Kelly calculator, Investment Kelly & portfolio optimiser

Classic Kelly criterion

Calculates the optimal position size for repeated decisions with a known win probability and defined profit and loss.

🎯 Input

Win rate (%)

Profit (%)

Risk (%)

Starting capital

€

📊 Result

Optimal position size

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Enter values and calculate

⚠️ Negative Kelly result: do not take this bet. The expected value is negative.

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Win rate

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Net payoff (b)

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Loss rate

🎲 Kelly simulator

times

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Wins

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Losses

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Win rate

Ending capital by position size (log scale)

Portfolio Kelly optimiser

Optimises multiple assets jointly using returns and a covariance matrix, distinguishing between interest-bearing cash and optional borrowing.

🧮 Portfolio Kelly optimiser

Optimises multiple risky assets jointly (default: MSCI World, Bitcoin, Gold, REITs). The optimiser uses covariances and correlations. The Inflation rate (as in the Investment tab, the default 2% inflation rate is deducted from each asset return, so Kelly uses real returns). Positive cash earns the cash return. With borrowing, net risky exposure can exceed 100% when the borrowing rate still makes leverage worthwhile.

€-Cash return p.a.

Inflation p.a.

Starting capital

€

Zeithorizont

Years

Allow borrowing Off

Off: net risky exposure is capped at 100%. On: full Kelly may exceed 100% (negative cash means borrowing) when the model still justifies the leverage at the borrowing rate. A high borrowing rate often pushes the optimum exactly to 100%.

Borrowing rate p.a.

Allow short positions Off

Off: long-only optimisation with wᵢ ≥ 0%. On: the optimiser may assign negative weights to individual assets. The restriction is applied directly in the optimisation.

Assets

Korrelationen

Position	Full Kelly	Half Kelly	Amount

Kelly allocation by position

Full Kelly is compared with the currently selected fractional Kelly level. Negative cash bars represent borrowing only when borrowing is enabled.

Expected logarithmic growth by net risky exposure

x-axis = net risky exposure as a percentage of equity. Full Kelly is at the calculated exposure and can exceed 100% when borrowing is enabled.

Risky Exposure

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Sum of risky-asset weights

Expected return

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inklusive Cashkomponente

Portfolio volatility

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from the covariance matrix

Cash / financing

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remaining weight

What happens to 50.000 € after 10 Jahren?

Expected ending capital (nominal) from the log-growth model at constant allocation. Purchasing power equals nominal value less inflation. Cash-only is shown for comparison.

Half Kelly

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Full Kelly

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Wealth path over the time horizon

The selected fractional Kelly level, full Kelly, and euro cash only are compared. The curve follows the expected logarithmic growth of each allocation.

🎲 Monte Carlo simulation

Runs automatically with the current portfolio parameters. It simulates correlated annual returns (multivariate normal from μ and Σ) at a fixed Kelly allocation. Change the number of runs or simulate again to draw fresh random paths.

times

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Median ending capital

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5. Percentile

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95. Percentile

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Loss rate

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Better than cash

Median ending capital by net risky exposure (log simulation)

x-axis = net risky exposure. Full Kelly is at the calculated exposure and can exceed 100%. Each point uses the same simulated multi-asset returns.

Wealth band over the years (simulation)

P10–P90 band and median for the currently selected fractional Kelly level, plus euro cash only.

Shorts vs. borrowing: These are two separate concepts. A negative asset weight is a short position. A negative cash weight means that the portfolio is leveraged with borrowing. The two options can now be enabled or disabled independently.

g(w) = r + wᵀ(μ − r1) − ½wᵀΣw | Borrowing from: 1ᵀw ≤ 1 | Shorts from: wᵢ ≥ 0

📖 Documentation

1 | Using the calculator

The calculator determines position size for repeated processes with known probabilities. It maximises the long-term growth rate of capital.

→Win probability (p): The probability that the scenario produces a positive return.
→Profit (b): The expected relative return in a winning outcome. Example: a €100 stake with a €10 profit equals 10%.
→Risk (a): The expected loss rate in a losing outcome. Example: a €100 stake with a €10 loss equals 10%.

The optimal position size is the share of total capital invested in each round. Values above 100% would theoretically require borrowing. The calculator caps the result at 100%.

2 | Definition & fundamentals

The Kelly criterion describes position size for repeated decisions. It maximises expected logarithmic wealth growth and requires estimated probabilities and payoffs.

3 | Risks & sensitivities

The model treats its variables as deterministic. In practice, probabilities and payoffs vary and can only be estimated.

The result is highly sensitive to small changes in the inputs. Overestimating the win probability can make the position too large and increase drawdown. That is why practitioners often use only part of the Kelly value, such as 50% (half Kelly).

4 | Economic usefulness

The model adjusts position size to the asset's volatility. At higher volatility, the calculated capital share is smaller.

Beispiel: Two scenarios have the same profit-to-loss ratio: Scenario A with +20 % Profit and −10 % Risk and Scenario B with +10 % Profit and −5 % Risk. For the more volatile Scenario A, the Kelly criterion halves the position size compared with Scenario B.

5 | Mathematical formulation

If the entire stake is not lost in a losing outcome (risk a < 100%), the Kelly criterion is:

      f* = (p / a) − (q / b)
    

f*Optimal position size (capital share as a decimal)
pwin probability (0 ≤ p ≤ 1)
qloss probability (q = 1 − p)
aLoss rate if the outcome fails
bReturn rate if the outcome succeeds

6 | Investment Kelly

For exposure up to 100%, the relevant benchmark rate is the cash return. The default euro cash return is 2.25% p.a. Above 100%, additional exposure is financed with borrowing rather than cash, so the calculator uses the separately entered borrowing rate.

g(f) = r + f(μ − r) − ½f²σ²   |   r = rCash for f ≤ 1, rBorrowing for f > 1

If borrowing is disabled, f ≤ 1 also applies. If it is enabled, the optimum may exceed 100%. A higher borrowing rate pushes the Kelly optimum down and can make exactly 100% exposure optimal.

Cash / financing is simply 1 − f: At f = 0.70, 30% of the portfolio remains in cash; at f = 1.00, 100% is invested and no cash remains; at f = 1.30, the cash position is −30%, meaning 30% additional borrowing.

7 | Portfolio Kelly

With multiple assets, the covariance matrix is used. The optimiser maximises the approximate logarithmic growth rate. As long as the cash weight is positive, the cash rate applies; when the cash weight is negative and borrowing is enabled, the borrowing rate applies to that portion.

g(w) = r + wᵀ(μ − r1) − ½wᵀΣw

Without borrowing, the optimisation is solved subject to Σw ≤ 1. With borrowing enabled, the calculator evaluates the cash region, borrowing region, and 100% boundary, then selects the solution with the highest modelled growth rate. With short positions disabled, the additional constraint for every asset is wᵢ ≥ 0. This long-only constraint is enforced directly in the optimisation; negative weights are not clipped afterwards. Fractional Kelly then scales the risky-asset weights.

Kelly criterion Calculator

Classic Kelly criterion

Investment Kelly

Portfolio Kelly optimiser