Classic Kelly criterion
Determines the optimal capital allocation for repeated decisions with a known win probability, gain and loss.
Investment Kelly
Derives the growth-optimal exposure to one asset from expected return, volatility, cash rate and an optional borrowing rate.
For a single investment with an assumed annual return and volatility. The inflation rate (2% by default) is subtracted from the expected return automatically, so Kelly sizing and the growth projection use the real return. Positive cash earns the default 2.25% p.a. (nominal). If leverage is allowed and exposure rises above 100%, the borrowing rate applies to the financed portion.
Expected ending capital from the Kelly log-growth model. Nominal in future euros; purchasing power in today’s euros with inflation deducted.
Portfolio Kelly optimiser
Optimises several assets jointly using returns and a covariance matrix, while distinguishing between interest-bearing cash and optional leverage.
Optimises any number of risky assets jointly (default: MSCI World, Bitcoin, gold and REITs). The optimiser uses covariances and correlations. As in the Investment tab, inflation (2% by default) is deducted from every asset return, so Kelly sizing uses real returns. Positive cash earns the cash rate; with leverage, net risky exposure can exceed 100% when the borrowing rate still makes the leverage worthwhile.
| Position | Full Kelly | Half Kelly | Amount |
|---|
Expected ending capital (nominal) from the log-growth model with a constant allocation. Purchasing power is nominal capital adjusted for inflation. Cash-only is shown for comparison.
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 a fresh sample.
1 | Using the calculator
The calculator determines the investment size for repeated processes with known probabilities. It maximises the long-term growth rate of capital.
- →Win rate (p): The probability that the scenario generates a positive return.
- →Gain (b): The expected relative return in the successful case. Example: €100 invested → €10 gain = 10%.
- →Risk (a): The expected loss rate in the negative scenario. Example: €100 invested → €10 loss = 10%.
2 | Definition & basics
The Kelly criterion describes the position size for repeated decisions. It maximises expected logarithmic wealth growth and therefore requires estimated probabilities and payoffs.
3 | Risks & sensitivities
The model assumes deterministic inputs. In practice, probabilities and payoffs vary and can only be estimated.
4 | Economic usefulness
The model adjusts the position size to the asset’s volatility. Higher volatility produces a smaller calculated capital allocation.
5 | Mathematical formulation
If the entire stake is not lost in the negative case (risk a < 100%), the Kelly criterion is:
- f*Optimal investment fraction (capital share as a decimal)
- pWin probability (0 ≤ p ≤ 1)
- qLoss probability (q = 1 − p)
- aLoss rate on failure
- bReturn rate on success
6 | Investment Kelly
For exposure up to 100%, the relevant alternative 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.
When leverage is disabled, f ≤ 1 also applies. When it is enabled, the optimum can exceed 100%. A higher borrowing rate lowers the Kelly optimum and can make exactly 100% exposure optimal.
Cash / financing is simply 1 − f: at f = 0.70, 30% of wealth is held in cash; at f = 1.00, 100% is invested and cash is 0%; at f = 1.30, the value 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; with a negative cash weight and leverage enabled, the borrowing rate applies to that portion.
Without leverage, the optimisation is solved under the constraint Σw ≤ 1. With leverage, the calculator checks the cash region, the borrowing region and the 100% boundary, then selects the solution with the highest modelled growth rate. When shorts are disabled, wᵢ ≥ 0 also applies to every asset. This long-only condition is solved directly during optimisation; negative weights are not clipped afterwards. Fractional Kelly then scales the risky-asset weights.