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fixed-income-sovereign

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Original

English
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Translation

Chinese

Fixed Income — Sovereign

固定收益——主权债券

Purpose

用途

Analyze government bonds including US Treasuries and sovereign debt. This skill covers bond pricing, yield curve construction, duration and convexity analytics, TIPS mechanics, forward rate derivation, and auction processes critical for interest rate risk management.
分析政府债券，包括US Treasuries和主权债务。本技能涵盖债券定价、收益率曲线构建、久期与凸性分析、TIPS机制、远期利率推导，以及对利率风险管理至关重要的拍卖流程。

Layer

层级

2 — Asset Classes
2 — 资产类别

Direction

适用方向

both
both

When to Use

适用场景

  • User asks about government bonds, Treasuries, or sovereign debt
  • User asks about yield curves, spot rates, or forward rates
  • User asks about interest rate risk, duration, or convexity
  • User asks about TIPS, breakeven inflation, or real yields
  • User asks about bond pricing or yield to maturity calculations
  • User asks about key rate duration or yield curve shape analysis
  • 用户询问政府债券、国债（Treasuries）或主权债务相关问题
  • 用户询问收益率曲线、即期利率或远期利率相关问题
  • 用户询问利率风险、久期或凸性相关问题
  • 用户询问TIPS、盈亏平衡通胀率或实际收益率相关问题
  • 用户询问债券定价或到期收益率（YTM）计算相关问题
  • 用户询问关键利率久期或收益率曲线形态分析相关问题

Core Concepts

核心概念

Bond Pricing

债券定价

The price of a bond is the present value of its future cash flows:
P = sum(t=1 to n) [C / (1+y)^t] + F / (1+y)^n
where C = coupon payment per period, y = yield to maturity per period, F = face value, n = total number of periods. For semi-annual bonds, divide the annual coupon by 2 and the annual yield by 2, and double the number of years to get n.
债券价格是其未来现金流的现值：
P = sum(t=1 to n) [C / (1+y)^t] + F / (1+y)^n
其中，C = 每期票息支付，y = 每期到期收益率，F = 面值，n = 总期数。对于半年付息债券，将年票息除以2，年收益率除以2，将年数翻倍得到n。

Yield to Maturity (YTM)

到期收益率（YTM）

The discount rate y that solves the bond pricing equation — the single rate that equates the bond's market price to the present value of all future cash flows. Assumes reinvestment of coupons at the YTM rate. It is the standard yield measure for bonds.
满足债券定价公式的贴现率y——使债券市场价格等于所有未来现金流现值的单一利率。假设票息以YTM利率再投资。它是债券的标准收益率指标。

Current Yield

当前收益率

Current Yield = Annual Coupon / Price. A simple income measure that ignores capital gains/losses and the time value of money.
当前收益率 = 年票息 / 价格。这是一个简单的收益指标，忽略了资本利得/损失和货币的时间价值。

Yield Curve: Spot Rates, Forward Rates, Par Curve

收益率曲线：即期利率、远期利率、平价曲线

The spot curve gives zero-coupon yields for each maturity. The par curve gives coupon rates at which bonds would price at par. Forward rates are implied future rates derived from spot rates. The three curves contain equivalent information and can be derived from one another.
即期曲线给出各期限的零息债券收益率。平价曲线给出债券按面值定价时的票息率。远期利率是从即期利率推导出来的隐含未来利率。这三条曲线包含等价信息，可相互推导。

Bootstrapping the Spot Curve

即期曲线的bootstrap法

Extract spot (zero-coupon) rates from par yields by starting at the shortest maturity and working outward. Each step uses previously derived spot rates to solve for the next spot rate.
从最短期限开始向外推导，从平价收益率中提取即期（零息）利率。每一步都使用之前推导的即期利率求解下一个即期利率。

Forward Rate

远期利率

The implied rate between two future dates derived from spot rates:
f(t1,t2) = [(1+s_t2)^t2 / (1+s_t1)^t1]^(1/(t2-t1)) - 1
where s_t1 and s_t2 are spot rates for maturities t1 and t2.
从即期利率推导的两个未来日期之间的隐含利率：
f(t1,t2) = [(1+s_t2)^t2 / (1+s_t1)^t1]^(1/(t2-t1)) - 1
其中，s_t1和s_t2是期限t1和t2的即期利率。

Duration (Macaulay)

麦考利久期（Macaulay Duration）

The weighted average time to receive cash flows, where weights are the present value of each cash flow as a proportion of the bond's price:
D_mac = (1/P) × sum(t × CF_t / (1+y)^t)
Measured in years. Longer maturity, lower coupon, and lower yield all increase duration.
收到现金流的加权平均时间，权重为每个现金流的现值占债券价格的比例：
D_mac = (1/P) × sum(t × CF_t / (1+y)^t)
以年为单位。期限越长、票息越低、收益率越低，久期越长。

Modified Duration

修正久期（Modified Duration）

D_mod = D_mac / (1 + y/m)
where m = number of coupon periods per year. Gives the approximate percentage price change for a 1 percentage point change in yield: dP/P ≈ -D_mod × dy.
D_mod = D_mac / (1 + y/m)
其中，m = 每年票息支付次数。用于估算收益率变动1个百分点时的近似价格变动百分比：dP/P ≈ -D_mod × dy。

Dollar Duration (DV01)

美元久期（DV01）

The dollar change in price for a 1 basis point change in yield:
DV01 ≈ -D_mod × P × 0.0001
Used for hedging — match DV01 exposures to immunize a portfolio against parallel rate shifts.
收益率变动1个基点时的美元价格变动：
DV01 ≈ -D_mod × P × 0.0001
用于对冲——匹配DV01敞口，使投资组合免受利率平行变动的影响。

Convexity

凸性

Measures the curvature of the price-yield relationship (second derivative):
C = (1/P) × sum(t(t+1) × CF_t / (1+y)^(t+2))
For option-free bonds, convexity is always positive — duration alone overstates losses and understates gains.
衡量价格-收益率关系的曲率（二阶导数）：
C = (1/P) × sum(t(t+1) × CF_t / (1+y)^(t+2))
对于不含期权的债券，凸性始终为正——仅用久期会高估损失、低估收益。

Price Change Approximation

价格变动近似值

ΔP/P ≈ -D_mod × Δy + 0.5 × Convexity × (Δy)²
The convexity term is a correction that becomes important for large yield changes.
ΔP/P ≈ -D_mod × Δy + 0.5 × Convexity × (Δy)²
凸性项是修正项，在收益率大幅变动时尤为重要。

TIPS (Treasury Inflation-Protected Securities)

TIPS（通胀保值债券）

Principal adjusts with CPI. The coupon rate is fixed but applied to the inflation-adjusted principal. Real yield = TIPS yield. Breakeven inflation = nominal Treasury yield - TIPS real yield. TIPS have a deflation floor that protects par value at maturity.
本金随CPI调整。票息率固定，但应用于通胀调整后的本金。实际收益率 = TIPS收益率。盈亏平衡通胀率 = 名义国债收益率 - TIPS实际收益率。TIPS设有通缩下限，在到期时保护面值。

Key Rate Duration

关键利率久期

Sensitivity to specific points on the yield curve (e.g., 2yr, 5yr, 10yr, 30yr). Allows analysis of non-parallel yield curve shifts such as steepening, flattening, or butterfly moves. Sum of key rate durations equals effective duration.
对收益率曲线上特定点（如2年期、5年期、10年期、30年期）的敏感度。允许分析非平行的收益率曲线变动，如陡峭化、扁平化或蝴蝶型变动。关键利率久期之和等于有效久期。

Key Formulas

关键公式

FormulaExpressionUse Case
Bond PriceP = sum C/(1+y)^t + F/(1+y)^nPrice from yield
Current YieldAnnual Coupon / PriceSimple income measure
Forward Ratef(t1,t2) = [(1+s_t2)^t2 / (1+s_t1)^t1]^(1/(t2-t1)) - 1Implied future rate
Macaulay Duration(1/P) × sum(t × CF_t / (1+y)^t)Weighted avg time to cash flows
Modified DurationD_mac / (1 + y/m)% price sensitivity to yield
DV01D_mod × P × 0.0001Dollar price change per 1bp
Convexity(1/P) × sum(t(t+1) × CF_t / (1+y)^(t+2))Curvature of price-yield curve
Price ChangeΔP/P ≈ -D_mod×Δy + 0.5×Convexity×(Δy)²Estimate price impact of rate move
公式表达式使用场景
债券价格P = sum C/(1+y)^t + F/(1+y)^n从收益率计算价格
当前收益率年票息 / 价格简单收益指标
远期利率f(t1,t2) = [(1+s_t2)^t2 / (1+s_t1)^t1]^(1/(t2-t1)) - 1隐含未来利率
麦考利久期(1/P) × sum(t × CF_t / (1+y)^t)现金流加权平均时间
修正久期D_mac / (1 + y/m)价格对收益率的百分比敏感度
DV01D_mod × P × 0.0001每1个基点的美元价格变动
凸性(1/P) × sum(t(t+1) × CF_t / (1+y)^(t+2))价格-收益率曲线的曲率
价格变动ΔP/P ≈ -D_mod×Δy + 0.5×Convexity×(Δy)²估算利率变动对价格的影响

Worked Examples

实例演算

Example 1: Price a 5-Year 4% Semi-Annual Coupon Bond at 5% YTM

示例1：按5% YTM为5年期4%半年付息债券定价

Given: Face = $1,000, coupon = 4% (semi-annual), YTM = 5%, maturity = 5 years Calculate: Bond price Solution: Semi-annual coupon = $1,000 × 4% / 2 = $20 Semi-annual yield = 5% / 2 = 2.5% Number of periods = 5 × 2 = 10 P = $20 × [(1 - (1.025)^(-10)) / 0.025] + $1,000 / (1.025)^10 P = $20 × 8.7521 + $1,000 × 0.7812 P = $175.04 + $781.20 = $956.24
The bond trades at a discount ($956.24 < $1,000) because the coupon rate (4%) is below the market yield (5%).
已知： 面值 = 1000美元，票息 = 4%（半年付息），YTM = 5%，期限 = 5年 计算： 债券价格 解答： 半年票息 = 1000美元 × 4% / 2 = 20美元 半年收益率 = 5% / 2 = 2.5% 期数 = 5 × 2 = 10 P = 20美元 × [(1 - (1.025)^(-10)) / 0.025] + 1000美元 / (1.025)^10 P = 20美元 × 8.7521 + 1000美元 × 0.7812 P = 175.04美元 + 781.20美元 = 956.24美元
该债券折价交易（956.24美元 < 1000美元），因为票息率（4%）低于市场收益率（5%）。

Example 2: Modified Duration and Price Change Estimate

示例2：修正久期与价格变动估算

Given: A bond with Macaulay duration = 4.5 years, YTM = 5% (semi-annual), price = $956.24, convexity = 22.5 Calculate: Estimated price change for a +50bp rate increase Solution: D_mod = 4.5 / (1 + 0.05/2) = 4.5 / 1.025 = 4.39 years ΔP/P ≈ -4.39 × 0.005 + 0.5 × 22.5 × (0.005)² ΔP/P ≈ -0.02195 + 0.000281 = -0.02167 = -2.167% ΔP ≈ -2.167% × $956.24 = -$20.72 New price ≈ $956.24 - $20.72 = $935.52
Duration alone would estimate -2.195%; the convexity correction reduces the estimated loss by about 3bp.
已知： 某债券麦考利久期 = 4.5年，YTM = 5%（半年付息），价格 = 956.24美元，凸性 = 22.5 计算： 收益率上升50个基点时的估算价格变动 解答： D_mod = 4.5 / (1 + 0.05/2) = 4.5 / 1.025 = 4.39年 ΔP/P ≈ -4.39 × 0.005 + 0.5 × 22.5 × (0.005)² ΔP/P ≈ -0.02195 + 0.000281 = -0.02167 = -2.167% ΔP ≈ -2.167% × 956.24美元 = -20.72美元 新价格 ≈ 956.24美元 - 20.72美元 = 935.52美元
仅用久期估算的变动为-2.195%；凸性修正将估算损失减少了约3个基点。

Common Pitfalls

常见误区

  • Confusing Macaulay and modified duration — Macaulay is in years, modified gives price sensitivity
  • Ignoring convexity for large yield changes — duration alone overstates losses and understates gains
  • Day count conventions (30/360 vs actual/actual) — Treasuries use actual/actual, corporates use 30/360
  • Clean price vs dirty price (accrued interest) — quoted prices exclude accrued interest, but settlement requires paying it
  • 混淆麦考利久期和修正久期——麦考利久期以年为单位，修正久期反映价格敏感度
  • 收益率大幅变动时忽略凸性——仅用久期会高估损失、低估收益
  • 计息天数惯例（30/360 vs 实际/实际）——国债使用实际/实际，公司债使用30/360
  • 净价与全价（应计利息）——报价不含应计利息，但结算时需支付

Cross-References

交叉引用

  • time-value-of-money (core plugin, Layer 0): discounting and present value fundamentals
  • fixed-income-corporate (wealth-management plugin, Layer 2): credit spreads over the sovereign curve
  • fixed-income-municipal (wealth-management plugin, Layer 2): muni-to-Treasury yield ratios
  • asset-allocation (wealth-management plugin, Layer 3): bonds as an asset class in portfolio construction
  • time-value-of-money（核心插件，层级0）：贴现与现值基础
  • fixed-income-corporate（财富管理插件，层级2）：主权曲线之上的信用利差
  • fixed-income-municipal（财富管理插件，层级2）：市政债与国债收益率比
  • asset-allocation（财富管理插件，层级3）：作为资产类别的债券在投资组合构建中的应用

Reference Implementation

参考实现

See 
scripts/fixed_income_sovereign.py
for computational helpers.
计算辅助工具请参见
scripts/fixed_income_sovereign.py