1/N 打败最优化：八条腿、九个组合、三十三年1/N beats the optimiser: eight legs, nine portfolios, thirty-three years

01那句话是怎么来的How the claim earned its reputation

组合管理的教科书都从同一个流程开始：估一份预期收益，估一份协方差矩阵，解一个最优化问题，读出权重。"1/N 打败最优化"说的是把这个流程整个绕开——把钱平摊给每一条腿。这不是民间智慧，它有一个很硬的出处。

Portfolio textbooks all begin the same way: estimate expected returns, estimate a covariance matrix, solve an optimisation problem, read off the weights. "1/N beats optimisation" says to skip the whole procedure and split the money evenly instead. This is not folk wisdom; it has a hard citation.

2009 年，DeMiguel、Garlappi 与 Uppal 在《Review of Financial Studies》上发表了《Optimal Versus Naive Diversification: How Inefficient Is the 1/N Portfolio Strategy?》。他们把当时常用的十四个最优化模型——样本均值方差、最小方差、带约束的版本、各种贝叶斯收缩——放在七个公开数据集上，与最简单的 1/N 等权逐一对比。结论一句话：把估计误差和换手成本算进去之后，没有哪个模型能稳定地赢过等权。

In 2009 DeMiguel, Garlappi and Uppal published "Optimal Versus Naive Diversification: How Inefficient Is the 1/N Portfolio Strategy?" in the Review of Financial Studies. They lined up fourteen optimisation models of the day — sample mean-variance, minimum variance, constrained variants, assorted Bayesian shrinkages — against plain 1/N across seven public datasets. The finding fits in one sentence: once estimation error and turnover costs are counted, none of the models reliably beats equal weighting.

原因不在"最优化"这三个字，在参数。均值方差最优化要求你把预期收益和协方差当成已知真值代入，而现实中两者都只能估。Michaud 在 1989 年就把这件事说破了：优化器会把估计误差当成信号来放大，它本质上是一台"估计误差最大化器"。1/N 之所以难被击败，恰恰因为它不估计任何东西——没有参数可以估错，也就没有误差可以放大。

The problem is not the word "optimisation", it is the parameters. Mean-variance optimisation asks you to plug in expected returns and a covariance matrix as if they were known; in practice both are estimates. Michaud said as much in 1989: the optimiser treats estimation error as signal and amplifies it — in effect, an error maximiser. What makes 1/N hard to beat is precisely that it estimates nothing: no parameter can be got wrong, so no error can be amplified.

但这句结论经常被引用成一个更大的判断——"优化没有用"。那不是原论文的意思。它测的是"用历史样本估参数"这一种做法，而 2009 年之后的二十年里，这块领域出现了收缩估计、贝叶斯先验、因子模型、稳健优化这一整批工具。所以本文要问的不是"优化有没有用"，而是三个更具体的问题。

But the line is often quoted as a broader verdict — "optimisation doesn't work". That is not what the paper says. It tested one practice: estimating parameters from a historical window. The twenty years since have produced shrinkage estimators, Bayesian priors, factor models and robust optimisation. So the question here is not whether optimisation works, but three narrower ones.

三问依次是：（一）在 8 条腿、32.7 年的样本上，"1/N 难被击败"还成立吗？（二）如果成立，是估计误差把优化器吃掉了，还是别的原因？（三）有没有哪一种优化器真的能稳定赢——以及在什么条件下赢。

The three questions, in order. One: on eight legs and thirty-two years, does "1/N is hard to beat" still hold? Two: if so, is the culprit estimation error, or something else? Three: is there any optimiser that wins reliably — and if so, when?

02数据与方法Data and method

样本刻意选宽：8 条腿——股票、长债（30 年期）、中债（10 年期）、短债（2 年期）、黄金、能源、金属、食品。本文的数据来自 Yahoo Finance 标普 500 全收益、美国财政部的国债收益率曲线（合成三条债券腿）、LBMA 伦敦金价，以及世界银行 Pink Sheet 的能源、金属与食品价格指数。这比《风险平价实测》的四条腿宽一倍，而且不是为了好看：论战本身研究的正是"资产数量上去之后 1/N 还成不成立"，只有把资产数拉到 8，问题才问得完整。

The universe is deliberately wide: eight legs — equities, 30-year bonds, 10-year bonds, 2-year bonds, gold, energy, metals and food. The data come from Yahoo Finance (S&P 500 total return), the US Treasury yield curve (synthesising the three bond legs), the LBMA London gold fix and the World Bank Pink Sheet for energy, metals and food. That is twice the four legs used in the risk-parity note, and not for show: the debate itself is about whether 1/N survives as the asset count rises, so the question is only complete at eight.

数据从 1990-02 到 2025-09 共 428 个月，前 36 个月用来做第一次估计，所以样本外从 1993-02 开始，共 392 个月（32.7 年）。所有方法的规则完全一样：每年 1 月用最近 36 个月的月度收益估一次参数、解一次权重，年内随价格漂移不再调整；交易按每条腿 1–5 个基点的单边成本扣掉。换句话说，下文所有的排序都不是"某个方法参数没调好"，而是同一套规则下估出来的。

The data run from February 1990 to September 2025, 428 months, with the first 36 used for the initial estimate, so the out-of-sample record starts in February 1993 and runs 392 months, 32.7 years. Every method runs identical rules: each January the parameters are re-estimated from the trailing 36 monthly returns and weights are re-solved once, then left to drift for the rest of the year; trading is charged at one to five basis points per leg, one way. In other words, nothing below is the result of badly tuned parameters — it is all the same rules, estimated.

这张表也解释了 1/N 的起点有多低。八条腿里年化最高的是股票 10.80%，最低的是食品 2.15%、短债 2.86%；等权拿到的是它们的平均，同时把股票那条腿 14.8% 的波动摊薄掉——它换来 6.4% 的组合波动，代价是年化只有 5.88%。1/N 从来不是"最优"，它只是"不需要估参数"，而这两件事在长期数据里经常是同一件事。

This table also shows how low 1/N's starting point is. Of the eight legs, equities earn the most at 10.80% a year and food the least at 2.15% (short bonds 2.86%); equal weighting gets their average while diluting equities' 14.8% volatility down to a 6.4% portfolio volatility, in exchange for 5.88% a year. 1/N was never "optimal"; it simply needs no parameters estimated — and over long samples those two things are often the same thing.

03第一层：32.7 年的成绩First layer: the thirty-three years

先把结果整张摊开。九条线都从 1993-02 的 1 元出发，终点从 2.36 元（最小方差·样本）到 10.47 元（最大夏普·收缩）；两条参照线落在 60/40 的 14.24 元与纯股的 28.48 元上。

Start by laying the whole result out. All nine lines begin at 1 on February 1993 and end between 2.36 (sample minimum variance) and 10.47 (shrunk max-Sharpe); the two reference lines land at 14.24 for the 60/40 and 28.48 for all-equities.

真正的看点在排序，不在高度。按夏普排，1/N 的 0.54 排在九个组合的第四位：三个优化器在它之上（风险平价 ERC 0.65、最大夏普·收缩 0.58、逆波动率 0.57），一个与它几乎完全相同（最大夏普·样本 0.54），四个在它之下（层次风险平价 0.44、最小方差·收缩 0.38、最大分散化 0.18、最小方差·样本 0.11）。而 1/N 做到这件事，没有估任何一个参数。

The story is the ordering, not the height. Ranked by Sharpe, 1/N's 0.54 comes fourth of nine: three optimisers sit above it (risk-parity ERC 0.65, shrunk max-Sharpe 0.58, inverse volatility 0.57), one matches it almost exactly (sample max-Sharpe 0.54) and four sit below (hierarchical risk parity 0.44, shrunk minimum variance 0.38, maximum diversification 0.18, sample minimum variance 0.11). And 1/N did it without estimating a single parameter.

如果改用年化收益来排，结论更难堪：只有两个优化器超过 1/N 的 5.88%——最大夏普·收缩 7.45% 与最大夏普·样本 6.23%。而 7.45% 是用 8.8% 的波动换来的，比 1/N 的 6.4% 高三分之一；它之所以赢，是因为它承担了更多风险，而不是因为它算得更准。用风险调整后的口径，这场比试只剩一个赢家——风险平价 ERC，而它恰恰是最不依赖收益预测的那一个。

Rank on raw return instead and it gets uglier: only two optimisers beat 1/N's 5.88% — shrunk max-Sharpe at 7.45% and sample max-Sharpe at 6.23%. And that 7.45% was bought with 8.8% volatility, a third more than 1/N's 6.4%: it wins by taking more risk, not by estimating better. On a risk-adjusted basis exactly one optimiser wins, risk-parity ERC — the one that never uses a return forecast at all.

04第二层：预期夏普是幻觉Second layer: the expected Sharpe is an illusion

第一层回答了谁赢谁输，还没回答为什么。先把每个优化器放到同一把尺子上量一次：横轴是它相对 1/N 的夏普差，右边的点赢、左边的点输，1/N 自己钉在零点上。

The first layer settles who wins. It does not yet say why. Begin by measuring every optimiser against the same ruler: the horizontal axis is its Sharpe advantage over 1/N, points to the right win, points to the left lose, and 1/N is pinned at zero.

问题出在每年 1 月那一步。优化器把权重交出来的时候，它同时交出一个判断：这个组合有多好。这个判断可以事后算出来——在它用的那段历史数据上，这个权重对应的夏普是多少。

The problem sits in that January step. When the optimiser hands back weights, it hands back a judgement along with them: how good this portfolio is. That judgement can be computed after the fact — what Sharpe those weights would have earned on the history the optimiser was looking at.

这解释了一个看起来矛盾的现象：优化器并不是"算错了"。在它看的那 36 个月里，1.74 是真的——如果那 36 个月会原样重来一次，那个组合确实会给出接近 1.74 的夏普。错的不是计算，是假设历史会重演。样本越小、资产越多，"历史重演"这个假设就越不可靠，而 36 个月估 8 条腿的均值和协方差，恰好落在最不可靠的那个区间里。

This explains an apparent contradiction: the optimiser is not miscalculating. Over the 36 months it was shown, 1.74 was true — had those months repeated exactly, that portfolio would indeed have earned roughly a 1.74 Sharpe. What is wrong is not the arithmetic but the assumption that the history repeats. The shorter the window and the more assets, the worse that assumption gets, and estimating eight legs' means and covariances from 36 months sits squarely in the unreliable zone.

05第三层：优化器到底买了什么Third layer: what the optimiser actually bought

如果优化器赢不了，那它一年到头在做什么？把每年的权重打开看，答案比"它算错了"具体得多：它把钱压在极少数几条腿上。

If the optimiser cannot win, what is it doing all year? Open up the weights and the answer is far more concrete than "it got the arithmetic wrong": it piles the money into a handful of legs.

最小方差·样本是最极端的一个：平均把 92.6% 的钱放在短债上，股票只剩 1.5%，长债与中债都是 0.0%。意思是它"最小化方差"的办法不是"分散"，而是"退回到最安全的那一条腿"——年化 2.67% 是数学上必然的结果，不是运气不好。逆波动率同理（短债 57.3%），层次风险平价（短债 45.2%）、风险平价 ERC（短债 47.9%）也都偏向同一侧。真正敢把钱分出去的只有 1/N 和最大夏普·收缩：1/N 每条腿 12.5%，最大夏普·收缩最重的股票 35.9%、最轻的金属 6.1%。
这张表还藏着一个反直觉的细节：短债的波动率只有 1.6%，但它对组合的用处也很小——1.6% 的波动意味着它几乎不贡献收益。"把风险降下来"和"把收益降下来"在这类方法里是同一次操作，不是两次。

Sample minimum variance is the extreme case: an average of 92.6% in short bonds, 1.5% in equities, and 0.0% in both long and intermediate bonds. Its way of "minimising variance" is not diversification but retreat into the single safest leg — and 2.67% a year is the arithmetic consequence, not bad luck. Inverse volatility does the same at 57.3%, hierarchical risk parity at 45.2%, risk-parity ERC at 47.9%. The only portfolios that genuinely spread the money are 1/N and shrunk max-Sharpe: 12.5% a leg for 1/N, and for shrunk max-Sharpe a high of 35.9% in equities against a low of 6.1% in metals.
The table hides one counter-intuitive detail: short bonds run 1.6% volatility, but that also makes them nearly useless to the portfolio — 1.6% volatility means almost no contribution to return. In this family of methods, cutting the risk and cutting the return are one operation, not two.

06收缩能不能救Can shrinkage rescue it

如果病根是"样本协方差不可靠"，那么对症的药是现成的：收缩估计。Ledoit-Wolf 的做法是把样本协方差朝着一个稳定的目标（本文用 μ·I，即平均方差乘单位阵）拉过去，拉多少由一个最优强度决定；均值那一侧则可以用 James-Stein 收缩。这两种方法都写在标准教科书里，也都比样本估计"更稳"。

If the disease is an unreliable sample covariance, the obvious medicine already exists: shrinkage. Ledoit-Wolf pulls the sample covariance toward a stable target — here μ·I, the average variance times the identity — by an optimally chosen intensity; on the mean side, James-Stein shrinkage does the equivalent job. Both are standard textbook material, and both are "more stable" than the raw sample estimate.

把协方差与均值两侧的收缩交叉组合，结论更清楚：协方差收缩有用——样本协方差 0.54、换成 Ledoit-Wolf 是 0.58；均值收缩只在协方差已经收缩之后才有一点用（0.58 → 0.60），而单独用均值收缩反而更差（0.54 → 0.53）。原因不难理解：协方差的估计是"多点平均"，几百个月的观测对 8×8 的矩阵来说虽然不够，但至少有信息；而预期收益的估计在月度尺度上本来就接近噪声，无论怎么收缩都很难从历史里提出可用的信号。这也解释了为什么风险平价（完全不碰均值）排第一，而所有涉及最大夏普（必须估均值）的方法都更差。

Crossing the two shrinkage choices makes the point sharper: shrinking the covariance helps — 0.54 on the sample estimate, 0.58 with Ledoit-Wolf; shrinking the mean only helps once the covariance is already shrunk (0.58 → 0.60), and on its own it hurts (0.54 → 0.53). The reason is not mysterious. A covariance estimate averages over many points, and a few hundred monthly observations carry some information about an 8×8 matrix; an expected-return estimate at monthly frequency is close to noise to begin with, and no amount of shrinkage can pull a usable signal out of it. That also explains why risk parity — which never touches the mean — ranks first, while everything that must estimate a mean does worse.

07换窗口、换资产数，结论都不动Change the window, change the universe — nothing moves

到这里有人会提出两个很自然的反驳：也许是 36 个月的窗口选错了；也许是资产太少。这两件事都能直接测。

Two natural objections arrive at this point: perhaps 36 months was the wrong window, or perhaps eight legs is too few. Both can be tested directly.

第一个测试是换估计窗口，从 12 个月换到 120 个月。为了让五个窗口公平比较，评估区间统一压到 2000-02 之后的 308 个月——否则窗口越长、起点越晚，"少跌的那一段"就越少。

The first test sweeps the estimation window from 12 months out to 120. To compare the five fairly, the evaluation period is compressed to the 308 months after February 2000 — otherwise the longer the window, the later the start and the less of the bad years any of them sees.

第二个测试是换资产数：从只有股票和长债的两条腿，一路加到八条腿。这一测的是论战的核心命题——1/N 的优势是不是"资产少"造成的假象。

The second test changes the number of assets, from a two-leg portfolio of equities and long bonds up to all eight. This gets at the core proposition of the debate: is 1/N's edge an artefact of a small universe?

08什么时候优化器真的有用When the optimiser actually pays

如果只看全样本，结论是"优化器基本没用"。但这个判断不公平，因为它把"有用"和"没用"的时段按时间长度一视同仁。真实的钱不是在 392 个月里平均赚出来的，而是在少数几个时点被决定的。所以最后一层测试把样本切成五个已知的环境。

Full-sample results say optimisers mostly do not pay. That verdict is unfair, because it weighs good times and bad by length. Real money is not earned evenly across 392 months; it is decided at a handful of moments. So the last test cuts the sample into five known environments.

前面用的是"整段"的视角。换成尺子再量一次：把 392 个月切成所有可能的滚动五年窗口（每个方法 333 个），看每个方法在这些窗口里相对 1/N 的超额分布。

Everything so far used the full stretch. Now change the ruler: slice the 392 months into every possible rolling five-year window (333 per method) and look at the distribution of each method's excess over 1/N.

09自己动手：拖动估计窗口Do it yourself: drag the estimation window

最后一步留给你自己。下面这张图不是回测结果，是一台可以转的旋钮：拖动估计窗口，三条净值曲线与整套指标会实时重算。1/N 那条线永远不会动——它没有窗口这个参数；这也正是全文的论点本身。

The last step is yours. What follows is not a result but a dial: drag the estimation window and three value curves plus a live panel of statistics are recomputed. The 1/N line never moves — it has no window parameter — and that is the whole argument of this note in one gesture.

10结论：排序、机制、条件Conclusion: ordering, mechanism, condition

回到开头的三个问题，答案分别是这样的。

Back to the three opening questions. The answers go like this.

第一，排序成立，而且比原论文更干净。八个优化器里四个明确输给 1/N、一个打平、三个赢，而赢的那三个里有两个（最大夏普·收缩、逆波动率）的优势只有 0.03–0.04 个夏普——在 392 个月上，这个差距没有统计意义。真正明显胜出的是风险平价 ERC（0.65 对 0.54），而它是这九个方法里唯一一个完全不使用收益预测的。换句话说：在本文的样本里，"更好的组合"来自"不提收益假设"，而不是"更准地估收益"。

First, the ordering holds, and more cleanly than in the original paper. Four of the eight optimisers clearly lose to 1/N, one ties and three win — and two of those three (shrunk max-Sharpe and inverse volatility) win by 0.03 to 0.04 of Sharpe, which over 392 months is not a statistically meaningful margin. The one clear winner is risk-parity ERC (0.65 against 0.54), the only method of the nine that uses no return forecast at all. Put differently: in this sample, a better portfolio came from making fewer return assumptions, not from estimating returns better.

第二，病根是估计误差，而且主要在均值那一侧。样本最大夏普每年算出来的夏普是 1.74，实现 0.54，落差 1.20；九个方法里"预期"与"实现"的相关性最好只有 0.04。收缩能修的是协方差（最小方差 0.11 → 0.38，最大夏普 0.54 → 0.58），修不了均值——而所有涉及最大夏普的方法都必须估均值，这才是它们排在后面的原因。

Second, the cause is estimation error, and mostly on the mean side. Sample max-Sharpe promised a 1.74 Sharpe each year and delivered 0.54, a gap of 1.20; across nine methods the best promise-to-delivery correlation is 0.04. Shrinkage repairs the covariance (minimum variance 0.11 → 0.38, max-Sharpe 0.54 → 0.58) but not the mean — and everything involving max-Sharpe must estimate a mean, which is why it ranks where it does.

第三，优化器的价值是有条件的，而且条件很具体。它在"某一条腿崩掉"的时段明显有用：2008 年最小方差只跌 0.7%（1/N 跌 24.6%），科网泡沫里层次风险平价 +20.6%、最小方差 +18.0%，2022 年 ERC 只跌 4.2%（60/40 跌 17.7%）。但它的代价写在另外两处：三年高通胀里八个优化器只有一个赚到钱，以及滚动五年窗口里七个方法的中位超额在零线左边。结论不是"优化器无用"，而是"优化器是一份针对单一资产崩塌的保险，保费是它在其余时间里的落后"。

Third, the optimiser's value is conditional, and the condition is specific. It clearly pays when a single leg collapses: minimum variance lost 0.7% in 2008 against 1/N's 24.6%, hierarchical risk parity made +20.6% and minimum variance +18.0% through the dot-com bust, and ERC lost only 4.2% in 2022 against the 60/40's 17.7%. But the premium shows up elsewhere: in three years of high inflation only one of the eight optimisers made money, and in rolling five-year windows seven of eight have median excess below zero. The finding is not that optimisers are useless; it is that they are insurance against one leg collapsing, and the premium is the ground they give up the rest of the time.

如果只从这篇拿走一句话，拿走这一句：在这个样本里，1/N 之所以难被击败，不是因为它更好，而是因为它不需要被估计——而"不需要估计"这一项优势，恰好与"资产越多"的直觉反着走：腿加得越多，估计的负担增长得越快，等权的相对优势反而越大。

If you take one sentence from this note, take this one: in this sample 1/N is hard to beat not because it is better but because it needs no estimating — and that advantage runs against the obvious intuition about adding assets: the more legs you add, the faster the estimation burden grows, and the larger equal weighting's relative edge becomes.

边界必须写清楚，因为它们全都朝着同一个方向。第一，全部资产都是美国资产，没有新兴市场、REITs 与外汇，而这三类恰恰是实践中"需要优化"的典型理由；第二，全部策略都是多头、不加杠杆、不做空，而真实的优化组合经常用杠杆与卖空来兑现更小的估计差异；第三，债券由收益率曲线合成（收益＝收益率 ÷ 12 − 久期 × 收益率变化），久期固定、忽略凸性；第四，黄金是现货裸价、没有租赁与仓储，商品是世界银行价格指数、不含展期收益——这一点会让商品腿的长期回报被低估；第五，1–5 个基点的单边成本是机构级的假设，个人投资者的实际成本更高，这会同时压低所有高换手方法；第六，本文只测了"用历史窗口估参数"这一种最朴素的做法，没有测因子模型、贝叶斯先验与稳健优化，所以结论的准确说法是"朴素优化不如等权"，不是"优化不如等权"。

The boundaries need stating, because they all lean the same way. One, every asset is a US asset; there are no emerging markets, REITs or currencies — exactly the exposures that motivate optimisation in practice. Two, every strategy is long-only, unlevered and short-free, while real optimised portfolios often use leverage and shorts to express smaller estimated differences. Three, bonds are synthesised from the yield curve (return = yield ÷ 12 − duration × change in yield) at fixed duration, with convexity ignored. Four, gold is spot with no lease or storage, and commodities are World Bank price indices without roll yield, which understates the long-run return of the commodity legs. Five, one to five basis points per leg is an institutional assumption; a private investor's real costs are higher, and that would penalise every high-turnover method at once. Six, this note tests only the plainest practice — estimating parameters from a historical window — and not factor models, Bayesian priors or robust optimisation, so the accurate claim is "naive optimisation loses to equal weighting", not "optimisation loses to equal weighting".