Fixed-Income Securities and Credit Analysis

Introduction and Context

Fixed-income securities are contractual claims under which an issuer promises to make specified payments to investors. The payments may be fixed, floating, inflation-linked, contingent, or derived from a pool of financial assets. Governments, corporations, financial institutions and special-purpose vehicles use these instruments to finance expenditure, investment and balance-sheet activity. Investors use them to generate income, preserve capital, manage liquidity, diversify risk and express views about interest rates or credit conditions.

Fixed-income investing is often described as conservative, but this description is incomplete. A bond portfolio is exposed to interest-rate risk, reinvestment risk, inflation risk, liquidity risk, call or prepayment risk, currency risk and credit risk. The central analytical task is therefore not merely to calculate a coupon yield. It is to decompose expected return into compensation for time value, risk, liquidity and optionality, and then determine whether that compensation is adequate.

This chapter develops a professional framework for valuing bonds, interpreting yield measures, constructing a term structure, measuring interest-rate sensitivity, analysing credit risk and implementing fixed-income portfolio strategies.

Learning Objectives

By the end of this chapter, the reader should be able to:

1. Classify major fixed-income instruments and identify their cash-flow characteristics.
2. Value coupon bonds, zero-coupon bonds and floating-rate instruments.
3. Distinguish current yield, yield to maturity, spot rates, forward rates and spread measures.
4. Apply duration, modified duration and convexity to estimate price sensitivity.
5. Explain immunization and its practical limitations.
6. Analyse credit spreads, default probability, recovery and credit ratings.
7. Describe securitized products and the risks created by pooling and tranching.
8. Select appropriate interest-rate and credit portfolio strategies.
9. Recognize regulatory, governance, ethical and technological issues in fixed-income markets.

1. Architecture of Fixed-Income Securities

A bond is defined by its promised cash flows, payment dates, seniority, collateral, embedded options and governing documentation. Common categories include:

A bond’s face value or par value is the amount conventionally repaid at maturity. The coupon rate is the annual coupon divided by par value. Coupon frequency determines the number of payments per year. A bond trading above par is at a premium; one trading below par is at a discount.

The distinction between promised and expected cash flows is fundamental. A defaultable bond may promise a coupon and principal but have lower expected payments because default can occur. A callable bond may also have uncertain cash flows because the issuer can refinance when rates fall.

2. Bond Valuation and Yield Measures

2.1 Present-value valuation

The value of a non-embedded-option bond equals the present value of its promised cash flows. With annual periods:

Here, P is the clean or dirty price depending on convention, C_t is the cash flow at date t, y_t is the discount rate applicable to that date, and n is the number of payment dates. The formula assumes deterministic cash flows and rates expressed per payment period. If a single yield is used for all dates, the valuation becomes:

For a conventional coupon bond with face value F, coupon rate c, and annual coupon C = cF,

This is an annual-payment special case. For m payments per year, use periodic coupon C = cF ÷ m, periodic yield i = y ÷ m when the quoted yield is a nominal annual rate convertible m times, and N = mn periods:

The coupon rate and yield must be stated on compatible compounding bases. A continuously compounded yield requires exponential discounting rather than the periodic formula.

2.2 Worked valuation example

An annual-pay bond has par value 1,000, coupon rate 6%, five years remaining and a yield to maturity of 7%. Its annual coupon is 60. Therefore:

The annuity factor is approximately 4.1002 and the discounted principal is approximately 712.99. Thus:

The bond trades below par because its 6% coupon is less than the market yield of 7%.

2.3 Clean price, dirty price and accrued interest

Between coupon dates, the buyer compensates the seller for coupon income earned since the previous payment. The dirty price is the full settlement price:

Under a simple actual-days convention, accrued interest may be approximated as:

Actual market conventions differ by jurisdiction and instrument. A valuation system must therefore specify day-count convention, settlement date, ex-coupon rules and payment calendars.

2.4 Yield measures

Current yield is annual coupon divided by market price:

It ignores capital gain or loss and is therefore incomplete for bonds bought away from par.

Yield to maturity (YTM) is the single discount rate that equates the present value of promised cash flows with the observed price, assuming the bond is held to maturity and coupons can be reinvested at that yield. It is an internal rate of return, not a guaranteed realized return. It assumes no default, a fixed schedule and a consistent compounding convention.

Yield to call replaces maturity with the call date and call price. It is particularly important when a premium bond is callable. Yield to worst is the lowest relevant yield among specified contractual redemption scenarios, subject to market convention. Holding-period return reflects actual purchase price, sale price, coupons and reinvestment; it may differ substantially from YTM.

2.5 Spot rates and forward rates

A spot rate is the annualized yield for a single payment at a particular maturity. The price of a zero-coupon security maturing at time t is:

where s_t is the spot rate for maturity t. With spot rates, a coupon bond is valued as:

This is theoretically preferable to discounting every cash flow at one YTM because different maturities may have different rates.

A one-period forward rate implied by annual spot rates satisfies the no-arbitrage relationship:

Thus:

For example, if the one-year spot rate is 4% and the two-year spot rate is 5%, the implied one-year forward rate beginning in year one is:

Forward rates are arbitrage-implied rates, not unbiased forecasts. They include risk premia and may differ from investors’ expectations.

3. The Term Structure of Interest Rates

The term structure describes how yields or spot rates vary with maturity. A yield curve may be upward sloping, flat, inverted or humped. It reflects expectations about future short-term rates, inflation, real economic activity, liquidity and risk premia.

Three influential explanations are:

• Expectations theory: long rates reflect expected future short rates, although the strict version excludes term premia.
• Liquidity-preference theory: investors require compensation for holding longer maturities, usually producing a positive term premium.
• Market-segmentation and preferred-habitat theories: institutional demand and supply differ across maturities, so yields are influenced by investor mandates and issuance patterns.

3.1 Term-structure models

The Nelson–Siegel model represents the yield curve using level, slope and curvature factors:

Here, τ is maturity, β₀ approximates long-run level, β₁ controls slope, β₂ controls curvature and λ controls the decay rate. The model is descriptive and parsimonious; it does not by itself provide an arbitrage-free dynamic pricing theory.

No-arbitrage models, such as the Ho–Lee, Hull–White and Heath–Jarrow–Morton families, impose consistency between the evolution of rates and observed bond prices. They are useful for derivatives and scenario generation but depend heavily on calibration, volatility assumptions and model specification.

4. Duration, Convexity and Immunization

4.1 Macaulay and modified duration

Macaulay duration is the present-value-weighted average time at which promised cash flows are received:

The units are years when t is measured in years and payments are annual. Modified duration measures approximate percentage price sensitivity to a change in yield:

For a small parallel change in yield:

The yield change must be expressed as a decimal. If modified duration is 4.5 and yield rises by 0.50 percentage points, estimated price change is −4.5 × 0.005 = −2.25%.

For m compounding periods per year, the periodic-yield version is:

with duration measured consistently in years and the annual yield convention clearly stated.

4.2 Convexity

Duration gives a first-order approximation. Convexity captures curvature in the price–yield relationship:

This annual-payment expression assumes a flat yield and deterministic cash flows. A second-order approximation is:

For a conventional option-free bond, positive convexity means that the price gain from a yield decline is generally greater than the price loss from an equal yield increase. Embedded options can create negative or state-dependent convexity.

4.3 Factors affecting duration

Duration generally increases with maturity and decreases with coupon rate and yield. A zero-coupon bond’s Macaulay duration equals its maturity because all cash flows occur at maturity. Floating-rate notes usually have low interest-rate duration near a reset date, but their prices can still fall because of credit-spread widening, basis risk or liquidity deterioration.

4.4 Immunization

Immunization seeks to protect a portfolio’s ability to meet a specified liability despite small parallel changes in interest rates. The classical single-liability conditions are:

1. Present value of assets equals present value of liabilities.
2. Asset duration equals liability duration.
3. Asset convexity is preferably at least as large as liability convexity.

These conditions work locally under a parallel shift, stable cash flows and reinvestment of interim cash flows. They are not a guarantee. Non-parallel curve movements, credit events, transaction costs, taxes, liquidity constraints, changing liability patterns and callable assets can invalidate the hedge.

A practical immunization programme rebalances periodically, monitors key-rate exposures and stress-tests curve twists, steepening, flattening and spread shocks. Key-rate duration measures sensitivity to a movement at a particular maturity point and is more informative than one aggregate duration when curve movements are non-parallel.

5. Credit Analysis

5.1 Credit spread decomposition

A credit spread is the yield difference between a risky bond and a benchmark of comparable maturity and convention. It compensates for more than expected default loss. A useful conceptual decomposition is:

This is an analytical approximation, not an identity. The benchmark may itself contain liquidity and term premia, and spread definitions vary between government, swap and asset-swap references.

Expected default loss can be approximated as:

Loss given default is:

For multiple periods, expected losses should be probability-weighted by the timing of default and discounted using an appropriate framework. Simple multiplication is most useful as a first-pass screening tool.

5.2 Fundamental credit analysis

A professional credit review combines quantitative and qualitative evidence.

Business risk includes industry cyclicality, competitive position, regulation, technological disruption, customer concentration and management quality.

Financial risk includes leverage, interest coverage, cash-flow stability, liquidity, refinancing needs, pension obligations and off-balance-sheet commitments. Common measures include:

These ratios are not interchangeable. EBITDA excludes interest, taxes, depreciation and amortization and may overstate cash available for debt service when capital expenditure or working-capital needs are high. Analysts should reconcile accounting earnings to operating cash flow and free cash flow.

Structural analysis examines seniority, collateral, guarantees, covenants, restricted-payment provisions, change-of-control clauses and legal entity exposure. Recovery prospects depend on asset values, priority, jurisdiction, costs and the speed of restructuring.

Market-based evidence includes credit default swap spreads, bond spreads, equity volatility and implied default indicators. Market signals are timely but may be distorted by liquidity, technical flows or risk aversion.

5.3 Ratings and their limitations

Rating agencies classify relative credit quality using letter grades and outlooks. Ratings support comparability and investment mandates, but they are opinions rather than guarantees. They may be lagging indicators, especially during rapid deterioration. Investors should not outsource due diligence to ratings; they should analyse issuer fundamentals, covenants, maturity walls and market pricing independently.

Ratings also raise governance concerns when institutions mechanically rely on them, when conflicts affect incentives, or when structured products receive ratings that obscure model and correlation risk. Regulatory frameworks in many jurisdictions restrict mechanistic dependence on external ratings and require internal risk assessment.

6. Securitized Products

Securitization transforms pools of loans or receivables into tradable claims. A typical structure transfers assets to a special-purpose vehicle, which issues securities supported by asset cash flows. A waterfall allocates collections to fees, senior notes, mezzanine notes and equity or residual interests.

Credit enhancement may be provided through overcollateralization, excess spread, reserve accounts, subordination, guarantees or insurance. Tranching redistributes risk rather than eliminating it. Senior investors receive priority but remain exposed to collateral deterioration, structural weaknesses, servicer failure and model error.

Mortgage-backed securities have distinctive prepayment risk. When interest rates fall, borrowers may refinance, causing investors to receive principal earlier and reinvest at lower rates. When rates rise, prepayments slow and the security extends, leaving investors exposed to high rates for longer. This creates negative convexity in many pass-through structures.

Collateralized loan obligations typically pool leveraged loans and issue several debt tranches plus equity. Their performance depends on borrower defaults, recovery, reinvestment, manager decisions, concentration limits and the subordination structure. Due diligence should cover data quality, servicing, legal true sale, trigger mechanics, stress assumptions and waterfall behaviour.

7. Interest-Rate and Credit Portfolio Strategies

7.1 Passive and liability-oriented strategies

A buy-and-hold strategy emphasizes income and contractual maturity, but it does not remove market or default risk. A ladder distributes maturities across dates, improving liquidity and reducing the risk of reinvesting all proceeds at one rate. A barbell concentrates in short and long maturities, while a bullet concentrates around a target maturity.

Duration matching aligns portfolio duration with an investment horizon or liability. Full cash-flow matching is stronger but often more expensive. Immunization is appropriate when liabilities are defined and small rate changes are the main concern.

7.2 Active rate strategies

Active managers may express views through:

• Yield-curve steepener or flattener positions.
• Duration extension when yields are expected to fall.
• Duration reduction when yields are expected to rise.
• Sector rotation across governments, corporates, agencies and securitized assets.
• Relative-value trades between bonds with similar risk but different pricing.

A portfolio’s risk should be decomposed into level, slope, curvature, spread, basis, liquidity and optionality exposures. Derivatives such as interest-rate swaps, futures and options can alter duration efficiently, but introduce collateral, counterparty, basis and roll risks.

7.3 Credit strategies

Credit strategies include carry and roll-down, investment-grade selection, high-yield allocation, distressed investing, pairs trades and hedging with credit default swaps. Carry is the income earned for holding a position, while roll-down is the price effect of a bond moving to a shorter point on an unchanged curve. Neither is risk-free: spreads can widen, curves can shift and issuers can default.

A sound process sets issuer, sector, rating, maturity, liquidity and loss limits. Stress tests should include recession, refinancing failure, commodity-price shocks, spread widening and correlated defaults.

8. Illustrative Case: Immunizing a University Liability

Background. A university must pay 10 million in three years to fund a contractual construction commitment. The liability is fixed in nominal terms. The investment committee is considering a portfolio of government bonds.

Problem. The committee wants to reduce the effect of interest-rate changes while preserving the required amount. A portfolio with present value 10 million and duration 2.2 years is proposed, while the liability has duration three years.

Analysis. The proposed assets are under-duration. If yields rise, the present value of assets will fall less than the liability value, but the portfolio may not grow sufficiently through reinvestment to meet the payment. If yields fall, the mismatch may also create a surplus or deficit depending on the path of rates. The committee should first match present value and then move asset duration toward three years.

Approach. The manager constructs a combination of two government bonds: a shorter bond and a longer bond. If the present values of the two positions are A_S and A_L, the duration target is:

Subject to:

Suppose the short bond has duration 1.5 and the long bond duration 6.0. Solving gives:

and:

The weighted duration is three years. The manager must still evaluate convexity, liquidity, tax, transaction costs and curve-shift scenarios. If the liability amount is inflation-linked or the bonds have credit exposure, the hedge is incomplete.

Outcome. The revised portfolio is approximately duration-matched for small parallel rate changes. Quarterly monitoring is required because time passes, yields change, bonds mature and the liability’s characteristics may change.

Learning points. Immunization is a risk-control discipline, not a one-time transaction. Present-value matching, duration matching, convexity review and scenario analysis must be integrated.

9. Risk Governance, Regulation and Technology

Fixed-income governance requires independent valuation, documented models, segregation of duties, best-execution procedures, approved counterparties and transparent limit monitoring. Valuation adjustments should reflect liquidity, credit deterioration and model uncertainty rather than relying blindly on vendor prices.

Regulatory requirements may cover capital, liquidity, large exposures, margin, clearing, disclosure and suitability. Securitized products require particular attention to risk retention, investor reporting, data standards and waterfall transparency. Fiduciaries must ensure that portfolio decisions are consistent with mandates and clients’ risk capacity.

Artificial intelligence can assist with document extraction, covenant screening, anomaly detection, issuer monitoring and scenario generation. However, models may inherit historical bias, misread legal language, overfit sparse default data or generate plausible but unsupported conclusions. Human review, audit trails, explainability, access controls and model validation remain essential. AI should augment, not replace, credit judgment and accountability.

10. Limitations and Best Practices

Important limitations include the following:

• YTM compresses a changing term structure into one number.
• Duration assumes small, usually parallel, yield changes.
• Convexity estimates can fail for large shocks or securities with options.
• Credit spreads mix default, liquidity, risk-premium and technical effects.
• Ratings can be delayed and may not capture sudden event risk.
• Historical default and recovery rates may not represent future regimes.
• Securitization models depend on assumptions about correlation, prepayment and servicing.

Best practice is to combine cash-flow valuation, market-implied measures, fundamental analysis and stress testing. Analysts should document assumptions, use independent price verification, analyse liquidity under stressed conditions, monitor concentration and update scenarios as macroeconomic and regulatory conditions evolve.

Chapter Summary

Fixed-income valuation begins with the present value of contractual cash flows. Coupon bonds can be priced using a common yield or, more precisely, maturity-specific spot rates. Forward rates are implied by no-arbitrage relationships but are not necessarily unbiased forecasts. Yield measures such as current yield and YTM summarize return differently and must be interpreted with their assumptions.

Duration estimates first-order interest-rate sensitivity, while convexity improves the approximation for larger changes. Immunization combines present-value and duration matching, preferably with convexity and key-rate analysis, but remains vulnerable to non-parallel shifts, changing cash flows and transaction costs. Credit analysis evaluates business, financial and structural risk; expected loss depends on default probability and recovery, while observed spreads also include liquidity and risk premia. Ratings are useful reference points but not substitutes for independent analysis.

Securitized products redistribute cash-flow and credit risk through pooling, waterfalls and tranching. Portfolio strategies range from ladders and liability matching to active curve, sector and credit trades. Effective practice integrates quantitative measurement, scenario analysis, governance, ethical judgment and careful use of technology.

Key Terms

Bond: A debt instrument requiring contractual payments.

Coupon rate: Annual coupon divided by face value.

Yield to maturity: Single discount rate equating price with promised cash flows.

Spot rate: Rate applying to a single cash flow at a specified maturity.

Forward rate: Rate implied for a future period by spot rates.

Term structure: Relationship between interest rates and maturities.

Clean price: Quoted bond price excluding accrued interest.

Dirty price: Full settlement price including accrued interest.

Macaulay duration: Present-value-weighted average time of cash flows.

Modified duration: Approximate percentage price sensitivity to yield.

Convexity: Curvature of the price–yield relationship.

Immunization: Strategy designed to protect liability funding from rate changes.

Credit spread: Yield difference between a risky bond and a benchmark.

Probability of default: Likelihood that an obligor fails to meet obligations.

Recovery rate: Proportion recovered after default.

Loss given default: One minus the recovery rate.

Securitization: Transformation of pooled assets into tradable securities.

Tranche: Security class with defined priority in a cash-flow waterfall.

Prepayment risk: Risk that principal is returned earlier than expected.

Roll-down: Price effect of a security moving toward shorter maturity on an unchanged curve.

Review Questions

Short Answer Questions

1. Distinguish clean price from dirty price.
2. Why does a bond sell below par when its coupon rate is below its required yield?
3. Define spot rate and forward rate.
4. What does modified duration measure?
5. State two limitations of credit ratings.

Long Answer Questions

1. Explain bond valuation using both a single YTM and a term structure of spot rates.
2. Discuss expectations, liquidity-preference and market-segmentation explanations of the yield curve.
3. Derive and interpret the duration–convexity approximation for bond price changes.
4. Explain the conditions, assumptions and limitations of classical immunization.
5. Describe how securitization reallocates credit, liquidity and prepayment risk.

Analytical / Application Questions

1. A five-year annual-pay bond has par value 1,000, a 5% coupon and a YTM of 6%. Calculate its price and explain whether it trades at a premium or discount.
2. A portfolio has modified duration 5.2 and convexity 48. Estimate its percentage price change if yield rises by 75 basis points, using the duration–convexity approximation.
3. An issuer’s probability of default is estimated at 3% over a relevant horizon and its recovery rate at 40%. Estimate first-pass expected loss and explain why the observed credit spread may differ from this estimate.

Multiple Choice Questions with Answers

Question 1

A bond with a coupon rate below its yield to maturity will normally trade:

A. At par

B. At a premium

C. At a discount

D. Without accrued interest

Correct Answer: C. At a discount

Question 2

Which rate discounts a single cash flow at a specified maturity?

A. Current yield

B. Spot rate

C. Coupon rate

D. Recovery rate

Correct Answer: B. Spot rate

Question 3

If modified duration is 4 and yield rises by 0.50%, the approximate percentage price change, ignoring convexity, is:

A. −2.00%

B. −0.50%

C. +2.00%

D. +0.125%

Correct Answer: A. −2.00%

Question 4

For an option-free bond, positive convexity means that:

A. Price is unrelated to yield

B. Price gains from a yield fall generally exceed price losses from an equal yield rise

C. Duration is always zero

D. Credit risk is eliminated

Correct Answer: B. Price gains from a yield fall generally exceed price losses from an equal yield rise

Question 5

Classical immunization generally requires asset and liability present values to be equal and their:

A. Coupon rates identical

B. Credit ratings identical

C. Durations matched

D. Maturities identical

Correct Answer: C. Durations matched

Question 6

Loss given default is equal to:

A. Default probability plus recovery rate

B. 1 minus recovery rate

C. Recovery rate divided by coupon rate

D. Yield minus duration

Correct Answer: B. 1 minus recovery rate

Question 7

Which risk is especially associated with mortgage-backed securities when interest rates fall?

A. Prepayment and reinvestment risk

B. Only currency risk

C. Elimination of convexity risk

D. Guaranteed extension

Correct Answer: A. Prepayment and reinvestment risk

Question 8

A forward rate derived from spot rates is best described as:

A. A guaranteed future realized rate

B. An arbitrage-implied rate

C. A bond’s coupon rate

D. A recovery estimate

Correct Answer: B. An arbitrage-implied rate

Question 9

Current yield is calculated as:

A. Annual coupon divided by market price

B. Market price divided by annual coupon

C. Face value divided by duration

D. Recovery rate divided by spread

Correct Answer: A. Annual coupon divided by market price

Question 10

Which item is generally included in a credit spread besides expected default loss?

A. Liquidity premium

B. Guaranteed profit

C. Coupon frequency only

D. Face value amortization only

Correct Answer: A. Liquidity premium