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.

---

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:

Convention	Used For	Rule
Actual/Actual (ICMA)	U.S. Treasuries, sovereign bonds	Actual days in period / actual days in coupon year
30/360	U.S. corporates, munis	Assume each month = 30 days, year = 360 days
Actual/360	Money market instruments, bank loans	Actual days elapsed / 360
Actual/365	Some government bonds, UK gilts	Actual days elapsed / 365

1.5 Bond Covenants and Embedded Options

Bond covenants are contractual restrictions or requirements embedded in the indenture:

• Affirmative covenants: Actions the issuer must take — maintain insurance, provide audited financial statements, preserve corporate existence.
• Negative covenants: Actions the issuer must not take without bondholder consent — issue additional debt above certain limits, pay dividends above thresholds, sell assets above certain amounts.

Covenants protect bondholders by reducing agency conflicts between shareholders and creditors. Violation of a covenant is a technical default, which may trigger acceleration of principal repayment even if no cash default has occurred.

Common embedded options include:

• Call provision: Issuer may redeem at par or a call premium before maturity (benefits issuer; disadvantages bondholder).
• Put provision: Holder may sell back to issuer at par on specified dates (benefits holder; disadvantages issuer).
• Conversion option: Holder may exchange bond for equity (benefits holder).
• Sinking fund: Issuer retires a portion of the outstanding bond each year — reduces refinancing risk for bondholder but introduces reinvestment risk.
• Make-whole call: Issuer redeems at the present value of remaining cash flows discounted at a benchmark spread (typically Treasury + 25–50 bps), which is usually above the call price — effectively prevents early call.

---

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).

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 at the specified call price.

Yield to Worst (YTW): The minimum of the YTM and all YTC values. Investors should evaluate YTW as the most conservative yield measure for callable bonds.

Bond Equivalent Yield (BEY): Converts yields of different payment frequencies to a semiannual basis for comparison. For a bond with a periodic yield of \( y \):

\[ BEY = 2 \times \left[ (1 + y_{\text{periodic}})^{1/2} - 1 \right] \text{ if converting from annual} \]

For money market instruments quoted on a discount basis (like U.S. Treasury bills):

\[ \text{Discount Yield} = \frac{F - P}{F} \times \frac{360}{t} \]\[ \text{BEY} = \frac{F - P}{P} \times \frac{365}{t} \]

2.4 Spread Measures

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

2.5 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.

\[ \text{Repo Interest} = \text{Purchase Price} \times \text{Repo Rate} \times \frac{\text{Days}}{360} \]

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.

General collateral (GC) repo uses any eligible Treasury security; special repo uses specific named securities (typically on-the-run Treasuries in high demand for delivery). Special repo rates are lower than GC rates because the borrower of the specific security derives additional value from receiving that particular bond.

The repo haircut (initial margin) is the percentage by which the collateral’s market value exceeds the cash lent, protecting the lender against price decline in the collateral:

\[ \text{Cash Lent} = \text{Collateral Market Value} \times (1 - \text{Haircut}) \]

---

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). The 2–10 year U.S. Treasury spread inverted before every recession since the 1970s.
• Flat: Yields are approximately equal across maturities. Often occurs during transitions between normal and inverted curves.
• Humped: Intermediate maturities yield more than both short and long maturities. Rare; can occur during complex monetary policy transitions.

3.2 Theories of the Term Structure

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.

Forward rates are implied rates for future periods embedded in the current spot rate structure.

3.4 Arbitrage-Free Valuation

The concept of arbitrage-free valuation extends the spot rate framework: any bond’s fair value should equal the sum of its cash flows discounted at the appropriate spot rates, not at a single flat yield. If a bond’s market price differs from its no-arbitrage value, dealers can profit by stripping (decomposing into zero-coupon bonds) or reconstituting (reassembling strips into a coupon bond).

The STRIPS (Separate Trading of Registered Interest and Principal of Securities) program in the U.S. Treasury market allows investors to strip Treasury coupon bonds into individual zero-coupon securities. This creates an observable spot rate curve from Treasury STRIPS prices, which is used as the foundation for arbitrage-free bond pricing models.

---

Chapter 4: Interest Rate Risk — Duration and Convexity

4.1 Macaulay Duration

4.2 Modified Duration and Dollar Duration

DV01 is the practitioner’s preferred risk measure for hedging because it directly translates yield movements into dollar P&L impact, making it easy to size hedges and communicate risk.

4.3 Duration Properties and Portfolio Duration

Key properties of duration:

1. Duration of a zero-coupon bond = Maturity.
2. Duration of a coupon bond < Maturity. The higher the coupon, the shorter the duration (more weight on early cash flows).
3. Duration generally increases with maturity (for fixed coupon rate) but at a decreasing rate and can decline for very long maturities at very low yields.
4. Duration decreases as coupon rate increases.
5. Duration decreases as yield increases (higher discount rate reduces the relative weight of distant cash flows).
6. Duration decreases as the bond approaches maturity (pull to par).

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

\[ D_{portfolio} = \sum_{i} w_i D_i, \quad w_i = \frac{V_i}{V_{total}} \]

The portfolio DV01 is the sum of individual DV01s, which makes it directly addable across positions — a crucial property for risk management.

4.4 Effective Duration

For bonds with embedded options, cash flows change as yields change (e.g., a callable bond is less likely to pay the distant coupon cash flows if it gets called when rates fall). Modified duration — which holds cash flows constant — is inappropriate.

Effective duration for a callable bond will be shorter than that of an otherwise identical non-callable bond, particularly when rates are low and the call option is near the money — the price appreciation is capped near the call price.

4.5 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.

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

Convexity is always positive for plain vanilla bonds — this means 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.

4.6 Key Rate Duration and Curve Risk

Key rate duration (also called partial duration) measures price sensitivity to changes in specific maturities on the yield curve, holding all other maturities constant. It captures the bond’s exposure to non-parallel shifts (twists, butterflies).

A barbell portfolio (concentrated at short and long maturities) and a bullet portfolio (concentrated at one intermediate maturity) can be structured to have identical portfolio duration, but very different key rate duration profiles. The barbell has large key rate durations at the short and long ends; the bullet has a large key rate duration at the intermediate point. Their P&L will differ whenever the curve twists rather than shifting in parallel.

---

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 Equilibrium Models

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

5.3 No-Arbitrage Models

Equilibrium models may not fit the currently observed yield curve exactly. No-arbitrage models incorporate a time-dependent drift, calibrated so the model perfectly fits the initial term structure.

5.4 Binomial Interest Rate Trees

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 (on-the-run Treasuries), then used to price options-embedded securities via backward induction.

At each node, the bond value is:

\[ V(t,j) = \frac{q_u \cdot V(t+1, j+1) + q_d \cdot V(t+1, j)}{1 + r(t,j)} + C(t+1) \]

where \(q_u\) and \(q_d\) are the risk-neutral probabilities (typically 0.5 each in a symmetric tree), \(r(t,j)\) is the short rate at node \((t,j)\), and \(C(t+1)\) is the coupon payment.

For a callable bond, at each node the issuer will call if the continuation value exceeds the call price:

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

The OAS is found by adding a constant spread to all short rates in the calibrated tree until the model price equals the market price. The OAS can then be compared across different callable bonds to identify relative value.

---

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 rates have fallen below the coupon rate enough that refinancing is advantageous.

\[ \text{Price}_{\text{callable}} = \text{Price}_{\text{non-callable}} - \text{Value of Call Option} \]

The call option value increases with:

• Higher yield volatility (greater probability of rates falling far below coupon)
• Lower current yields (call is closer to being in-the-money)
• More time remaining to the first call date (more time for rates to fall into the money)
• Higher coupon rate relative to current market yields

Negative convexity and price compression: When rates fall, the callable bond’s price increase slows as it approaches the call price. This creates negative convexity in the region where the call is in or near the money:

Rate Environment	Callable Bond Behavior
Rates rise sharply	Duration extends (like non-callable); large price decline
Rates fall sharply	Duration compresses; price appreciation capped near call price
Rates stable	Earns coupon plus OAS return

Call protection period: The period during which the bond cannot be called. Longer call protection benefits the holder and reduces the option value to the issuer, so callable bonds with longer call protection typically have lower yields than those with shorter protection.

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 (lower-yielding):

\[ \text{Price}_{\text{putable}} = \text{Price}_{\text{non-putable}} + \text{Value of Put Option} \]

Putable bonds are particularly useful to investors concerned about rising rates — the put provides a price floor. They also protect against credit deterioration of the issuer (the holder can put the bond back if the issuer’s credit deteriorates). Issuers accept the lower yield in exchange for access to investors who would not otherwise purchase long-term bonds without downside protection.

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 (shares per $1,000 face).

\[ \text{Conversion Value} = \text{Conversion Ratio} \times \text{Current Stock Price} \]\[ \text{Conversion Premium} = \frac{\text{Market Price of Convertible} - \text{Conversion Value}}{\text{Conversion Value}} \]

The convertible’s value has two components:

• Bond floor (investment value): The PV of cash flows discounted at the yield of a comparable non-convertible bond — provides downside protection.
• Equity call option: The right to convert into stock.

As equity prices rise, the convertible trades more like equity (delta approaches 1). As equity falls, the bond floor provides support and it trades more like a straight bond. This asymmetric payoff profile — equity upside, bond downside protection — makes convertibles attractive to certain hedge fund strategies (convertible arbitrage: long convertible, short the underlying stock).

---

Chapter 7: SOFR, Swaps, and Interest Rate Derivatives

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 unsecured — was the world’s most widely referenced interest rate benchmark for decades, with hundreds of trillions of dollars of contracts referencing USD LIBOR alone. Manipulation scandals (the 2012 Barclays LIBOR scandal) and the thin actual transaction base underlying the rates led regulators globally to mandate a transition to alternative reference rates.

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 — approximately $1 trillion in daily volume. Key differences from LIBOR:

Feature	LIBOR	SOFR
Basis	Unsecured interbank lending (survey-based)	Secured (Treasury-collateralized) repo transactions
Tenor	O/N, 1W, 1M, 3M, 6M, 12M	Overnight only (Term SOFR rates are derived)
Credit risk embedded	Yes (bank credit risk)	No (near risk-free)
Backward vs. forward	Forward-looking	Backward-looking (compounded in arrears)

Term SOFR rates (1-month, 3-month) are published by CME Group based on SOFR futures markets and are used in loan documentation and floating-rate notes where borrowers need to know interest due at the start of a period.

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 the floating rate.
• The fixed-rate receiver receives fixed and pays floating.

No principal is exchanged — only net interest payments. The notional principal serves only as the reference for computing cash flows.

Swap Rate: The fixed rate that makes the present value of fixed cash flows equal to the present value of expected floating cash flows — making the swap’s initial value zero:

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

where \(d_t = \frac{1}{(1+s_t)^t}\) is the discount factor and \(\tau_t\) is the day-count fraction for period \(t\).

Valuation after inception: After rates change, the swap has positive or negative value. The value to the fixed-rate receiver is:

\[ V_{\text{receive-fix}} = \sum_{t} d_t \times (R_{fix} - f_t) \times \tau_t \times N \]

where \(f_t\) is the current forward rate for period \(t\), \(R_{fix}\) is the original fixed rate, and \(N\) is the notional.

Applications of interest rate swaps include:

• Duration management: Receive-fixed swaps increase portfolio duration (like buying long bonds). Pay-fixed swaps reduce duration.
• Liability management: Corporations with floating-rate debt synthetically convert to fixed by entering a pay-fixed, receive-floating swap.
• Speculative positioning: Hedge funds express views on the level and shape of the curve using swap spreads (swap rate minus Treasury yield).

7.3 SOFR Futures

CME Group lists futures on SOFR in two primary formats:

1-Month SOFR Futures: Priced as 100 minus the simple average of daily SOFR for the contract month. Used to hedge short-term SOFR-linked exposures (monthly loan resets, commercial paper).

3-Month SOFR Futures: Priced as 100 minus the average daily SOFR over the contract quarter. A strip of consecutive 3-month SOFR futures implies the full forward curve of overnight rates, equivalent to the OIS (overnight indexed swap) forward curve. Portfolio managers use SOFR futures strips to hedge floating-rate loan portfolios.

Duration of SOFR Futures: A 3-month SOFR futures contract settles to the average of daily SOFR, so its DV01 is approximately $25 per contract per basis point (based on a $1M notional at 90 days/360 days × $1M × 0.0001).

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.

The coupon formula for a SOFR-linked FRN is:

\[ \text{Coupon}_t = N \times (\text{SOFR}_t + \text{Spread}) \times \tau_t \]

FRNs may have a floor on the coupon (typically at 0%) to protect investors against negative reference rates, or a cap to protect the issuer. These embedded options give rise to small duration adjustments.

7.5 Caps, Floors, and Swaptions

Interest Rate Cap: A series of call options on the floating rate (caplets), each paid if SOFR (or another reference rate) exceeds a strike rate on the reset date. Caps protect floating-rate borrowers against rising rates.

Interest Rate Floor: A series of put options on the floating rate (floorlets), each paid if SOFR falls below a strike rate. Floors protect floating-rate lenders (investors in FRNs) against falling rates.

Swaption: An option to enter an interest rate swap at a specified future date. A payer swaption gives the holder the right to enter as fixed-rate payer; a receiver swaption gives the right to enter as fixed-rate receiver. Swaptions are used by mortgage lenders to hedge prepayment risk (when rates fall, prepayments accelerate, and lenders effectively receive early principal back at a bad time — they hedge this with payer swaptions).

---

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), removing them from the originator’s balance sheet.
2. The SPV issues MBS backed by the mortgage pool, with a credit guarantee from Fannie Mae, Freddie Mac, or Ginnie Mae (for agency MBS).
3. A mortgage servicer collects monthly payments and passes them through to investors, net of a servicing fee.

Agency MBS carry an explicit or implicit government guarantee of timely payment, eliminating credit risk. The dominant risk in agency MBS is prepayment risk.

8.2 Prepayment Risk and the PSA Convention

Homeowners have an implicit option to prepay their mortgages at any time. This prepayment option creates significant complexity in MBS valuation:

• When rates fall, refinancing accelerates → call risk (investors receive principal at a bad time).
• When rates rise, refinancing slows → extension risk (investors hold below-market-rate mortgages longer than expected).

8.3 MBS Valuation: OAS and WAL

Because MBS cash flows are uncertain (dependent on prepayments), standard YTM calculations are inadequate. The proper valuation approach uses Monte Carlo simulation of interest rate paths, generating prepayment scenarios for each path, and computing the spread (OAS) that equates the average present value across paths to the current price.

Weighted Average Life (WAL): The average time to receipt of principal payments (both scheduled amortization and prepayments), weighted by the dollar amount of each payment. WAL is used instead of maturity for MBS because actual maturity depends on prepayments.

\[ WAL = \sum_{t=1}^{T} t \times \frac{\text{Principal Payment}_t}{\text{Total Principal}} \]

WAL is shorter at higher PSA speeds (faster prepayment returns principal sooner) and longer at lower PSA speeds.

8.4 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, to appeal to different investor types.

Sequential-pay CMOs: Tranches are paid down in order (A, B, C, Z). Tranche A receives all principal until paid off; then tranche B receives principal, and so on. Short tranches receive principal first and have short WAL; long tranches have long WAL but more extension risk. Z-tranches (accrual bonds) receive no cash flows until earlier tranches are paid off, then receive both accrued interest and principal.

PAC (Planned Amortization Class) bonds: A PAC schedule specifies a fixed principal payment schedule that is maintained across a wide band of prepayment speeds (the PAC band, e.g., 100–250 PSA). Support (companion) tranches absorb excess prepayments above the band or extend to cover shortfalls below the band. Within the PAC band, investors receive highly predictable cash flows.

IO and PO Strips:

Instrument	Receives	Rate Sensitivity
IO (Interest Only)	Interest portion only	Rises with rates (slower prepayments preserve interest income)
PO (Principal Only)	Principal portion only	Falls with rates (faster prepayments return principal sooner at high PV)

IO strips have negative effective duration — they gain value when rates rise. This makes them powerful hedging instruments for fixed income portfolios against rising rate risk, but they are complex and volatile.

---

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.

• Default risk: Probability the issuer cannot pay (PD).
• Recovery risk: Uncertainty about recovery post-default. LGD (Loss Given Default) = 1 − Recovery Rate. Historical recovery rates vary by seniority: senior secured ~65%, senior unsecured ~40%, subordinated ~25%, equity ~0%.
• Spread risk: Credit spreads can widen (prices fall) even without default — mark-to-market loss.
• Downgrade risk: Rating downgrade forces selling by constrained investors, amplifying price declines.

Expected Loss:

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

where EAD = Exposure at Default (face amount of the bond).

9.2 Credit Ratings

Rating agencies (Moody’s, S&P, Fitch) assign credit ratings summarizing issuer creditworthiness. Rating methodology examines:

• Business risk: Industry dynamics, competitive position, revenue stability, geographic diversification.
• Financial risk: Leverage ratios (Debt/EBITDA, Debt/Equity), coverage ratios (Interest Coverage = EBIT/Interest Expense), cash flow generation, liquidity.
• Management and governance: Track record, financial policy conservatism.
• Structural issues: Seniority of the specific issue, security/collateral, covenants.

S&P / Fitch	Moody’s	Category	Typical Spread Range (bps over Treasury)
AAA	Aaa	Highest quality	5–30
AA	Aa	High quality	20–60
A	A	Upper medium grade	40–120
BBB	Baa	Lower medium grade	80–250
BB	Ba	Speculative / High Yield	200–450
B	B	Highly speculative	350–700
CCC	Caa	Substantial credit risk	700–1500+
D	C	Default	N/A

9.3 Credit Spread Dynamics

Credit spreads are cyclical and vary with macroeconomic conditions:

• Economic expansion: Spreads typically narrow (issuers more profitable, lower default probability).
• Recession / financial stress: Spreads widen dramatically (higher default risk, flight-to-quality into Treasuries).
• Sector-specific events: Spread widening in the energy sector (oil price crash), financial sector (banking crisis), and retail (e-commerce disruption) illustrate how sector dynamics dominate individual credit selection.

The credit spread curve plots spreads against maturity for a given issuer or rating category. Investment-grade credit curves tend to be upward sloping (more spread at longer maturities, reflecting greater uncertainty over long horizons). Distressed high-yield issuers sometimes show inverted credit curves (near-term default risk dominates, so short-dated spreads are very wide while long-dated spreads are somewhat lower because the market assumes resolution — either recovery or default — before long maturities are reached).

9.4 Structural Credit Models

The structural approach to credit modeling, pioneered by Merton (1974), treats equity as a call option on the firm’s assets and debt as a put option. A firm defaults when the value of its assets falls below the face value of debt at maturity.

9.5 Reduced-Form Credit Models

Reduced-form (intensity-based) models do not model the firm’s balance sheet directly. Instead, they model default as a Poisson arrival process with a stochastic intensity (hazard rate) \(\lambda_t\). The survival probability to time \(T\) is:

\[ Q(T) = E\left[\exp\left(-\int_0^T \lambda_t\,dt\right)\right] \]

The credit risky bond price is:

\[ P = \sum_{t} d_t \times Q_t \times C_t + \text{Recovery value terms} \]

Reduced-form models are more tractable for calibration to observable credit spreads and CDS prices. The hazard rate \(\lambda\) is directly linked to the credit spread \(s\) by:

\[ s \approx \lambda \times LGD \]

9.6 Credit Default Swaps (CDS)

In a single-name CDS:

• The protection buyer pays a periodic fee (the CDS spread or running premium) to the protection seller.
• If a credit event (default, bankruptcy, restructuring) occurs, the protection seller pays the buyer the loss: (1 − Recovery Rate) × Notional.

CDS markets provide real-time credit pricing that is often more liquid and transparent than the underlying bond market. Key applications:

• Hedging: Bondholders buy CDS protection to hedge credit risk while maintaining interest rate exposure.
• Speculation: Selling CDS protection is equivalent to a synthetic long position in credit.
• Arbitrage: Basis trading exploits differences between CDS spread and bond spread.

The CDS-bond basis = CDS spread − bond spread. A positive basis means the CDS is more expensive than the bond spread implies; a negative basis means the bond appears cheaper (higher spread) than the CDS. Negative basis strategies (buy bond + buy CDS protection) lock in a spread greater than the CDS premium — effectively a risk-free arbitrage in theory, but subject to funding risk in practice.

---

Chapter 10: Inflation-Linked Bonds

10.1 Overview of Inflation-Linked Bonds

Inflation-linked bonds (ILBs) adjust their principal and/or coupons for realized inflation, providing investors with a real return guarantee. The most prominent example is the U.S. Treasury Inflation-Protected Security (TIPS).

10.2 TIPS Structure and Mechanics

TIPS principal is indexed to the Consumer Price Index (CPI-U). The principal is adjusted daily by the CPI ratio:

\[ \text{Adjusted Principal}_t = \text{Face Value} \times \frac{CPI_t}{CPI_{\text{issue date}}} \]

The coupon is paid on the inflation-adjusted principal, so actual coupon dollars rise with inflation. At maturity, the investor receives the greater of the inflation-adjusted principal or the original face value (deflation floor).

Real Yield: The yield on TIPS, net of inflation. If a TIPS has a real yield of 1.5%, the investor earns 1.5% above the CPI inflation rate (compounding approximately).

Breakeven Inflation Rate:

\[ \text{Breakeven Inflation} = \text{Nominal Treasury Yield} - \text{TIPS Real Yield} \]

10.3 Real vs. Nominal Yield Analysis

The Fisher equation relates nominal yields, real yields, and expected inflation:

\[ (1 + r_{\text{nominal}}) = (1 + r_{\text{real}}) \times (1 + E[\pi]) \times (1 + \text{inflation risk premium}) \]

Approximating: \( r_{\text{nominal}} \approx r_{\text{real}} + E[\pi] + \text{inflation risk premium} \)

The inflation risk premium compensates nominal bond investors for uncertainty about future inflation — it is embedded in nominal yields but not TIPS yields. Thus the breakeven rate slightly overstates the market’s inflation forecast by the size of this premium.

Duration of TIPS: TIPS have positive duration (sensitivity to changes in real yields). However, they also have negative sensitivity to changes in inflation expectations (when inflation expectations rise, TIPS prices rise). This makes TIPS useful for portfolios seeking protection against inflationary surprises.

---

Chapter 11: Bond Futures and Basis

11.1 Treasury Bond Futures

U.S. Treasury bond futures (CBT) are standardized contracts for the future delivery of a Treasury bond. The futures contract specifies:

• A basket of deliverable Treasury bonds (those with at least 25 years to maturity or first call, for the long-bond contract).
• A conversion factor system to equate the prices of different deliverable bonds.
• A “standard” notional of $100,000 face value.

11.2 Conversion Factors and Cheapest-to-Deliver

Because any of several Treasury bonds can be delivered, the seller of the futures contract chooses which bond to deliver — the cheapest-to-deliver (CTD) bond. Each deliverable bond has a conversion factor (CF) approximately equal to the price at which the bond would trade if its yield were 6% (the standard futures coupon):

\[ \text{Invoice Price} = \text{Futures Price} \times CF + \text{Accrued Interest} \]

The CTD bond is the bond that minimizes the delivery cost:

\[ \text{CTD} = \arg\min_i \left( \text{Cash Price}_i - \text{Futures Price} \times CF_i \right) \]

The basis of a deliverable bond is:

\[ \text{Basis} = \text{Cash Price} - \text{Futures Price} \times CF \]

11.3 Using Futures for Duration Management

Bond futures are the most efficient instrument for adjusting portfolio duration:

\[ N_{\text{futures}} = \frac{D_{\text{target}} \times V_{\text{portfolio}} - D_{\text{current}} \times V_{\text{portfolio}}}{D_{\text{futures}} \times V_{\text{futures}}} \]

where \(D_{\text{futures}}\) is the modified duration of the CTD bond and \(V_{\text{futures}}\) is the futures price times the face value per contract.

---

Chapter 12: Fixed Income Portfolio Management

12.1 Portfolio Management Mandates and Objectives

Fixed income portfolio managers operate under various mandates:

• Total return: Maximize risk-adjusted return relative to a benchmark (e.g., Bloomberg U.S. Aggregate Bond Index). Success is measured by tracking error (standard deviation of return difference vs. benchmark) and information ratio (excess return / tracking error).
• Liability-Driven Investing (LDI): Align asset duration and cash flows to match liabilities (pension obligations, insurance policy reserves). The primary goal is minimizing the funded status volatility — changes in the surplus (asset value minus liability PV).
• Absolute return: Earn a positive return regardless of benchmark performance (hedge funds, alternative credit).

12.2 Immunization

Immunization protects a bond portfolio against interest rate changes by matching its duration to the investment horizon (for a single liability) or to the duration of liabilities (for multiple liabilities).

Classical single-period immunization (Redington, 1952) conditions for a portfolio to be immunized against small parallel yield shifts:

1. PV matching: Portfolio value ≥ PV of liability.
2. Duration matching: Portfolio duration = Liability duration.
3. Convexity condition: Portfolio convexity ≥ Liability convexity (ensures a surplus under any parallel shift, not just no-change).

Limitations of classical immunization:

• Only works for parallel shifts — non-parallel shifts (twists, butterflies) can create immunization gaps.
• Must be rebalanced continuously as durations change with time and yield shifts.
• Does not address multiple liabilities or reinvestment risk precisely.

12.3 Liability-Driven Investing (LDI)

LDI is an extension of immunization for pension funds and insurance companies with complex, long-dated liabilities. Key LDI concepts:

Funded Status: \(\text{Surplus} = \text{Asset Value} - \text{Liability PV}\). A fully funded plan has surplus ≥ 0.

Duration Extension: Many pension plans are structurally short duration (assets shorter than liabilities). LDI programs extend asset duration using long-dated bonds and receive-fixed swaps to better match liability duration.

Key Rate Duration Matching: Matching not just total duration but the profile of key rate durations across the curve — ensuring that twists in the yield curve do not create unexpected surplus volatility.

Hedge Ratio: The fraction of interest rate risk on the liability side that is hedged by the asset portfolio:

\[ \text{Hedge Ratio} = \frac{\text{DV01 of Assets}}{\text{DV01 of Liabilities}} \]

A 100% hedge ratio means the portfolio is fully interest-rate hedged (changes in rates do not change the funded status). A 70% hedge ratio means 30% of rate risk remains, which may be appropriate if the plan sponsor has a positive view on rates or wants to avoid the cost of full hedging.

12.4 Barbell vs. Bullet Strategies

A bullet portfolio concentrates holdings around a single target maturity. A barbell portfolio holds securities at both short and long maturities. Both can be constructed with the same portfolio duration, but they have very different profiles:

Feature	Bullet	Barbell
Convexity	Lower	Higher (more spread duration)
Yield	Slightly higher (fewer reinvestment options)	Slightly lower
Reinvestment risk	Moderate	Higher (short-end proceeds must be reinvested)
Performance under parallel shift	Similar to barbell	Similar to bullet
Performance under yield curve flattening	Underperforms	Outperforms (long end gains)
Performance under yield curve steepening	Outperforms	Underperforms

A butterfly strategy combines a long intermediate position (bullet) with short positions at the short and long ends — used to express a view that the intermediate portion of the yield curve is cheap (or to express curvature views). Conversely, a reverse butterfly (short intermediate, long short and long ends) expresses a view that the curve will flatten.

12.5 Active Fixed Income Strategies

Active fixed income managers generate alpha by taking deliberate deviations from the benchmark:

Duration bets: Extending duration when expecting rates to fall; shortening when expecting rates to rise. Duration management is the highest-impact decision in most fixed income mandates.

Curve positioning: Overweighting parts of the curve expected to outperform. A “flattener” trade (long long-end, short short-end) profits when the curve flattens. A “steepener” does the opposite.

Sector rotation: Rotating between government bonds, investment-grade corporates, high-yield, MBS, and TIPS based on relative value and macro views.

Credit selection: Identifying individual issuers whose credit quality is improving faster than reflected in current spreads (or avoiding deteriorating credits before the market reprices them).

Carry strategies: Buying higher-yielding bonds and funding them at lower short-term rates, profiting from the carry (the “roll down” as a bond moves down the yield curve toward lower yields over time).

---

Chapter 13: Structured Products and the Global Financial Crisis

13.1 Collateralized Debt Obligations (CDOs)

Collateralized Debt Obligations (CDOs) pool a portfolio of debt instruments (corporate bonds, bank loans, or other structured products) and issue tranched securities against the pool. 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 lower tranches are exhausted).

Tranche	Typical Rating	Coupon Spread	Loss Absorption
Senior	AAA	20–60 bps	Absorbs losses after subordinate tranches
Mezzanine	BBB to BB	200–500 bps	After equity tranche
Equity (first-loss)	Not rated	Residual return	Absorbs first losses

The key variable in CDO pricing is default correlation: if defaults are uncorrelated, the law of large numbers ensures that the senior tranche faces very low loss probability. But if defaults are highly correlated (e.g., all assets lose value simultaneously in a systemic crisis), the diversification benefit disappears and the entire structure fails.

13.2 Lessons from the 2007–2008 Financial Crisis

CDOs backed by subprime mortgage securities (MBS-CDOs, including “CDO-squared”) were assigned AAA ratings based on models that:

1. Underestimated default correlation across mortgages in different regions (housing was treated as uncorrelated regionally, which proved incorrect when national price declines occurred).
2. Used historical data from a period (2000–2006) of steadily rising home prices — a poor basis for tail-risk calibration.
3. Were subject to model gaming by structurers who selected assets to optimize ratings while maximizing yield.

When U.S. home prices declined nationally in 2007–2008, default rates on subprime mortgages surged simultaneously across regions, correlation proved far higher than modeled, and subordinate tranches were quickly wiped out — including tranches that had received investment-grade ratings. Senior tranches of MBS-CDOs, even those rated AAA, suffered severe losses. The opacity and complexity of the structures amplified uncertainty, causing wholesale freezing of credit markets.

---

Chapter 14: Fixed Income Risk Management Frameworks

14.1 Value at Risk (VaR) for Fixed Income

Value at Risk (VaR) estimates the maximum loss on a portfolio over a given time horizon at a given confidence level (e.g., 1-day 99% VaR = the loss exceeded only 1% of trading days).

For a fixed income portfolio, VaR can be computed:

• Parametric (Delta-Normal): Uses portfolio DV01 × yield volatility (assuming normally distributed yield changes).

\[ VaR = MD \times P \times \sigma_y \times z_{\alpha} \times \sqrt{h} \]

where \(\sigma_y\) is daily yield volatility, \(z_\alpha\) is the normal quantile (2.326 for 99%), and \(h\) is the holding period in days.

• Historical simulation: Re-values the portfolio using historical yield changes from the past 250–500 trading days.
• Monte Carlo simulation: Generates thousands of random yield curve scenarios using a model, re-values the portfolio under each, and reads off the loss distribution.

14.2 Stress Testing

VaR fails to capture tail scenarios beyond the confidence level. Stress tests evaluate portfolio losses under extreme but plausible scenarios:

• Historical stress scenarios: 1994 rate shock (Fed hikes 300 bps in one year), 2008 credit crisis (spreads widen by 500+ bps for investment-grade; 2000+ bps for high yield), 2020 COVID-19 pandemic (rapid rate cuts + spread widening).
• Hypothetical scenarios: Instantaneous parallel shifts of ±100 bps, ±200 bps; curve steepening of 100 bps (2s10s spread moves by 100 bps); credit spread widening of 200 bps across all sectors.

Regulators require banks to conduct stress tests under scenarios specified by supervisors (e.g., Federal Reserve’s DFAST in the U.S.; EBA stress tests in Europe).

14.3 Duration Hedging with Swaps and Futures

For a portfolio with total DV01 = \(X\) per basis point, a perfect hedge requires offsetting positions with total DV01 = \(X\):

Using futures: Buy or sell \(N\) futures contracts where \(N = \text{Portfolio DV01} / \text{Futures DV01}\).

Using swaps: Enter a swap with notional \(N_{\text{swap}}\) such that \(N_{\text{swap}} \times MD_{\text{swap}} / 10{,}000 = \text{Portfolio DV01}\).

Partial hedges: Managers may hedge 50–80% of duration risk to retain some interest rate exposure while limiting tail losses — a common approach in liability-driven mandates.

---

Summary and Key Formulas

---

---

Chapter 15: Active Yield Curve Strategies

15.1 Total Return Framework

Active fixed income management begins with decomposing the expected total return on a bond holding over a given horizon into its constituent sources. For a bond held for \(h\) periods and then sold:

\[ \text{Total Return} = \frac{\text{Coupon Income} + \text{Reinvestment Income} + \text{Price Change}}{\text{Beginning Price}} \]

Each term has a different dependence on the yield environment:

• Coupon income is contractually fixed for straight bonds. It is the dominant return source for short horizons and high-coupon bonds.
• Reinvestment income depends on the rates at which interim coupons are reinvested. Over long horizons this component becomes significant — at 6% YTM held for 20 years, reinvestment income contributes roughly 40% of total dollar return.
• Price change depends on the yield at the end of the horizon relative to the yield at purchase. This component dominates for short holding periods and long-duration bonds.

15.2 Rolldown Return and Riding the Yield Curve

When the yield curve is upward sloping and assumed to remain stable, a bond that is held for a period “rolls down” the curve — as time passes, its remaining maturity shortens, and (if the curve is stable) its yield drops toward the lower short-end yields. This capital gain from roll-down is the rolldown return.

15.3 Carry and Roll in Coupon Bond Portfolios

For coupon-paying bonds, the carry-and-roll framework combines two sources of static return (assuming no yield change):

\[ \text{Carry} = \text{Coupon Yield} - \text{Repo Rate (financing cost)} \]\[ \text{Roll-down} \approx -D_{Mod} \times \Delta y_{\text{roll}} \]

where \(\Delta y_{\text{roll}}\) is the yield change implied by moving down the curve by the holding period (a negative number on a normal curve — yield falls as maturity shortens, so the sign convention produces a positive price change).

15.4 Curve Positioning: Flatteners, Steepeners, and Butterflies

Active managers express views on the shape of the yield curve using relative-value trades that are approximately duration-neutral at the portfolio level.

Yield Curve Flattener: Long the long end, short the short end (or duration-equivalent positions). Profits when the spread between long and short yields narrows. A duration-neutral flattener buys \(\$DV01\) of the long-end bond and sells the same \(\$DV01\) in the short-end bond, so the combined position has near-zero sensitivity to parallel shifts.

Yield Curve Steepener: The mirror trade — long the short end, short the long end. Profits when the 2s10s or 2s30s spread widens. Steepeners are popular when the market expects the Fed to cut rates (short end rallies) while inflation concerns keep long yields elevated.

Butterfly Trade: A combination of a long position in the “belly” (intermediate maturities) and short positions in the “wings” (short and long ends). A butterfly profits if the intermediate portion of the curve outperforms both wings — the curve becomes more humped.

Trade	Long	Short	View
Flattener	10yr / 30yr	2yr / 5yr	Curve will flatten
Steepener	2yr / 5yr	10yr / 30yr	Curve will steepen
Butterfly (long belly)	5yr / 7yr	2yr + 10yr	Belly cheapens less
Reverse Butterfly	2yr + 10yr	5yr / 7yr	Belly to cheapen

15.5 Sector Rotation and Relative Value

Beyond duration and curve trades, active managers rotate among fixed income sectors based on relative value assessments:

Common relative value signals:

• Spread vs. historical range: If a BBB corporate trades at 150 bps G-spread vs. a 5-year average of 100 bps and fundamentals are stable, the bond appears cheap.
• Spread vs. peers: Two issuers with similar ratings and financials trading at different spreads — the wider issuer may be a buy if the gap is not justified by fundamentals.
• On-the-run vs. off-the-run: Off-the-run Treasuries yield slightly more than on-the-run bonds with similar maturity. Investors who can hold to maturity or tolerate lower liquidity can capture this spread.
• Swap spread: The spread between the par swap rate and the same-maturity Treasury yield. Negative swap spreads (swap rate below Treasury) in long maturities, which have occurred in recent years, reflect technical supply-demand imbalances and create relative value opportunities.

---

Chapter 16: Advanced Duration Topics

16.1 Key Rate Duration — Full Worked Example

Key rate durations (KRDs) describe a bond’s price sensitivity to changes in the yield curve at specific maturities (key rates), holding all other rates constant. The standard key rates for U.S. Treasury analysis are the 0.25-year, 0.5-year, 1-year, 2-year, 3-year, 5-year, 7-year, 10-year, 20-year, and 30-year points.

For each key rate \( k \), the KRD is:

\[ KRD_k = \frac{P_{k,-} - P_{k,+}}{2 \times P_0 \times \Delta y} \]

where \(P_{k,\pm}\) are the bond prices when only the \(k\)-maturity rate is shifted by \(\pm \Delta y\), and all other rates are held constant. The sum of all KRDs equals the effective duration:

\[ D_{eff} = \sum_k KRD_k \]

16.2 Convexity as an Asymmetric Hedge

Convexity creates a natural “asymmetric” benefit in portfolios. Algebraically:

\[ \frac{\Delta P}{P} \approx -D_{Mod} \cdot \Delta y + \frac{1}{2} \cdot C \cdot (\Delta y)^2 \]

The convexity term is always positive (for long positions in non-callable bonds), regardless of the sign of \(\Delta y\). This means that for equal upward and downward yield moves, the bond gains more in a rally than it loses in a selloff — a desirable property.

16.3 Dollar Duration, DV01, and Hedging Mechanics

In practice, fixed income risk managers work in dollar terms. The dollar duration of a position is:

\[ \text{Dollar Duration} = D_{Mod} \times P \times \text{Face Value} / 100 \]

For a $10M face position in a bond priced at 98 with MD = 7.2:

\[ \text{Dollar Duration} = 7.2 \times \$9{,}800{,}000 = \$70{,}560{,}000 \]

This means a 100 bps parallel yield increase causes an approximate $70,560,000 loss on the position (7.2% of $9.8M).

The DV01 (Dollar Value of a Basis Point) of this position:

\[ DV01 = \$70{,}560{,}000 / 10{,}000 = \$7{,}056 \text{ per basis point} \]

To hedge this exposure with 10-year Treasury futures (DV01 per contract ≈ $63.50 assuming futures price = 109-00 and CTD MD = 7.8, with conversion factor 0.8890):

Effective DV01 per futures contract = CTD DV01 per contract / CF. Approximate approach: each 10-year note futures contract controls $100,000 face. Futures DV01 ≈ (futures price / 100) × $100,000 × MD\(_{CTD}\) / 10,000 × CF = (109/100) × $100,000 × 7.8 / 10,000 × (1/0.8890) ≈ $95.70 per contract…

More precisely, the standard approach: number of contracts to short = Portfolio DV01 / Futures DV01 per contract.

If CTD bond has DV01 per $100,000 face = $75.00, and the CF = 0.8890, then the futures DV01 = $75.00 × CF = $66.68 per contract. Number of contracts to short = $7,056 / $66.68 ≈ 106 contracts.

---

Chapter 17: Fixed Income Market Microstructure and Institutions

17.1 The Over-the-Counter Bond Market

Unlike equity markets with centralized exchanges and public quote dissemination, the majority of bond trading occurs in over-the-counter (OTC) dealer markets. Key features:

• Dealer intermediation: Large broker-dealers (primary dealers in the U.S. Treasury market include Goldman Sachs, JPMorgan, Citi, and approximately 22 others) make markets by standing ready to buy (bid) and sell (offer) bonds. The bid-ask spread is the dealer’s compensation for providing liquidity and bearing inventory risk.
• Quote-driven vs. order-driven: Bond markets are largely quote-driven (dealers post prices) rather than order-driven (investors submit orders to a central book). This means large institutional trades require negotiation rather than simply hitting a visible price.
• Transparency: Post-trade transparency in U.S. corporate bonds improved significantly with FINRA’s TRACE (Trade Reporting and Compliance Engine) system, which disseminates trade prices within 15 minutes. Pre-trade transparency (bid and offer quotes) remains limited compared to equity markets.

17.2 Primary Dealer System and Treasury Auctions

In the United States, the Treasury sells new debt through regular auctions. Key formats:

Single-Price (Dutch) Auction: All successful bidders pay the same clearing yield (the highest yield — lowest price — at which the full auction amount is covered). This format was adopted by the U.S. Treasury in 1998 for all coupon-bearing securities.

Bid-to-Cover Ratio: Total bids received divided by the amount auctioned. A strong auction has a bid-to-cover ratio above 2.5×, indicating robust demand. A weak auction (bid-to-cover below 2.0×) can cause the on-the-run yield to spike and creates concession — the market must sell off to attract buyers.

17.3 Bond Market Liquidity and Liquidity Premiums

Liquidity risk is a distinct and significant risk factor in fixed income markets. Less liquid bonds must offer higher yields — a liquidity premium — to compensate investors for:

1. Higher transaction costs (wider bid-ask spreads).
2. Price impact risk (inability to sell quickly without moving the market).
3. Funding liquidity risk (margin calls or redemptions may force sales at distressed prices).

During the 2008 financial crisis, the corporate bond market nearly froze — bid-ask spreads for investment-grade bonds widened from 5–10 bps to 50–100 bps; high-yield bonds became untradeable at any reasonable price. The resulting forced selling created a self-reinforcing liquidity spiral.

The flight-to-quality phenomenon describes how investors shift from risky, illiquid assets (corporate bonds, high-yield, MBS) into safe, liquid assets (on-the-run Treasuries) during periods of financial stress. This simultaneously widens credit spreads and reduces Treasury yields — the two effects reinforce each other to create large swings in risk asset prices.

---

Chapter 18: Quantitative Topics — Duration Gaps and Immunization in Practice

18.1 Duration Gap Analysis for Financial Institutions

Banks and insurance companies manage a duration gap — the mismatch between the duration of assets and liabilities. For a balance sheet with assets of value \(A\) and duration \(D_A\), and liabilities of value \(L\) and duration \(D_L\) (equity \(E = A - L\)):

\[ \Delta E \approx -(D_A - D_L \cdot L/A) \times A \times \Delta y \]

The term \((D_A - D_L \cdot L/A)\) is the duration gap. A bank with a positive duration gap (assets longer than liabilities in dollar-duration terms) loses equity value when rates rise.

18.2 Multiple Liability Immunization

For portfolios with multiple future liabilities at different horizons (e.g., an insurance company with policyholder obligations in years 3, 7, 10, and 20), classical single-period immunization is insufficient. Multi-period immunization requires:

1. PV of assets ≥ PV of all liabilities (discounted at the current spot rate curve).
2. Duration of assets = Dollar-duration-weighted average of liabilities.
3. Asset cash flows span the full range of liability dates — the portfolio must have cash flows on both sides of each liability date to ensure immunization.
4. Portfolio convexity ≥ Liability convexity (ensures a surplus under any parallel shift).

A common approach is cash flow matching — structuring asset maturities and coupon payments to exactly match each liability outflow. Cash flow matching is the strictest form of immunization (no rebalancing needed) but typically results in a higher-cost portfolio because the exact matching may require purchasing bonds at premium prices.

18.3 Interest Rate Sensitivity Across the Yield Curve: A Recap

The table below summarizes how different fixed income instruments respond to a 100 bps parallel yield increase:

Instrument	Approx. MD	Price Change (100 bps up)	Notes
3-month T-bill	0.25	−0.25%	Minimal rate risk
2-year Treasury note	1.9	−1.9%	Short duration
5-year Treasury note	4.3	−4.3%	Moderate rate risk
10-year Treasury note	8.2	−8.2%	Core bond market
30-year Treasury bond	18.0	−18.0%	Highest rate risk
5-year IG corporate	4.2	−4.2% (rate) + spread widening	Credit + rate risk
Agency MBS (30yr FNMA)	4.5–6.0	−4 to −6% (but negative convexity)	Prepayment risk
Callable corporate	3.0–5.0	Less than non-callable (call caps price gain)	Negative convexity
TIPS (10yr)	8.0	−8% (real yield)	Inflation hedge
Floating-rate note	0.1–0.3	~0%	Resets to market
IO Strip (30yr MBS)	−5 to −15	+5 to +15%	Negative effective duration

The IO strip is the only common fixed income instrument with significantly negative effective duration — it gains value when rates rise because slower prepayments preserve the interest income stream for longer.

---

Chapter 19: Practice Problems and Review

19.1 Worked Problem Set — Bond Pricing and Yields

19.2 Conceptual Review Questions

The following questions test understanding at the depth expected in AFM 425 examinations:

1. Explain why a bond’s Macaulay duration is always less than or equal to its maturity, and under what condition does equality hold.

2. A bond portfolio manager claims that “as long as my portfolio’s modified duration equals my investment horizon, I am perfectly immunized against interest rate changes.” Identify two conditions under which this statement fails, and explain why.

3. The U.S. yield curve inverts (2-year yield exceeds 10-year yield). Interpret this under (a) the pure expectations hypothesis and (b) the liquidity preference theory. Which interpretation is more consistent with the empirical regularity that curve inversions precede recessions?

4. Explain the difference between the Z-spread and OAS. For a callable bond in a low-rate environment, which spread is larger? Why?

5. An MBS IO strip has negative effective duration. Construct an intuitive argument (without calculus) for why this is the case.

6. A pension fund has assets with total DV01 of $800,000 and liabilities with total DV01 of $1,200,000. Calculate the hedge ratio and describe the practical steps to bring it to 100%.

7. Describe the mechanics of a repo transaction and explain why the repo rate for on-the-run Treasuries is lower than the general collateral (GC) repo rate.

8. In the Merton structural credit model, holding asset value constant, what happens to the credit spread as (a) leverage increases, (b) asset volatility increases, (c) the time horizon lengthens for a highly leveraged firm?

---