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
• Don't deploy capital simply because it's available
• But do not treat "wait" as a free, self-justifying default — see the cash mandate below.

The budgeted cash mandate (added 2026-06-28)

The 2026-06-28 external audit identified the line that previously closed this section — "Cash is not idle — it is optionality" — as a doctrinal escape hatch: it made an all-cash book unfalsifiable, because every market state (up, down, flat) could be re-described as virtuous patience. A process that mechanically produces 100% cash and then defines 100% cash as success cannot detect its own over-conservatism. For 33 days the kit held 100% cash and called it discipline; the audit called it a closed loop. This mandate replaces the escape hatch with an accountable one.

Cash is a position with an opportunity cost, and that cost must be defended, not assumed.

• Cash ≤ 30%: normal. No special justification required.
• Cash 30–70%: permitted, but the weekly review must carry a one-sentence defended reason — a specific, falsifiable claim about the opportunity set, not a mood. Acceptable: "The five cheapest watchlist names each sit >20% above their recalibrated triggers; I expect that to still be true in a quarter." Not acceptable: "Nothing looks cheap" / "staying patient."
• Cash > 70% for more than two consecutive weeks: this is a flagged state, not a resting state. It triggers a written cash post-mortem in the weekly optimization note answering: (a) is the opportunity set genuinely barren, or is the ruler miscalibrated (re-run the cheapest 3 names through margin-of-safety-pricing and check whether they trigger under the recalibrated single-discount bands)? (b) what observable market change would move me off the sidelines, and by when? (c) what has the cash posture cost vs. SPY/RPV/RPG since it began (read the Shadow-Book and Performance numbers — the real ones, not the frozen-favorable ones)?

The test is falsifiability: if there is no market state in which the doctrine would say "you are being too cautious," the doctrine is broken. The cash post-mortem exists to manufacture that state on a schedule.

The probe-book mandate — buying information (added 2026-06-28)

A kit that has taken zero positions has a sample of zero and therefore cannot know whether it works — every calibration instrument downstream is starved of data (the audit found the Shadow Book and Near-Miss Ledger built but empty). To break the zero-sample trap, the kit maintains a small probe book:

• Whenever ≥1 name carries a continue/quality verdict and sits within ~10% of its (recalibrated) trigger, the kit may open a Probe position (0.5–1.5%) explicitly framed as buying information, not conviction — marked in Transactions with rationale probe: information.
• The standing target: be invested in at least the top 1–3 ranked names once any of them is within probe range, rather than waiting for the full margin of safety on a Core-sized position. A probe is sized so that being wrong costs almost nothing and being right teaches the kit how its own ruler behaves against live outcomes.
• Probes are not exempt from the gates — they need a thesis (a stub is acceptable for a probe), a kill criterion, and a sizing line. They are exempt from the "wait for full MoS" reflex, because their purpose is calibration, not return maximization.
• A probe that the thesis later upgrades to Core is scaled up per the normal build pattern; a probe whose kill criterion fires is exited like any position. Either way, the kit now has realized data instead of another month of prose about hypothetical discipline.

Patience is still a virtue. But patience that has produced zero trades, zero data, and an empty scoreboard for a month is indistinguishable from paralysis — the kit said so itself in 04-intellectual-virtues. The probe book is how patience earns the right to call itself patience.

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