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The Mathematics of the Kelly Criterion

The Kelly Criterion is a formula that mathematically derives the optimal bet size to maximize long-term asset growth. It is expressed as f* = (bp - q) / b, where f* is the optimal investment fraction, b is the profit multiple on a win, p is the win probability, and q is the loss probability (1 - p).

For example, with a 60% win rate, a win paying 1.5x the stake, and a loss costing 1x the stake, f* = (1.5 x 0.6 - 0.4) / 1.5 = 0.33. Investing 33% of assets is mathematically optimal. Investing more sharply increases ruin risk; investing less reduces the growth rate.

The essence of the Kelly Criterion is maximizing the geometric mean return (compound growth rate), not the arithmetic mean. This distinction is critical because it accurately reflects compounding in long-term investing. For products like 3x ETFs whose prices are formed through daily compounding, the Kelly framework is especially well-suited.

Applying Kelly to 3x ETFs

To apply the Kelly Criterion to 3x ETFs, calculate the win rate and payoff ratio from monthly return data. For TQQQ over 2015-2025, the win rate (fraction of positive-return months) was 62%, the average winning return was +18.5%, and the average losing return was -14.2%.

Plugging into the Kelly formula: b = 18.5 / 14.2 = 1.30, p = 0.62, q = 0.38, giving f* = (1.30 x 0.62 - 0.38) / 1.30 = 0.33. The mathematically optimal allocation to TQQQ is 33% of the portfolio.

However, full Kelly (applying f* directly) is too risky in practice. The Kelly Criterion assumes no ruin, but real investors need psychological resilience and living expenses. From a maximum-drawdown perspective, full Kelly with 33% in TQQQ would produce a -24% portfolio-level drawdown during a 2022-type decline.

Kelly Calculations for TQQQ and SPXL

For SPXL (S&P 500 3x), monthly data shows a 65% win rate, +12.8% average win, and -11.5% average loss. Kelly gives f* = (1.11 x 0.65 - 0.35) / 1.11 = 0.34, nearly identical to TQQQ's 33%.

SOXL (semiconductor index 3x) has a 58% win rate, +24.3% average win, and -18.7% average loss, yielding f* = 0.28. Higher volatility produces a lower optimal allocation - evidence that the Kelly Criterion properly accounts for risk (variance).

When combining multiple 3x ETFs, a multivariate Kelly Criterion incorporating correlations is needed. TQQQ and SPXL have a correlation of 0.85, so holding both provides limited diversification. TQQQ and TMF (long-term Treasury 3x) have a correlation of -0.35, and combining them raises the Kelly-optimal total allocation.

Full Kelly vs. Half Kelly vs. Quarter Kelly

Full Kelly (f* = 33%) is the theoretical optimum but can produce maximum drawdowns exceeding -50%. Few investors can psychologically endure this, and panic selling becomes likely.

Half Kelly (f*/2 = 16.5%) maintains 75% of full Kelly's growth rate while roughly halving maximum drawdown. It offers the best risk-return balance and is recommended for most investors. During the 2022 decline, portfolio-level losses would have been limited to approximately -12%.

Quarter Kelly (f*/4 = 8%) is a conservative allocation. Growth rate drops to 44% of full Kelly, but maximum drawdown is contained to approximately -6%. Suitable for 3x ETF beginners or investors with low risk tolerance.

The practical recommendation is to choose between half and quarter Kelly based on investment experience and psychological resilience. Investors who have previously endured -20%+ drawdowns without selling can use half Kelly; others should start with quarter Kelly.

Portfolio Context - Core-Satellite Strategy

In a core-satellite framework, 80-90% of the portfolio forms the core (index funds, bonds, and other low-risk assets) while the remaining 10-20% is the satellite (high-risk assets like 3x ETFs). This structure aligns with the half-to-quarter Kelly range.

The core generates stable returns while the satellite 3x ETF boosts overall portfolio performance. If the core returns +7% annually and the satellite (15% allocation) returns +30%, the portfolio return is +10.5% - a +3.5% alpha over the core-only +7%.

Even if the satellite is completely wiped out, portfolio losses are limited to 15%. This 'non-fatal even in the worst case' design is the key to safely utilizing 3x ETFs. From a compounding perspective, the core's stable growth absorbs satellite losses and maintains long-term asset growth.

Recommended Allocations by Age and Asset Level

20s (assets under 5 million yen, large human capital): 20-25% allocation to 3x ETFs is feasible. Even a total loss can be recovered through future labor income (human capital), making aggressive risk-taking rational.

30s (assets 10-30 million yen): 15-20% is the guideline. Reduce to 10% if family formation or home purchase is imminent. Human capital remains substantial but liquidity for life events is also needed.

40s (assets 30-50 million yen): 10-15% is appropriate. As asset levels grow, the same percentage represents larger absolute risk. 15% of 30 million yen is 4.5 million - a psychologically significant amount if lost entirely.

50s and beyond (assets over 50 million yen): Limit to 5-10%. Time until retirement is short, limiting recovery capacity. Design a glide path that gradually reduces 3x ETF allocation to zero by age 60.

Working Backward from Maximum Acceptable Loss

To size positions with numbers rather than intuition, work backward from maximum acceptable loss. First define in monetary terms how much you can lose in 3x ETFs without impacting your lifestyle. If 2 million yen is the limit, and assuming a -80% maximum drawdown for 3x ETFs, the investment cap is 2.5 million yen.

Check whether this 2.5 million yen aligns with the Kelly Criterion as a percentage of total assets. With 20 million yen in total assets, it represents 12.5% (close to quarter Kelly). With 10 million yen, it is 25% (close to half Kelly).

Set the maximum acceptable loss only after securing six months of living expenses. Never expose emergency funds to 3x ETF risk. Maximizing compounding requires long-term holding, and being forced to sell due to a cash shortfall must be avoided at all costs.

Practical approaches to risk management are covered in several titles on Amazon.