Position Sizing and Kelly

How much to invest in each idea. Sizing turns analytical insight into portfolio outcome; even great analysis is wasted on poorly-sized positions. The deep-value agent uses Kelly-derived logic as a frame, then sizes conservatively below the Kelly optimum.

The Kelly Criterion

The Kelly criterion T2 solves for the fraction of bankroll that maximizes the expected logarithm of wealth — i.e., the long-run geometric growth rate. For a discrete bet:

f = (bp − q) / b*

Where:

• f* = fraction of bankroll to bet
• b = net odds received (a $1 bet that wins $b net of the stake)
• p = probability of winning
• q = probability of losing = 1 − p

In the simple form often quoted in practitioner texts T2:

f ≈ edge / odds*

Higher edge → larger position. Wider odds (larger possible loss per unit gain) → smaller position. The two work in tension.

Generalizing to continuous returns — a clarification

Many practitioner treatments quote a "continuous Kelly" formula as:

f = (μ − r) / σ²

Where μ is expected return, r is the risk-free rate, σ² is variance. This is not strictly Kelly. It is the Markowitz/Merton mean-variance optimum T2. It equals the Kelly optimum only under the restrictive assumption of log utility with lognormal returns. For equities, where return distributions are fat-tailed and asymmetric, treating (μ − r)/σ² as the Kelly fraction will tend to overstate the optimal size T2.

The honest position: Kelly's exact result applies to bets with known discrete payoff distributions. For an equity position, we use Kelly logic as a frame — bigger edge warrants more capital, bigger variance warrants less — but we do not pretend to compute an exact f* from imprecise inputs.

Why pure Kelly is too aggressive

Even when correctly computed, Kelly is the optimal long-run growth rate at the cost of severe short-run volatility T2:

• Pure Kelly positions can show 50%+ drawdowns even when the edge is genuine, because the criterion accepts large interim losses in exchange for log-optimal long-run growth.
• Probability and edge are estimated imperfectly. Mauboussin notes that even a modest overestimate of edge leads to materially destructive over-sizing under full Kelly T2.
• Real-world bankrolls have constraints (regulatory, redemption, career) that pure Kelly assumes away.

The practitioner adjustment is fractional Kelly — typically 25-50% of the computed Kelly size — which trades some long-run geometric growth for materially lower drawdown and protection against estimation error T2. AlphaSteve uses this fractional-Kelly framing rather than computing f* directly.

The deep-value sizing structure

A practical structure, calibrated to deep-value characteristics. The tier and limit numbers below are <span class="tier-cal" title="our calibration; informed by Klarman's conservative-concentration approach in *Margin of Safety* (1991), Buffett's 5-position high-conviction frame in the Berkshire letters, and Greenblatt's *You Can Be a Stock Market Genius* (1997) on small-portfolio focused concentration">AS-cal</span>.

Conviction tiers

Each holding falls into a tier:

A typical portfolio: 3-5 Core 1, 5-8 Core 2, 8-15 Mid, several Probe. Total 20-30 positions <span class="tier-cal" title="">AS-cal</span>.

Hard limits <span class="tier-cal" title="">AS-cal</span>

• Single position: no more than 15% at cost (capped to prevent ruinous concentration)
• Single industry: no more than 30%
• Single country exposure (for non-Tier-1): no more than 25%
• Concentrated risk drivers: monitored across positions

These are floor protections, not optimization constraints.

Sizing inputs

1. Margin of safety

Bigger discount to intrinsic value → bigger size. A 50% discount at high conviction warrants more capital than a 25% discount.

2. Confidence interval width

Narrower confidence interval on intrinsic value → bigger size. A regulated utility with $50 ± $5 intrinsic value can be sized larger than a cyclical with $50 ± $25.

3. Permanent loss probability

Higher tail risk → smaller size. A position with 5% probability of going to zero should be smaller than one with 0.5%.

4. Liquidity

Lower liquidity → smaller size (sometimes capped to ensure liquidity at exit).

5. Correlation with other holdings

A position correlated with several existing holdings should be sized smaller (or replace one of them).

6. Cycle position

Cyclical names at trough can be sized somewhat larger because asymmetric upside; the same names at peak should be sized small or absent.

A simple sizing decision

For each new position, work through:

1. Intrinsic value range (from valuation work): e.g., $40-60, midpoint $50
2. Current price: e.g., $30
3. Margin of safety: 40%
4. Confidence in intrinsic value: high (narrow range)
5. Permanent loss probability: low (asset floor at $25)
6. Conviction tier: Core 1

Position size starts at 8%. Adjustments:

• Highly correlated with existing positions → reduce
• Material liquidity constraint → reduce
• Specific tail-risk concern → reduce

Final size: 6-8%.

For an opposite case (modest MoS, wider range, higher tail risk, less liquid): Position size starts at 2%. Probably ends at 1-1.5%.

Building positions over time

Deep-value positions are rarely sized to target on day one. Reasons:

• The price may continue falling (averaging down at greater discount)
• The thesis develops with continued work and confirmation
• Earnings or events may provide better entry points
• Liquidity constraints may require splitting the buy

Typical pattern:

• Initial entry at 30-50% of target size
• Add as price weakens (with thesis re-validation)
• Reach target size over weeks to months

The discipline: set entry points and target size in advance, not in response to short-term price action. Mechanical execution against pre-set plan beats discretionary chasing.

Trimming and exiting

The mirror of building:

• Trim as price approaches central value (e.g., 50% trim at 80% of central value)
• Exit fully near central value
• Optional: hold a residual through value if business is improving and re-rating fully

Exit triggers also include:

• Kill criterion fires (immediate exit)
• Better idea on the same balance sheet of attention (reallocate)
• Thesis weakening (reduce or exit)

The opportunity cost discipline

Every position holds a slot. New ideas compete with existing positions for that slot.

The discipline: a new idea must be clearly better than the weakest existing position to displace it. "Clearly better" means:

• Wider margin of safety, or
• Lower tail risk, or
• Better correlation diversification, or
• Higher conviction with similar margin of safety

Mechanical re-prioritization of the portfolio against new ideas keeps capital deployed in the best opportunities.

When Kelly logic suggests sizing larger than your discipline allows

Sometimes the analysis suggests a very high-confidence, wide-margin-of-safety position that Kelly would size at 20-30% of portfolio. The discipline:

• Take the lower discipline-allowed size
• Recognize that you are leaving expected return on the table for risk reduction
• Be willing to forego optimization for survival

The deep-value defeat is not "didn't capture the full Kelly optimum on one trade" — it is "lost a lot of money on one trade that turned out wrong despite high conviction." The asymmetry favors caution.

When you have many great ideas

A rare and excellent problem. The discipline:

• Spread capital across all of them rather than over-concentrating in one
• Accept that you may underweight any single one relative to its merit
• Maintain hard limits even when multiple positions could each be sized larger

In stress periods when many great opportunities appear, take advantage but maintain discipline.

When you have few or no great ideas

Equally important. The discipline:

• Don't manufacture conviction
• Carry meaningful cash
• Wait
• Continue researching but don't deploy capital simply because it's available

The patience reservoir is part of the position-sizing framework. Cash is not idle — it is optionality.

Output

Sizing logic for any position includes:

1. Conviction tier assigned
2. Initial position size target
3. Build pattern (initial entry, add points)
4. Sell discipline (trim and exit points)
5. Hard rule compliance check
6. Correlation check with existing portfolio
7. Adjusted final size with rationale

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): 247-257.
• Thorp, E. (2006). "The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market." In Handbook of Asset and Liability Management (S. Zenios & W. Ziemba, eds.).
• MacLean, L. C., Thorp, E. O., & Ziemba, W. T., eds. (2011). The Kelly Capital Growth Investment Criterion. World Scientific.
• Poundstone, W. (2005). Fortune's Formula. Hill and Wang. — accessible long-form history of Kelly in finance.
• Mauboussin, M. J. (2006). More Than You Know. Columbia Business School Publishing. — practitioner treatment of probabilistic sizing and the cost of overconfidence.
• Klarman, S. (1991). Margin of Safety. HarperBusiness. — conservative-concentration discipline informing the AS-cal tier structure.
• Greenblatt, J. (1997). You Can Be a Stock Market Genius. Simon & Schuster. — small-portfolio focused concentration.

This file synthesizes named primary sources. Position-size tiers, industry/country caps, and the 20-30 position guideline are tagged <span class="tier-cal" title="">AS-cal</span> and are AlphaSteve's own calibrations, revisable as the calibration tracker accumulates outcomes.

Linked

• permanent-capital-loss — sizing protects against this
• tail-risk-and-fat-tails — sizing accounts for this
• margin-of-safety-pricing — input to sizing
• base-rates — calibration of conviction
• cycle-positioning — sizing for cycle position
• 05-decision-framework — sizing implements decisions
• sources-policy — citation discipline applied here