Sources and References

Primary textbook — Tuckman, B., & Serrat, A. (2022). Fixed Income Securities: Tools for Today’s Markets, 4th ed. Wiley. Supplementary — CFA Institute (2023). Fixed Income Analysis, 5th ed. Wiley. Fabozzi, F. J. (2012). Fixed Income Mathematics: Analytical and Statistical Techniques, 4th ed. McGraw-Hill. Online resources — SIFMA (sifma.org); Bank for International Settlements (bis.org); Federal Reserve Economic Data (FRED); CME Group learning resources; ISDA (isda.org); CFA Institute fixed income learning materials.

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Chapter 1: Overview of Fixed Income Markets

1.1 What Are Fixed Income Securities?

Fixed income securities are financial instruments that obligate the issuer to make specified cash payments to the holder on defined dates. The most familiar form is the coupon-paying bond, where the issuer promises periodic coupon payments and repayment of principal (face value) at maturity. The “fixed income” label, while traditional, is somewhat misleading — the universe includes floating-rate notes, inflation-linked bonds, mortgage-backed securities, and complex derivatives — all of which may have highly variable cash flows.

The global fixed income market is enormous: as of the early 2020s, the total outstanding amount of debt securities globally exceeded $130 trillion USD, dwarfing the global equity market capitalization. Issuers include national governments, supranational organizations (World Bank, IMF), municipalities, financial institutions, and corporations.

1.2 Market Segments and Participants

Fixed income markets are organized into several segments:

Segment	Issuers	Instruments
Government / Sovereign	Federal governments	Treasury bonds, bills, notes; STRIPS
Agency	GSEs (Fannie Mae, Freddie Mac), Crown corps	Agency bonds, MBS
Municipal	State, provincial, local governments	GO bonds, revenue bonds
Corporate Investment-Grade	BBB-/Baa3 and above rated companies	Bonds, medium-term notes
Corporate High Yield	Below investment-grade companies	High-yield (“junk”) bonds
Structured Products	SPVs backed by asset pools	MBS, ABS, CLOs, CDOs
Money Market	Governments, financial institutions	T-bills, commercial paper, repos

Key market participants include central banks (monetary policy), commercial banks (liquidity management, proprietary trading), insurance companies and pension funds (liability matching, long-duration buyers), mutual funds and ETFs (retail and institutional intermediation), and hedge funds (relative value, macro, and credit strategies).

1.3 Primary and Secondary Markets

Fixed income securities are issued in the primary market through:

• Public offerings: Registered with regulators, available to all investors. Governments use auction mechanisms (single-price or multiple-price auctions for Treasury securities).
• Private placements (Rule 144A in the U.S.): Sold to qualified institutional buyers, with less disclosure.
• Medium-Term Note (MTN) programs: Allow issuers to sell notes continuously off a shelf registration, customizing maturities and structures.

The secondary market trades outstanding securities. Most fixed income trading occurs over-the-counter (OTC) — directly between dealers and institutional clients — rather than on organized exchanges. Market makers (dealers) quote bid-ask spreads and stand ready to buy or sell. Electronic platforms (MarketAxess, Tradeweb) have increased transparency in corporate bond markets, though they remain less transparent than equity exchanges.

1.4 Bond Conventions and Terminology

Day count conventions govern how accrued interest is calculated: Actual/Actual is standard for U.S. Treasuries; 30/360 is used for corporate and municipal bonds; Actual/360 is used for money market instruments.

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Chapter 2: Bond Valuation

2.1 Present Value of Cash Flows

The fundamental bond pricing equation states that the price of a bond equals the present value of all future cash flows, discounted at the appropriate yield:

\[ P = \sum_{t=1}^{n} \frac{C}{(1+y)^t} + \frac{F}{(1+y)^n} \]

where \( P \) is the bond price, \( C \) is the periodic coupon payment, \( F \) is the face value, \( n \) is the number of periods to maturity, and \( y \) is the periodic yield. For a bond with semiannual coupons and a stated annual yield \( Y \):

\[ P = \sum_{t=1}^{2T} \frac{C/2}{\left(1 + Y/2\right)^t} + \frac{F}{\left(1 + Y/2\right)^{2T}} \]

where \( T \) is the number of years to maturity.

Using the annuity formula, the coupon component can be simplified:

\[ P = \frac{C}{y} \left[ 1 - \frac{1}{(1+y)^n} \right] + \frac{F}{(1+y)^n} \]

2.2 Price-Yield Relationship

The price-yield relationship is inverse and convex — as yields rise, prices fall, and the fall is smaller than the corresponding rise when yields fall by the same amount (this asymmetry is convexity, discussed in Chapter 4). Three key pricing relationships:

• If coupon rate = yield: bond prices at par (P = F)
• If coupon rate > yield: bond prices at a premium (P > F)
• If coupon rate < yield: bond prices at a discount (P < F)

As a bond approaches maturity, its price converges to face value regardless of the coupon rate — the pull to par effect.

2.3 Yield Measures

Current Yield: Annual coupon / market price. A simple but incomplete yield measure that ignores the time value of money and capital gain/loss from price-to-par convergence.

\[ \text{Current Yield} = \frac{\text{Annual Coupon}}{\text{Market Price}} \]

Yield to Maturity (YTM): The discount rate that equates the present value of all future cash flows to the current price. YTM is the IRR of the bond investment. It assumes reinvestment of coupons at the YTM rate — an unrealistic assumption that makes YTM an imprecise measure of realized return.

Yield to Call (YTC): For callable bonds, the yield calculated assuming the bond is called at the first call date. Relevant when yields have declined enough that the call is likely.

Yield to Worst (YTW): The minimum of the YTM and all YTC values. Relevant for bonds with embedded call options.

Spread Measures: Fixed income yields are often quoted as spreads over a benchmark:

• G-spread: Spread over the interpolated government yield at the same maturity.
• I-spread: Spread over the interpolated swap rate.
• Z-spread (Zero-Volatility Spread): The constant spread added to each spot rate on the Treasury curve such that the present value of cash flows equals the market price. The Z-spread removes the distortion of assuming a flat yield curve.
• OAS (Option-Adjusted Spread): The Z-spread adjusted for the value of embedded options. For a callable bond, OAS = Z-spread − Call Option Value. OAS represents the spread attributable solely to credit and liquidity risk, not optionality.

2.4 Repos (Repurchase Agreements)

A repurchase agreement (repo) is a short-term collateralized borrowing where one party sells securities and simultaneously agrees to repurchase them at a higher price on a future date. The implicit interest rate is the repo rate. Repos are fundamental to fixed income market functioning:

• Dealers use repos to finance their bond inventories cheaply.
• Investors use reverse repos to earn interest on excess cash with high-quality collateral.
• Short sellers borrow specific securities (special repos) to deliver against short sales.

The term structure of repo rates reflects credit quality of the collateral, term, and whether the security is “on special” (in high demand for delivery). General collateral (GC) repo uses any eligible security; special repo uses specific named securities (often newly issued on-the-run Treasuries).

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Chapter 3: The Yield Curve

3.1 The Yield Curve and Its Shapes

The yield curve (or term structure of interest rates) plots yields to maturity against maturity for bonds of similar credit quality. The government (sovereign) yield curve is the benchmark reference. Four principal shapes are observed:

• Normal (Upward Sloping): Long-term yields exceed short-term yields. Most common historically — reflects expectations of future growth and inflation, and a liquidity/term premium.
• Inverted (Downward Sloping): Short-term yields exceed long-term yields. Often precedes economic recessions (historically a reliable leading indicator).
• Flat: Yields are approximately equal across maturities.
• Humped: Intermediate maturities yield more than both short and long maturities.

3.2 Theories of the Term Structure

Several competing theories explain the shape of the yield curve:

3.3 Spot Rates and Forward Rates

Spot rates are the yields on zero-coupon bonds (or equivalent strips) for each maturity. The spot rate curve (zero-coupon curve) is derived from the par yield curve by bootstrapping:

Using the 1-year par rate \( r_1 \) directly as the 1-year spot rate, then solving for the 2-year spot rate \( s_2 \) from the 2-year par bond pricing equation:

\[ P_2 = \frac{c/2}{1 + s_1/2} + \frac{c/2 + 100}{(1 + s_2/2)^2} = 100 \]

Bootstrapping proceeds iteratively, using each previously computed spot rate to extract the next.

Forward rates are implied rates for future periods embedded in the current spot rate structure. The \( n \times m \) forward rate \( f(n,m) \) is the rate implied for an investment from period \( n \) to period \( n+m \):

\[ \left(1 + s_{n+m}\right)^{n+m} = \left(1 + s_n\right)^n \times \left(1 + f(n,m)\right)^m \]

The relationship between spot rates and forward rates reflects the no-arbitrage condition: investing for \( n+m \) periods at the spot rate must equal rolling over investments at successive forward rates (under the pure expectations hypothesis).

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Chapter 4: Interest Rate Risk — Duration and Convexity

4.1 Dollar Duration and Modified Duration

Duration is a measure of a bond’s price sensitivity to changes in interest rates. Modified Duration (MD) estimates the percentage change in price for a given change in yield:

\[ \Delta P \approx -MD \times \Delta y \times P \]\[ MD = \frac{1}{P} \sum_{t=1}^{n} t \cdot \frac{C_t}{(1+y)^t} \times \frac{1}{1+y} \]

or equivalently:

\[ MD = \frac{D_{Mac}}{1+y} \]

where \( D_{Mac} \) is Macaulay Duration — the weighted average time to receipt of cash flows, with weights equal to the present value fraction of each cash flow.

Dollar Duration (DV01 / PVBP): The dollar change in price for a 1 basis point (0.01%) change in yield:

\[ DV01 = -\frac{MD \times P}{10000} \]

DV01 (also called Price Value of a Basis Point or PVBP) is the practitioner’s preferred risk measure for hedging, as it directly translates yield movements into dollar P&L impact.

4.2 Duration Properties

Key properties of duration:

1. Duration of a zero-coupon bond = Maturity.
2. Duration of a coupon bond < Maturity.
3. Duration increases with maturity (for fixed coupon rate) but at a decreasing rate.
4. Duration decreases as coupon rate increases (higher coupons bring forward more cash flow weight).
5. Duration decreases as yield increases (higher discount rate reduces the relative weight of distant cash flows).

For a portfolio of bonds, the portfolio duration is the market-value-weighted average of the individual bond durations:

\[ D_{portfolio} = \sum_i w_i D_i \]

where \( w_i = V_i / V_{total} \) is the portfolio weight of bond \( i \).

4.3 Convexity

The duration approximation is linear, but the true price-yield relationship is curved (convex). Convexity captures the curvature — the second-order effect of yield changes on price:

\[ \Delta P \approx -MD \times \Delta y \times P + \frac{1}{2} \times Convexity \times (\Delta y)^2 \times P \]\[ Convexity = \frac{1}{P(1+y)^2} \sum_{t=1}^{n} \frac{t(t+1) \cdot C_t}{(1+y)^t} \]

Convexity is always positive for plain vanilla bonds: this means that the actual price decline from a yield increase is less than the duration-based estimate, and the actual price rise from a yield decrease is more than the duration-based estimate. All else equal, more convexity is desirable.

4.4 Portfolio Duration Targeting and Immunization

Immunization is the strategy of constructing a bond portfolio so that its present value is protected against changes in interest rates. For a liability with a fixed future payment, classical immunization requires:

1. Duration matching: Set portfolio duration equal to the duration of the liability.
2. Funding: Ensure portfolio value ≥ present value of the liability.
3. Convexity condition: Portfolio convexity ≥ liability convexity ensures a surplus under any parallel shift in rates.

Cash flow matching (dedication): A more conservative approach — match each liability cash flow exactly with an asset cash flow, eliminating reinvestment risk entirely but typically at higher cost.

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Chapter 5: Interest Rate Models

5.1 The Need for Interest Rate Models

Duration and convexity are adequate risk measures for small, parallel shifts in the yield curve. However, they do not:

• Account for non-parallel curve moves (twists, butterflies)
• Price bonds with embedded options (callable bonds, putable bonds, MBS prepayments)
• Value interest rate derivatives (caps, floors, swaptions)

Interest rate models provide a dynamic, probabilistic description of how yields evolve over time, enabling the valuation of these more complex instruments.

5.2 One-Factor Models

One-factor models assume that a single source of uncertainty (typically the short-term interest rate \( r \)) drives the entire term structure. Popular one-factor models:

\[ dr = \kappa(\theta - r) dt + \sigma dW \]

This is an Ornstein-Uhlenbeck process with mean-reversion. \( \kappa \) governs mean-reversion speed, \( \theta \) is the long-run mean rate, and \( \sigma dW \) is the diffusion (random shock). The Vasicek model produces closed-form bond prices and is analytically tractable, but allows negative interest rates.

\[ dr = \kappa(\theta - r) dt + \sigma\sqrt{r} \, dW \]

The \( \sqrt{r} \) diffusion term ensures that rates cannot become negative (if parameters satisfy the Feller condition \( 2\kappa\theta > \sigma^2 \)). CIR also has closed-form bond prices.

Ho-Lee Model (1986): A no-arbitrage model that fits the initial term structure exactly by allowing the drift to be a time-dependent function calibrated to market prices. More flexible than equilibrium models but less parsimonious.

Hull-White Model (1990): An extension of Vasicek that incorporates a time-dependent drift to match the initial yield curve, and optionally a time-dependent volatility function. The Hull-White model is widely used in practice for calibration to observable market prices of caps and swaptions.

5.3 Binomial Tree Models for Bond Pricing

Binomial interest rate trees discretize the evolution of the short rate into up and down moves over successive time steps. The tree is calibrated to market prices of observable instruments (typically on-the-run Treasuries), then used to price options-embedded securities.

At each node, the bond’s value is computed by backward induction:

\[ V(t,j) = \frac{1}{2} \left[ V(t+1, j+1) + V(t+1, j) \right] \times \frac{1}{1 + r(t,j)} + C(t+1) \]

For a callable bond, at each node the issuer will call if the model price exceeds the call price; thus the effective value at each node is the minimum of the hold value and the call price:

\[ V_{\text{callable}}(t,j) = \min\left( V_{\text{non-callable}}(t,j), \, \text{Call Price} \right) \]

The OAS is then found by adding a constant spread to all short rates in the calibrated tree until the model price equals the market price.

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Chapter 6: Bonds with Embedded Options

6.1 Callable Bonds

A callable bond grants the issuer the right to redeem the bond before maturity at specified call prices on or after specified call dates. Issuers exercise calls when interest rates have fallen sufficiently below the coupon rate that refinancing is advantageous.

\[ \text{Price of Callable Bond} = \text{Price of Non-Callable Bond} - \text{Value of Call Option} \]

Because the call option is held by the issuer (and works against the bondholder), the callable bond must offer a higher yield (lower price) than an otherwise identical non-callable bond. The call option value increases with:

• Higher yield volatility (more likely for rates to move far below the coupon)
• Lower current yields (call is closer to being in-the-money)
• Longer call protection period (shorter effective call option term = lower value)

Negative convexity: Callable bonds exhibit negative convexity when rates fall (the option delta approaches 1 and the bond’s price appreciation is capped near the call price). This is also called price compression. The yield spread between callable and non-callable bonds is the compensation investors demand for bearing this negative convexity.

6.2 Putable Bonds

A putable bond grants the holder the right to sell the bond back to the issuer at par on specified dates. Putable bonds are more valuable to investors (and thus lower-yielding) than non-callable bonds:

\[ \text{Price of Putable Bond} = \text{Price of Non-Putable Bond} + \text{Value of Put Option} \]

Putable bonds are useful for investors concerned about rising rates: the put provides a floor on price. Issuers accept the higher cost (lower yield) in exchange for the ability to issue longer-duration debt to investors who would otherwise not accept long-term exposure.

6.3 Convertible Bonds

A convertible bond allows the holder to convert the bond into shares of the issuer’s common stock at a specified conversion ratio. The convertible’s value has two components:

• Bond floor (Investment value): The value of the bond ignoring the conversion option — the present value of cash flows discounted at the yield of an equivalent non-convertible bond.
• Conversion value: The current equity price × conversion ratio.

The convertible typically trades at a conversion premium over both components. As equity prices rise, the convertible trades more like equity; as equity falls, it trades more like debt (the bond floor provides downside protection).

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Chapter 7: SOFR Futures, Swaps, and Other Fixed Income Securities

7.1 The LIBOR to SOFR Transition

The London Interbank Offered Rate (LIBOR) — a benchmark rate at which large banks stated they could borrow from each other — was the world’s most widely referenced interest rate benchmark for decades. Manipulation scandals and the thin actual transaction base led to its phase-out, with most LIBOR tenors discontinued after June 2023.

SOFR (Secured Overnight Financing Rate) is the U.S. dollar reference rate that replaced USD LIBOR. SOFR is derived from actual transactions in the U.S. Treasury repo market — a deep, highly liquid market with approximately $1 trillion in daily volume. Unlike LIBOR, SOFR is a secured rate (collateralized by Treasuries) and is an overnight rate. Term SOFR rates (1-month, 3-month, 6-month) are published by CME Group and are used for loan documentation.

7.2 Interest Rate Swaps

An interest rate swap is an agreement between two counterparties to exchange cash flows based on a notional principal amount. In a standard (vanilla) fixed-for-floating swap:

• The fixed-rate payer pays a fixed rate and receives floating rate payments.
• The fixed-rate receiver receives fixed and pays floating.

The floating rate is reset periodically based on a reference rate (SOFR, or historically LIBOR). No principal is exchanged — only net interest payments.

Swap rate: The fixed rate that makes the swap’s present value equal to zero at inception (fair value). The swap rate is equivalent to the par rate on a bond with the same payment schedule — connecting the swap market to the bond market.

\[ \text{Swap Rate} = \frac{1 - d_n}{\sum_{t=1}^{n} d_t \cdot \tau_t} \]

where \( d_t \) is the discount factor for period \( t \) and \( \tau_t \) is the length of the period.

Applications of interest rate swaps include:

• Duration management: A fixed-income portfolio manager can increase (decrease) duration by entering a receive-fixed (pay-fixed) swap, without buying or selling bonds.
• Liability management: A corporation with floating-rate debt can synthetically convert it to fixed using a pay-fixed, receive-floating swap.
• Speculative/macro positioning: Hedge funds use swaps to express views on the level and shape of the yield curve.

7.3 SOFR Futures

CME Group lists futures on SOFR in two primary formats:

1-Month SOFR Futures: Cash-settled contracts reflecting the simple average of SOFR published over the contract month. Useful for hedging monthly loan resets.

3-Month SOFR Futures: Cash-settled at 100 minus the average of daily SOFR over the contract quarter. The price of a 3-month SOFR future implies the expected average overnight SOFR rate over the future delivery quarter. A strip of consecutive 3-month SOFR futures can be used to construct an implied forward curve or to hedge floating-rate exposures across multiple periods.

7.4 Floating-Rate Notes (FRNs)

Floating-rate notes pay coupons that reset periodically based on a reference rate (SOFR + spread). Because the coupon adjusts with market rates, the FRN’s price stays close to par between reset dates. Duration of an FRN is approximately equal to the time to the next coupon reset — typically very short, making FRNs attractive to investors seeking low interest rate risk with credit-risk exposure to the issuer.

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Chapter 8: Mortgage-Backed Securities

8.1 The Securitization Process

Mortgage-backed securities (MBS) are created by pooling individual mortgage loans, then issuing securities backed by the pool’s cash flows. The securitization process:

1. A mortgage originator (bank, mortgage company) originates loans and sells them to a Special Purpose Vehicle (SPV).
2. The SPV issues MBS backed by the mortgage pool.
3. Investors receive pro-rata shares of mortgage payments (principal + interest) passed through by a servicer.

In the U.S., agency MBS issued by Fannie Mae, Freddie Mac, or Ginnie Mae carry an explicit or implicit government guarantee of timely payment of principal and interest, eliminating credit risk. The dominant risk in agency MBS is prepayment risk.

8.2 Prepayment Risk

Homeowners have an implicit option to prepay their mortgages at any time (refinancing when rates fall, or moving and selling the property). This prepayment option creates significant complexity in MBS valuation:

• When rates fall, prepayment speeds accelerate (refinancing). Investors receive their principal back when reinvestment rates are low — call risk (similar to callable bonds).
• When rates rise, prepayment speeds slow as refinancing becomes unattractive. Investors are stuck receiving below-market coupons — extension risk.

This dual exposure is sometimes called negative convexity: MBS tend to extend in duration when rates rise and shorten when rates fall — the opposite of what investors typically want.

Prepayment speed conventions:

• Single Monthly Mortality (SMM): The fraction of the outstanding balance prepaid in a single month.
• Conditional Prepayment Rate (CPR): The annualized prepayment rate. CPR = 1 − (1 − SMM)^12.
• PSA (Public Securities Association) Standard: A benchmark prepayment model where 100% PSA assumes CPR starting at 0.2% in month 1, increasing by 0.2% per month to 6% CPR by month 30, then remaining constant. Pools prepaying faster than this benchmark are quoted as “150 PSA,” etc.

8.3 CMOs and Structured MBS

Collateralized Mortgage Obligations (CMOs) restructure the cash flows from an agency MBS pool into multiple tranches with different maturities and risk profiles:

• Sequential-pay CMOs: Tranches are paid down in order. Short-duration tranches receive all principal until paid off before longer tranches begin receiving principal.
• PAC (Planned Amortization Class) bonds: A PAC schedule specifies a fixed principal payment schedule over a range of prepayment speeds. Support (companion) tranches absorb excess prepayments or shortfalls, providing the PAC with more stable cash flows.
• IO (Interest Only) and PO (Principal Only) strips: IOs receive only the interest portion of mortgage payments; POs receive only principal. IOs gain value when rates rise (prepayments slow, preserving the interest income stream); POs gain value when rates fall (prepayments accelerate, returning principal sooner). These strips are used for hedging.

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Chapter 9: Credit Risk

9.1 Sources and Dimensions of Credit Risk

Credit risk is the risk that a bond issuer fails to make promised payments in full and on time. It encompasses:

• Default risk: The probability that the issuer defaults.
• Recovery risk: The uncertainty about how much investors recover after a default (the recovery rate, \( 1 - LGD \), where LGD = Loss Given Default).
• Spread risk: The risk that credit spreads widen even without a default, reducing the mark-to-market value of the bond.
• Downgrade risk: The risk of a credit rating downgrade, which increases required yields and causes price declines.

Expected Loss (EL) from credit risk:

\[ EL = PD \times LGD \times EAD \]

where PD = probability of default, LGD = loss given default (1 − recovery rate), and EAD = exposure at default.

9.2 Credit Ratings

Rating agencies (Moody’s, S&P, Fitch) assign credit ratings that summarize their assessment of an issuer’s creditworthiness. The investment-grade / high-yield boundary (Baa3/BBB-) is particularly significant because many institutional investors are restricted to investment-grade bonds by mandate. A downgrade below investment grade (“fallen angel”) causes selling pressure from constrained investors, often exacerbating price declines.

S&P / Fitch	Moody’s	Category
AAA	Aaa	Highest quality
AA	Aa	High quality
A	A	Upper medium grade
BBB	Baa	Lower medium grade
BB	Ba	Speculative
B	B	Highly speculative
CCC/CC/C	Caa/Ca/C	In or near default
D	C	Default

9.3 Credit Spreads and Credit Default Swaps

The credit spread is the additional yield investors demand above the risk-free rate to compensate for credit risk (and liquidity risk, which is often bundled into the spread). Credit spreads are observed in corporate bond markets and trade dynamically based on macroeconomic conditions, sector fundamentals, and individual issuer news.

Credit Default Swaps (CDS) are derivative contracts that transfer credit risk. In a single-name CDS:

• The protection buyer pays a periodic fee (the CDS spread or premium) to the protection seller.
• If a credit event (default, bankruptcy, restructuring) occurs, the protection seller compensates the buyer for the loss (par minus recovery value on the reference entity’s bonds).

CDS allow investors to isolate and trade credit risk independently of interest rate risk. Key applications:

• Hedging: A bondholder buys CDS protection to eliminate credit risk while retaining interest rate exposure.
• Speculation: An investor sells CDS protection to earn income, effectively taking a synthetic long credit position.
• Arbitrage: Traders exploit price discrepancies between CDS spreads and bond spreads (“basis trading”).

The CDS-Bond basis = CDS spread − bond spread. Theoretically zero in a perfect market; in practice, it fluctuates due to funding costs, counterparty risk, and contract specification differences.

9.4 CDOs — Collateralized Debt Obligations

Collateralized Debt Obligations (CDOs) are structured credit products that pool a diversified portfolio of debt instruments (corporate bonds, bank loans, or even other structured products) and issue tranched securities against the pool, much as CMOs do for mortgages.

The tranching creates credit enhancement for senior tranches: losses are allocated bottom-up (equity/first-loss tranche absorbs losses first; mezzanine tranches absorb losses next; senior AAA tranches absorb losses only if the lower tranches are wiped out). This credit enhancement allows senior CDO tranches to achieve AAA ratings despite the underlying pool containing sub-investment-grade assets.

CDOs played a central role in the 2007–2008 global financial crisis. Mortgage-backed CDOs (particularly those backed by subprime MBS) were assigned AAA ratings based on models that underestimated the correlation of defaults across mortgages in different regions. When home prices declined nationally — a scenario the models assigned near-zero probability — the credit enhancement proved insufficient, and senior tranches suffered severe losses, amplifying the financial crisis. This episode underscored the importance of understanding model assumptions, particularly correlation assumptions in structured credit.