负债驱动的资产配置：换一个分母，排序就变了Liability-driven allocation: change the denominator and the ranking changes

01负债是一个会动的分母The liability is a moving denominator

资产配置的教科书流程是：估一份资产的预期收益与协方差，解一个最优化问题，最大化风险调整后的收益。这个流程里没有「谁的钱」这个问题——组合是自成一体的，涨得多就是好。

The textbook allocation process estimates asset returns and a covariance matrix, solves an optimisation, and maximises risk-adjusted return. Nowhere in it does the question "whose money is this" appear — the portfolio stands alone, and more is better.

保险公司与养老金的处境不一样。它们手上的资产对应着一笔确定的、以本币计价的未来支付：保单到期要赔、养老金按月要发。这笔支付流有一个现值，叫做负债；资产除以负债叫做 funding ratio（偿付能力比率）。真正要管的不是资产本身，是这两者之差。

Insurers and pension funds are different. Their assets stand behind a contractual stream of future payments in domestic currency: claims at maturity, monthly benefits. That stream has a present value — the liability — and assets divided by it is the funding ratio. What has to be managed is not the assets but the difference between the two.

这个区别的第一层含义，就足以推翻很多直觉：负债不是常数。它的现值随利率反向变动，变动的幅度由久期决定。本文把负债建模为一笔 20 年期定额年金，按当月的 10 年期美国国债收益率平折现。下图把它拆开：左图是利率与年金现值因子的关系，右两幅是样本内负债现值与修正久期的实际路径。

The first implication of that difference is enough to overturn several intuitions: the liability is not a constant. Its present value moves inversely with rates, and the size of the move is set by duration. Here the liability is a 20-year level annuity discounted flat at that month's 10-year Treasury yield. The figure below opens it up: the left panel maps yield to annuity factor; the right two panels show the actual path of liability value and modified duration through the sample.

所以本文要回答的是三件事：（一）把目标从资产换成盈余，组合的排序会怎么变？（二）如果会变，变的是哪一项指标——收益、波动，还是回撤？（三）给定当前 funding ratio，能拿多少风险是算得出来的吗？

Three questions follow. One: if the objective moves from assets to surplus, how does the ranking change? Two: if it changes, which statistic changes — return, volatility, or drawdown? Three: given today's funding ratio, is the tolerable amount of risk computable?

02数据与方法Data and method

样本与站上其他研究笔记共用同一份公开数据管线：6 条资产腿——股票（标普 500 全收益）、长债（30 年期国债收益率合成，久期 15）、中债（10 年期，久期 7.5）、短债（2 年期，久期 1.9）、黄金（LBMA 伦敦金价）、商品（世界银行能源 / 金属 / 食品价格指数的等权合成）。数据从 1990-02 到 2025-09，前 36 个月用来做第一次估计，样本外从 1993-02 开始，共 392 个月（32.7 年）。原 8 条腿里的三条商品腿在这里合成一条，好让讨论集中在久期上。

The sample shares the public data pipeline used by the other notes: six asset legs — equities (S&P 500 total return), long bonds (synthesised from the 30-year Treasury yield, duration 15), 10-year bonds (duration 7.5), 2-year bonds (duration 1.9), gold (LBMA fix) and commodities (an equal-weighted composite of the World Bank energy, metals and food indices). The data run from February 1990 to September 2025, with the first 36 months used for the initial estimate, so the out-of-sample record starts in February 1993 and runs 392 months, 32.7 years. The three commodity legs of the eight-leg universe are merged into one here, to keep the discussion on duration.

负债的构造是全文最要紧的一段，所以单独写清楚：一笔恒定到期的 20 年期定额年金，按当月的 10 年期国债收益率平折现，修正久期用麦考利久期除以 1+y 得到——这一步必须做，否则和资产端债券腿的口径对不上（站上债券腿的月度收益是「票息 − 久期 × Δy」，那个久期就是修正久期）。

The liability construction is the load-bearing paragraph of this note, so it gets its own: a constant-maturity 20-year level annuity, discounted flat at that month's 10-year Treasury yield, with modified duration computed as Macaulay duration divided by one plus the yield. That last step is not optional — without it the liability is not on the same footing as the bond legs, whose monthly return is "carry − duration × Δy" with that duration being a modified one.

资产端有七个组合，覆盖从「完全不管负债」到「完全按负债优化」：

Seven portfolios sit on the asset side, spanning from "ignore the liability entirely" to "optimise straight through it":

需要说明三个口径。第一，久期匹配不是固定权重：它每个月重算一次，只用长债与短债两腿，把资产久期对准当月的负债久期，所以它的长债权重会在 50%–60% 之间移动。第二，盈余最优化用的是「资产收益减负债收益」后的切点组合，每年 1 月用最近 36 个月重解一次，只做多、权重和为 1。第三，所有组合都从 funding ratio 100% 出发，并像站上其他笔记一样扣掉 1–5 个基点的单边成本。

Three conventions need stating. First, duration matching is not a fixed weight: it is recomputed monthly using only long and short bonds, targeting that month's liability duration, so its long-bond weight moves between roughly 50% and 60%. Second, surplus optimisation uses the tangency portfolio on "asset return minus liability return", re-solved each January on the trailing 36 months, long-only with weights summing to one. Third, every portfolio starts from a funding ratio of 100%, and all of them pay one to five basis points per leg, one way, as elsewhere on this site.

03机制：久期缺口就是盈余风险The mechanism: the duration gap is the surplus risk

把资产和负债放到同一把尺子上，只需要一个定义：盈余收益 s 满足 (1+s) = (1+r_资产) ÷ (1+r_负债)。当 r 都是小量时，它近似等于两者之差。

Putting assets and liabilities on one ruler takes a single definition: the surplus return s satisfies (1+s) = (1+r_asset) ÷ (1+r_liability), which for small r is approximately the difference between the two.

现在假设利率瞬时变动 Δy。资产端动 −D_A·Δy，负债端动 −D_L·Δy，于是盈余动 −(D_A − D_L)·Δy。也就是说：盈余的风险不来自资产波动，来自两边久期之差。久期缺口是 9 年，利率动 1 个百分点，funding ratio 就动 9%。

Now let rates move by Δy. The asset side moves −D_A·Δy, the liability −D_L·Δy, so the surplus moves −(D_A − D_L)·Δy. In other words: surplus risk does not come from asset volatility; it comes from the difference between the two durations. A nine-year gap means a one-point rate move moves the funding ratio nine per cent.

把这条关系画出来，是一条 U 形线。下面这个实验只做一件事：只用长债与短债两腿，把资产久期从 0 年一点一点调到 15 年，其他什么都不变。

Drawn out, that relationship is a U. The experiment below does one thing only: using just long and short bonds, walk asset duration from zero up to fifteen years, changing nothing else.

一个必须写下来的细节：这七个组合里，只有久期匹配和盈余最优化是「按负债设计」的——前者的缺口是 0，后者的缺口是 −1.65 年，因为盈余最优化选出来的权重平均压在 10 年期国债上（89.6%），而 10 年期国债的久期（7.5）略低于负债（9.0）。其余五个组合的久期都是各自资产端逻辑的副产品，没有人替负债考虑过。

One detail has to be written down: only two of the seven are designed around the liability. Duration matching sits at a zero gap; surplus optimisation sits at −1.65 years because the optimiser put an average of 89.6% into 10-year bonds, whose duration of 7.5 falls a little short of the liability's 9.0. The other five portfolios have whatever duration their asset-side logic happened to produce, with nobody answering to the liability.

0432.7 年：七条 funding ratio 路径Thirty-two years: seven funding-ratio paths

现在把七个组合放进同一段历史，起点都是 funding ratio 100%。看两条线就够：一条是终点，一条是路径最差的时候跌到哪里。

Now run all seven through the same history, each starting from a funding ratio of 100%. Two things matter: where each ends up, and how far it fell on the way.

这张表里藏着全文的转折点。如果只看资产端，结论和站上其他笔记一样：1/N 等权的夏普 0.71 最高，久期匹配的 0.26 最低。但把视线移到右边四列，排序立刻不一样了——久期匹配的盈余波动是 2.15%（第二低），funding ratio 最大回撤是 −2.4%（最低）；1/N 等权的盈余波动是 6.96%，盈余回撤 −24.4%。

The turning point of this note is inside this table. In asset space the answer matches the other notes on this site: 1/N leads on Sharpe at 0.71 and duration matching trails at 0.26. Move to the four right-hand columns and the ordering breaks: duration matching's surplus volatility is 2.15%, second lowest, and its maximum funding-ratio drawdown is −2.4%, the lowest. 1/N's surplus volatility is 6.96% and its surplus drawdown −24.4%.

还有一处值得单独指出：久期匹配的盈余年化只有 3.51%，比纯股的 10.00% 低 6.5 个百分点，但这是它自己选的——Exhibit 2 已经说明，久期越长盈余收益越高，久期匹配为了把波动压到最低，主动放弃了那部分期限溢价。这不是一个「跑不赢」的组合，而是一个明确定价的保险。

One more thing deserves its own sentence: duration matching earns 3.51% a year on surplus, 6.5 points below all-equities at 10.00% — but that was its choice. Exhibit 2 already showed that surplus return rises with duration, and duration matching gives up that term premium on purpose to minimise volatility. This is not a portfolio that failed to keep up; it is an explicitly priced insurance policy.

05换一个纵轴，排序就换了一遍Change the vertical axis and the ranking changes with it

把七个组合分别画在两张图上——左边是资产空间（横轴资产波动、纵轴资产年化），右边是盈余空间（横轴盈余波动、纵轴盈余年化）。同样的七个点，位置完全不同。

Plot the same seven portfolios in two spaces: on the left asset space (volatility against return), on the right surplus space (surplus volatility against surplus return). Same seven dots, completely different places.

把这张图拆成一句话：资产空间衡量的「效率」，是每单位资产波动换来多少资产收益；盈余空间衡量的，是每单位盈余波动换来多少盈余收益。这两件事在数学上不等价，因为负债本身也在动。而负债的波动是负相关的——利率下行使负债上升，同时使长债上涨——所以一个在资产端看起来很「保守」的组合（久期匹配，资产端夏普 0.26），在盈余端可以很有效率。

Compressed into one sentence: asset space measures how much asset return each unit of asset volatility buys; surplus space measures how much surplus return each unit of surplus volatility buys. These are not equivalent, because the liability moves too. And the liability is negatively correlated with the thing that hedges it — falling rates raise the liability and lift long bonds at the same time — so a portfolio that looks conservative on the asset side (duration matching, asset Sharpe 0.26) can be highly efficient on the surplus side.

06资产跌三成，盈余只跌 2.4%Assets fell 30%; the surplus fell 2.4%

上一节用的是波动，这一节换一个更贴近决策的指标：最大回撤。资产端的最大回撤和 funding ratio 的最大回撤，是两个几乎不相关的数字。

The last section used volatility; this one uses the statistic that actually drives decisions: maximum drawdown. An asset-side drawdown and a funding-ratio drawdown are two nearly unrelated numbers.

为什么会这样？因为久期匹配手里只有长债与短债，而长债在 2021-2023 那段加息里跌了将近一半——资产端它当然是差生。但同一段时间负债也跌了，而且久期缺口是零，两边等量下跌，funding ratio 几乎不动。这就是「资产端跌得难看、盈余端却没事」的完整解释：不是资产没有风险，是这份风险与负债的风险方向一致、数量相同。

Why does this happen? Because duration matching holds only long and short bonds, and long bonds lost nearly half their value through the 2021–2023 hiking cycle — on the asset side it is plainly the worst pupil. Over the same window, though, the liability fell too, and with a zero duration gap the two declines cancel almost exactly, leaving the funding ratio intact. That is the full explanation of "ugly on assets, untouched on surplus": the risk is not absent, it is the same risk, in the same direction, in the same size, as the liability's.

07利率动一下，谁的 funding ratio 会动When rates move, whose funding ratio moves

久期缺口是抽象的，把它翻译成投资委员会的语言就是一张情景表：利率动 1 个百分点，我的偿付能力会变成什么样。下表是两边都按修正久期的一阶近似算出来的瞬时冲击，久期匹配那一行是 0。

A duration gap is abstract; translated into the language of an investment committee it becomes a scenario table: if rates move a point, what happens to my funding position? The table below applies the first-order duration approximation on both sides instantaneously; the duration-matched row is zero.

这张表最实用的地方在于它是可以反过来用的：如果一家保险公司的偿付能力预算是「利率冲击下 funding ratio 波动不超过 5%」，那么把缺口控制在 5 年以内就能满足；「不超过 3%」对应 3 年以内。久期缺口是一个可以直接写进风险限额的数字，而资产端波动率不能。

What makes this table useful is that it runs backwards: if an insurer's solvency budget is "funding-ratio impact under a rate shock no worse than 5%", then a gap inside five years satisfies it; 3% corresponds to three years. A duration gap is a number that can be written directly into a risk limit; asset volatility cannot.

08什么时候可以加风险When you are allowed to take more risk

前面几节说的是「久期应该调准」，但没有回答另一个问题：调准之后，剩下的风险预算应该放多少到股票上？这不是偏好问题，它可以算：给定起始 funding ratio，找出让「一年内跌破 100%」的历史概率落在容忍度以内的最大股票权重。

The earlier sections say the duration should be matched; they do not answer the next question: once it is matched, how much of the remaining risk budget belongs in equities? That is not a matter of taste — it can be computed. Given a starting funding ratio, find the largest equity weight whose historical one-year probability of falling below 100% stays inside tolerance.

三档容忍度给出三条完全不同的曲线，而它们的方向一致：容忍度高一点、funding ratio 富余一点，就可以多拿风险。100% 那一行三档都只能给 0–5% 的股票权重——这是一个很反直觉但很实际的结论：刚好够付的机构，在数学上几乎没有资格承担股票风险。而要等到 funding ratio 上到 130% 以上，股票权重才变得自由。

Three tolerances produce three different curves pointing the same way: tolerate a little more shortfall risk, or hold a little more surplus, and you can carry more equity. In the 100% row all three allow only 0–5% equities — a counter-intuitive but practical result: an institution that is exactly fully funded has, mathematically, almost no licence to own equity risk. Equity weights only become free once the funding ratio is above roughly 130%.

09同一段历史，五种利率环境One history, five rate environments

全样本的平均值会掩盖一件重要的事：这些组合的相对表现高度取决于利率往哪个方向走。把 32.7 年按 10 年期收益率的走势切成五段，看每段里 funding ratio 累计变化了多少。

Full-sample averages hide something important: the relative performance of these portfolios depends heavily on which way rates went. Cut the 32.7 years into five stretches by the path of the 10-year yield, and track how each funding ratio moved.

再换一把尺子：不看整段，看所有可能的滚动五年窗口（每个组合 333 个），把每个窗口的盈余年化收益画成区间。

Change the ruler once more: instead of whole episodes, look at every possible rolling five-year window (333 per portfolio) and plot each one's annualised surplus return as a range.

10自己动手：拖动股票权重与久期目标Do it yourself: drag the equity weight and the duration target

最后一步留给你自己。下面这台旋钮有两个开关：股票权重（其余放进债券袖子）与久期目标（债券袖子把资产久期对准几年级别的负债）。两条线会实时重算：深色是盈余最优化与久期匹配所对应的「负债对齐」档，浅色是六条腿等权。把久期从 0 拖到 15，你会看到 funding ratio 的曲线从「抖」变成「直」，再从「直」变回「抖」。

The last step is yours. Two dials: the equity weight (the rest goes into the bond sleeve) and the duration target (the liability maturity the sleeve aims at). Two curves recompute live — a pale 1/N line and the dark line for the selected duration target. Drag the target from 0 to 15 and watch the funding-ratio path go from jagged to straight and back to jagged again.

11结论：换分母，换排序，换风险预算Conclusion: a new denominator, a new ranking, a new risk budget

回到开头的三个问题。

Back to the three opening questions.

第一，排序确实会变，而且变得很彻底。资产端夏普排名前四的组合里，有三个在盈余端掉出前四；资产端垫底的久期匹配（0.26）在盈余端排第二低波动。用盈余夏普排，1/N 等权从第一掉到第六。这不是因为久期匹配变好了，而是因为衡量它的那把尺子换了——而换尺子恰恰是负债驱动投资的第一步。

First, the ranking does change, and thoroughly. Three of the top four asset-only portfolios drop out of the surplus top four, while duration matching — dead last on assets at 0.26 — sits second lowest on surplus volatility. Ranked by surplus Sharpe, 1/N falls from first to sixth. Not because duration matching improved, but because the ruler changed — and changing the ruler is precisely the first move in liability-driven investment.

第二，变的主要是回撤，其次才是波动。七个组合的资产端最大回撤跨 −8.8% 到 −50.9%（5.8 倍），盈余端跨 −2.4% 到 −58.9%（24 倍）。久期匹配的资产端回撤是 −30.1%，盈余端只有 −2.4%。同一份风险，在两张报表上的读数可以差一个数量级——这是本文最实用的一句话。

Second, what changes most is the drawdown, then the volatility. Asset drawdowns span −8.8% to −50.9% (a factor of 5.8); surplus drawdowns span −2.4% to −58.9% (a factor of 24). Duration matching drew down 30.1% in assets and 2.4% in surplus. The same risk can read an order of magnitude differently on the two statements — the most practical sentence in this note.

第三，风险预算算得出来，而且答案比直觉保守。funding ratio 100% 时，即使不持有任何股票，一年内跌破 100% 的历史概率仍有 5%；115% 时才允许 55% 的股票，130% 允许 80%。久期缺口是可以写进限额的数字，股票权重应该是 funding ratio 的函数，而不是投资委员会的偏好。

Third, the risk budget is computable, and the answer is more conservative than intuition. At a funding ratio of 100%, even with no equities at all the historical probability of falling short within a year is 5%; only at 115% does 55% equities fit inside that tolerance, and 80% needs 130%. A duration gap is a number you can write into a limit, and the equity weight should be a function of the funding ratio rather than a committee preference.

边界必须写清楚。第一，负债是合成的、单一利率平折现，真实的保险负债有分期限的贴现曲线、退保与续保选择权、以及新业务带来的久期漂移——本文的恒定到期假设是一个简化，它让负债变成一个干净的久期工具，也让结论更容易读，但不能替代真实的精算模型。第二，资产端只有六条腿、只做多、不加杠杆，没有利率互换、没有通胀挂钩债券——真实的负债驱动组合主要靠这两样工具做精细对冲。第三，全部为美国资产与美元负债，没有汇率维度。第四，全部为历史回测，含 1–5 个基点的单边交易成本，不含税费、融资成本与保证金。第五，「跌破 100% 的概率」是用 392 个月的经验频率估计的，样本内只包含了两个完整的加息周期，估计的尾部比真实情况乐观。

The boundaries need stating. One, the liability is synthetic and discounted off a single yield; a real insurance liability has a term structure of discount rates, lapse and renewal options, and duration drift from new business. The constant-maturity assumption is a simplification that turns the liability into a clean duration instrument and makes the results readable, but it is not a substitute for an actuarial model. Two, the asset menu is six legs, long-only and unlevered, with no interest-rate swaps and no inflation-linked bonds — the two tools a real liability-driven portfolio leans on most for fine hedging. Three, everything is US assets against a dollar liability, with no FX dimension. Four, these are historical backtests with one to five basis points per leg, excluding taxes, funding costs and margin. Five, the "probability of falling below 100%" is estimated from 392 monthly observations containing only two complete hiking cycles, so its tail is optimistic relative to reality.