Fixed-Income Analysis and Management

Introduction and context

Fixed-income securities are contractual claims that promise specified or formula-based cash flows. They include government and corporate bonds, money-market instruments, mortgage-backed securities, asset-backed securities, and derivatives used to manage interest-rate and credit exposures. Although the cash flows of a conventional bond appear certain, its market value is not: price changes when discount rates, expected inflation, liquidity, taxation, or credit conditions change.

For the Chartered Finance Manager, fixed-income analysis has two complementary purposes. First, it estimates value and risk. Second, it converts those estimates into portfolio decisions: security selection, benchmark construction, hedging, liability matching, and active management. This chapter develops those functions from bond mathematics through duration, convexity, credit analysis, securitization, immunization, liability-driven investing (LDI), and active strategies.

Learning objectives

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

1. Price fixed-income securities using discounted cash flow principles.
2. Distinguish coupon rate, current yield, yield to maturity, spot rates, forward rates, and yield spread.
3. Explain the term structure of interest rates and the principal theories that describe it.
4. Measure interest-rate sensitivity using duration and convexity.
5. Evaluate credit, liquidity, optionality, and structured-product risks.
6. Construct immunized and liability-driven portfolios.
7. Compare active fixed-income strategies and assess their governance implications.

1. Bond cash flows and valuation

A plain-vanilla bond normally pays periodic coupons and returns principal at maturity. If face value is F, annual coupon rate is c, and payments occur m times per year, each coupon is:

Here, c is expressed as a decimal and C has the same currency unit as F. The formula assumes a fixed coupon and equal payment intervals.

If all cash flows are discounted at a single periodic yield y/m, the price of a bond with n payment periods is:

P is the clean or dirty price only after accounting for accrued interest conventions. The expression gives the full (dirty) value immediately after a coupon date. C is the periodic coupon, F is principal, y is the nominal annual yield compounded m times per year, and n is the number of remaining periods. The equation assumes no default, fixed cash flows, and a flat discount curve.

For a zero-coupon bond, C = 0, so:

A more general and theoretically preferable approach discounts each cash flow at its corresponding spot rate s_t:

where CF_t is the cash flow at time t and s_t is the annual effective spot rate for that maturity. If rates are continuously compounded, the denominator is replaced by e^(s_t t), with t measured in years. Discounting must use rates consistent with the cash-flow currency, compounding convention, and timing.

Worked example: price and yield measures

An illustrative five-year bond has face value 1,000, annual coupon rate 6%, and annual payments. If its yield to maturity is 7%, its price is:

Since (1.07)^−5 ≈ 0.7130, the annuity component is approximately 246.1 and the principal component is approximately 713.0. Thus:

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

Its current yield is:

Current yield ignores the capital gain or loss produced by convergence toward par. Yield to maturity (YTM) is the discount rate that equates the present value of promised cash flows with the observed price. It assumes coupons are reinvested at the YTM, the bond is held to maturity, and no default occurs. These assumptions make YTM a useful summary measure, not a guaranteed realized return.

The relationship between price and yield is inverse and nonlinear. A premium bond has coupon rate greater than its YTM; a discount bond has coupon rate below its YTM; and a bond at par has coupon rate equal to its YTM, subject to identical compounding conventions.

Clean price, dirty price, and accrued interest

Between coupon dates, the buyer compensates the seller for coupon income earned since the previous payment. Dirty price is the full invoice price:

Accrued interest depends on the coupon amount and the day-count fraction. Market conventions differ, including Actual/Actual, 30/360, and Actual/360. Analysts must not compare quoted yields without confirming settlement date, day-count basis, compounding, and ex-coupon rules.

2. Yield measures and spread analysis

Important yield measures include:

For a callable bond, the issuer owns an option to redeem early. The investor therefore receives a lower yield than an otherwise identical noncallable bond, and price appreciation is limited when rates fall. A putable bond gives the investor an option and generally commands a higher price.

A credit spread is the yield difference between a risky bond and a benchmark of comparable maturity and convention. It compensates for expected default loss, unexpected credit risk, liquidity risk, taxation, and risk premia. It is not identical to expected default loss.

3. The term structure of interest rates

The term structure is the relationship between yields and maturities for otherwise comparable instruments. A yield curve may be upward sloping, flat, inverted, or humped. It reflects expectations, risk premia, liquidity preferences, supply and demand, inflation expectations, and central-bank policy.

The pure expectations hypothesis states that a long-term rate approximately reflects expected future short rates. Under a simple two-period annual framework, the one-year forward rate for year two is:

where s_1 and s_2 are one- and two-year spot rates, and f_{1,2} is the annual effective forward rate for the second year. The relationship assumes arbitrage-free pricing and consistent compounding.

The liquidity-preference theory adds a positive term premium for holding longer maturities. The preferred-habitat theory emphasizes that investors and issuers may prefer particular maturity sectors but can move when compensation is adequate. These theories are not mutually exclusive: observed curves can reflect expected monetary policy together with maturity-specific risk premia.

Bootstrapping spot rates

Coupon bonds can be used to infer spot rates sequentially. The first spot rate is obtained from a short zero-coupon instrument. A later coupon bond’s price is then reduced by the present value of earlier coupons, and the remaining principal-plus-final-coupon cash flow determines the next spot rate. This method requires accurate prices, synchronized settlement, and a specified interpolation method between observed maturities.

Curve construction is a model-risk issue. Different interpolation methods can produce different forward rates and valuations, especially for long-dated or option-sensitive securities. A professional process documents instruments, liquidity screens, pricing time, interpolation, compounding, and validation controls.

4. Duration and convexity

Macaulay duration

Macaulay duration is the present-value-weighted average time until receipt of a bond’s cash flows:

For m payments per year, t is measured in payment periods and duration in periods. In years, divide the result by m. CF_t is the cash flow, P is the full price, and y is the nominal annual yield. The formula assumes fixed promised cash flows and a single yield used for discounting.

Modified duration

Modified duration measures first-order price sensitivity to yield changes:

For a small parallel change Δy in annual nominal yield:

Δy must be expressed as a decimal. Thus, a modified duration of 6 implies an approximate 6% price decline for a 1-percentage-point yield increase, provided the change is small and the curve shifts in parallel.

For a portfolio, market-value weights produce an approximate duration:

where w_i is security i’s market-value weight and ∑w_i = 1. Portfolio duration does not fully capture key-rate, spread, convexity, or nonlinear option risk.

Convexity

Because the price-yield curve is curved, duration becomes less accurate for larger yield changes. Convexity measures the second-order effect:

A discrete approximation for annual payments is:

The approximation assumes annual compounding and fixed cash flows. The duration-convexity estimate is:

Positive convexity means that, for equal upward and downward yield changes, the price gain from a decline exceeds the price loss from an increase. Callable bonds may exhibit negative convexity over ranges where the call option becomes valuable.

Worked duration interpretation

Suppose a bond has modified duration 5.8 and convexity 42. If yield rises by 0.50% (Δy = 0.005), the estimated return is:

The estimated price decline is approximately 2.85%, rather than the duration-only estimate of 2.90%. The result is an approximation; actual performance also includes coupon income, carry, roll-down, spread movement, and any changes in expected cash flows.

5. Credit risk and fixed-income research

Credit risk is the possibility that an issuer or counterparty fails to meet contractual obligations or that its perceived credit quality deteriorates. Its components include:

• Default risk: failure to pay interest or principal.
• Recovery risk: uncertainty about value recovered after default.
• Migration risk: rating or perceived-quality changes without default.
• Spread risk: market repricing of credit compensation.
• Settlement and counterparty risk: failure in transaction performance.

Expected loss can be represented, in a simplified one-period framework, as:

PD is probability of default, LGD is loss given default as a fraction of exposure, and EAD is exposure at default. The expression is a risk estimate, not a complete bond price model. It excludes unexpected loss, spread volatility, liquidity, and dependence across borrowers.

Credit research combines quantitative and qualitative evidence: leverage, interest coverage, cash-flow stability, refinancing needs, covenant protection, collateral, industry structure, management quality, ownership, legal ranking, and macroeconomic sensitivity. Analysts distinguish senior secured, senior unsecured, subordinated, and hybrid claims because priority affects recovery.

Rating agencies provide useful common language but do not replace independent analysis. Ratings may lag market information, and conflicts, model limitations, and rating-based investment mandates require governance. Internal limits should address issuer, group, country, sector, maturity, liquidity, and downgrade exposure.

6. Securitized products

Securitization pools financial assets and issues securities whose cash flows depend on the pool. Mortgage-backed securities (MBS) are backed by mortgages; asset-backed securities (ABS) may be backed by auto loans, credit-card receivables, leases, or student loans.

A typical structure uses tranching. Senior tranches receive priority, mezzanine tranches absorb losses after equity but before senior notes, and the equity tranche bears first loss. Credit enhancement may include overcollateralization, excess spread, reserve accounts, subordination, and guarantees.

The principal risks are more complex than those of a bullet corporate bond:

• Prepayment risk: borrowers repay early, often when rates fall, shortening an MBS’s life.
• Extension risk: repayments slow when rates rise, lengthening life and increasing exposure.
• Contraction risk: cash returns sooner than expected, creating reinvestment risk.
• Correlation and model risk: defaults may become concentrated in stress conditions.
• Servicer, legal, operational, and liquidity risk.

Weighted-average life is not the same as maturity. For a security with principal cash flows Principal_t, weighted-average life is:

Here t is measured in years and principal cash flows are nonnegative. Prepayment assumptions must be stress-tested rather than treated as forecasts with certainty. Option-adjusted spread analysis is often preferred for securities containing material embedded options.

7. Immunization and liability-driven investing

Classical immunization seeks to protect a portfolio’s value against small parallel shifts in interest rates while meeting a known liability. A basic single-liability immunization requires:

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

If the liability present value is L and its duration is D_L, the portfolio is designed so:

and

where P_A and D_A are asset present value and duration. These conditions are local: they protect against small parallel changes, not arbitrary curve movements, defaults, inflation changes, or changes in liability behavior.

For multiple liabilities, key-rate duration is more informative. It measures sensitivity to a shift at a particular maturity node while other nodes are held constant. A liability-driven investor may use long government bonds, swaps, futures, inflation-linked bonds, credit assets, and cash to separate interest-rate hedging from return generation.

LDI begins with liability mapping: amount, timing, currency, inflation indexation, legal priority, liquidity needs, and confidence intervals. The hedge ratio may be defined as the proportion of liability interest-rate sensitivity hedged by assets or derivatives. A high hedge ratio reduces funded-status volatility but can reduce expected return and create collateral or basis risk.

Rebalancing is essential because duration changes as time passes, yields move, cash flows arrive, and liabilities evolve. Derivatives introduce margin, collateral, counterparty, legal, and liquidity requirements. The 2022 United Kingdom gilt-market stress illustrated that leveraged LDI can create forced selling when collateral demands rise rapidly; governance therefore must include liquidity stress testing and contingency funding.

8. Active fixed-income management

Active management seeks returns beyond a benchmark or controls risk differently from a passive portfolio. Common strategies include:

• Duration positioning: overweighting or underweighting interest-rate exposure based on macroeconomic and curve views.
• Yield-curve positioning: expressing steepener, flattener, butterfly, or maturity-rotation views.
• Sector rotation: shifting among government, investment-grade, high-yield, securitized, and emerging-market debt.
• Credit selection: identifying improving or mispriced issuers.
• Spread trading: exploiting relative value between bonds, swaps, or sectors.
• Carry and roll-down: earning coupon and potential price appreciation as a bond moves down an upward-sloping curve.
• Index and liquidity strategies: exploiting inclusion, issuance, or forced-flow effects.

Total return should be decomposed into income, curve exposure, spread movement, currency, optionality, trading, and residual effects. Performance attribution helps determine whether results came from intended skill or unintended beta. Tracking error, information ratio, drawdown, turnover, transaction costs, and liquidity-adjusted returns should be monitored.

A portfolio’s information ratio is commonly expressed as:

Active return is portfolio return minus benchmark return, and tracking error is the standard deviation of active returns. The ratio is meaningful only when the benchmark, horizon, return frequency, and risk estimation method are appropriate.

Passive strategies are not risk-free. A bond index may be weighted by debt outstanding, thereby allocating more to heavily indebted issuers. Index replication can also be difficult where securities are illiquid or frequently called. Enhanced indexing combines benchmark control with limited active positions.

9. Case study: illustrative pension fund LDI redesign

Background. The fictional Meridian University Pension Fund has 500 million currency units of nominal liabilities with an effective duration of 14 years. Its assets are 35% short government bonds, 40% investment-grade credit, 15% equities, and 10% cash. The asset portfolio duration is 7 years. The trustees are concerned that rising yields will reduce asset values differently from liability values and that a liquidity shock could force sales.

Problem. The fund has a duration mismatch and substantial equity risk. Credit assets may lose value when spreads widen, even if government yields decline. Cash is available, but the fund has not measured collateral requirements under derivative hedges.

Analysis. The investment team maps liability cash flows and estimates key-rate durations at 5-, 10-, 20-, and 30-year nodes. It finds that the liability is concentrated at long maturities, while the asset portfolio is concentrated in shorter maturities. A parallel duration comparison alone understates the mismatch. Stress tests include a 200-basis-point rate rise, a 150-basis-point credit-spread widening, equity decline, rapid inflation, and a collateral call.

Approach. The trustees approve a staged redesign. Long government bonds and receive-fixed interest-rate swaps increase long-end hedge exposure. Investment-grade credit is retained but subject to issuer and sector limits. A liquidity sleeve is increased to cover projected twelve-month benefit payments and stressed collateral needs. Equity exposure is reduced gradually, avoiding forced transactions. The policy specifies hedge ratios, eligible collateral, counterparty standards, rebalancing triggers, and escalation procedures.

Outcome. The asset duration becomes closer to liability duration, and key-rate mismatches decline. The fund gives up some expected return and accepts basis risk between swap and bond curves, but funded-status volatility becomes more manageable. The trustees receive quarterly attribution separating rates, spreads, equity, currency, carry, and implementation effects.

Learning points. LDI is not simply a purchase of long bonds. It is an integrated process involving liability measurement, scenario analysis, leverage control, liquidity planning, governance, and continuous rebalancing.

10. Limitations, technology, and governance

Fixed-income models depend on assumptions about rates, defaults, prepayments, recovery, liquidity, and correlations. A model can be internally precise yet economically wrong. Sensitivity analysis should therefore accompany point estimates. Scenario analysis, historical stress, reverse stress testing, and independent price verification are core controls.

Technology improves curve construction, cash-flow projection, electronic execution, portfolio attribution, and anomaly detection. Machine-learning systems may identify nonlinear relationships or abnormal prices, but they can embed historical biases, fail under regime change, and obscure economic reasoning. AI-generated recommendations require human validation, explainability, data lineage, access controls, and documented model-risk ownership.

Ethical and regulatory responsibilities include fair valuation, truthful performance reporting, best execution, suitability, segregation of duties, avoidance of conflicts, appropriate use of ratings, and accurate disclosure of leverage and liquidity. Climate and transition risks may alter credit spreads, default probabilities, collateral values, and stranded-asset risk; they should be analyzed economically rather than reduced to unsupported scores.

Best practice is to maintain a documented investment process: define objectives and constraints; establish benchmarks; validate prices and models; measure duration, key-rate, convexity, spread, and liquidity risk; stress-test collateral; set limits; attribute performance; and review assumptions after material market events.

Chapter Summary

Fixed-income valuation discounts contractual cash flows using yields or, more rigorously, maturity-specific spot rates. Coupon rate, current yield, YTM, forward rates, and spreads answer different questions and must be interpreted with their assumptions. Bond prices move inversely and nonlinearly with yields; modified duration provides a first-order sensitivity estimate, while convexity improves estimates for larger movements and highlights the effect of embedded options.

The term structure reflects expectations, risk premia, liquidity preferences, and preferred habitats. Credit analysis must consider default, recovery, migration, spread, liquidity, and counterparty risk. Securitized products add prepayment, extension, tranching, correlation, servicing, and model risks. Immunization matches asset and liability present value and duration, while LDI extends the approach through key-rate hedging, derivatives, liquidity management, and governance.

Active fixed-income management uses duration, curve, sector, credit, carry, roll-down, and relative-value decisions. Sound evaluation requires benchmark discipline, attribution, transaction-cost analysis, and stress testing. Across all strategies, model limitations, leverage, collateral, liquidity, ethical conduct, and regulatory controls are as important as valuation mathematics.

Key Terms

Bond pricing: Present-value valuation of promised or expected bond cash flows.

Coupon rate: Contractual annual coupon divided by face value.

Yield to maturity: Discount rate equating a bond’s price with promised cash flows.

Spot rate: Discount rate applicable to one maturity cash flow.

Forward rate: Rate implied for borrowing or lending during a future period.

Term structure: Relationship between interest rates and maturities.

Yield spread: Yield difference between a security and a reference instrument.

Clean price: Quoted bond price excluding accrued interest.

Dirty price: Full settlement price including accrued interest.

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

Modified duration: Approximate percentage price sensitivity to yield changes.

Convexity: Second-order sensitivity of price to yield changes.

Credit risk: Risk of loss from deterioration or failure of an obligor.

PD: Probability of default.

LGD: Loss given default.

Securitization: Pooling assets and issuing claims supported by their cash flows.

Prepayment risk: Risk that borrowers repay earlier than projected.

Immunization: Matching asset and liability value and interest-rate sensitivity.

LDI: Investment focused on managing assets relative to liability risks.

Tracking error: Volatility of portfolio returns relative to a benchmark.

Review Questions

Short Answer Questions

1. Why does a conventional bond’s price fall when its required yield rises?
2. Distinguish clean price from dirty price.
3. What does modified duration measure?
4. Identify two risks specific to mortgage-backed securities.
5. State the basic conditions for single-liability immunization.

Long Answer Questions

1. Explain bond pricing using spot rates and compare it with pricing using a single YTM.
2. Discuss the expectations, liquidity-preference, and preferred-habitat theories of the term structure.
3. Explain duration and convexity, including their assumptions and limitations.
4. Analyze the principal risks and structural protections in securitized products.
5. Compare passive, enhanced-index, and active fixed-income management.

Analytical / Application Questions

1. A bond has modified duration 7 and convexity 60. Estimate its percentage price change for a 75-basis-point yield increase using the duration-convexity approximation. Explain why the estimate may differ from the realized return.
2. A pension fund’s liabilities have duration 16 while assets have duration 9. Design a diagnostic and restructuring process using key-rate durations, liquidity stress tests, and governance controls.
3. A callable corporate bond offers a smaller spread than a comparable noncallable bond. Explain how the embedded call affects valuation, duration, convexity, and the appropriate spread measure.

Multiple Choice Questions with Answers

Question 1

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

A. A premium to par

B. A discount to par

C. Exactly par

D. A price unrelated to its coupon

Correct Answer: B. A discount to par

Question 2

Which measure is the present-value-weighted average timing of a bond’s cash flows?

A. Current yield

B. Macaulay duration

C. Credit spread

D. Convexity

Correct Answer: B. Macaulay duration

Question 3

If modified duration is 4 and yield rises by 1%, the duration-only estimated price change is:

A. +4%

B. +0.04%

C. −4%

D. −0.04%

Correct Answer: C. −4%

Question 4

Which risk is especially associated with mortgage-backed securities?

A. Prepayment risk

B. Inventory turnover risk

C. Goodwill impairment risk

D. Pension service-cost risk

Correct Answer: A. Prepayment risk

Question 5

In the simplified expected-loss expression, PD represents:

A. Price duration

B. Probability of default

C. Portfolio discount

D. Principal duration

Correct Answer: B. Probability of default

Question 6

A constant spread added to each spot rate to match a risky bond’s price is called the:

A. Coupon spread

B. Z-spread

C. Bid-ask spread

D. Recovery spread

Correct Answer: B. Z-spread

Question 7

A basic single-liability immunization requires asset duration to be:

A. Zero

B. Greater than twice liability duration

C. Equal to liability duration

D. Equal to the coupon rate

Correct Answer: C. Equal to liability duration

Question 8

Positive convexity implies that, for equal-sized yield movements:

A. Price gains from falling yields can exceed price losses from rising yields

B. Price never changes

C. Price losses always exceed gains

D. Duration becomes zero

Correct Answer: A. Price gains from falling yields can exceed price losses from rising yields

Question 9

Which strategy expresses a view that the yield difference between two maturities will change?

A. Yield-curve positioning

B. Cash indexing

C. Face-value accounting

D. Coupon clipping

Correct Answer: A. Yield-curve positioning

Question 10

An information ratio is commonly calculated as active return divided by:

A. Face value

B. Coupon rate

C. Tracking error

D. Macaulay duration

Correct Answer: C. Tracking error