sharpe

Worked numeric example, even-money bet. Suppose your model gives p = 0.6 for an even-money market, so b = 1. Full Kelly gives fstar = 2 times 0.6 minus 1 = 0.2 or 20 percent of bankroll. With c = 0.5, fractional Kelly yields f = 0.1 or 10 percent of bankroll. With c = 0.25, f = 0.05 or 5 percent. Converting to a $1000 simulated bankroll means stakes of $200, $100, and $50 respectively for c values of 1, 0.5, and 0.25 Kelly notes by Sharpe

Interpreting the output means understanding sensitivity. A small error in p matters more for higher f values; half-Kelly often captures a large share of long-run growth while cutting volatility, so it is a pragmatic compromise when your p estimate might be noisy Long-term capital growth analysis and you can analyze this uncertainty to see how estimation noise affects ruin probabilities.

Inputs you need, p odds b and your chosen c

Minimal inputs cover your forecast probability p, the net odds b you would receive, and your chosen scaling c. Add bankroll size if you want absolute stake sizes instead of fractions. Record assumptions and run sensitivity tests by varying p and c to see how stake recommendations change Edward Thorp paper

Worked numeric example for an even-money bet

Using the $1000 bankroll example above, emphasize that stake sizes scale linearly with bankroll. If the challenge environment has minimum or maximum position sizes, convert f into allowed stake units and apply a position cap to avoid an oversized exposure. These practical conversions are often the step where modelers make operational errors, so double-check unit conversions and caps Kelly notes by Sharpe

Interpreting the output stakeholder size and edge sensitivity

When you test different c values, observe how expected long-run growth decreases slowly at first while volatility falls faster. That pattern explains why moderate scaling often yields much of the growth benefit with less short-run pain. Use simulations to quantify those trade-offs for your specific forecast distribution Kelly theory and practice

Fractional Kelly for portfolios and multiple predictions

When you have many correlated predictions or betting opportunities, the single-bet formula generalizes to a portfolio problem. The Kelly-optimal weight vector in a multi-asset setting is proportional to the inverse covariance matrix times the expected return vector, often written as Sigma inverse times mu. Here Sigma is the covariance matrix of returns or outcomes and mu is the vector of expected edges or returns. That proportionality is the direct multi-asset analogue of the single-bet fraction and it shows how correlation and variance reshape optimal allocations Kelly theory and practice and is discussed in a recent journal article.

Fractional Kelly in a portfolio context is simply scaling the entire full-Kelly weight vector by c. Practically that means reducing leverage and raising effective risk aversion while preserving the same relative weights across bets implied by Sigma inverse times mu. The scaling keeps the growth-optimal direction but limits magnitude, which helps when covariance estimates are noisy or when platform rules restrict leverage Funded Plays

Real-world complications change implementation details. Covariance estimates are sensitive to sample length, correlations between similar prop bets can inflate portfolio risk, and position limits or maximum exposure rules on challenge platforms may force additional caps. All of these factors make a smaller c attractive in practice Long-term capital growth analysis

Kelly portfolio weights and the Sigma inverse mu rule

In plain language, Sigma inverse mu means you increase allocations to bets with high expected edge and low correlation with others, and you reduce or avoid bets that add variance without enough expected return. The matrix inverse step captures the diversification trade-offs that the single-bet formula cannot express Kelly theory and practice

What scaling by c means for leverage and risk aversion

Scaling by c works like moving a risk aversion dial: lower c reduces both leverage and expected volatility. For many practitioners choosing c between 0.25 and 0.5 is a practical way to protect against estimation noise and extreme drawdowns while still aiming for positive long-run growth Long-term capital growth analysis

Practical limits correlation estimation noise and leverage constraints

Keep in mind that highly correlated bets can create concentrated risk even if each individual stake is modest. Estimation noise in Sigma or mu can lead to unstable weights, so regularization, shrinkage, or using a conservative c are common mitigations. Also verify that any simulated leverage fits within platform rules before applying it in a challenge account Kelly theory and practice

How to choose the scaling factor c, rules and trade-offs

Choosing c is a risk-management decision. Many practitioners find that half-Kelly is a reasonable default because it captures a substantial portion of the theoretical growth while materially reducing volatility and drawdown severity. That practical recommendation comes from comparisons between full and fractional Kelly behavior under estimation error and is commonly used as a starting point for further testing Long-term capital growth analysis

How should I choose the kelly fraction when my probability estimates are noisy?

Choose a conservative fraction c, run simulations or resampling to measure drawdowns, and set position caps; many practitioners use c between 0.25 and 0.5 as a starting point.

When might you select a smaller c, such as 0.25 or lower? Use a smaller c if your edge estimates are noisy, your model has limited historical data, your forecast process is nonstationary, or you face strict drawdown or position limits that amplify the cost of a single large mistake. Lower c values act as a robustness buffer against overfitting and unmodeled tail risk Kelly theory and practice

Here is a short decision checklist: estimate uncertainty in p, run stress tests or simple Monte Carlo resampling to see likely drawdowns, set an absolute position cap on single bets, and pick c consistent with your drawdown tolerance and platform rules. If you are running challenge-style evaluations, align c with the progression criteria and consistent performance goals rather than chasing maximum theoretical growth Edward Thorp paper and review how evaluations work on challenge platforms.

Why half-Kelly is common and what it buys you

Half-Kelly often provides roughly three quarters of the long-run growth rate of full Kelly while reducing volatility notably. That trade-off is attractive when probabilities are estimated with error because it lessens the penalty from misestimation without abandoning the growth-oriented framework entirely Long-term capital growth analysis

When to use smaller c 0.25 or lower

Choose lower c values in environments with short or noisy historical data, frequent model revisions, or when single losses have outsized consequences for your evaluation standing. Lower c is essentially an explicit way to pay for robustness when you suspect your inputs are uncertain Kelly theory and practice

A basic decision checklist for picking c

Checklist items: quantify forecasting uncertainty, run scenario simulations, impose an absolute cap on single-position exposure, choose conservative c for correlated or high-variance bets, and document the rationale so you can review performance objectively during challenges Edward Thorp paper

Common mistakes and real-world pitfalls to avoid

One frequent error is overconfidence in p. If you treat forecast probabilities as exact rather than noisy estimates, full Kelly stakes can be far too large and lead to damaging drawdowns. Fractional Kelly reduces that sensitivity and buys robustness in the face of estimation error Long-term capital growth analysis

Another common pitfall is applying full Kelly as a blunt rule without position caps or consideration of correlated bets. When several predictions are similar or driven by the same model error, aggregate exposure can become much larger than intended. Simple mitigations include absolute caps and stress testing via resampling or Monte Carlo Kelly theory and practice

A final operational mistake is misunderstanding zero or negative fstar outcomes. Negative fstar means the math recommends not taking a positive stake, and forcing a bet in that case is usually a losing approach. Respecting sign and cap rules is a low-cost safeguard against this error Classic Kelly derivation

Applied example, using fractional Kelly in sports prediction challenges

To make the ideas concrete, translate your model edge into a probability p for each prop or market you plan to enter. If your model outputs expected return, convert it into win probability using a reasonable mapping or calibration step so you can apply the binary or even-money formulas. This conversion is an essential practical step when bringing Kelly sizing into sports-prediction workflows Kelly theory and practice

Funded Plays Challenges page

Open a calculator template

Example multi-bet scenario in a simulated challenge. Suppose you have three independent prop predictions with p values of 0.6, 0.55, and 0.58 for even-money payouts and a $10,000 simulated bankroll. Full Kelly fractions would be 20 percent, 10 percent, and 16 percent respectively; half-Kelly would halve those fractions. Allocating half-Kelly stakes reduces total portfolio volatility and helps meet challenge-style drawdown rules that penalize large single-event losses Edward Thorp paper

When bets are correlated, adjust expectations. Correlation between props means that aggregate exposure can be higher than naive addition suggests. Use covariance-aware portfolio calculations or simply reduce c to limit aggregate risk. In structured evaluation programs, consistent medium-sized stakes often outperform infrequent large bets because consistency aligns with progression criteria and drawdown constraints Long-term capital growth analysis

Remember that fractional Kelly is a sizing tool within a process. It does not guarantee qualification or rewards in any challenge; outcomes remain dependent on forecast quality, discipline, and compliance with platform rules. Use fractional Kelly as part of a documented, testable strategy rather than a promise of success Kelly theory and practice

Key takeaways and next steps for testing fractional Kelly

Summary checklist: compute fstar from your forecasts, select a conservative c, cap single-position exposure, backtest or simulate to measure likely drawdowns, and adjust c based on observed performance and tolerance for risk Edward Thorp paper

Suggested experiments: run simple Monte Carlo resampling of your forecast errors, compare full, half, and quarter Kelly in simulated runs, and track metrics such as median time to a target, worst drawdown, and volatility. Use results to pick a durable c before live use Long-term capital growth analysis

Further reading: start with the original Kelly paper for the core idea, read practitioner notes and modern theory for portfolio extensions, and consult applied literature for stress-testing methods. Fractional Kelly is a risk-management tuning knob, not a guarantee of outcomes, and it works best when combined with disciplined forecasting and testing Kelly 1956 paper and see related posts on our blog.

What is the kelly fraction in plain terms?

The kelly fraction is a scaled version of the full-Kelly stake where you multiply the theoretical fstar by a factor c between 0 and 1 to reduce volatility and drawdown risk.

How do I pick a reasonable c value?

Start with conservative values such as 0.25 to 0.5, estimate your forecast uncertainty, run simple simulations, and adjust c to match your drawdown tolerance and platform rules.

Can fractional Kelly guarantee profits in challenges?

No. Fractional Kelly manages stake size and risk but does not guarantee outcomes; results depend on forecast accuracy, discipline, and compliance with challenge rules.

Fractional Kelly is a pragmatic compromise between theoretical growth and real-world risk. It gives you a discipline for sizing stakes that acknowledges forecast uncertainty and operational limits. Treat c as a tuning parameter: test it, stress it, and document why a chosen value fits your forecasting process and the rules of any challenge you enter.