Which frequency nets the most
The premium barely changes with frequency — it comes from the assets' volatility, not from your trading. Costs do change: each rebalance is smaller but there are more of them, so total cost climbs with the square root of frequency. That is why there is a real optimum instead of "more is better".
| Frequency | Gross premium | Cost drag | Net premium | After horizon |
|---|
Two different "bonuses" — and why they disagree
The textbook figure benchmarks against the weighted average of what each asset compounds on its own. A real buy-and-hold investor does better than that, because the winner is allowed to grow into a bigger share of the portfolio. The honest premium is the second number, and it grows with horizon.
Volatility you can harvest vs volatility that just costs you
The same volatility that produces a rebalancing premium is what causes volatility drag on a leveraged position — harvested at fixed weights it pays, compounded with leverage it bleeds. Measure the inputs before trusting the output: get each asset's realised volatility from the volatility calculator and the pair's correlation from the correlation calculator, since a correlation of one wipes the premium out entirely. When it is time to actually place the trades, the portfolio rebalancing calculator turns target weights into orders, and the fee drag calculator shows what those repeated trades cost over time.
The math
A continuously rebalanced portfolio grows at Σwᵢμᵢ − σ_p²⁄2, where σ_p² is the portfolio variance. Holding the assets separately grows at Σwᵢ(μᵢ − σᵢ²⁄2). Subtract and the drifts cancel, leaving the classic rebalancing bonus = ½[Σwᵢσᵢ² − σ_p²], which for two assets collapses to ½·w·(1−w)·(σ_A² + σ_B² − 2ρσ_Aσ_B) — half the weight product times the variance of the difference in returns. Zero at ρ = 1, maximal at ρ = −1.
That benchmark flatters rebalancing, so the headline number here uses a different one. Both the rebalanced portfolio at interval Δt and the untouched portfolio over the full horizon H are priced by numerically integrating E[ln(w·e^X + (1−w)·e^Y)] over the correlated lognormal returns (a 61×61 Simpson grid on the two standard normals), then annualised. Buy-and-hold gets no bonus but does get drift: the winner compounds into a larger weight, so over long horizons an untouched portfolio converges on its best asset's growth rate. The gap between the two curves is the premium you actually capture.
Costs use the expected drift in weight between rebalances. Weight wanders with standard deviation w(1−w)·σ_diff·√Δt, so the average amount moved each time is √(2⁄π) of that, and the annual drag is frequency × turnover × cost — which grows as √frequency. Everything here is an expectation under lognormal returns with constant volatility and correlation, so it is a planning tool, not a backtest: real crypto has fat tails, correlations that spike in stress, and drifts nobody knows in advance.