This formula is the true secret of Buffett!

Use the Kelly criterion to build your own long-term compounding machine.
1. Compound interest is the fundamental driver of wealth growth, but the real world is full of volatility and uncertainty.
2. The essence of the Kelly formula is to optimize capital allocation in an uncertain world to achieve maximum long-term growth via compounding.
3. The "Enhanced Kelly Strategy" is the mathematical foundation of Buffett's success, balancing higher efficiency and lower risk through special business models.
4. Operating in a form of "Anti-Kelly" strategy is the reason behind most investment failures.
5. The Kelly formula is ineffective for gamblers, because gambling is a negative expected value game; the optimal strategy is not to participate.
6. The most important investment principle: don't gamble, avoid excessive leverage, and never suffer permanent capital loss.

Part One: When the market begins to crash, people once again praise Buffett's seasoned judgment.

There’s a saying: over the past few decades, Buffett has performed modestly most of the time, even lagging behind the market (of course, consistently beating the market with such large capital is inherently difficult). But whenever the market crashes, he recovers and even surpasses the index.

This time seems no different; he reduced his positions early on, sold most of his Apple holdings, and now holds hundreds of billions in cash.

In contrast, many superstar investors, while gaining headlines, often can't withstand a big wave.

The most memorable example I recall is Ellis's story:
A star fund manager achieved an annualized return of 16.5% from 1990 to 2005, significantly surpassing the S&P 500’s 11.5%. But the following three years, returns plummeted to -23.7%.

Although time-weighted returns seem to outperform the market slightly, actual investors suffered from buying heavily at market peaks, leading to losses—totaling $3.6 billion in client losses. The fund manager and fund company profited, but investors were deeply trapped.

In other words, even if you outperform the S&P 500 for 15 years in total, overall you are still losing money, especially on the high-priced entries.

Ellis is likely referring to Bill Miller. Miller created a myth of beating the market for 15 consecutive years with the Lehman Brothers Value Trust, but was heavily impacted during the 2008 financial crisis, with fund NAV dropping about 55-60%, far exceeding the S&P 500’s approximate 37% decline in the same period.

Although Miller later demonstrated remarkable insight in investing in Amazon and Bitcoin, he failed to reproduce his former glory.

Buffett and Miller form a striking contrast. Buffett’s ability to perform averagely in calm markets but demonstrate extraordinary value during crises validates that long-term investment success depends more on avoiding large drawdowns than on maximizing short-term gains.

This also explains why many short-term shining stars in investing may ultimately falter during major market corrections. They often rely heavily on strategies effective only within certain market environments, lacking true full-cycle risk management.

The comparison of these two investing masters perfectly illustrates a core truth in investing:

Success depends more on avoiding big mistakes than on chasing astonishing returns.

Buffett’s career proves that investors who withstand major market declines are often able to beat those superstar fund managers shining during bull markets.

One key difference between them is:

Buffett cares about win rate;

Miller cares about odds.

This seemingly subtle difference hides behind a common yet powerful mathematical principle—the Kelly formula.

Part Two: In the above comparison, Miller’s approach can be seen as a form of "Anti-Kelly," while Buffett employs an "Enhanced Kelly" strategy.

The Kelly formula was originally created by John Kelly, a scientist at Bell Labs, to solve communication problems, and later proved to be the basis for optimal capital allocation:

f* = (bp - q) / b

Where f* is the optimal bet proportion, b is the odds, p is the probability of success, and q = 1-p is the failure probability.

Some articles suggest that the Kelly formula emphasizes "win rate" over "odds." Here are two investment opportunities for comparison:

1. A 10x odds chance with only 10% probability of success; optimal stake is just 1%;
2. A 0.5x odds chance (earn 1, lose 2) with 80% success probability; optimal stake can be up to 40%.

Based on this, some argue that classical investment theory favors high-probability investments, believing that increasing the probability of success is more important.

Applying the Kelly formula’s focus on probability (win rate) over odds makes it very difficult to achieve high returns practically because:

- Probabilities are hard to estimate accurately,
- Investment is not a repeated, dice-throwing game, so outcomes cannot be easily verified.

However, the reality is different: the Kelly formula itself has no bias; it considers both success probability and odds to determine the optimal stake.

Regarding probability estimates, we can only use Bayesian "subjective probabilities."

This comparison raises the key topic I want to explore:
Value investing, which can take larger positions, allowing for better long-term compounding returns.

Let’s examine the relationship between value investing and the Kelly formula:
From the Kelly perspective, value investing seeks a relatively high success rate (p), while accepting moderate odds (b). By searching for undervalued stocks below intrinsic value, value investors create a "win rate advantage."

The Kelly formula shows that when success probability is high, even with moderate odds, larger positions are supported. This aligns well with value investing:

- Typical win rate for value investing: 65-75%
- Odds usually range from 0.5 to 2 times
- The Kelly-optimized position size often falls between 20-40%
In contrast, growth or speculative investments tend to have low success rates but high odds, leading Kelly to recommend smaller positions.

Why does supporting larger positions in value investing help long-term compounding?
1. Drawdown control
Larger positions require managing downside risk.
Value investing reduces large declines via a margin of safety, allowing sustained larger exposure without forced liquidation during market drops.
2. Time advantage
Larger positions leverage the power of compounding more fully.
Suppose two strategies have the same expected return (e.g., 15%. Over time, Strategy A exposes more capital to growth, yielding higher geometric mean returns.
- Strategy A: high win rate, moderate odds, Kelly suggests 40%
- Strategy B: low win rate, high odds, Kelly suggests 10%
3. Capital efficiency
Idle capital has high opportunity costs.
Supporting larger positions improves capital utilization.

Part Three: Of course, compared to the simplicity of the Kelly formula, real-world situations are much more complex. Several key gaps exist between theory and practice:
1. Difficulty in estimating probabilities
Success rate in investing is hard to estimate precisely. Value investors try to improve estimates through qualitative and quantitative analysis, but subjectivity remains.
2. Non-independence of events
Kelly assumes bets are independent; but in reality, stock markets are highly correlated. Systematic risks can impact multiple value positions simultaneously.
3. Time frame differences
Value investing usually takes a long time to realize, which means funds are "locked in" and may miss other opportunities.
4. Psychological factors
The real challenge: maintaining large positions or even increasing them during market panic—psychologically very difficult.
Buffett’s success hinges on creatively overcoming these limitations:

- Using insurance float as low-cost or negative-cost capital to expand the investable base
- Developing systematic methods to identify high success probability opportunities
- Keeping enough cash and psychological resilience during market panic to increase positions
The often-cited "high success rate, high confidence, circle of competence, safety margin" in value investing actually serves these strategies. Because wealth growth ultimately is a mathematical game.

Therefore, Buffett’s essence is an "Enhanced Kelly Strategy."

Part Four: Returning to Miller and other investors’ "Anti-Kelly" approach:
The situations include:
1. "Anti-Kelly"—Miller experienced fund redemptions after crashes, preventing him from repairing the fatal damage to his geometric mean.
This is exactly opposite Buffett’s contrarian approach of holding cash to buy in dips.

2. "Anti-Kelly" also describes typical investors.
Most notably, Cathie Wood’s fund clients—most of their money poured in during her peak years, but suffered huge losses during downturns. Her annualized returns are modest, but the destruction of wealth can be severe.

These star fund managers, at times when their strategy’s expectation is highest )high bp(, manage the least funds; when expectations decline or turn negative, they manage the most.

This is like a gambler betting small when odds are favorable, but risking everything when odds turn worse. Not only star fund managers, but most ordinary investors unconsciously practice this "reverse Kelly"—buying heavily at market peaks )low expectations(, panicking and selling at lows )high expectations(.

3. The biggest danger of "Anti-Kelly": leverage.
Kelly emphasizes only risking a part of your capital each time. Leveraging amplifies risk, and a big market move can cause permanent losses.
Recently, some fund managers have been using leverage; this may be driven by incentive structures, as everyone looks at annual performance leaders, fueling gambling behavior.
Or society’s valuation system focuses on very short-term "arithmetic mean," not the long-term "geometric mean."

4. For investors who care about odds, position size may follow "Anti-Kelly."
For example, a fund manager wanting better returns might buy more innovative tech stocks. But based on Kelly calculation, high odds with low success probability should lead to smaller bets.

To improve returns, the manager increases capital efficiency by buying more similar tech stocks.

Here’s the problem: although each stock’s position might seem calculated per Kelly, combined they often violate Kelly’s proportionality principle because:

- Kelly derivation assumes investments are independent.
- When the manager buys multiple similar stocks in the same sector or affected by the same macro factors, their returns are correlated.
- This correlation increases the overall risk, especially during downturns.
- Excessive concentration can cause large drawdowns, severely damaging geometric returns.
- For example, a star fund that earns 70% in one year, then loses 40% the next, will wipe out previous gains.

This is the core reason many star managers shine during booms but collapse during crises—they pursue odds-focused strategies that seem clever but actually fall into the "Anti-Kelly" trap.

5. The time mismatch of "Anti-Kelly"
Kelly implicitly assumes a long-term horizon where maximum growth can be achieved. But in reality, investors and fund managers face short-term performance pressure and liquidity constraints.

During downturns, many can't sustain long enough to wait for recovery due to:

- Retirement needs
- Education expenses
- Psychological endurance

This "short-sightedness" contradicts the premise of long-term Kelly growth.

Peter Bernstein states that volatility is the hidden killer for long-term investors, especially during withdrawals.
He introduced the concept of "sequence risk":

Compare two retirement portfolios, both starting with $1 million, withdrawing $65,000 annually, with an average annual return of 7%:
- Portfolio A: 7% return for first 15 years, then 7% for last 15, leaving ~$950,000 after 30 years
- Portfolio B: 2% for first 15 years, then 12% for last 15, depleting the funds

Despite identical arithmetic averages, early market performance crucially impacts the longevity of retirement funds.

Bernstein’s sequence risk links deeply to Kelly:

Kelly aims to maximize geometric mean, which assumes stable capital flow. When withdrawals are needed at specific times, the sequence’s importance becomes critical.

Miller’s fund faced this "double anti-Kelly" trap:

- Managed most funds during market declines (violating Kelly’s capital allocation)
- Experienced mass redemptions (violating Kelly’s time horizon assumption)

When forced to sell assets at lows to meet redemptions, geometric returns suffer irreparably.

Buffett, through Berkshire Hathaway’s structure, cleverly avoids this trap:

As a holding company, not a mutual fund, he doesn’t face short-term redemption pressures, allowing him to hold or even buy more during panic, thereby fully enjoying Kelly’s advantages over long periods.

The lesson of sequence risk: true Kelly strategies must incorporate capital flow and timing considerations.

For individual investors, this means adjusting strategies based on cash needs—reducing portfolio volatility as you approach withdrawal time—even if that sacrifices some long-term expected return.

Moreover, continuous cash inflows are vital: building sources of passive income or ongoing savings can significantly enhance long-term growth.

6. Institutional "Anti-Kelly" structures
Modern financial institutions often incentivize "Anti-Kelly" behaviors:

- Performance evaluations and bonuses favor short-term results
- Emphasis on attribution fosters style purity but high correlation
- Separation of decision-making and risk management reduces holistic thinking
- Relative benchmarks create herding, increasing systemic risks

Particularly, hedge funds’ "2 and 20" fee model (2% management + 20% performance) creates asymmetric incentives:

- Success yields large profits for managers
- Failures mostly burden investors

This structure encourages excessive risk-taking—"Anti-Kelly" strategies.

Humans’ loss aversion, in terms of asymmetric gains and losses, is actually quite wise:

- Gains are often limited or fixed
- Losses can be large and irreversible

But for fee-based managers, losses are less feared because they don’t bear the downside—only the upside.

Increasingly, managers are attempting to optimize their structures.

7. Lack of mechanisms to handle Black Swan events
Nassim Taleb argues: extreme losses are far more damaging than extreme gains; tail risks are underestimated.
He often uses "Bankruptcy Theory" to explain:

Suppose a fund manager earns an average excess return of 3% annually over 15 years.
If a single 40% loss occurs, a return of about 67% is needed to recover.
Analysis of hedge fund data (2006–2009) shows that over 30% of "star funds" shut down due to large losses from single events.

A critical flaw in the traditional Kelly approach is assuming probability distributions are known and stable.

In fact, Taleb points out that markets are full of "unknown unknowns"—unpredictable, unquantifiable, but impactful events—Black Swans. These make standard Kelly application dangerous.

When Miller held large financial positions and faced the 2008 crisis, he encountered tail risks that Kelly couldn’t account for adequately.

No historical data or probability model can accurately predict the severity of a subprime crisis. Relying on "known" distributions leads to systematic errors.

Taleb’s "Antifragile" concept aligns with Kelly’s goal: survival and growth over the long term.

The difference is that Taleb emphasizes conservative strategies when facing unknown distributions. He recommends designing portfolios that benefit from volatility and chaos, rather than only optimizing under known risks.

In this view, Buffett’s investing shows remarkable "Antifragile" traits:

- Always maintaining ample cash to seize opportunities during chaos
- Strictly avoiding complex, poorly understood financial products
- Picking companies that survive and thrive in recessions
- Building long-term, low-turnover portfolios
From this perspective, strategies to counter Black Swan risks include:

- Additional buffers against extreme events (e.g., using "Half-Kelly" instead of full Kelly)
- Constructing antifragile portfolios ("keep powder dry")
- Focusing on risk composition rather than just expected returns
- Maintaining humility toward unknown risks
Taleb’s insights combined with Kelly offer a comprehensive risk management framework:

While seeking optimal capital allocation, acknowledge the fundamental limitations of probability estimates, and leave ample safety margins for unpredictable events.

Part Five: Buffett’s "Enhanced Kelly Formula" reveals seven secrets:

1. Long-term compounding first
Buffett’s core goal aligns with Kelly’s: maximize long-term geometric growth. This is the common goal for all investors.
But achieving this is hard, because short-term performance—sometimes even over years—relies more on arithmetic averages, which are easier to evaluate psychologically.
Buffett’s strength is that he truly doesn’t care about being criticized as "not capable."
2. High success rate in value investing
Buffett insists on value investing, pursuing high win rates, which allows him to allocate larger positions within Kelly’s framework, holding for longer to better realize compounding.
3. "Float" as capital growth driver
Buffett leverages Berkshire Hathaway’s insurance float and cash flows from investments, continuously increasing investable capital—beyond Kelly’s assumption of fixed principal.
For ordinary investors, this suggests maintaining some source of steady cash flow to grow capital.
4. Sacrificing short-term arithmetic for long-term geometric
Buffett sometimes sacrifices short-term book profits (arithmetic mean), holding large cash to buy at lower prices during downturns, thus optimizing long-term geometric returns.
For those aiming to be champions, doing this is very difficult.
5. Using probability to enhance Kelly’s efficacy
Buffett’s "margin of safety" has precise mathematical meaning in Kelly:

Buying below intrinsic value increases success probability )p( and reduces failure probability )q(.

In Kelly’s formula f* = (bp - q) / b, this dual effect significantly raises the optimal stake.

A classic example is Buffett’s 1964 investment in American Express:

When the "oil crisis" caused price collapse, he invested 40%, eventually gaining 300%. Margin of safety enabled larger deployments in high-certainty opportunities.

The "buy good companies" philosophy provides a practical framework for the most difficult part of Kelly—probability estimation. By translating abstract probabilities into business analysis, Buffett can:

a. Assess business stability )impact p(: focus on fundamentals, not market volatility, turning "win rate" into an analyzable business question.

b. Analyze economic moats )extend return horizon(: choose companies with lasting competitive advantages, enhancing forecast reliability, reinforcing Kelly’s "multiple plays" effect.

c. Calculate discounted cash flows )determine b(: analyze financials and management quality to forecast future cash flows, providing more accurate odds parameters for Kelly.

This framework converts market probabilities into business judgments, greatly improving Kelly’s practical use.

Buffett’s margin of safety isn’t static; it adjusts dynamically as understanding of the company deepens. He continually learns and recalibrates his probability estimates, embodying the iterative nature of value investing.

6. Win rate advantage leads to concentrated investing
Buffett’s concentrated strategy extends Kelly’s application in high success probability contexts. When confident of an advantage, moderate concentration yields higher long-term returns. This is based on three principles:

a. Circle of competence and information advantage: invest only within areas you deeply understand, satisfying Kelly’s premise—accurately estimating probabilities and odds. Inside your circle, expect more precise expectations than the market.

b. Moats and forecast reliability: durable competitive advantages both boost expected returns and reduce estimation errors in success probability.

"Wide moats and high confidence"—this certainty directly strengthens Kelly’s effectiveness.

c. Rational concentration: as Munger said, "Diversification is protection against ignorance, not a strategy."

This aligns with Kelly’s logic—allocate more capital where your advantage is greatest. Buffett’s highly concentrated portfolio is the ultimate expression of Kelly’s optimal capital allocation under high confidence.

Concentration isn’t just about capital allocation; it’s a test of your research ability and a vote of confidence in your judgment. Buffett’s holdings reflect his conviction in his deep analysis and high-confidence assessments.

7. Long-term perspective and mean reversion
Buffett systematically optimizes Kelly’s success probability parameter based on the market’s mean reversion.

This long-term approach involves four key elements:

a. Mean reversion offers a probabilistic edge: market prices fluctuate around intrinsic value, providing a statistical advantage—when prices are far below value, reversion probability increases. This allows Kelly to operate with less perfect forecasts.

b. High-confidence thresholds: Buffett only invests when confidence is very high, adjusting the probability estimates within Kelly to account for uncertainty.

Munger said, "We only act when the probability is high enough," avoiding the common Kelly pitfall—over-investment due to estimation errors.

c. Dynamic capital allocation: "Be fearful when others are greedy, and greedy when others are fearful"—this describes Kelly’s application aligned with market sentiment. During low valuation, increase bets; during high valuation, reduce.

The 2008 Goldman Sachs investment exemplifies this principle.

d. Tax effects and Kelly multiplier: Buffett’s long-term holdings defer taxes, effectively providing an interest-free loan to Kelly’s calculation.

Unrealized gains aren’t taxed, so the capital remains in play, boosting growth—assuming a 20% tax rate, long-term holdings increase Kelly’s multiplier by roughly 25%.

He famously said, "My favorite holding period is forever," which is more than a philosophical statement; it’s a precise application of tax mathematics—delaying gains to compound without immediate tax impact.

Long-term holding aligns with "time is the friend of good businesses."

As intrinsic value grows over time, long-term holding not only benefits from mean reversion but also from continuous value creation, amplifying positive expectations in Kelly’s application.

Part Six: How should ordinary investors apply these principles after understanding Buffett’s "Enhanced Kelly"?
Here are five practical tips to avoid "Anti-Kelly" traps and build a more resilient long-term investment strategy:
1. Build your personal "Circle of Competence," and stick to probability advantage investing
Action: List 3-5 industries or companies you truly understand. Limit your investments within this scope.

If your background is medicine, healthcare stocks might be within your circle;

If you’re a software engineer, certain tech companies may be within your expertise.

Kelly’s principle: accurate probability estimates depend on understanding. Investing outside your circle is like betting blindly—violating Kelly’s core principle.

Buffett said: "Risk comes from not knowing what you’re doing." Only within your circle can you truly gain a probabilistic edge.

Beware of pitfalls: hot sectors, FOMO, overconfidence in your judgment.

Remember Miller’s lesson: heavy leverage or concentration in your knowledge domain during crises can lead to catastrophic losses.

2. Implement a "Half-Kelly" conservative approach, leaving a margin of safety
Action: For any Kelly-calculated optimal position, only deploy half or less.

For example, if your calculation suggests 30% of assets, limit to 15%.

Kelly’s principle: estimates of probabilities and returns are always uncertain. The "Half-Kelly" strategy buffers against estimation errors, preventing overexposure and potential disaster.

Taleb’s research shows that conservative strategies tend to produce better long-term results when facing unknown distributions.

Beware of overconfidence, tail risks, underestimating Black Swan events. Think about "what’s the worst that can happen" rather than "what’s the best."

3. Layer your capital to match different time frames and goals
Action: Divide funds into three layers, e.g.,—

- Short-term needs (1-3 years): maintain liquidity and low risk;

- Medium-term goals (3-10 years): moderate allocation in stable assets;

- Long-term (10+ years): allocate in stocks and high-risk assets using Kelly principles.

Kelly’s insight: Bernstein’s "sequence risk" shows that timing of returns affects ultimate wealth significantly.

Only funds not needed for the short term can fully benefit from Kelly’s long-term geometric growth maximization.

Beware of pitfalls: investing essential living funds in high-risk assets; being forced to sell good assets at a loss; misaligning time horizons with investment strategies.

4. Create a cash flow engine to continuously grow your principal
Action: Develop a personal "float"—via side businesses, monetizing skills, royalties, or small ventures—to generate steady income streams, regularly adding new funds to investments.

Even small, sustained inflows can significantly enhance long-term compounding.

Kelly’s principle: Buffett’s insurance float expanded his capital base; individual investors can similarly benefit from continuous cash inflows.

Steady cash flows enable buying during downturns, not just selling, optimizing Kelly’s benefits.

Beware of pitfalls: relying solely on salary; neglecting passive income opportunities; overspending rather than investing. Remember Buffett’s frugal lifestyle and high savings rate.

5. Implement "Crash Readiness" plans
Action: Prepare in advance for market crashes, including—

- Reserve 10-20% cash as "ammunition";

- List 5-10 quality companies you want to buy at lower prices and define reasonable entry points;

- Set clear stepwise buy-in triggers, e.g., allocate one-third of cash when down 20%, another third at 30% decline.

Most importantly: document these plans during calm markets and follow them strictly during panic.

Kelly’s insight: market panic often offers the highest expected value but is psychologically hardest to act upon.

Pre-planned "Crash Plans" help overcome fear, applying Kelly’s dynamic principles—adding more when expectations are high.

This was key to Buffett’s large-scale buying in 2008 and 2020 crises.

Beware of pitfalls: herd behavior during panic; lack of cash reserves; absence of specific plans leading to missed opportunities.

These five tips embody not only the mathematics of Kelly but also address practical execution and psychological challenges.

Together, they form a "Enhanced Kelly Strategy" accessible to ordinary investors—helping you survive, thrive, and achieve long-term wealth growth like Buffett.

Final thoughts
Wealth growth fundamentally is about compounding.
But the reality is turbulent, discontinuous, uneven—full of uncertainty.
The Kelly formula calculates how to achieve compounding in an uncertain world.
In practice, it’s hard to compute precisely, but it offers a way to frame and think about risk and return.
This article’s main point: Buffett’s investment philosophy is largely centered around Kelly principles—even if he doesn’t explicitly perform these calculations.
Proving this is simple:
Kelly’s core is mathematical optimization—balancing risk and reward to maximize long-term growth.
Buffett, through his unique business models, has created an "Enhanced Kelly" that’s more efficient and less risky.
Most investment failures can be traced to "Anti-Kelly" strategies.
Buffett’s approach emphasizes predictability—good companies tend to generate strong free cash flow.

As a result, this cash flow is continually reinvested in Omaha, fueling Buffett’s bets.

Kelly was originally used to solve signal-to-noise problems in information transmission.
In a sense, investing is about signals and noise.

Later, Kelly was applied in casinos by Thorp, and then to investments.

But in fact, Kelly is ineffective for gamblers, because they’re playing a negative expected value game. In such cases, the best move is:

Don’t bet.

The philosophical core of Kelly is:
It reminds us to focus on long-term, stable growth rather than short-term gains and losses.

It highlights risk management—maintaining appropriate position sizes and cautious attitudes to avoid catastrophic losses from single adverse outcomes.
Additionally, Kelly teaches the importance of information utilization—making better decisions based on understanding outcomes.

In a broader sense, Kelly advocates patience and wisdom in facing uncertainty, using scientific methods to manage risk and seize opportunities.

In summary:
Don’t gamble, don’t leverage excessively, and never suffer permanent capital loss.