bet-sizing

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Bet Sizing

头寸规模确定

Purpose

目的

Provides frameworks for determining how much capital to allocate to individual positions within a portfolio. Covers the Kelly criterion, fractional Kelly, risk budgeting, liquidity-based sizing, and conviction weighting. Proper bet sizing is critical — even a portfolio of good ideas can fail with poor sizing.

为投资组合内单个头寸的资金分配比例提供框架，涵盖Kelly criterion、fractional Kelly、风险预算、基于流动性的头寸规模确定以及信念权重法。合理的头寸规模至关重要——即便投资组合的标的选择得当，糟糕的头寸规模策略也可能导致失败。

Layer

层级

4 — Portfolio Construction

4 — 投资组合构建

Direction

适用方向

prospective

前瞻性

When to Use

适用场景

Determining the appropriate size for a new position
Applying the Kelly criterion to a bet or investment with estimable odds
Allocating a risk budget across active positions
Setting maximum position sizes based on liquidity or risk limits
Sizing positions proportional to conviction and edge
Deciding the optimal number of positions in a concentrated portfolio
Scaling position sizes with volatility changes

确定新头寸的合适规模
将Kelly criterion应用于可估算概率的投注或投资
在活跃头寸间分配风险预算
根据流动性或风险限制设置最大头寸规模
根据信念强度和优势确定头寸规模
决定集中型投资组合的最优头寸数量
随波动率变化调整头寸规模

Core Concepts

核心概念

Kelly Criterion (Discrete)

Kelly Criterion（离散型）

For a binary bet with payoff odds b, win probability p, and loss probability q = 1-p:

f* = (b*p - q) / b

where f* is the optimal fraction of wealth to wager. The Kelly criterion maximizes the expected logarithm of wealth (geometric growth rate) over repeated bets.

Properties:

f* = 0 when edge = 0 (no bet when there is no advantage)
f* < 0 when negative edge (the formula tells you to bet the other side)
f* > 0 only when b*p > q (positive expected value)

对于具有赔付赔率b、获胜概率p、失败概率q=1-p的二元投注：

f* = (b*p - q) / b

其中f*是应投注的财富最优比例。Kelly criterion可最大化重复投注下财富的预期对数增长率（几何增长率）。

特性：

当优势为0时，f*=0（无优势时不应投注）
当存在负优势时，f*<0（公式建议反向投注）
仅当bp > q（正期望值）时，f>0

Kelly Criterion (Continuous / Investment)

Kelly Criterion（连续型/投资类）

For a normally distributed investment return with expected excess return mu-r_f and variance sigma^2:

f* = (mu - r_f) / sigma^2

This gives the fraction of total wealth to allocate. For example, an asset with 8% expected excess return and 20% volatility: f* = 0.08 / 0.04 = 2.0 (200% of wealth — implying leverage).

对于服从正态分布、预期超额收益为mu-r_f、方差为sigma²的投资回报：

f* = (mu - r_f) / sigma²

该公式给出应分配的总财富比例。例如，某资产的预期超额收益为8%、波动率为20%，则f*=0.08/0.04=2.0（即投入200%的财富，意味着使用杠杆）。

Fractional Kelly

Full Kelly sizing is theoretically optimal but practically too aggressive because:

It assumes perfect knowledge of probabilities and payoffs
It produces large drawdowns (the expected drawdown of full Kelly is significant)
Estimation error in parameters can turn optimal into catastrophic

Practical approach: use a fraction of Kelly, commonly:

Half Kelly (f/2):* Achieves 75% of the growth rate with substantially lower variance and drawdown risk
Third Kelly (f/3):* Even more conservative; appropriate when parameter uncertainty is high
Quarter Kelly (f/4):* Suitable for highly uncertain estimates

The key insight: the growth rate curve is flat near the peak. Reducing from full Kelly to half Kelly only sacrifices 25% of growth but reduces risk dramatically.

完全Kelly规模在理论上是最优的，但在实际应用中过于激进，原因如下：

它假设概率和赔付赔率是完全已知的
会导致大幅回撤（完全Kelly策略的预期回撤幅度很大）
参数估算误差可能使最优策略变为灾难性的策略

实用方案：使用分数Kelly，常见类型包括：

*Half Kelly（f/2）**：可实现75%的增长率，同时大幅降低方差和回撤风险
*Third Kelly（f/3）**：更为保守，适用于参数不确定性较高的场景
*Quarter Kelly（f/4）**：适用于估算结果高度不确定的情况

核心见解：增长率曲线在峰值附近较为平缓。从完全Kelly调整为Half Kelly仅损失25%的增长率，但风险显著降低。

Risk Budgeting

风险预算

Allocate risk (not capital) across positions. The total risk budget is the maximum acceptable portfolio risk (e.g., 10% VaR or 5% tracking error).

VaR-based budgeting:

Total VaR budget: e.g., $1M at 95% confidence
Allocate across positions: Position VaR_i <= allocated VaR_i
Position VaR = w_i * sigma_i * z_alpha * Portfolio Value

Tracking error budgeting (for active managers):

Total active risk budget: e.g., 4% tracking error
Allocate across bets: each active bet consumes a portion of tracking error
Size active positions so that sum of risk contributions equals total risk budget

在头寸间分配风险（而非资金）。总风险预算是可接受的最大投资组合风险（如95%置信水平下的10% VaR或5%跟踪误差）。

基于VaR的预算：

总VaR预算：例如，95%置信水平下100万美元
在头寸间分配：单个头寸VaR_i ≤ 分配的VaR_i
头寸VaR = w_i * sigma_i * z_alpha * 投资组合价值

基于跟踪误差的预算（适用于主动型基金经理）：

总主动风险预算：例如4%跟踪误差
在各投注间分配：每个主动投注消耗部分跟踪误差
调整主动头寸规模，使风险贡献总和等于总风险预算

Maximum Position Sizes

最大头寸规模

Hard limits on individual positions to prevent concentration risk:

Liquidity-based limits:

Position < X% of average daily volume (ADV) — common limits: 10-25% of ADV
Ensures ability to exit within a reasonable time frame (e.g., 5-10 trading days)

Risk-based limits:

Position risk contribution < X% of portfolio volatility (e.g., max 10% of portfolio risk)
Single position < X% of portfolio value (common: 5% for diversified, 10% for concentrated)

Regulatory/mandate limits:

Mutual fund: no more than 5% in a single name (diversified fund) or 25% (non-diversified)
Index tracking: weight cannot deviate from benchmark by more than specified amount

对单个头寸设置硬性限制，以防止集中度风险：

基于流动性的限制：

头寸规模 < 日均成交量（ADV）的X%——常见限制为ADV的10-25%
确保能在合理时间内平仓（如5-10个交易日）

基于风险的限制：

头寸风险贡献 < 投资组合波动率的X%（如最大不超过投资组合风险的10%）
单个头寸 < 投资组合价值的X%（分散型组合通常为5%，集中型组合通常为10%）

监管/委托限制：

共同基金：分散型基金单个标的持仓不得超过5%，非分散型基金不得超过25%
指数跟踪：权重与基准的偏差不得超过规定范围

Conviction Weighting

信念权重法

Size positions proportional to the strength of the investment thesis:

High conviction (largest positions): Strong edge, deep research, multiple confirming factors
Medium conviction: Solid thesis but some uncertainty or limited information
Low conviction (smallest positions): Early-stage idea, limited edge, or purely diversification-motivated

Framework: Score each position on edge strength (1-5) and certainty (1-5). Size proportional to the product: edge * certainty.

头寸规模与投资逻辑的强度成正比：

高信念（最大头寸）：显著优势、深入研究、多重验证因素
中信念：逻辑坚实但存在一定不确定性或信息有限
低信念（最小头寸）：早期想法、优势有限或仅为分散风险

框架：对每个头寸的优势强度（1-5分）和确定性（1-5分）打分，头寸规模与两者的乘积（优势×确定性）成正比。

Optimal Number of Positions

最优头寸数量

Trade-off between diversification and conviction:

Concentrated (10-20 positions): High conviction, deep research. Each position is 5-10% of the portfolio. Appropriate when the manager has genuine skill and edge.
Diversified (50-100 positions): Lower conviction per position but broader risk reduction. Each position is 1-3%. Appropriate for systematic or factor-based strategies.
Very diversified (100+): Index-like. Risk comes from factor tilts, not individual positions.

在分散化和信念强度间权衡：

集中型（10-20个头寸）：高信念、深入研究。每个头寸占投资组合的5-10%。适用于基金经理具备真正能力和优势的场景。
分散型（50-100个头寸）：单个头寸信念较低，但风险分散更广。每个头寸占投资组合的1-3%。适用于系统化或因子驱动策略。
高度分散型（100+个头寸）：类似指数。风险来自因子倾斜，而非单个头寸。

Volatility Scaling

波动率调整

Adjust position sizes inversely with volatility to maintain consistent risk per position:

Adjusted size = Target risk / Current volatility

When volatility doubles, position size halves, keeping the dollar risk constant. This is a core principle in managed futures and risk-targeting strategies.

与波动率成反比地调整头寸规模，以保持每个头寸的风险一致：

调整后规模 = 目标风险 / 当前波动率

当波动率翻倍时，头寸规模减半，从而保持美元风险恒定。这是管理期货和风险目标策略的核心原则。

Anti-Martingale (Kelly-like) Sizing

反鞅（类Kelly）规模

Increase position sizes after gains (wealth grows, so Kelly fraction applied to larger base) and decrease after losses. This contrasts with martingale strategies (doubling down after losses) which can lead to ruin.

Kelly naturally implements anti-martingale sizing: bet a constant fraction of current wealth, so absolute bet size grows with wealth and shrinks with losses.

盈利后增加头寸规模（财富增长，Kelly比例应用于更大的基数），亏损后减少头寸规模。这与鞅策略（亏损后加倍投注）形成对比，后者可能导致破产。

Kelly准则自然实现了反鞅规模：投注当前财富的固定比例，因此绝对投注规模随财富增长而增加，随亏损而减少。

Key Formulas

核心公式

Formula	Expression	Use Case
Kelly (Discrete)	f* = (b*p - q) / b	Binary bet sizing
Kelly (Continuous)	f* = (mu - r_f) / sigma^2	Investment position sizing
Half Kelly	f = f* / 2	Practical conservative sizing
Growth Rate at Kelly	g* = (mu - r_f)^2 / (2*sigma^2)	Maximum geometric growth
Growth Rate at f	g(f) = f(mu - r_f) - f^2sigma^2/2	Growth rate for any fraction
Volatility-Scaled Size	w = target_risk / sigma_i	Constant risk per position
Position VaR	VaR_i = w_i * sigma_i * z_alpha * V	Position-level risk

公式	表达式	适用场景
Kelly（离散型）	f* = (b*p - q) / b	二元投注规模确定
Kelly（连续型）	f* = (mu - r_f) / sigma^2	投资头寸规模确定
Half Kelly	f = f* / 2	实用保守型规模确定
Kelly下的增长率	g* = (mu - r_f)^2 / (2*sigma^2)	最大几何增长率
任意比例f下的增长率	g(f) = f(mu - r_f) - f^2sigma^2/2	任意比例下的增长率
波动率调整后规模	w = target_risk / sigma_i	单个头寸风险恒定
头寸VaR	VaR_i = w_i * sigma_i * z_alpha * V	头寸层面风险

Worked Examples

示例计算

Example 1: Kelly Criterion for a Discrete Bet

示例1：离散型Kelly准则的投注规模

Given:

Win probability: p = 55%
Loss probability: q = 45%
Even-money payoff: b = 1 (win $1 for every $1 wagered)

Calculate: Optimal bet size

Solution:

f* = (b*p - q) / b = (1 * 0.55 - 0.45) / 1 = 0.10 / 1 = 10%

Interpretation: Wager 10% of current wealth on each bet. This maximizes long-run geometric growth.

Practical adjustment (half Kelly): f = 10% / 2 = 5% — achieves 75% of the maximum growth rate with much lower drawdown risk.

Full Kelly expected drawdown: the probability of losing 50% of wealth at some point is substantial. Half Kelly dramatically reduces this tail risk.

已知条件：

获胜概率：p=55%
失败概率：q=45%
等额赔付：b=1（每投注1美元，获胜则盈利1美元）

计算： 最优投注规模

解决方案：

f* = (b*p - q) / b = (1 * 0.55 - 0.45) / 1 = 0.10 / 1 = 10%

解读：每次投注当前财富的10%，可最大化长期几何增长率。

实际调整（Half Kelly）：f = 10% / 2 = 5% —— 可实现75%的最大增长率，同时大幅降低回撤风险。

完全Kelly的预期回撤：在某一时刻损失50%财富的概率很高。Half Kelly可显著降低这种尾部风险。

Example 2: Continuous Kelly for an Investment

示例2：连续型Kelly准则的投资头寸规模

Given:

Expected excess return (mu - r_f): 8%
Volatility (sigma): 20%

Calculate: Kelly-optimal allocation

Solution:

f* = (mu - r_f) / sigma^2 = 0.08 / (0.20)^2 = 0.08 / 0.04 = 2.00 (200%)

This implies 200% allocation (2x leverage), which is extremely aggressive.

Practical adjustments:

Half Kelly: 100% (no leverage, fully invested)
Third Kelly: 67% allocation
Quarter Kelly: 50% allocation

Given that the 8% expected return and 20% volatility are estimates with significant uncertainty, half Kelly (100%) or less is prudent. The growth rate curve is:

Full Kelly: g* = 0.08^2 / (2 * 0.04) = 8% per year
Half Kelly: g(1.0) = 1.0 * 0.08 - 1.0^2 * 0.04/2 = 6% per year (75% of maximum)
Quarter Kelly: g(0.5) = 0.5 * 0.08 - 0.5^2 * 0.04/2 = 3.5% per year (44% of maximum)

已知条件：

预期超额收益（mu - r_f）：8%
波动率（sigma）：20%

计算： Kelly最优分配比例

解决方案：

f* = (mu - r_f) / sigma^2 = 0.08 / (0.20)^2 = 0.08 / 0.04 = 2.00（200%）

这意味着投入200%的资金（即使用2倍杠杆），这一策略极具攻击性。

实际调整方案：

Half Kelly：100%（无杠杆，全额投资）
Third Kelly：67%的资金分配
Quarter Kelly：50%的资金分配

考虑到8%的预期收益和20%的波动率均为估算值，存在显著不确定性，因此Half Kelly（100%）或更低比例更为谨慎。增长率曲线如下：

完全Kelly：g* = 0.08^2 / (2 * 0.04) = 8% 每年
Half Kelly：g(1.0) = 1.0 * 0.08 - 1.0^2 * 0.04/2 = 6% 每年（为最大增长率的75%）
Quarter Kelly：g(0.5) = 0.5 * 0.08 - 0.5^2 * 0.04/2 = 3.5% 每年（为最大增长率的44%）

Common Pitfalls

常见误区

Full Kelly is too aggressive for practical use — estimation errors in probabilities and payoffs can lead to over-betting and ruin; always use fractional Kelly
Kelly assumes known probabilities and payoffs — in reality these are estimated with significant error, making full Kelly dangerous
Kelly maximizes log wealth (geometric growth rate), which may not match an investor's actual utility function or risk tolerance
Ignoring liquidity constraints: Kelly-optimal size may exceed what the market can absorb without impact
Correlation between positions: the single-asset Kelly formula does not account for portfolio effects; positions with correlated risk collectively require smaller sizing
Survivorship bias in parameter estimation: historical win rates may overstate future edge
Not adjusting for regime changes: edge and volatility are time-varying

完全Kelly在实际应用中过于激进——概率和赔付赔率的估算误差可能导致过度投注和破产；始终使用fractional Kelly
Kelly准则假设概率和赔付赔率已知——但实际上这些都是估算值，存在显著误差，使得完全Kelly具有危险性
Kelly准则最大化对数财富（几何增长率），这可能与投资者的实际效用函数或风险承受能力不匹配
忽视流动性约束：Kelly最优规模可能超出市场可吸收的范围，从而产生冲击成本
头寸间的相关性：单资产Kelly公式未考虑投资组合效应；具有相关风险的头寸整体需要更小的规模
参数估算中的生存偏差：历史胜率可能高估未来优势
未适应市场环境变化：优势和波动率随时间变化

Cross-References

交叉引用

risk-return (Layer 1): expected return and volatility as key inputs to Kelly sizing
diversification (Layer 4): tension between concentration (large bets) and diversification (many small bets)
asset-allocation (Layer 4): bet sizing operates within the asset allocation framework
rebalancing (Layer 4): positions drift from target sizes and require rebalancing
quantitative-valuation (Layer 3): valuation-based edge estimates feed into conviction weighting

risk-return（层级1）：预期收益和波动率是Kelly规模确定的关键输入
diversification（层级4）：集中（大额投注）与分散（小额多投注）之间的矛盾
asset-allocation（层级4）：头寸规模确定在资产配置框架内运作
rebalancing（层级4）：头寸规模会偏离目标，需要再平衡
quantitative-valuation（层级3）：基于估值的优势估算为信念权重法提供输入

Reference Implementation

参考实现

See

scripts/bet_sizing.py

for computational helpers.

详见

scripts/bet_sizing.py

中的计算工具。