Educational sheet: why not simply maximise growth (Kelly)

All the research sheets

Translated from the Spanish original. Where they differ, the Spanish one holds.

• For: every Horalis app · Reviewed: 3 October 2026
• Type: educational. It describes no calculation of any app: it explains why no Horalis app proposes weights or "the mix that grows most".

What the Kelly criterion says

John L. Kelly (1956) looked for how much to bet, of what you have, so that your money grows as fast as possible in the long run. His answer: the fraction that maximises the average growth of the logarithm of wealth. Betting less grows more slowly; betting more does too, and with much larger falls as well. Betting twice that fraction leaves growth at zero (more exactly, at that of the risk-free asset).

For investing, with an asset that yields an average μ with a swing σ and a risk-free asset that yields r (all in continuous rates), Merton's version (1969) gives:

Kelly fraction f* = (μ − r) / σ²

growth with a fraction f: G(f) = r + f·(μ − r) − f²·σ²/2

maximum growth: G(f*) = r + (μ − r)² / (2σ²)

Here μ = log(1 + g) + σ²/2, with g the typical growth (the one Horalis Growth uses).

What would come out with our assumptions

Horalis Growth's assumptions (see the growth and default value sheets); the risk-free asset is the savings account (1.5 % minus 2 % inflation: −0.49 % real) or, if one has to borrow, a loan at 2 % real (some 4 % before inflation):

(Real yearly growth, as effective rates.)

Kelly asks for leverage. With our data, the "optimal" fraction is to put between 1.25 and 3.4 times your money in shares, borrowing the rest. That always happens when the share premium is large against its variance.

Why professionals use prudent fractions, or do not use it

1. We do not know μ. With 35 years of data, the standard error of the average is σ/√35 ≈ 3 points a year, and f* inherits that error divided by σ²: ±0.95 (one standard deviation). With 125 years, ±0.5. That is: f* = 2.19 could be 0.3 or 4. And overshooting is much worse than falling short: with 2·f* growth drops to that of the risk-free asset, and beyond that, to losses.
2. The falls are enormous. With full Kelly, the probability that your money ever falls to half is 50 %; with half Kelly, 12.5 % (with a fraction c of Kelly, the probability of ever falling to a fraction x of the start, against what the risk-free asset would give, is x^(2/c − 1)). With f* = 2.19, a year like 2008 in the World (−40.8 %) takes 89 % of the money.
3. The model is not the world. Kelly assumes lognormal returns, continuous rebalancing and unlimited borrowing at the risk-free rate. In reality there are fat tails, jumps, margin calls that force selling at the worst moment, borrowing costs and taxes.
4. Not everyone wants to maximise the logarithm. Kelly is optimal for someone with logarithmic risk aversion and an infinite horizon. Samuelson (1979) criticised it for finite horizons; whoever needs the money on a date, or cannot bear losing half, prefers less.
5. That is why those who use it (Thorp, for example) bet a fraction of Kelly (half or less), which loses little growth and greatly reduces the falls: with half Kelly, from 7.39 % to 5.36 % in the starting plan, and the probability of falling to half, from 50 % to 12.5 %. Many managers do not use it at all and work with risk budgets, liabilities and concrete horizons.

Why Horalis does not recommend weights

• It would be advice. Proposing to a person how much to put in what, according to their figures, comes close to a personalised recommendation, which in the EU is reserved for authorised firms (see inform, do not advise).
• Our parameters do not stretch that far. As the table shows, an "optimum" calculated with 35 years of data is almost noise, and the same method that produces it asks to take on debt.
• What we do instead: show consequences with data. A mix shows how much it could grow, how much it moved and its worst year; the "What if…?" shows the effect of changing something; the templates (100 %, 80/20, 60/40) are textbook starting points, never "the best for you". The decision is always that of whoever uses the app.

Sources

• Kelly, J. L. (1956), "A New Interpretation of Information Rate", Bell System Technical Journal 35(4): 917–926.
• Merton, R. C. (1969), "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case", Review of Economics and Statistics 51(3).
• Samuelson, P. A. (1979), "Why We Should Not Make Mean Log of Wealth Big Though Years to Act Are Long", Journal of Banking & Finance 3(4).
• Thorp, E. O. (2006), "The Kelly Criterion in Blackjack, Sports Betting and the Stock Market", in Handbook of Asset and Liability Management, vol. 1, Elsevier.
• MacLean, L. C., Thorp, E. O. and Ziemba, W. T. (2010), "Long-term capital growth: the good and bad properties of the Kelly and fractional Kelly capital growth criteria", Quantitative Finance 10(7): 681–687 (doi:10.1080/14697688.2010.506108); and (eds., 2011) The Kelly Capital Growth Investment Criterion: Theory and Practice, World Scientific.
• The data and assumptions, from Horalis Growth's sheets (3 October 2026).

Validation

There is no app calculation to validate. The table's figures come from the formulas above with the averages and swings of packages/seed-kit/src/data/asset-returns/ (period 1988–2022) and the starting plan; anyone can redo them with a calculator.

History

• 2026-10-03: first sheet.
• 2026-10-08: the correct title and journal of the article by MacLean, Thorp and Ziemba (2010); "Good and bad properties of the Kelly criterion" is the title of the chapter of the 2011 book.
• 2026-10-09 (phase A2): the table above was calculated with world shares (MSCI World), which the app no longer uses (its data are not open), and with the starting plan's swing of then (17.82 %; today, that of US shares, 16.39 %). It is left as an example: the conclusion (Kelly gives leverage the app does not recommend) does not change.