Fixed Income

Study Guide

Fixed-Income Instrument Features

A bond is a contractual agreement between an issuer and a bondholder. The key features are detailed in the bond indenture (the legal contract).

• Issuer: The entity borrowing the money (e.g., corporation, government).
• Par Value (Face Value): The principal amount repaid at maturity.
• Coupon Rate: The annual interest rate paid on the par value.
  • Fixed-Rate: Coupon rate is constant.
  • Floating-Rate: Coupon rate resets periodically based on a reference rate.
• Maturity Date: The date when the issuer repays the par value.
• Covenants: Rules placed on the issuer to protect bondholders.
  • Affirmative Covenants: Actions the issuer must perform (e.g., make payments, maintain assets).
  • Negative Covenants: Actions the issuer cannot perform (e.g., take on excessive debt, sell key assets).
• Contingency Provisions (Embedded Options):
  • Call Option: Issuer has the right to repay the bond before maturity. Benefits the issuer.
  • Put Option: Bondholder has the right to demand repayment before maturity. Benefits the bondholder.
  • Convertible Bond: Bondholder has the right to convert the bond into a specified number of the issuer's common shares.

Fixed-Income Cash Flows and Types

The structure of a bond determines its cash flow pattern.

• Sinking Fund Provision: Requires the issuer to retire a portion of the bond issue each year. This reduces credit risk for bondholders.
• Floating-Rate Note (FRN): A bond whose coupon payments are based on a reference rate (e.g., SOFR) plus a fixed spread. The coupon payment is $Reference\ Rate + Quoted\ Margin$.

Fixed-Income Issuance and Trading

• Primary Market: Where new bonds are issued and sold to investors for the first time.
  • Public Offering: Sold to the general public through an underwritten offering.
  • Private Placement: Sold directly to a small group of qualified investors.
• Secondary Market: Where existing bonds are traded among investors. Most bond trading occurs in decentralized, dealer-based over-the-counter (OTC) markets.
• Settlement: The process of transferring funds and securities after a trade. For government and corporate bonds, settlement is often T+2 or T+1.

Fixed-Income Markets for Corporate Issuers

Corporations issue debt to finance operations and growth.

• Commercial Paper: Short-term, unsecured debt with maturities up to 270 days.
• Medium-Term Notes (MTNs): Can have a range of maturities, offered continuously by the issuer's agent.
• Bonds and Notes: Longer-term debt instruments.
• Seniority Ranking (Priority of Claims):
  1. Secured Debt: Backed by specific collateral (e.g., mortgage bonds). Highest priority.
  2. Unsecured Debt (Debentures): Backed only by the issuer's creditworthiness.
     • Senior Unsecured: Highest priority among unsecured debt.
     • Subordinated Unsecured: Lower priority than senior debt.

Fixed-Income Markets for Government Issuers

• Sovereign Governments: National governments (e.g., U.S. Treasury). Considered the highest credit quality in their domestic currency.
  • U.S. Treasury Securities: T-Bills (<1 year), T-Notes (2-10 years), T-Bonds (>10 years).
  • Treasury Inflation-Protected Securities (TIPS): Par value adjusts with inflation.
• Non-Sovereign Governments: State, provincial, or municipal governments.
• Quasi-Government Bonds: Issued by entities created by a government but not having the full faith and credit of the government (e.g., Fannie Mae, Freddie Mac).
• Supranational Agencies: Issued by organizations that operate globally (e.g., World Bank, IMF).

Fixed-Income Bond Valuation: Prices and Yields

The price of a bond is the present value of its future cash flows (coupons and par value), discounted at the market discount rate (yield).

Valuation Formula: $P = \sum_{t=1}^{N} \frac{C}{(1+r)^t} + \frac{FV}{(1+r)^N}$ Where:

• $P$ = Price of the bond
• $C$ = Coupon payment per period
• $FV$ = Future Value (Par Value)
• $r$ = Market discount rate (yield) per period
• $N$ = Number of periods

Price-Yield Relationship: The relationship between a bond's price and its yield is inverse.

• If Coupon Rate > Yield, the bond trades at a premium (Price > Par).
• If Coupon Rate < Yield, the bond trades at a discount (Price < Par).
• If Coupon Rate = Yield, the bond trades at par (Price = Par).

Clean vs. Dirty Price:

• Clean Price: The quoted price of a bond, excluding accrued interest.
• Dirty Price (Full Price): The actual price paid, which includes accrued interest. $Dirty\ Price = Clean\ Price + Accrued\ Interest$

Yield and Yield Spread Measures for Fixed-Rate Bonds

• Yield-to-Maturity (YTM): The internal rate of return (IRR) on a bond, assuming it is held to maturity and all coupons are reinvested at the YTM. It is the single discount rate that equates the bond's price to its cash flows.
• Yield-to-Call (YTC): The YTM assuming the bond is called by the issuer on the first possible call date.
• Yield-to-Worst (YTW): The lowest of the YTM, YTC, YTP, etc. for all possible call/put dates.

Yield Spreads: A measure of the additional yield a bond offers over a benchmark.

Yield and Yield Spread Measures for Floating-Rate Instruments

An FRN's coupon is $Reference\ Rate + Quoted\ Margin$.

• Quoted Margin (QM): The spread specified in the bond indenture.
• Required Margin / Discount Margin (DM): The spread required by the market for a bond with similar risk.
• Valuation:
  • If QM = DM, the FRN is priced at par.
  • If QM > DM, the FRN is priced at a premium.
  • If QM < DM, the FRN is priced at a discount.

The Term Structure of Interest Rates: Spot, Par, and Forward Curves

• Spot Rate Curve: A series of yields-to-maturity on zero-coupon bonds of different maturities.
• Par Curve: A series of yields-to-maturity for coupon-paying bonds priced at par.
• Forward Curve: A series of forward rates, which are interest rates for a future period agreed upon today.
  • The relationship between a 1-year spot rate ($S_1$), a 2-year spot rate ($S_2$), and the 1-year forward rate in 1 year ($f_{1,1}$) is: $(1+S_2)^2 = (1+S_1)(1+f_{1,1})$
• Bootstrapping: The process of deriving the spot rate curve from the par curve, one maturity at a time.

Interest Rate Risk and Return

Interest rate risk is the risk that a bond's price will decline due to rising interest rates.

• Macaulay Duration: The weighted-average time to receipt of a bond's cash flows.
• Modified Duration: A measure of the bond's price sensitivity to changes in its YTM. It is the approximate percentage price change for a 1% change in yield. $\Delta P \approx -D_{mod} \times \Delta YTM$
• Price Value of a Basis Point (PVBP): The absolute change in a bond's price for a 1 basis point (0.01%) change in yield.
• Convexity: A measure of the curvature of the price-yield relationship. It improves the price change estimate provided by duration alone. Bonds with positive convexity have more upside price potential and less downside price risk.

Yield-Based Bond Duration Measures and Properties

Modified Duration: $D_{mod} = \frac{D_{mac}}{1 + \frac{YTM}{k}}$ Where:

• $D_{mac}$ = Macaulay Duration
• $YTM$ = Yield-to-Maturity
• $k$ = Number of coupon periods per year

Properties of Duration:

• Duration is inversely related to the coupon rate and YTM.
• Duration is positively related to the time to maturity (for bonds at or above par).
• A zero-coupon bond's duration is equal to its time to maturity.
• Portfolio Duration: The weighted average of the durations of the individual bonds in the portfolio.

Yield-Based Bond Convexity and Portfolio Properties

Duration provides a linear estimate of a bond's price change, while the actual price-yield relationship is curved (convex). Convexity adjusts for this curvature.

Price Change Estimation with Duration and Convexity: $\frac{\Delta P}{P} \approx [-D_{mod} \times \Delta YTM] + [\frac{1}{2} \times C \times (\Delta YTM)^2]$ Where:

• $D_{mod}$ = Modified Duration
• $C$ = Convexity
• $\Delta YTM$ = Change in Yield-to-Maturity

The convexity term is always positive for a standard (option-free) bond, meaning it adds to the price when yields fall and reduces the price loss when yields rise.

Curve-Based and Empirical Fixed-Income Risk Measures

• Effective Duration: A measure of interest rate sensitivity used for bonds with embedded options (e.g., callable bonds). It measures price change based on a parallel shift in the benchmark yield curve, not the bond's own YTM. $D_{eff} = \frac{P_{-} - P_{+}}{2 \times P_0 \times \Delta Curve}$ Where $P_{-}$, $P_{+}$, and $P_0$ are the bond prices after the curve shifts down, up, and the initial price, respectively.
• Key Rate Duration: Measures a bond's price sensitivity to a change in a specific point (a "key rate") on the yield curve, holding other rates constant. It helps identify sensitivity to non-parallel shifts (e.g., steepening or flattening).

Credit Risk

Credit risk is the risk of loss resulting from the borrower (issuer) failing to make full and timely payments of principal and interest.

• Default Risk: The probability that the issuer will fail to meet its obligations.
• Credit Spread Risk: The risk that a bond's price will decline due to an increase (widening) of its credit spread.
• Expected Loss: The potential loss from a credit event. $Expected\ Loss = Probability\ of\ Default\ (PD) \times Loss\ Given\ Default\ (LGD)$
• Credit Ratings: Assess an issuer's creditworthiness.
  • Investment Grade: Baa3/BBB- or higher.
  • High Yield (Junk): Ba1/BB+ or lower.

Credit Analysis for Government Issuers

Analysis of sovereign debt focuses on the issuer's ability and willingness to pay. Key factors include:

1. Institutional Effectiveness and Political Risk: Stability of government, rule of law, transparency.
2. Economic Prospects: Economic growth potential, diversity of the economy.
3. International Investment Position: Foreign currency reserves, external debt levels.
4. Fiscal Flexibility: Government's ability to raise revenue and control spending; debt-to-GDP ratio.
5. Monetary Flexibility: Credibility of monetary policy, exchange rate regime.

Credit Analysis for Corporate Issuers

Analysis of corporate debt focuses on the issuer's ability to generate cash flow to service its debt.

• The Four Cs of Credit Analysis:
  1. Capacity: Ability to make payments (analyzed via financial ratios).
  2. Collateral: Quality of assets backing the debt.
  3. Covenants: Terms and conditions of the lending agreement.
  4. Character: Management's integrity and operational track record.
• Key Financial Ratios:
  • Leverage Ratios: Measure debt levels (e.g., Debt/Capital, Debt/EBITDA).
  • Coverage Ratios: Measure ability to service debt payments (e.g., EBIT/Interest Expense, EBITDA/Interest Expense).

Fixed-Income Securitization

Securitization is the process of transforming illiquid financial assets into marketable securities.

The Process:

1. A seller/originator (e.g., a bank) pools assets like mortgages or auto loans.
2. The seller transfers these assets to a legally separate entity, a Special Purpose Vehicle (SPV).
3. The SPV issues asset-backed securities (ABS) to investors, which are backed by the cash flows from the pooled assets.
4. A servicer collects payments from the underlying assets and passes them to the SPV for distribution to investors.

Asset-Backed Security (ABS) Instrument and Market Features

ABS are securities backed by a pool of assets other than mortgages. Common underlying assets include auto loans, credit card receivables, and student loans.

• Credit Enhancement: Mechanisms to improve the credit quality of the ABS.
  • Internal:
    • Credit Tranching (Subordination): Creating multiple bond classes (tranches) with different payment priorities. Senior tranches are paid first and have the highest credit quality, while subordinated (junior) tranches absorb initial losses.
    • Overcollateralization: The value of the collateral pool is greater than the par value of the securities issued.
  • External: Guarantees from third parties.

Mortgage-Backed Security (MBS) Instrument and Market Features

MBS are securities backed by a pool of mortgage loans.

• Mortgage Pass-Through Security: The simplest form of MBS. Investors receive a pro-rata share of all principal and interest payments from the underlying mortgage pool.
• Prepayment Risk: The primary risk for MBS investors. It is the uncertainty that homeowners will repay their loans early (e.g., when refinancing).
  • Contraction Risk: When interest rates fall, homeowners refinance, and the MBS principal is returned faster than expected. Investors must reinvest this principal at lower rates.
  • Extension Risk: When interest rates rise, homeowners are less likely to prepay, and the MBS principal is returned slower than expected. Investors are locked into a lower-yielding security.
• Collateralized Mortgage Obligations (CMOs): Securities that redistribute the cash flows (and prepayment risk) from a pass-through MBS into different tranches, each with a different risk profile.