约束的代价：多资产组合里，每个"不许超"都在收费 Every constraint has a price

01会议上的那句话The sentence in the meeting

"单资产不超过 20%，久期不超过 2 年，月度换手控制在 25% 以内，新兴市场先别碰。" "No single asset above 20%, duration under two years, monthly turnover below 25%, and stay out of emerging markets."

这几句话在投委会里从不引起争论，因为每一条听起来都合理，都是风控语言。但没人说出它们的价格。结果通常是：模型给出的权重被手工"修一修"，一年后跟基准比，说不清是判断对了还是被约束磨平了。 Nobody argues with these sentences — each sounds prudent. But nobody says what they cost. The weights get hand-adjusted, and a year later nobody can tell whether the view paid off or the constraint sanded it away.

这篇想做的就是把价格算出来：在同样的风险预算下，每条约束、以及它们合在一起，究竟拿走多少预期收益。 This note prices them: at a fixed risk budget, how much expected return does each constraint — and all of them together — actually take?

读的时候只需要抓住一件事：本文所有的"代价"都是相对量——相对于 8% 的波动预算、相对于一个已经合规的基准组合、相对于当下的约束清单。换一组前提，数字会完全不一样，但比较的方法是一样的，这也是我最希望读者带走的部分。下面每一节都先给结论，再讲为什么。 One thing to hold on to: every cost here is relative — to an 8% volatility budget, to a compliant benchmark, and to the current list of constraints. Change the premises and the numbers change; the way of comparing them does not.

02先把"无约束最优"亮出来Look at the unconstrained optimum first

实验的统一口径是：波动不超过 8%，在此前提下最大化预期收益。这个口径比"最大化夏普"更接近机构实务（风险预算先定，收益尽量争取），也避免了"全押现金"这类退化解。 The common yardstick: maximise expected return subject to volatility ≤ 8%. This mirrors institutional practice — the risk budget comes first — and avoids degenerate solutions.

先看左边这列：模型把六成资金压在一个资产上。这不是因为它更了解信用债，而是因为它对预期收益的估计太自信——这正是 Michaud 1989 说的"优化器是误差放大器"。[1] The left column is the point: 64% in one asset, not because the model knows credit better, but because it trusts its own point estimates.[1]

怎么读表 1

看两列之间的差别，比看两列的绝对值更有信息量。第一，信用债从 63.8% 被压到 20%——那条上限切断的正是模型最爱下注的地方。第二，股票从 0% 被动升到 20%：模型原本根本不想要股票，是上限把资金"挤"过去的，这属于约束的副作用，不是模型的观点。第三，久期从 2.55 年降到 1.83 年，下降的部分来自久期上限，而不是优化器主动选择更短的久期。这三处变化合起来，才是"46.8 bps"的来历。 Read the difference between the columns, not the levels. Credit is capped from 63.8% to 20% — the cap cuts exactly where the optimiser wanted to bet. Equities rise from 0% to 20% because the cap pushes money there: a side effect, not a view. Duration falls from 2.55 to 1.83 years because of the duration cap, not because the optimiser prefers it.

03每条约束的价格The price of each constraint

• 单资产上限 20%：27.9 bps——四项里最贵的一组，因为它直接切断"集中押注"的能力。 Single-asset cap 20%: 27.9 bps — the most expensive, because it removes concentration precisely where the optimiser wants it.
• 月度换手上限 25%：13.5 bps——它的价格取决于你从哪里出发。基准组合与最优解差得越远，这条约束越贵。 Turnover cap 25%: 13.5 bps — priced relative to the starting portfolio; the further the optimum, the costlier the cap.
• 禁投清单（去掉新兴市场）：10.1 bps——删掉一个资产等于删掉一维自由度，价格不高但不可逆。 Exclusion list: 10.1 bps — deleting an asset removes a degree of freedom: cheap here, but irreversible.
• 久期上限 2 年：7.0 bps——最便宜的一条，因为它只在优化器想把久期拉长时才咬人。 Duration cap 2y: 7.0 bps — the cheapest, because it only bites when the optimiser wants duration.

把 27.9 bps 翻译成钱

bps 这个词容易让人麻木，换成钱会清楚很多：对一个 10 亿港元的组合，27.9 bps ≈ 每年 279 万港元的预期收益；四条约束加总 46.8 bps ≈ 468 万港元。要提醒两点：这是预期收益的差，不是每年必然发生的损失，某些年份完全可能因为约束而少亏；而且它衡量的是"理论代价"，在样本很短、参数不可信的场景里，同一组约束的净效果可能是正的（见第 6 节）。 bps numbs the senses; money does not. On a HKD 1bn book, 27.9 bps is about HKD 2.79m of expected return a year, and all four constraints together about HKD 4.68m. Two caveats: this is a difference in expected return, not a guaranteed annual loss, and in short-sample settings the same constraints can be worth more than they cost (section 6).

04约束之间会互相遮蔽Constraints mask each other

05收紧上限是非线性的Tightening is not linear

06但样本短的时候，约束是保险With short samples, constraints are insurance

上面全是"用真实参数算出来的代价"。现实中你并不知道真实参数，只能从历史里估。这时结论会翻转一部分——这就是 Michaud 与 Jagannathan & Ma 之间的张力：一位说约束有害，另一位说"错误的约束反而有用"。[1][2] Everything above is priced with true parameters. In practice you estimate them from history — where a different result appears: Michaud says constraints hurt, Jagannathan & Ma show wrong constraints often help.[1][2]

这解释了一个常见困惑：为什么"回测里约束拖累业绩"，而"实盘里加了约束反而更稳"。两个说法都对——只是在用不同的样本长度说话。 This resolves a common paradox: constraints look like a drag in backtests and like ballast in live portfolios. Both are right — they are just measured on different sample lengths.

07所以我在实际工作里怎么做What I actually do

1. 先报价格，再谈约束。任何约束提案都附一句"在我们 8% 风险预算下约值 X bps"——这句话让讨论从偏好变成取舍。 Price it before you debate it. Every constraint proposal carries a line: about X bps at our risk budget.
2. 用包裹定价，不用逐条定价。报告"逐条加入"和"逐条移除"两个方向——只报一个方向会系统性高估约束的代价。 Price the package, not the items. Report both add-one and remove-one margins; a single direction overstates the cost.
3. 约束要按"它挡住什么"来设计，而不是按习惯。挡集中度用上限，挡误差放大用收缩与上限，挡流动性用换手，每家机构的约束清单应该长短不一。 Design constraints by what they block. Caps block concentration, shrinkage and caps block error amplification, turnover limits block liquidity risk.
4. 先查可行性再上会。1/n 这道小学算术能挡掉一部分提案（n 是可选资产数）。 Check feasibility first. The arithmetic of 1/n, with n the number of eligible assets, kills some proposals before the meeting.
5. 样本短时，把约束当成不确定性预算来管：与其纠结"要不要加 20% 上限"，不如问"我凭什么相信 64% 这个数字"。[3] With short samples, treat constraints as an uncertainty budget. The real question is not the cap; it is why anyone believes 64%.[3]

08局限与反方观点Limits and the other side

• 参数是我设的。预期收益、波动与相关矩阵都是假设，换一组数字，bps 会变。这篇文章的价值在方法（怎么定价、怎么读边际），不在具体数值。 The parameters are mine. Change them and the bps change. The contribution is the method, not the numbers.
• 没有交易成本与税。换手上限的真实价值部分来自成本节省，本文的换手代价只算收益维度，因此低估了换手上限的合理性。 No costs or taxes. Part of the value of a turnover limit is the cost it saves, which is outside this measurement.
• 约束的价值有很大一部分是"防人"而不是"防模型"。禁投清单、单资产上限常来自合规与委托要求，这时候谈 bps 只是让代价可见，不是否定约束。 Much of a constraint's value is governance, not modelling. Pricing them makes the cost visible without arguing them away.
• 反方：为什么不干脆别用优化器？1/N 在文献里常常打赢样本内最优的均值方差组合，而且不需要估计参数。这确实是本文结论的自然延伸——如果你的参数估计不够好，约束或等权是同一件事的两种说法。[4] The other side: why use an optimiser at all? 1/N often beats in-sample mean-variance with no parameter estimation.[4]

三个常见误解

• "约束越少越好。"不对。本文第 6 节显示，估计窗口越短，约束带来的样本外保护越明显（10 分位从 −0.02 提到 +0.18）。约束多不等于差，约束与你的参数质量不匹配才差。 "Fewer constraints is better." No: with short estimation windows, constraints improved the 10th-percentile outcome from −0.02 to +0.18. What hurts is a constraint set that does not match the quality of your estimates.
• "代价不大就无所谓。"46.8 bps 听起来很小，但它对抗的是夏普 0.5–0.6 这个量级的策略——它相当于吃掉约 8%–9% 的预期超额收益。在长期复利里，这类"看起来很小"的数字才是最终决定成败的那一批。 "The cost is too small to care about." 46.8 bps against a Sharpe of 0.5–0.6 is roughly 8–9% of expected excess return — the kind of small number that compounds into the final answer.
• "算这个是为了删掉约束。"不是。合规、委托与治理类约束不该被 bps 论证掉；把它们定价的目的是让取舍变得可见——知道价格之后，会议讨论的就不再是"我觉得"，而是"我们愿意为这条约束付多少"。 "Pricing constraints is an argument for removing them." No. Mandate and governance constraints should not be argued away by bps; pricing them makes the trade-off visible so the discussion becomes "how much are we willing to pay for this rule".

09我的判断My take

10来源（链接均于 2026-09-29 核查）Sources (links verified 2026-09-29)

1. Michaud, R. O. (1989). The Markowitz Optimization Enigma: Is 'Optimized' Optimal? Financial Analysts Journal. doi.org/10.2469/faj.v45.n1.31
2. Jagannathan, R., & Ma, T. (2003). Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps. Journal of Finance. doi.org/10.1111/1540-6261.00580 · 免费工作论文版：free working paper: NBER w8922
3. Black, F., & Litterman, R. (1992). Global Portfolio Optimization. Financial Analysts Journal. doi.org/10.2469/faj.v48.n5.28
4. DeMiguel, V., Garlappi, L., & Uppal, R. (2009). Optimal Versus Naive Diversification. Review of Financial Studies. doi.org/10.1093/rfs/hhm075
5. Clarke, R., de Silva, H., & Thorley, S. (2011). Minimum-Variance Portfolio Composition. Journal of Portfolio Management. doi.org/10.3905/jpm.2010.2010.1.009
6. arXiv (2026-07). Fragility of Minimum-Variance Portfolios. arxiv.org/abs/2607.18624
7. arXiv (2026-09). Decision-Focused Learning for Mean-Variance Portfolio Optimization via KKT-Based Reformulation. arxiv.org/abs/2609.21427
8. 本文实验代码：tools/make_figs_constraints.py（固定种子；模拟设定见开头「口径」）。

11分享版摘要Shareable summary

同样 8% 的波动预算下，四条典型机构约束一共拿走 46.8 bps/年 —— 单资产上限 20%（27.9）、久期上限 2 年（7.0）、月度换手上限 25%（13.5）、禁投清单（10.1）。但不是简单相加：四条单价的算术和是 58.5 bps，合起来只有 46.8，因为约束之间会互相遮蔽。

更反直觉的一半：约束的价格不是常数。估计窗口只有 24 个月时，无约束组合样本外超额夏普的 10 分位是 −0.02，加了约束反而是 +0.18；样本拉长到 240 个月，两者收敛。约束买到的，其实是对自己估计误差的保险。

12引用本文Cite this note

Aylon (2026). 约束的代价：多资产组合里，每个"不许超"都在收费. Notes, v1.0. https://aayloo.github.io/notes/cost-of-constraints/

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