Fixed Income — Deep Reference

Fixed income is the largest asset class by notional outstanding on planet Earth. As of mid-2025 global debt securities exceed $140 trillion (BIS data); USD interest-rate derivatives outstanding exceed $500 trillion notional. The discipline encompasses sovereign and corporate cash bonds, mortgage-backed securities, agency and municipal debt, asset-backed paper, credit default swaps, interest-rate swaps, caps and floors, swaptions, and the term-structure models that price them all consistently. This note covers the modern stack: Treasury curve construction by bootstrapping and parametric fitting; duration and convexity at every level of refinement; option-adjusted spreads for callable, putable, and prepayable instruments; MBS prepayment modeling; corporate credit-spread decomposition; interest-rate derivative pricing under SABR, Black-76, and term-structure models from Vasicek through HJM and LMM; the post-LIBOR RFR regime; credit derivatives from single-name CDS through synthetic CDOs; and the electronification of cash bond trading via TRACE, MarketAxess, and Tradeweb.

See also

• derivatives-and-quant-finance
• options-pricing-deep
• structured-products-deep
• portfolio-construction-and-risk-deep
• corporate-finance-and-markets
• market-microstructure-and-hft

1. Treasury curve construction

The US Treasury yield curve is the foundation of dollar interest-rate markets. Construction starts from observable instrument prices and produces a continuous discount function $D(t)$ such that the present value of a unit cash flow paid at time $t$ equals $D(t)$.

Bootstrapping is the iterative method: start from the shortest instrument (T-bills), solve for $D(t_1)$, then sequentially solve for $D(t_2),D(t_3),\dots$ from longer instruments using already-determined earlier discount factors. For semiannual-coupon Treasuries the bootstrap of the par-coupon curve $c_i$ at maturities $T_i$ yields zero-coupon prices via:

$100=c_i{\sum }_{j:T_j\le T_i}D(T_j)+100\cdot D(T_i)$

The bootstrap requires perfect-fit assumption — every instrument prices to its quoted level. Choices among nearly-equivalent instruments (e.g., on-the-run vs off-the-run 10Y Treasuries) introduce $\pm 1-3$ bp curve dispersion that real desks track as the on-the-run premium.

Cubic spline interpolation between bootstrapped knot points produces a smooth discount function. Tension splines (Hagan-West 2006) reduce wiggle and oscillation. Monotone convex interpolation (Hagan-West 2006) preserves forward-rate positivity — important for arbitrage-free pricing of forwards and OIS swaps.

Nelson-Siegel (1987) and the Nelson-Siegel-Svensson (1994) extension provide parametric yield-curve fits. The instantaneous forward rate under NSS:

$f(t)={\beta }_0+{\beta }_1{e}^{-t/{\tau }_1}+{\beta }_2\frac{t}{{\tau }_1}{e}^{-t/{\tau }_1}+{\beta }_3\frac{t}{{\tau }_2}{e}^{-t/{\tau }_2}$

with parameters $({\beta }_0,{\beta }_1,{\beta }_2,{\beta }_3,{\tau }_1,{\tau }_2)$. Used by the Federal Reserve Board, ECB, BoE, and Treasury Bureau of Fiscal Service for official curves. The 2018 Fed Gurkaynak-Sack-Wright dataset publishes daily NSS Treasury curves since 1961.

Diebold-Li 2006 is a dynamic Nelson-Siegel with time-varying $\beta$‘s evolving as VAR(1) — popular in macro forecasting because it isolates level, slope, and curvature factors.

2. Zero, par, forward, and instantaneous-forward curves

Four equivalent curve representations cover the same information:

• Zero (spot) curve $z(T)$: continuously compounded yield on a zero-coupon bond maturing at $T$. $D(T)=e^{-z(T)T}$.
• Par curve $c(T)$: coupon on a newly issued par-priced bond at maturity $T$.
• Forward curve $F(T_1,T_2)$: agreed forward rate for the period $[T_1,T_2]$ from today.
• Instantaneous forward curve $f(t)$: $f(t)=-\partial \ln D(t)/\partial t={\lim }_{\Delta \to 0}F(t,t+\Delta )$.

Conversions: $z(T)=\frac{1}{T}{\int }_0^{T}f(s)ds$; the forward $F(T_1,T_2)=\frac{1}{T_2-T_1}\ln \frac{D(T_1)}{D(T_2)}$ for continuous compounding.

Curve steepness ($z(10Y)-z(2Y)$) and butterfly ($2\cdot z(5Y)-z(2Y)-z(10Y)$) are the canonical curve-shape trades. The August 2019 - October 2022 US 2s10s inversion (deepest at -107 bp on July 2023) preceded the 2024 recession watch; the dis-inversion completed September 2024 ahead of the Fed’s 50 bp cut.

3. Duration — Macaulay, modified, effective, key-rate

Macaulay duration is the weighted average time of cash flows, weighted by present value:

$D_{\text{Mac}}=\frac{{\sum }_i{t}_i\cdot \text{PV}(C_i)}{{\sum }_i\text{PV}(C_i)}$

Modified duration is the percentage price sensitivity to a parallel yield shift:

$D_{\text{mod}}=-\frac{1}{P}\frac{\partial P}{\partial y}=\frac{D_{\text{Mac}}}{1+y/m}$

for compounding frequency $m$. Dollar duration \D = P \cdot D_\text{mod} / 100$ gives the dollar PnL per 1 bp parallel move (also called DV01 or PV01).

Effective duration is the empirical price sensitivity computed by repricing the bond at $y+\Delta y$ and $y-\Delta y$:

$D_{\text{eff}}=\frac{P(y-\Delta y)-P(y+\Delta y)}{2P\Delta y}$

Effective duration captures option-adjusted sensitivity for callable bonds, MBS, and convertibles where cash flows change with yields. For pure non-callable Treasuries, Macaulay $\approx$ modified $\approx$ effective.

Key-rate duration (Ho 1992) measures sensitivity to shocks at specific maturity points — typically 3M, 6M, 1Y, 2Y, 3Y, 5Y, 7Y, 10Y, 20Y, 30Y. The sum of key-rate durations equals modified duration for a parallel shift. KRD decomposition is the standard tool for non-parallel curve risk management at fixed-income desks.

Convexity $C=\frac{1}{P}\frac{{\partial }^{2}P}{\partial y^2}$ captures second-order curvature. The price change for a yield move $\Delta y$ to second order:

$\frac{\Delta P}{P}\approx -D_{\text{mod}}\Delta y+\frac{1}{2}C(\Delta y{)}^2$

Convexity is positive for plain bonds (price-yield is convex), negative for MBS in low-rate environments (extension risk), and varies non-monotonically for callable bonds.

4. Option-Adjusted Spread (OAS)

For bonds with embedded options — callables, putables, prepayable MBS, contingent convertibles — the static yield spread to Treasuries (Z-spread) overstates the true credit/liquidity premium because it ignores option value. The Option-Adjusted Spread is the constant spread that, when added to every node of a term-structure model’s tree or grid, reproduces the bond’s market price under the optimal exercise policy:

$P_{\text{market}}={\mathbb{E}}^{\mathbb{Q}}\left[{\sum }_t{C}_t\cdot e^{-{\int }_0^t(r_s+\text{OAS})ds}\cdot 1_{\text{not-called}}\right]$

OAS depends on the term-structure model used (Hull-White vs BDT vs LMM) and on assumed prepayment behavior for MBS — making cross-vendor OAS comparison non-trivial. Bloomberg OAS, BlackRock Aladdin OAS, and Yield Book OAS routinely diverge by 5-15 bp on the same MBS pool.

Z-spread, asset-swap spread (ASW), CDS-bond basis, and OAS are the four canonical credit-spread measures; the CDS-bond basis (CDS spread minus ASW of the underlying bond) is a major dealer P&L driver and a slow-motion crisis indicator (inverted hard in March 2020, September 2022 gilt crisis, March 2023 SVB/CS episode).

5. MBS prepayment modeling

US agency MBS — Ginnie Mae (full faith), Fannie Mae and Freddie Mac (conservatorship since September 2008) — represent the second-largest sector of US fixed income after Treasuries, at approximately $13 trillion outstanding (2025). Prepayment risk dominates pricing.

PSA (Public Securities Association) prepayment benchmark ramps from 0.2% CPR (Conditional Prepayment Rate) at month 1 to 6% CPR at month 30, then stays flat at 6% CPR. “100 PSA” matches this profile; “200 PSA” is twice the speed; “0 PSA” is no prepayment. The market quotes prepayment forecasts as multiples of PSA.

Conditional Prepayment Rate (CPR) is the annualized fraction of the remaining balance that prepays in a given month. Monthly SMM (Single-Monthly Mortality): $\text{SMM}=1-(1-\text{CPR}{)}^{1/12}$.

Andrew Davidson & Co. (AD&Co) prepayment models — proprietary econometric models calibrated to monthly loan-level data, covering refinance incentive (the spread between current mortgage rate and the loan’s note rate), turnover (housing-market activity), seasoning (loan age), burnout (declining responsiveness for pools that have already had multiple refi waves), and seasonality. AD&Co is the dominant non-bank prepayment model provider; competitors include Yield Book (LSEG), BlackRock Aladdin, Bloomberg PMOD, and ICE Mortgage Technology.

Endogenous (econometric) prepayment models — Schwartz-Torous 1989 Journal of Finance introduced the proportional-hazard form $h(t)=h_0(t)\cdot \exp ({\beta }^{\prime }{X}_t)$ where $X_t$ includes refi incentive, seasoning, and macro factors. The Stanton 1995 model adds option-based optimal refinance behavior. Optimal-exercise prepayment models treat the borrower as a rational option-exerciser — but empirical CPRs deviate substantially from optimal exercise, with transaction costs and media effects (visible refi opportunity windows) explaining the deviation.

deWitt prepayment model is another well-known proprietary model used by mortgage hedge funds.

The 2003 refi wave (30Y mortgage rate dropped to 5.21% from over 8% in 2000), the 2009-2012 HARP refi waves (Home Affordable Refinance Program), the 2020-2021 COVID refi wave (mortgage rates touched 2.65% in January 2021), and the 2022-2024 lock-in effect (rates jumped from 3% to 7%+ — borrowers refused to give up their sub-3% mortgages, depressing housing transactions and prepayment speeds to multi-decade lows) are the four most important regime episodes for MBS prepayment modeling.

6. Agency vs non-agency MBS and CMOs

Agency MBS — issued or guaranteed by Ginnie Mae (full faith, FHA/VA loans), Fannie Mae, or Freddie Mac (GSEs in conservatorship since 2008, implicit guarantee). To-Be-Announced (TBA) trading is the dominant secondary market: investors trade a generic 30Y FN 5.5 coupon for a specified settlement date without specifying which pools deliver, settling under good-delivery rules. TBA liquidity makes agency MBS the second-most-liquid market behind Treasuries.

Non-agency MBS — jumbo loans (above conforming limit, $766,550 in 2024 most areas), Alt-A, subprime, and pre-2008 private-label. Non-agency outstanding collapsed from $2.4T peak in 2007 to under $500B by 2025 after the GFC; the new issuance market revived modestly post-2013 with prime jumbo and now-rebuilt subprime/non-QM segments.

Collateralized Mortgage Obligations (CMOs) redistribute the cash flows of an underlying MBS pool into tranches with different prepayment-risk profiles:

• Sequential pay — Class A receives all principal until paid off, then Class B, then Class C.
• PAC (Planned Amortization Class) — receives a fixed-schedule principal payment within a PAC band (e.g., 100-250 PSA); other tranches (companion/support) absorb prepayment variability.
• TAC (Targeted Amortization Class) — like PAC but with a single-speed target.
• IO (Interest Only) — receives only interest; bullish on slow prepayments.
• PO (Principal Only) — receives only principal; bullish on fast prepayments. The classic 1994 mortgage market crash hit PO holders as rates spiked and prepayments collapsed.
• Floater / Inverse Floater — coupons tied to LIBOR (now SOFR) with the inverse floater absorbing the residual.
• Z-bonds (Accrual bonds) — accrue interest until earlier tranches pay off, then receive both interest and principal.

CMO design lets banks slice the MBS prepayment risk spectrum to match investor risk preferences. Z-bond convexity hedging drove the famous 1994 PIMCO and Askin Capital blow-ups; Whole Pool CMO arbitrage between the sum of tranches and the underlying pool is a continuous dealer-side opportunity.

7. Corporate credit spreads

Investment-grade corporate bonds (rated BBB-/Baa3 or above) trade at spreads typically 50-200 bp over Treasuries; high-yield (junk, BB+/Ba1 and below) trade at 300-1000 bp. Spread is decomposed into:

$\text{Spread}=\text{Expected loss}+\text{Risk premium}+\text{Liquidity premium}+\text{Tax + other}$

Empirically (Elton-Gruber-Agrawal-Mann 2001, Huang-Huang 2012) only 20-40% of investment-grade spreads is explained by expected default loss; the remainder is the credit-risk premium and liquidity premium (the credit-spread puzzle).

Z-spread: the parallel shift to the spot Treasury curve that, when added to discount factors, reproduces the bond’s market price.

Asset-swap spread (ASW): the floating-rate margin over the swap curve that funds a synthetic floating-rate version of the bond. ASW vs Z-spread is determined by basis-swap pricing.

OAS: applied to callable corporate bonds — corrects Z-spread for embedded call optionality.

CDS-bond basis: CDS spread on the same name and maturity minus ASW of the bond. In equilibrium near zero — sustained dislocations indicate dealer balance-sheet constraints, repo market dysfunction, or CDS contractual specifications (cheapest-to-deliver bond option).

Investment-grade indices: Bloomberg US Corporate Investment Grade (formerly Lehman/Barclays Agg Corp), ICE BofA US Corporate Index (C0A0). High-yield: Bloomberg US Corporate High Yield, ICE BofA US HY Master II. Spreads compressed to multi-decade tights in 2021 (HY OAS at 311 bp Sep 2021), widened to 2008 levels March 2020, widened to 540 bp October 2022, and re-tightened through 2023-2024 to 280-330 bp range.

8. Interest rate derivatives — caps, floors, swaps, swaptions

Interest rate swap (IRS) — exchange of fixed-rate cash flows against floating cash flows over a schedule. The vanilla USD IRS pays fixed semi-annually on a 30/360 basis vs floating SOFR compounded daily on an Actual/360 basis. Pre-2023 the floating leg referenced 3M LIBOR; the SOFR transition replaced ~$250T of LIBOR notional with SOFR equivalents.

The fair fixed rate equals the par-swap rate:

$s=\frac{{\sum }_i{\tau }_i^{\text{float}}{F}_{i}D(T_i)}{{\sum }_j{\tau }_j^{\text{fixed}}D(T_j)}$

with $F_i$ the forward floating rate over the $i$-th period, ${\tau }_i$ accrual fractions, and $D$ discount factors from the OIS-discounted curve.

OIS (Overnight Index Swap) — floating leg is the geometric compounding of an overnight index (SOFR, €STR, SONIA, TONA, SARON) over the accrual period. Post-GFC, OIS rates are the canonical risk-free rate; OIS discounting is the standard for collateralized derivatives.

Cap / floor — portfolio of caplets (European calls on the forward rate) or floorlets (European puts). Each caplet is priced under Black 1976 with a strike, expiry-tenor implied vol from the SABR cube, and discounting via the appropriate forward measure:

$\text{Caplet}=N\tau D(T_{i+1})\left[F_{i}N(d_1)-KN(d_2)\right]$

with $d_1=\frac{\ln (F_i/K)+{\sigma }^2{T}_i/2}{\sigma \sqrt{T_i}}$, $d_2=d_1-\sigma \sqrt{T_i}$.

Swaption — option to enter a swap. European swaption (single expiry, expiry on $T$, swap from $T$ to $T+M$). Black-76 in the annuity (swap) measure ${\mathbb{Q}}^A$ where the swap rate is a martingale:

$\text{Payer swaption}=N\cdot A_T\cdot \left[sN(d_1)-KN(d_2)\right]$

with $A_T={\sum }_j{\tau }_{j}D(T_j)$ the swap annuity factor.

The swaption-vol cube (expiry × tenor × strike) is the standard rates-vol structure, marked under SABR with per-(expiry, tenor) parameters $(\alpha ,\beta ,\rho ,\nu )$. The cube has approximately $11\times 11\times 7=847$ liquid nodes for USD; daily delta-vega-skew-convexity hedging happens against the cube.

Bermudan swaption — option to enter a swap on any of a discrete schedule of dates. The natural callable-debt hedging instrument and a major LMM application via Longstaff-Schwartz or parameterized exercise (Andersen 1999).

CMS (Constant Maturity Swap) — floating leg paying the swap rate of a fixed tenor (e.g., 10Y CMS rate). CMS spread options payoff $\max (s_1-s_2-K,0)$ where $s_1,s_2$ are CMS rates at different tenors — requires convexity adjustment because the CMS rate is not a martingale under the payoff measure.

9. SOFR, SONIA, €STR, TONA, SARON — RFR transition

The LIBOR scandal (2008-2012 rate manipulation by panel banks; Barclays, UBS, RBS, Deutsche Bank, and others fined billions; criminal convictions of Tom Hayes among others) led to the regulator-driven LIBOR phase-out.

• USD LIBOR overnight, 1W, 2M, 12M: ceased 31 December 2021.
• USD LIBOR 1M, 3M, 6M: synthetic continuation, full cessation 30 June 2023.
• GBP, JPY, CHF, EUR LIBOR all tenors: ceased 31 December 2021.

Successor RFRs (overnight, secured or unsecured, backward-looking):

• USD: SOFR (Secured Overnight Financing Rate) — NY Fed since 2018, Treasury repo-based.
• GBP: SONIA (Sterling Overnight Index Average) — reformed BoE since 2018, unsecured.
• EUR: €STR (Euro Short-Term Rate) — ECB since 2019, unsecured.
• JPY: TONA (Tokyo Overnight Average Rate) — BoJ.
• CHF: SARON (Swiss Average Rate Overnight) — SIX, secured.

CME Term SOFR (since 2021) is a forward-looking term SOFR backed by SOFR futures, used for select cash-product applications (syndicated loans, business credit). ISDA fallback protocol (October 2020) for legacy contracts cascades to compounded-in-arrears SOFR plus the ISDA spread adjustment (historical median LIBOR-SOFR spread, fixed at fallback trigger).

Post-2023 SOFR is the global USD reference rate. SOFR spikes at quarter-ends (especially 2024 and 2025 year-end traced to dealer balance-sheet constraints and SLR-driven Treasury holdings) drive non-trivial basis dynamics with Fed Funds; SOFR-Fed Funds basis swap is now a liquid product.

10. Term-structure models

The arc from short-rate models through HJM and LMM:

Vasicek 1977 — first short-rate model:

$d{r}_t=\kappa (\theta -r_t)dt+\sigma d{W}_t$

Gaussian distribution, possibly-negative rates, closed-form bond prices. Empirically misfit because constant $\theta$ doesn’t replicate the initial term structure.

Cox-Ingersoll-Ross (CIR) 1985 — $d{r}_t=\kappa (\theta -r_t)dt+\sigma \sqrt{r_t}d{W}_t$. Non-central chi-squared distribution, strictly non-negative rates (under Feller $2\kappa \theta \ge {\sigma }^2$), closed-form bond prices. Used in academic asset-pricing literature.

Hull-White 1990 — Vasicek with time-dependent mean reversion $\theta (t)$ calibrated to fit the initial term structure:

$d{r}_t=(\theta (t)-a{r}_t)dt+\sigma d{W}_t$

Affine, analytically tractable. Closed-form zero-coupon bonds, caplets, floorlets, European swaptions. Two-factor Hull-White ($G2++$) adds a second factor for richer correlation structure. Standard for callable bonds, mortgage OAS, and many exotic-rate products at second-tier desks.

Black-Karasinski 1991 — $d\ln r_t=(\theta (t)-a\ln r_t)dt+\sigma d{W}_t$. Lognormal rates (positive by construction); no closed form, requires trinomial-tree numerical solution. Popular pre-2008 for credit-cum-rate hybrids.

Black-Derman-Toy (BDT) 1990 — binomial-tree-fitted short rate model. Calibrates step-by-step to fit initial yields and caplet vols. Used historically; largely superseded by Hull-White and LMM.

Heath-Jarrow-Morton (HJM) 1992 — canonical no-arbitrage description of the full term structure. Models the instantaneous forward rate curve $f(t,T)$ directly:

$df(t,T)=\alpha (t,T)dt+\sigma (t,T)d{W}_t$

with the HJM no-arbitrage drift restriction $\alpha (t,T)=\sigma (t,T)\cdot {\int }_t^T\sigma (t,u)du$ under the risk-neutral measure. Subsumes Vasicek, CIR, Hull-White, Black-Karasinski as special cases. Generic HJM is non-Markov; Markov reductions require restricted volatility structures (separable $\sigma \to$ Hull-White 1F, two-factor HW, Cheyette quasi-Gaussian models used for exotic mortgage pricing).

LIBOR Market Model (LMM/BGM) — Brace-Gatarek-Musiela 1997, Miltersen-Sandmann-Sondermann 1997, Jamshidian 1997. Models discretely-compounded forwards $L_k(t)$ directly under each forward measure, with Black-style caplet pricing for free by construction. Calibrated to the SABR cube. Standard rates-exotic engine for thirty years.

11. Credit derivatives — single-name CDS

A credit default swap (CDS) is a contract where the protection buyer pays a periodic premium (coupon, conventionally 100 bp or 500 bp under the 2009 ISDA Big Bang protocol) and receives a payment $(1-R)\cdot N$ if a reference entity defaults, where $R$ is the recovery rate (40% senior unsecured is the LCDS default convention).

Pricing under the reduced-form / intensity-based approach (Jarrow-Turnbull 1995, Duffie-Singleton 1999): default is the first jump of a Cox process with stochastic hazard rate ${\lambda }_t$. The survival probability:

$Q(\tau >T)={\mathbb{E}}^{\mathbb{Q}}\left[\exp \left(-{\int }_0^T{\lambda }_{s}ds\right)\right]$

The CDS fair spread is approximately:

$s\approx \lambda \cdot (1-R)$

for flat hazard and recovery. Bootstrap is straightforward: given CDS spreads at standard tenors (6M, 1Y, 2Y, 3Y, 5Y, 7Y, 10Y) and assumed recovery, solve for piecewise constant hazard rates that reprice the curve. ISDA standard-model conventions (Markit/IHS Markit since 2009, now S&P Global Market Intelligence) standardize the calculation.

CDS-spread quoting post-2009 Big Bang: trades execute as a fixed 100 bp or 500 bp running coupon plus an upfront payment that adjusts to the market spread. ISDA model converts spread to upfront via the standard hazard/recovery convention.

Capital-structure arbitrage: CDS spread implies a default probability; the same default probability via Merton’s 1974 structural model implies an equity-implied default probability. When these diverge, the trade is to buy stock and CDS protection (or short stock and sell CDS), going to convergence. Moody’s KMV (now Moody’s Analytics) commercialized the structural approach in the 1990s.

12. Credit indices — CDX, iTraxx

Major CDS indices (rolled every March and September; the roll dates):

• CDX.NA.IG — 125 North American investment-grade names, 5Y is the benchmark.
• CDX.NA.HY — 100 North American high-yield names.
• iTraxx Europe Main — 125 European investment-grade names.
• iTraxx Crossover — 75 European sub-IG names.
• iTraxx Asia ex-Japan — 50 Asian IG names.
• iTraxx Japan — 50 Japanese names.

Index trading is highly liquid; index options (payer / receiver swaptions on the index) trade with bid-asks 10-30 bp of upfront. The 2008 GFC, 2011 European sovereign crisis, 2020 COVID, 2022 European energy crisis, March 2023 SVB / CS, and the late-2024 dispersion-trade unwind are the canonical CDX/iTraxx stress events.

The CDS-bond basis between CDS index spread and the weighted average bond spread of the underlying constituents is a major dealer P&L driver — inverted hard in March 2020 (basis to -200 bp) and September 2022 (gilt crisis basis to -150 bp).

13. CDO and tranche valuation

Synthetic CDOs are portfolio credit derivatives on baskets of 125 (CDX) or 100 (iTraxx) names, tranched into super-senior, senior, mezzanine, and equity. Standard tranches on CDX.NA.IG: 0-3%, 3-7%, 7-10%, 10-15%, 15-30%, 30-100%.

The one-factor Gaussian copula (Li 2000, Journal of Fixed Income) prices tranches via a single correlation parameter $\rho$. Asset values:

$V_i=\sqrt{\rho }M+\sqrt{1-\rho }{Z}_i$

with $M$ a systemic factor and $Z_i$ idiosyncratic, both standard normal. Name $i$ defaults if $V_i<{\Phi }^{-1}(p_i)$ where $p_i$ is the term-default probability from the CDS curve. Base correlation (the correlation parameter that reprices a tranche as the equity-loss-attachment portion of a 0-$K$% strike) is the practitioner standard for quoting tranche values.

The 2007 mass mark-downs of AAA super-senior CDO tranches and the cross-correlation failure of the Gaussian copula were a defining feature of the GFC. Bespoke tranches had a quiet renaissance 2017-2021 and a smaller revival post-2023 with the regulatory clean-up of CLO and CMBX markets.

14. TRACE and the electronification of corporate bond trading

TRACE (Trade Reporting and Compliance Engine) — FINRA-operated transaction reporting system for OTC corporate, agency, and securitized debt. Mandatory since 2002 for IG and HY corporate bonds; expanded to agency, MBS, ABS, and structured products through 2010-2020. TRACE reports trade-by-trade volume, price, and counterparty type (D2D dealer-to-dealer, D2C dealer-to-customer) with a 15-minute reporting lag (60 minutes for some asset classes). TRACE data is the academic and industry workhorse for liquidity, transaction-cost, and price-impact research.

Electronification:

• MarketAxess (founded 2000, Rick McVey CEO until 2023) — dominant electronic platform for IG and HY corporate bond trading via the Open Trading all-to-all protocol. Approximately 40% of US IG corporate volume traded electronically by 2024 (MarketAxess plus competitors). MarketAxess’s Bid/Ask Composite (CP+) and proprietary algos drive the price formation for active issuers.
• Tradeweb (founded 1996, public since 2019, controlled by LSEG since 2018) — dominant in Treasuries, EU government bonds, MBS TBA, and interest-rate swaps. Tradeweb FX, Tradeweb Mortgages, and Tradeweb Interest Rates each handle multi-trillion-dollar daily volumes.
• Bloomberg BondHub / BVAL — fixed-income evaluations and execution; BVAL is the benchmark pricing source for IG and HY at most asset managers.
• TruMid — newer electronic platform for IG and HY corporates, all-to-all protocols, growing share.
• LSEG / Refinitiv FXall — FX trading platform with rates-swap module via the Tradeweb integration.

RFQ (Request for Quote) is the dominant OTC corporate bond protocol: an investor asks multiple dealers for a quote, executes against the best. All-to-all (any participant can post a quote or take liquidity) is the newer protocol that has narrowed bid-asks substantially in the most-active IG names.

15. Repo market

The repurchase agreement (repo) market is the secured funding backbone of fixed income. A repo is the sale of a security with a commitment to repurchase at a later date at a higher price; the spread is the repo rate. Treasury repo notional outstanding exceeds $5T daily.

• Tri-party repo — settlement via BNY Mellon or JPMorgan as third-party agent. Standard for institutional cash investors (MMFs).
• General Collateral Financing (GCF) repo — DTCC-cleared blind broker repo where the collateral is any Treasury within a class.
• Specific (special) repo — repo on a particular CUSIP; specials trade rich (lower repo rate) when the CUSIP is in scarce supply (e.g., on-the-run 10Y after auctions, deliverable for futures expiry).
• Bilateral non-cleared repo — dealer-to-customer; subject to CSA-style documentation.

The DTCC FICC centrally clears GCF and bilateral DVP repo; the proposed expansion of FICC clearing to all Treasury repo (SEC final rule December 2023, phased through 2025-2026) is the largest structural change to the repo market in two decades.

Repo rate spikes: September 17, 2019 SOFR/Repo spike to 10% in the secured overnight market — Fed intervened with standing repo operations restored. March 2020 dash-for-cash. Year-end balance-sheet windows (2024, 2025) showed repeated SOFR rallies of 25-50 bp.

16. Treasury market structure

US Treasuries are the largest single market by daily volume ($700B-$1T daily). Issuance schedule: 4W/8W/13W/17W/26W bills, 2Y/3Y/5Y/7Y/10Y/20Y/30Y notes and bonds, TIPS (Treasury Inflation-Protected Securities) at 5Y/10Y/30Y, FRNs (2Y floating). Total marketable debt $28T (end-2025).

Primary dealers — 24 institutions (2024) authorized to bid in Treasury auctions and trade with NY Fed: Bank of America Securities, Barclays, BMO Capital Markets, BNP Paribas Securities, Cantor Fitzgerald, Citigroup, Daiwa Capital Markets America, Deutsche Bank Securities, Goldman Sachs, HSBC Securities, J.P. Morgan Securities, Jefferies, Mizuho Securities, Morgan Stanley, NatWest Markets, Nomura Securities International, RBC Capital Markets, Santander US Capital Markets, Scotia Capital, Société Générale, TD Securities, UBS Securities, Wells Fargo, and BMO.

On-the-run — the most recently issued Treasury at a given maturity. Trades at a liquidity premium (lower yield, 2-10 bp) to off-the-run. The on-the-run / off-the-run spread is the classic relative-value Treasury trade.

Treasury futures (CME) — 2Y (TU), 5Y (FV), 10Y (TY), Ultra-10 (TN), Long Bond (US), Ultra-Long (UB). The cheapest-to-deliver (CTD) bond minimizes the loss on physical delivery; CTD determination is a delicate trade-level calculation involving the conversion factor and the gross-vs-net basis.

Treasury basis trade — long cash Treasury vs short Treasury futures, financing the cash position in repo. Hedge funds run massive basis books (estimated $700B-$1T notional in 2024 per BIS); the March 2020 dash-for-cash unwind of basis trades severely amplified Treasury-market dysfunction, prompting Fed Treasury purchases of $1T over six weeks.

17. Inflation derivatives

TIPS principal adjusts with CPI-U; coupons are fixed real rates. The breakeven inflation rate is $y_{\text{nominal}}-y_{\text{TIPS}}$, indicating market-implied inflation expectations.

Inflation swaps — zero-coupon inflation swap pays $(I_T/I_0{)}^{1/T}-(1+s_{\text{swap}})$ at maturity, with $I$ a published price index (US CPI-U, UK RPI, EUR HICPxT). Pricing under the Jarrow-Yildirim 2003 model (HJM for nominal, real, and inflation factors) or under the simpler forward-CPI model with normal/lognormal forward dynamics.

Year-on-year (YoY) inflation swap — pays $(I_{T_i}/I_{T_{i-1}}-1-s_{\text{YoY}})\cdot \tau$ on each schedule date. Calibration to the YoY swap curve is nontrivial because of seasonality in monthly CPI prints.

Inflation caps and floors — options on YoY or zero-coupon inflation. Critical for pension liabilities and inflation-linked annuities.

US inflation derivatives have a smaller market than UK (RPI swaps are the dominant LDI hedging tool for UK pensions). The September 2022 UK gilt crisis was triggered by mass margin calls on LDI inflation-hedging programs.

18. Convertible bonds

Convertible bonds are corporate bonds with embedded call options on the issuer’s stock. Modeling decomposes into a straight bond plus a long equity call:

$\text{CB price}=\text{Bond floor}+\text{Equity option}$

Pricing under Tsiveriotis-Fernandes 1998 PDE (bond and equity factor with credit-spread separation) or under one-factor lattice models with deterministic credit spread. Vega, delta, and credit-spread sensitivity (omega) are the three core CB Greeks. Convertible arbitrage — long CB, short stock for delta, possibly hedge credit with CDS — was a major hedge-fund strategy 2003-2008 (Citadel, Highbridge, Fortress) and rebuilt after 2010 with smaller AUM.

Mandatory convertibles, PIPEs (Private Investment in Public Equity), death-spiral convertibles (toxic structures with discount-to-VWAP conversion, banned for many issuers) round out the family.

19. Sovereign and emerging-market debt

Sovereign credit pricing combines hard-currency (USD/EUR-denominated) vs local-currency debt with distinct risk dynamics. Hard-currency: pure credit risk against USD/EUR cash flows. Local-currency: credit risk plus FX risk plus inflation risk.

EMBI Global (Bloomberg, formerly JPMorgan) — the benchmark dollar-denominated emerging-market sovereign index, market-cap weighted, OAS to Treasuries. GBI-EM Global Diversified — local-currency emerging-market sovereign index. CEMBI — emerging-market corporate dollar bonds.

Distressed sovereign episodes recurring 2020-2025: Argentina (default May 2020, restructured), Lebanon (default March 2020), Sri Lanka (default May 2022, restructured 2023), Ghana (default December 2022, restructured 2024), Zambia (default November 2020, restructured 2024). Common features: bilateral negotiations (Paris Club, China non-Paris Club bilateral creditors), private creditor exchanges, IMF program conditionality.

20. Pension and insurance liability hedging

UK and EU defined-benefit pension funds run massive Liability-Driven Investment (LDI) programs hedging the duration and inflation profile of their liabilities. Standard structure: long-dated swaps (30Y-50Y), inflation swaps (UK RPI), and gilt repo financing.

The September 2022 UK gilt crisis — Truss-Kwarteng mini-budget triggered a sterling crisis and a multi-standard-deviation move in 30Y gilt yields, generating margin calls on LDI hedges that forced gilt selling and a self-reinforcing spiral. BoE intervened with emergency gilt purchases ($65B initial; total $19B used). The episode prompted UK pension regulator (TPR) and FCA guidance restricting LDI leverage and requiring liquidity buffers.

US insurance industry runs a parallel program for variable annuities (GMDB, GMIB, GMAB, GMWB embedded options) hedged with equity index puts, futures, and interest-rate swaps. The 2008 GFC and 2020 COVID stressed these programs hard; Athene, Apollo, and KKR have built large reinsurance pools to absorb similar liabilities (sidecar reinsurance, Bermuda-domiciled).

21. Notable firms and people

• PIMCO (Pacific Investment Management, founded 1971 by Bill Gross, Allianz-owned since 2000) — Newport Beach, CA; $2T AUM 2024; dominant in global core fixed income.
• BlackRock Fixed Income — Larry Fink’s New York firm; $10T total AUM with $2.5T+ in fixed income.
• Vanguard Fixed Income — Valley Forge, PA; passive index dominant; $2T fixed income AUM.
• Fidelity Fixed Income — Boston; active core funds.
• Western Asset Management — Pasadena, CA; subsidiary of Franklin Resources since 2020.
• DoubleLine Capital — Jeffrey Gundlach’s firm, founded 2009, MBS specialist.
• TCW Group — Los Angeles; $200B+ AUM; MBS and structured credit specialist; Tad Rivelle was longtime CIO.
• Loomis Sayles — Dan Fuss was the legendary lead PM; deep value bond shop.
• Bridgewater All Weather — risk-parity strategy that’s effectively a long-duration bond and inflation overlay.
• Citadel Fixed Income, Millennium Fixed Income, Balyasny Fixed Income — multi-strat platforms with large rates and credit pods.
• Bondholders’ Steering Committees — informal coordinating bodies for distressed sovereign and corporate restructurings (Lazard, Houlihan Lokey, Rothschild as advisors).

21b. Municipal bonds

The US municipal market (“munis”) is the third-largest US fixed-income market by outstanding ($4.2T, 2025), with approximately 50,000 distinct issuers (states, cities, counties, special districts, agencies, hospitals, universities). Key features:

• Tax exemption: interest on most muni bonds is exempt from federal income tax and often from state income tax for in-state residents (the “double-exempt” or “triple-exempt” trades). Taxable munis (Build America Bonds 2009-2010, taxable advance-refunding deals post-2017 TCJA) form a smaller but growing segment.
• General Obligation (GO) bonds: backed by the full faith, credit, and taxing power of the issuer. Voter approval typically required.
• Revenue bonds: backed by specific project revenues (toll roads, water/sewer, airport, university dorms). Most issuance is revenue.
• Conduit bonds: issued by a public conduit issuer on behalf of a private borrower (hospital systems, charter schools, private universities). Borrower credit drives pricing.
• Insured bonds: bond insurance from MBIA, Ambac, Assured Guaranty, BAM (Build America Mutual). Insurance penetration collapsed from ~50% pre-2008 to under 10% post-2008 after the monoline insurer downgrades.
• VRDOs (Variable Rate Demand Obligations): floating-rate munis with put options to remarketing agents. SIFMA index is the standard reset benchmark.

Tax-equivalent yield: $y_{\text{taxable equiv}}=y_{\text{muni}}/(1-t)$ for marginal tax rate $t$. The crossover for a high-income California resident (combined federal+state ~50%) makes a 3% muni equivalent to a 6% taxable bond.

MSRB EMMA (Electronic Municipal Market Access) is the SEC-mandated muni disclosure repository. MSRB Rule G-14 trade reporting mirrors TRACE for munis since 2005.

2013 Detroit bankruptcy (Chapter 9, the largest US municipal bankruptcy by debt) and 2017 Puerto Rico PROMESA debt restructuring ($74B of debt across multiple instrumentalities) are the canonical modern muni distress cases. Puerto Rico’s Title III restructuring under PROMESA concluded in March 2022.

Pension underfunding is the dominant long-term muni credit risk — Illinois, New Jersey, Connecticut, Pennsylvania, and Kentucky have the deepest unfunded pension liabilities relative to revenue. Annual ADC (Actuarially Determined Contribution) shortfalls are the canonical credit signal.

21c. Commercial paper and short-term funding markets

Commercial paper (CP): short-term unsecured debt (1-270 days) issued by corporates, financial institutions, and ABCP conduits. Outstanding ~$1.3T (2024). Two segments:

• Financial CP (banks, finance companies): largest segment historically; Section 3(a)(3) exemption from securities registration for 270-day-or-shorter issuance with proceeds used for current transactions.
• Non-financial CP (industrial corporates): smaller but high-quality issuer base — Apple, Microsoft, Johnson & Johnson, Procter & Gamble, Toyota, Volkswagen, Coca-Cola.
• ABCP (Asset-Backed Commercial Paper): backed by trade receivables, credit cards, auto loans. Largest sub-market historically; outstanding collapsed from $1.2T peak (2007) to $300B (2024) after the GFC ABCP conduit run.

CP money-market funds: large institutional purchasers (Federated Hermes, Fidelity, Goldman Sachs Asset Management, JPMorgan Asset Management, BlackRock). Post-2014 SEC money-market reforms (floating NAV for institutional prime MMFs, gates and fees) substantially reduced institutional-prime MMF appetite, shifting CP issuance financing to government MMF channels.

March 2020 CP crisis: ABCP and corporate CP markets froze in mid-March 2020 as dealers withdrew. Fed launched the Commercial Paper Funding Facility (CPFF) on March 17, 2020 (revived from 2008-09) to backstop the market. Spreads to OIS widened from 5 bp to 100+ bp; PRIME MMF outflows triggered Fed AMLF revival. Market stabilized within weeks of CPFF launch.

21d. Bank loans and the leveraged loan market

Leveraged loans are senior secured floating-rate loans to sub-investment-grade corporate borrowers, typically syndicated by a lead-arranger bank and sold down to institutional investors (CLOs, mutual funds, separately-managed accounts). Outstanding US leveraged loans $1.4T (2024).

• Term Loan B (TLB): 5-7Y maturity, minimal amortization (1% per year typical), institutional-friendly. The dominant tranche by volume.
• Term Loan A (TLA): shorter maturity, fuller amortization, bank-held.
• Revolver: undrawn commitment held by banks.

Reference rate: SOFR + spread (Term SOFR most common post-2023). Spreads vary 200-700 bp depending on credit quality and capital structure.

LSTA (Loan Syndications and Trading Association) sets market conventions and standard documentation. Loan trading settles T+7 in the secondary market — substantially longer than bond settlement.

Covenant-lite loans (without maintenance financial covenants) dominate post-2014; covenant-lite share of US leveraged-loan issuance reached 95% (2024). Less protection for lenders; faster path to distressed exchanges.

Distressed exchange and “creditor-on-creditor violence”: 2020-2024 saw a wave of out-of-court restructurings using uptier exchanges (some lenders agreeing to subordinate other lenders in exchange for priority new-money or additional terms) — J.Crew, Serta Simmons, Boardriders, TriMark, Wesco, Robertshaw cases. Litigation through 2024-2025 has clarified contractual limits on these maneuvers.

22. Pitfalls — production lessons

• Day-count conventions: Actual/360 for USD MMM, 30/360 for USD bonds, Actual/Actual for Treasuries, Actual/365 for GBP, 30E/360 for European corporates. Wrong day-count is the single most common source of trade-booking errors.
• Settlement conventions: T+1 for US Treasuries (was T+1, briefly T+2 mid-1990s, back to T+1 in 2024 for non-equity products); T+2 for corporates (now T+1 from May 28, 2024); T+1 for MBS TBA; T+3 for many municipals historically.
• Curve interpolation choice matters: cubic spline vs monotone convex vs linear-in-rates produces 0.5-3 bp curve dispersion at intermediate tenors. Stick to one convention firm-wide.
• OIS-Libor basis residual: post-LIBOR, SOFR-Fed-Funds basis, OIS-Treasury basis, and cross-currency bases remain active risk factors. Treat them as separate curves, not residuals.
• Cheapest-to-deliver shifts in Treasury futures around auctions and Fed meetings create 1-5 tick CTD changes that mark wrong if not tracked daily.
• MBS prepayment-model uncertainty: switching from one vendor model to another can shift OAS by 5-15 bp; the model risk is real and should be reserved.

Further reading

• Frank J. Fabozzi, 2021, Bond Markets, Analysis, and Strategies, 10th edition.
• Bruce Tuckman and Angel Serrat, 2022, Fixed Income Securities: Tools for Today’s Markets, 4th edition.
• Damiano Brigo and Fabio Mercurio, 2006, Interest Rate Models — Theory and Practice, 2nd edition.
• Riccardo Rebonato, 2002, Modern Pricing of Interest-Rate Derivatives.
• Robert McDonald, 2013, Derivatives Markets, 3rd edition.
• John Hull, 2023, Options, Futures, and Other Derivatives, 11th edition.
• Andrew Davidson and Anthony Sanders, 2007, Securitization: Structuring and Investment Analysis.
• Frank Fabozzi (ed.), 2016, The Handbook of Mortgage-Backed Securities, 7th edition.
• Lakhbir Hayre (ed.), 2001, Salomon Smith Barney Guide to Mortgage-Backed and Asset-Backed Securities.
• Darrell Duffie and Kenneth Singleton, 2003, Credit Risk: Pricing, Measurement, and Management.
• David Lando, 2004, Credit Risk Modeling: Theory and Applications.

22b. Sovereign and central-bank actions affecting fixed-income markets

• Fed dot plot and SEP: Federal Reserve Summary of Economic Projections quarterly; the dot-plot of FOMC participants’ rate forecasts is closely watched. 2024-2025 saw rapid revisions as inflation persistence surprised relative to mid-2023 expectations.
• Fed QT (Quantitative Tightening): balance-sheet runoff at $60B/month Treasuries plus $35B/month MBS through mid-2024; slowed to $25B/month Treasuries in June 2024. Affects Treasury supply-demand and MBS spreads materially.
• ECB APP / PEPP unwinding: Asset Purchase Program reinvestment ended July 2023; PEPP reinvestment ended December 2024. Driving sovereign-spread normalization between Germany and Italy.
• BoJ YCC exit: Yield Curve Control on 10Y JGBs was abandoned in March 2024 after a decade. JGB yields moved significantly higher; cross-currency basis with USD and EUR shifted materially.
• BoE QT: Bank of England gilt sales (active sales since 2022) total over $100B through 2024.
• PBoC and BoJ FX market interventions: PBoC daily USD/CNY fix manages yuan; BoJ intervened materially in 2022 and 2024 to defend yen.

23. Bond portfolio attribution and total-return calculation

Fixed-income portfolio attribution decomposes total return into:

• Yield: coupon plus pull-to-par effect. For a bond bought at $95 with 5% coupon and 1Y to maturity, yield contribution is approximately 5% + 5% = 10% (coupon + accretion).
• Roll-down: as time passes, the bond moves down the yield curve. For an upward-sloping curve, roll-down contributes positive return.
• Duration return: $-D_{\text{mod}}\times \Delta y$ from parallel curve shift.
• Curve return: from non-parallel curve shifts (steepening, flattening, twist), captured via key-rate durations.
• Spread return: change in option-adjusted spread (for credit) or OAS (for MBS).
• Currency return: for foreign-denominated bonds.

Campisi attribution (1999, Journal of Performance Measurement) is the canonical fixed-income attribution framework, decomposing into yield, treasury return, spread return, and selection.

Sector rotation: investment-grade vs high-yield, financials vs industrials, US vs EUR vs EM. Bridge between top-down macro positioning and bottom-up security selection.

Yield-curve trades: 2s5s steepener (long 5Y, short 2Y duration-neutral); butterfly (long wings, short belly); barbell vs bullet duration matching.

24. CDS settlement and ISDA standard model

When a credit event occurs (failure to pay, bankruptcy, restructuring under the relevant ISDA definitions), CDS contracts settle either physically (deliverable bond against par) or cash (auction price determined by ISDA-coordinated auction).

ISDA auction (since 2009 Big Bang): two-stage Dutch auction determines the cash settlement price. Held within 30 days of credit event. Recent significant auctions: Russia sovereign (June 2022, 56% recovery), SoftBank-rumored credit events (no auction), Credit Suisse AT1 (no CDS auction because AT1s were structurally subordinated), Yes Bank (2020 partial), Hertz (2020 auction).

Cheapest-to-deliver option: the protection buyer (in physical settlement) chooses among the deliverable obligations, naturally selecting the cheapest. This CTD option is part of the CDS basis vs the underlying bond pricing.

Succession: when reference entities reorganize (spin-offs, mergers), ISDA’s Determinations Committee determines the succession event and how the CDS reference obligation succeeds.

ISDA Determinations Committees: regional DCs (Americas, EMEA, APAC) make binding credit-event determinations. Cases like Argentina 2014 (partial-default debate), Greece 2012 (PSI restructuring), and Venezuela 2017 (sovereign distress) generated extensive DC deliberation.

Adjacent

• stochastic-calculus
• monetary-economics-and-banking
• macroeconomics-foundations
• securities-regulation-deep
• real-estate-finance
• insurance-and-actuarial
• accounting-foundations
• cuda-triton-gpu-programming