Fixed Income

General Knowledge

Roles of Fixed Income in Portfolio

Classify the Fixed-Income Mandates

Types of Lia

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TypesAmount is KnownTIming is KnownExample
Type I11Traditional Bond (without options)
Type II1 Callable bond (已知amount,不知道时间,可提前赎回), Term Life-time Insurance 寿险
Type III 1Float Rate Note, Inflation Related Bond (金额不确定,但是有确定到期)
Type IV  DB Plan, Property & Casualty Insurance 财险金额和时间都不确定

Type I: An advantage to knowing the size and timing of cash flows is that yield duration statistics—that is, Macaulay duration, modified duration, money duration, and PVBP—can be used to measure the interest rate sensitivity of the liabilities.

With Type II, III, and IV liabilities, a curve duration statistic known as effective duration is needed to estimate interest rate sensitivity. This statistic is calculated using a model for the uncertain amount and/or timing of the cash flows and an initial assumption about the yield curve.

Fixed-Income Returns 5个部分




Leveraged Portfolio Return

rp=Portfolio ReturnPortfolio Equity=ri+BE×(rIrB)

自有资金的return = 本身的return, r_i + leverage * (diff between 自己的return - borrowed return)

Leverage Effects on Duration


提升 Leverage 的方式
  1. Derivatives

    • Futures:

      • if i increase, then price decrease, FP also decrease

    • Swap:

      • Fixed-rate payer: long float short fix 因为 float 不影响duration,所以short fixed 会减少 duration。此时 i 提升, duration 负,则 value increase

      • Fixed-rate receivers: long fix short float 为duration增加。此时 i 提升,duration 为正,value 减少

  1. Repo

    1. Repo Margin: A 给 B asset,值100; B 给 A cash,为95 。 此时 B 少给的 5 为 Repo Margin,B少付的相当于是A给B的保证金

    2. Repo Rate: 结束后,A 给 B cash,为 97;B 把 Asset = 100 还给 A。那么97-95=2,多出的 2 为 repo rate,相当于借款利息

  2. Structural Financial Instruments (implicit)

    • inverse floator

    • embedded leverage

  3. Securtities Lending

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    Rebate Rate = Collateral Earning Rate - Securitiy Lending Rate

Risks of Leverage


如果实现 Capital Loss 可以减税

如果是 tax exempt account,那么可以直接 直接实现 capital gain,否则考虑实现 capital loss 来抵税

Short-term Capital Gain tax > Long term Capital Gain

Capital Loss 能抵减 Capital Gain, 不能抵减 Coupon

Liability Based Mandate

  1. Duration Matching

  2. Cash Flow Matching

  3. Contingent Immunisation

Bonds earn: (1) Coupon (2) Reinvestment (3) Price Changes

其中 Coupon 是固定的,但是 Reinvestment 和 Price Changes 受interest rate变动

因为 if interest rate increase, then (1) reinvestment return increase (2) price changes negative 所以,如果(1) & (2) 可以抵消,那么Duration matching 达成, Immunisation

So, we need to let


买 zero-coupon bond 最好,因为不涉及不用考虑中间 coupon reinvest的问题,但是 zero-coupon bond 只有短期。要匹配长期,只能买 coupon bearing fixed income bond

Duration Matching (Classic Immunisation 免疫策略)

保证 能earn IRR / Cash Flow Yield

For Single Lia Immunisation

Make price risks and reinvestment risks cancel each other.

如何 做 immunisation immunistion

  1. Portfolio Duration = Liability Duration <- price risks

  2. PV of Port = PV of Lia <- reinvestment risks

  1. minimuse Convexity, in order to minimise structural risks

Horizon 指的是 liability duration 负债的duration = 负债的到期日

Dealing with the Structural risk / Immunisation Risks:

Structural Risk (non-parallel steepening and flattening twists) yield curve 非平行移动会导致 structural risk 即

For Multi-Liabilities Immunisation

  1. PVA=PVLia

  2. DurationA=DurationLia

    • 1.2 可以合并为 Money Duration (BPV) 一样 or PVBP 一样

  3. (增加一条 for multi-lia )RangeA>RangeLia 资产的周期要能覆盖 lia port的周期,为了保证最后一期有足够的现金流能 cover liability payments

Cash Flow Matching

Cash Flow matching 是最nb的,可以避免 risk from non-parallel shifts in Yield Curve

先 cover 最后一笔负债,从后往前一层一层的剥离。先匹配长期的bond 是因为 买 asset 匹配长期的 lia 的时候,asset也会产生 interim cf,那么在匹配短期的时候,这些cf inflow将被合并考虑

, where PA,CA are for the bond A, 5-year. and bond B is 4-year, bond C is 3-year, etc

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Why not buy back and retire the liability early (tender offer)?

Duration Matching




  3. ConvexityA>ConvexityL 不同于 single 此时要求 convexity 比 lia 的更大,这样才增在non-parallel moves 是 cover CF

Cash Flow matching 和 Duration matching 的区别:具体见下表



Contingent Immunisation 用

Allow active management for the surplus amount of assets over liability 用surplus 的部分做 active management,其他正常部份 immunisation

Derivatives Overlay 用于 Cover the Duration Gap

前三个(1) duration matching (2) CF matching (3) contingent immunisation是用来构建组合。但是 Derivatives Overly 不是用于构建,而是用来 adjust. Rebalance the immunisation portfolio to keep it on its target duration.

Asset BPV×ΔAsset Yields+Hedge BPV×Δ Hedge YieldLia BPV×ΔLia Yield



Using Derivatives to adjust the duration of liability portfolio 用于调整 duration 不用于构建 asset portfolio

Hedge Ratio

Risks in LDI

  1. Model Risks: model assumptions are wrong

  2. Interest Rate Risks: DurA=DurL, duration explains (95%) most of price movement of bonds

  3. Yield Curve Risks: non-parallel shifs of the yield curve.

    How to deal with the non-parallel shifts of the yield curve: Minimise the dispersion of CF can mitigate this risks.

  4. Spread Risks: YTM=BaseYield(TreasuryYield)+Spread

    • yield on high-quality corporate bond are less volatilty than more liquid treasuries

  5. Counterparty Credit Risks

  6. Collateral Exhaustion Risks 由于 collateral不足,导致被平仓的risks

  7. Liquidity Risks

Total Return Mandates (Index Based)



因为 bond liquidity 比 equity 的低,所以mimic index bonds by purchasing the same bonds 可能比较难以操作。 所以 mimic Risk Factors

Risks Factors:

  1. Interest Rate Risks: exposure to parallel shift in the Yield Curve. Measured by Portfolio Duration

  2. Spread Risks: exposure to changes in spreads between Treasuries and non-Treasuries. Measured by Spread Duration.

    • (YTM = Benchmark Yield + Spread), so spread 涨1%带来的YTM 增加与 Benchmarked涨1%带来的YTM增加一致。thus spread duration = portfolio duration,as 都通过YTM影响 price

    • P.S.国债没有 spread

  3. Yield Curve Risks: exposure to a twist in the Treasury Yield Curve. Measured by Key Rate Duration & PV of Distribution of CF

  4. Credit Risks: exposure to downgrades and defaults. Measured by contribution to duration by credit rating

  5. Optionality Risks: exposure to changes in CF due to call/put features. Measured by Portfolio Delta**

Matching a FI Portfolio to an Index

FX market is difficult to track, because of size and breadth, wid array of security characteristics, and unique issuance and trading pattern.


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A bond’s yield spread includes both credit and liquidity risk. Liquidity risk depends on both market conditions and the specific supply-and-demand dynamics of each fixed-income security.


Below-investment-grade securities are affected more by changes in spread than by changes in general interest rate and exhibit stronger ocrrelations with equity markets.

Portfolio Duration in TOtal Return Mandates

Risks Factors for primary indexing

Bond Market Liquidity

Yield & Liquidity are negative correlated

Laddered Portfolio (convexity 在 Bullet 和 Barbell 之间)

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CF spreads and is diversified across the life of the bond.

The laddered approach provides both diversification over time and liquidity.

P.S. dispersion and convexity are positively correlated see equation.

Sample text: A laddered portfolio has lower convexity and dispersion than a barbell portfolio but more than a bullet portfolio, given comparable duration and cash flow yields. Lower convexity and dispersion are desirable aspects in liquidity management. In a laddered portfolio, there is always a bond close to redemption enhancing liquidity. As bonds mature, the final coupon and principal are available for distribution or can be reinvested in a long-term bond at the back of the ladder.

Yield Curve Strategy

Primary Yield Curve Risks Factors



Evaluating Yield Curve Strategies

Total Expected Return:


Coupon Return: coupon + reinvestment

  1. Rolldown Return: when Δy, how much ΔP

  2. Return: when Δy changes, how much ΔP

  3. Credit Loss

  4. FX G/L

Forward Rate Bias

Forward rate bias is defined as an observed divergence from interest rate parity conditions under which active investors seek to benefit by borrowing in a lower-yield currency and investing in a higher-yield currency.

Investors tend to favor unhedged fixed-income investments in higher-yielding currencies that are sometimes enhanced by borrowing in lower-yielding currencies.

Credit Strategy

Spread curve differs in differnt macro environments

High Yield bond and Low Yield bond behave different at differnt econ environment.

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Types of Spread

Fixed Rate Bonds Spread

Float Rate Bond Spread

The notion of discount margins is applied to floating-rate securities, which pay coupons based on an underlying benchmark such as Libor plus a spread. The value of FRNs is based on uncertain future cash flows as it is not known at purchase what the coupons will be as they fluctuate based on changes in the underlying index. The quoted margin on an FRN is the spread the bond pays above or below the underlying index. The quoted margin reflects the return required to compensate investors for the credit risk they take on when investing. If an issuer’s credit risk worsens or improves, the market’s perceptions of the quoted margin will change too. The difference between the quoted margin and the spread the market now requires for the FRN to trade at par is the discount margin.

MRR is based on current MRR and therefore implies a flat forward curve. 因为不知道未来的 MRR 是多少,所以会假设 flat curve,未来的 MRR 等于现在的MRR

Impacts of Yield Spreads on Portfolio Return

Refer to the basic rate


Similarly for Spread


However, for lower-rated bond, it is the percentage of spread change ΔSpreadSpreadwould have impact on the price, not absolute basis.对于低评级bond,对价格有影响的是 spread 变动的百分比,而不 spread 变动的数值

Credit Strategy


Bottom-up Credit Strategy

Transaction Cost

For buyer: Size×(TradePriceBid+Ask2)

For Seller: Size×(Bid+Ask2TradePrice)

Top-Down Credit Strategy

Choose Sector, broader sector 大类

  1. Asscess credit quality

  2. Sector allocation

Factor-Based Credit Strategy

FactorRationaleMeasures Used
CarryExpected Return measure if PoD or Aggregate Risk Premioum is unchangedOAS
DefensiveEmpirical Research suggests that safer low-risk assets deliver higher risk-adjusted returnsMarket-based leverage, gross profitability, and low duration
MomentumBonds with higher recent returns outperform those with lower recent returnsTrailing six-moth excess bond and equity returns
ValueLow market value versus fundamental value indicates greater than expected returnBond Spread less default probability measure, which includes rating, duration, and excess return volatility

ESG: (1) screen negative or positive, (2) filter charactireistics and invest the best-in-class, (3) directly fund , green bond


Liquidity Risks and Tail Risks

Liquidity Risks

Liquidity depends on 5 factors: (1) issuer's quality (2) creidt quality (3) issue frequency (4) issue size (5) maturity

Measure of Secondary Market Liquidty Risks

Tail Risks

Three methods:

Synthetic Credit Strategy: CDS

CDS buyer 买 CDS 相当于买了保险,会在每期付"保费" fixed payment,为固定金额 1% or 5%。

CDS Spread 为实际上值多少钱,如实际上值 1.75%,但是每期只固定付 1%,那么就要在inception 付 upfront premium

Fixed CDS Coupon 为标准化的 investment-grade = 1%, high-yield bond = 5% 。所以 实际的情况与标准的差值将作为 upfront fee 在前期支付,差值折现后求和,为 CDS quoted price。

CDS is quoted on a Issuer's CDS Spread = PV of difference between CDS Spread & Fixed Coupon

CDS Price

CDS Price=NP×[1+(FixedCouponCDS Spread)×Eff.Spread.DurCDS]

CDS Risk Position

Sample Text: Selling protection on the CDX index is a “long” credit spread risk position, 因为预期credit rating会提升,所以sell CDS protect是去赚钱。while purchasing protection on the CDX Financials subindex is a “short” credit spread risk position, leaving the investor with a long index position without exposure to financial reference entities in the CDX index.

Spread Curve CDS Strategy

Same for the yield curve movement。如果expected steepening of the investment-grade credit spread curve,相当于credit spread curve twists, 长期 spread提升,短期变小。长期提升意味着 长期credit变差,所以 long long-term

a synthetic credit roll-down strategy involves selling protection using a single-name CDS contract for a longer maturity. 因为卖protection 收到更多 CDS spread,到短期PoD变小了,收到的Spread将会变小

Global Credit Strategy