Fixed-Income Analysis and Valuation

Introduction and Context

Fixed-income securities are contractual claims under which an issuer promises specified or determinable cash flows to investors. Governments, financial institutions and corporations use bonds to finance expenditure, investment and refinancing. Investors use them to generate income, preserve capital, manage liquidity and diversify risk. Unlike equity, a bond normally gives its holder no ownership claim; its value depends primarily on promised cash flows, interest rates, time, liquidity and the issuer’s ability to pay.

Fixed-income analysis is therefore both a valuation discipline and a risk-management discipline. The analyst must determine what a bond should be worth, identify the risks embedded in that value, and construct portfolios whose sensitivity to rates, inflation and credit events is consistent with investment objectives.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Value coupon, zero-coupon, floating-rate and perpetuity bonds.
2. Distinguish price, coupon rate, current yield, yield to maturity and spot rates.
3. Explain the term structure of interest rates and derive forward rates.
4. Measure interest-rate sensitivity using duration and convexity.
5. Explain immunisation and its limitations.
6. Analyse credit spreads, ratings and default probabilities.
7. Evaluate fixed-income portfolio strategies and governance issues.

1. Bond Cash Flows and Pricing

A bond’s contractual cash flows consist of periodic interest (coupon) payments and repayment of principal at maturity. If a bond has face value F, annual coupon rate c, m coupon payments per year, and n years to maturity, its periodic coupon is C = Fc/m and the number of periods is N = mn. With a yield to maturity y compounded m times annually, the price is:

Here P is the clean or invoice price before any accrued-interest adjustment, C and F are measured in currency units, and y is an annual nominal yield. The formula assumes deterministic contractual cash flows, no default, and a constant yield for all payment dates. It is a present-value equation: each cash flow is discounted at the same periodic rate.

For a zero-coupon bond, C = 0:

For an annual coupon bond, m = 1. A perpetuity with constant annual coupon C and no maturity has value, provided y > 0:

A bond trades at par when its coupon rate equals its required yield, at a premium when its coupon rate exceeds the required yield, and at a discount when its coupon rate is lower. Holding all else constant, bond prices move inversely to yields. This inverse relationship follows directly from present-value discounting, not from a separate behavioural rule.

Clean Price, Dirty Price and Accrued Interest

Between coupon dates, the buyer compensates the seller for interest earned since the last coupon. The dirty price, or invoice price, is:

Under a simple 30/360 convention, accrued interest may be approximated as:

Actual bond contracts use specified day-count conventions, such as Actual/Actual or 30/360. Settlement date, ex-coupon rules and market convention must therefore be checked before calculating a transaction price.

Worked Example 1: Price and Yield

An annual-pay bond has face value 1,000, coupon rate 8%, five years remaining and a required yield of 10%. Its annual coupon is 80. The price is:

The bond trades below par because its 8% coupon is below the 10% required yield. If the market price were known instead, the yield to maturity would be the discount rate that equates the present value of promised cash flows to that price. Except for simple cases, this rate is obtained numerically.

2. Yield Measures and Return Analysis

The coupon rate is the annual coupon divided by face value. It is fixed by the contract and does not describe the investor’s current return. Current yield is:

For the bond in Example 1, current yield is 80/924.18 ≈ 8.66%. It excludes capital gain or loss and therefore differs from total expected return.

Yield to maturity (YTM) is the internal rate of return that equates the bond price with the present value of promised cash flows, assuming that:

• the bond is held to maturity;
• the issuer makes all promised payments; and
• interim coupons can be reinvested at the YTM.

These assumptions are often unrealistic. YTM is consequently a standard valuation convention, not a guaranteed realised return.

For a callable bond, yield to call measures the return if the issuer exercises a specified call. Yield to worst is generally the lowest relevant yield among maturity, call and other permitted redemption scenarios, subject to market convention. Floating-rate notes are commonly valued from the reference rate plus or minus a quoted spread; their prices are especially sensitive to changes in credit spreads and reset conventions.

Holding-period return for a bond bought at P₀ and sold at P₁, with coupons received during the period, is:

The formula is an ex post accounting return. It does not by itself specify reinvestment assumptions or separate the effects of rate movements, credit changes and liquidity.

3. Spot Rates, Forward Rates and the Term Structure

A spot rate is the annualised yield applicable to a single cash flow paid at a particular future date. Let s_t be the annual effective spot rate for maturity t years. The discount factor is:

If a bond’s cash flows are valued using maturity-specific spot rates, its no-arbitrage value is:

This is generally preferable to discounting every cash flow at one YTM because the term structure need not be flat.

A forward rate is an interest rate agreed today for borrowing or lending over a future interval. With annual effective spot rates, the one-year forward rate from year t to year t+1 is:

Thus:

Forward rates are implied rates under no-arbitrage; they are not necessarily unbiased forecasts of future spot rates. Risk premia, liquidity premia and investor preferences can cause expected future short rates to differ from forwards.

Bootstrapping

The zero-coupon or spot curve can be estimated by bootstrapping coupon-bond prices. The shortest maturity supplies the first discount factor. A longer bond’s price, after subtracting the present value of earlier coupons discounted using previously estimated factors, identifies the discount factor for its final payment. The procedure requires consistent prices, cash-flow dates, settlement conventions and a sufficiently liquid instrument set.

The term structure is often described as upward sloping, downward sloping, flat or humped. Explanations include expectations theory, liquidity-preference theory and market-segmentation theory. Expectations theory links long rates to expected future short rates; liquidity-preference theory adds a maturity-risk premium; market segmentation emphasises supply and demand within maturity sectors. No single explanation fits all markets and periods.

4. Interest-Rate Risk: Duration and Convexity

Macaulay and Modified Duration

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

The time t is measured in coupon periods. In years, divide the result by m. Modified duration measures approximate percentage price sensitivity to a change in annual nominal yield:

For a small parallel yield change Δy, with Δy expressed as a decimal:

The negative sign expresses the inverse price–yield relation. Duration is a first-order approximation and is most reliable for small yield changes and option-free bonds.

Worked Example 2: Duration Estimate

Suppose a bond has modified duration of 4.8 and its yield rises by 0.50%, or 0.005. The estimated price change is:

The estimated price decline is 2.4%. If the bond’s price is 980, the estimated loss is 23.52, giving an estimated new price of 956.48 before considering convexity and other effects.

Convexity

Convexity improves the duration approximation by incorporating curvature. A discrete annual-compounding expression is:

For other compounding conventions, the corresponding derivative must be used consistently. The second-order price approximation is:

For ordinary option-free bonds, convexity is usually positive: the price rises more for a yield decline than it falls for an equal yield increase. Callable bonds may exhibit negative or reduced convexity when interest rates fall because the call option becomes more valuable to the issuer.

Duration and convexity describe market-value sensitivity, not credit loss, liquidity loss or model error. Key-rate duration is often more informative than a single duration because it measures sensitivity to changes at particular maturities rather than assuming a parallel shift in the entire curve.

5. Immunisation and Liability Management

Classical immunisation seeks to protect a portfolio’s value against small, parallel interest-rate changes while meeting a known liability. A basic single-liability strategy matches:

• present value of assets and liabilities;
• Macaulay duration of assets and liabilities; and
• sufficient convexity, preferably asset convexity greater than liability convexity.

If asset and liability present values are equal and durations are matched, the first-order effects of a small parallel rate change approximately offset. Immunisation is not permanent: time passes, coupons are received, yields move non-parallelly and credit conditions change. Portfolios must therefore be rebalanced.

For multiple liabilities, practitioners use cash-flow matching, duration matching, key-rate matching or an optimisation model. Cash-flow matching reduces reinvestment and market-value uncertainty but may be expensive. Duration matching is cheaper and more flexible but depends on assumptions about yield-curve movements.

6. Credit Risk, Ratings and Default Probabilities

Credit risk is the possibility that an issuer or counterparty fails to make contractual payments or that its credit quality deteriorates. It includes default risk, downgrade risk, spread risk, recovery risk and migration risk.

A credit spread is the yield difference between a risky bond and a comparable benchmark, usually adjusted for maturity, currency, liquidity and embedded options. A spread is not a pure default probability. It may compensate investors for expected loss, unexpected loss, liquidity, taxation, risk aversion and model uncertainty.

Credit ratings from agencies classify relative credit quality using symbols such as investment-grade and speculative-grade categories. Ratings are opinions, not guarantees, and can lag rapidly changing information. Internal credit analysis should examine leverage, interest coverage, cash-flow stability, refinancing needs, collateral, covenants, industry conditions, governance and management quality.

A simplified expected-loss relationship is:

PD is probability of default over a specified horizon, LGD is loss given default as a proportion of exposure, and EAD is exposure at default. All variables must refer to the same horizon and be defined consistently. If recovery rate is R, then LGD is approximately 1 − R, subject to seniority, costs and timing.

For a one-period risky bond with promised payoff F, risk-neutral valuation can be represented simply as:

where q is a one-period risk-neutral default probability, R is the recovery fraction, and r is the risk-free rate. This is a stylised model: actual valuation requires default timing, stochastic recovery, risk premia and possibly coupon payments. Historical or physical PDs are not automatically interchangeable with risk-neutral PDs implied by market prices.

Credit Analysis and Governance

A disciplined process combines quantitative and qualitative evidence. Analysts should test base, adverse and severe scenarios; identify covenant headroom; monitor rating migrations and market spreads; and distinguish issuer liquidity from solvency. Concentration limits, independent approval, documented overrides and clear escalation procedures are essential. Ethical practice requires accurate disclosure of valuation assumptions, conflicts of interest, model limitations and uncertainty around ratings.

7. Fixed-Income Portfolio Strategies

Passive and Liability-Driven Strategies

A buy-and-hold strategy emphasises contractual income and controlled turnover, but it remains exposed to default, reinvestment and liquidity risk. Indexing seeks to replicate a bond index, though sampling, transaction costs, cash flows and illiquidity create tracking error.

Liability-driven investing aligns assets with the timing, amount and risk characteristics of liabilities. It is common for pension funds, insurers and institutions with predictable payment obligations. Surplus risk should be measured as the change in asset value relative to liability value, not merely as the volatility of the asset portfolio.

Active Strategies

Active managers may use:

• Riding the yield curve: purchasing securities expected to roll into a shorter, lower-yield maturity segment, assuming the curve remains sufficiently stable.
• Yield-curve positioning: overweighting or underweighting maturities based on a rate view.
• Sector rotation: shifting among government, corporate, securitised and municipal sectors.
• Credit selection: seeking compensation for mispriced default and spread risk.
• Barbell strategy: combining short and long maturities, often increasing convexity.
• Bullet strategy: concentrating maturities around a target date.
• Swap strategies: exchanging similar-risk securities when relative value appears attractive.

Active decisions must be evaluated after transaction costs, taxes, liquidity effects and risk-adjusted performance. Leverage and derivatives can magnify both gains and losses and require collateral, counterparty and legal documentation controls.

8. Illustrative Case: Meridian Pension Fund

Background

Meridian Pension Fund expects to pay 50 million currency units in five years. Its assets include government bonds, investment-grade corporate bonds and cash. The investment committee is concerned that a rise in rates will reduce bond prices, while a credit downturn could widen corporate spreads.

Problem

The fund’s portfolio has market-value duration of 3.2 years, while the five-year liability has duration close to 5 years. The assets therefore have greater sensitivity to a rate rise? No: because asset duration is shorter, asset value falls less than the liability value when rates rise, but the relationship is not sufficient to guarantee surplus protection. The mismatch also leaves the fund exposed to curve-shape changes.

Analysis and Approach

The committee first separates risks: government-rate exposure, corporate spread exposure, liquidity risk and reinvestment risk. It then establishes a liability benchmark and calculates present-value and key-rate exposures. A portion of the government allocation is extended to raise asset duration toward the liability duration. Corporate exposure is capped by issuer and sector, and the highest-risk names are replaced with stronger credits. Five-year key-rate exposure is matched more closely rather than relying only on aggregate duration.

Stress tests consider parallel rate changes, steepening, flattening, a 150-basis-point credit-spread widening and a recessionary default scenario. The fund retains liquid government securities to meet near-term payments and rebalances when duration or credit limits breach policy bands.

Outcome and Learning Points

The revised portfolio reduces first-order surplus sensitivity, but it does not eliminate risk. Rates may move non-parallelly, corporate spreads may widen independently of government yields, and defaults may occur. The case demonstrates that immunisation is a continuing governance process, not a one-time calculation. It also shows why duration, key-rate exposure, credit limits, liquidity and scenario analysis must be considered together.

9. Limitations and Contemporary Developments

Bond models can fail when cash flows are uncertain, markets are illiquid or correlations change. Callable and mortgage-backed securities require option-adjusted analysis because investors face prepayment and negative-convexity risk. Inflation-linked bonds require real-rate and inflation-breakeven analysis. Floating-rate instruments reduce duration risk but do not remove credit or basis risk.

Technology supports curve construction, portfolio optimisation, scenario generation, electronic execution and surveillance. Machine-learning systems may identify nonlinear patterns, but they can overfit historical data, reproduce biased ratings or obscure economic rationale. AI should therefore support—not replace—human validation, independent model governance, explainability, data-quality controls and compliance review.

Regulatory practice generally emphasises fair valuation, best execution, capital and liquidity adequacy, suitability, stress testing, independent risk management and transparent disclosure. Exact obligations vary by jurisdiction and instrument. Professionals must consult current applicable rules rather than relying solely on textbook conventions.

Chapter Summary

Fixed-income valuation discounts contractual cash flows using yields or, more precisely, maturity-specific spot rates. Coupon rate, current yield and YTM answer different questions and must not be confused. Spot rates generate discount factors, while forward rates are implied future-period rates under no-arbitrage; neither necessarily predicts realised future rates.

Duration provides a first-order estimate of price sensitivity to yield changes, and convexity improves that estimate for larger changes. Immunisation matches asset and liability value and interest-rate sensitivity, but requires continuing rebalancing and does not remove credit, liquidity or non-parallel yield-curve risk. Credit analysis combines ratings, financial analysis, spreads, default probabilities and recovery assumptions. Portfolio strategies range from buy-and-hold and indexing to yield-curve, sector, credit and liability-driven approaches. Sound practice integrates quantitative models with scenario analysis, governance, ethical disclosure and professional judgement.

Key Terms

Bond: A debt security representing a contractual claim on an issuer.

Coupon rate: Contractual annual coupon divided by face value.

Face value: Principal amount repaid according to the bond contract.

Clean price: Bond price excluding accrued interest.

Dirty price: Transaction price including accrued interest.

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

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

Forward rate: Rate implied today for a future borrowing or lending interval.

Discount factor: Present value of one currency unit received at a future date.

Term structure: Relationship between interest rates and maturities.

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.

Immunisation: Asset–liability technique intended to reduce interest-rate risk.

Credit spread: Yield premium over a comparable benchmark.

Probability of default: Likelihood of failure to meet contractual obligations.

Loss given default: Proportion of exposure lost after default and recovery.

Recovery rate: Fraction of exposure recovered after default.

Key-rate duration: Sensitivity to a change at a particular maturity point.

Liability-driven investing: Portfolio management centred on liability characteristics.

Review Questions

Short Answer Questions

1. Why does a bond’s price generally fall when its required yield rises?
2. Distinguish coupon rate, current yield and yield to maturity.
3. What is the purpose of bootstrapping a spot-rate curve?
4. State the relationship between modified duration and approximate price change.
5. Why is a credit spread not identical to expected default loss?

Long Answer Questions

1. Explain bond valuation using spot rates and compare it with single-rate YTM valuation.
2. Discuss Macaulay duration, modified duration and convexity, including their assumptions and limitations.
3. Explain classical immunisation and why it requires continuing rebalancing.
4. Evaluate the role of ratings, financial analysis and market spreads in credit assessment.
5. Compare passive, liability-driven and active fixed-income portfolio strategies.

Analytical / Application Questions

1. A five-year annual-pay bond has face value 1,000, coupon rate 6% and yield 8%. Calculate its price and explain why it trades at a discount.
2. A portfolio has modified duration 6.2 and value 20 million. Estimate the first-order change in value if yields rise by 0.75%. What important effects does this estimate omit?
3. An issuer has exposure at default of 10 million, probability of default of 3% and loss given default of 45%. Calculate expected loss and discuss why the result should not be treated as a complete credit-spread estimate.

Multiple Choice Questions with Answers

Question 1

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

A. A premium to face value

B. A discount to face value

C. Exactly face value

D. A price unrelated to yield

Correct Answer: B. A discount to face value

Question 2

Which measure includes the assumption that coupons are reinvested at the calculated rate?

A. Coupon rate

B. Current yield

C. Yield to maturity

D. Recovery rate

Correct Answer: C. Yield to maturity

Question 3

If the annual effective spot rates for one and two years are 5% and 6%, the one-year forward rate from year 1 to year 2 is approximately:

A. 5.00%

B. 6.00%

C. 7.01%

D. 11.00%

Correct Answer: C. 7.01%

Question 4

For a small parallel increase in yields, a bond with modified duration 4 will experience approximately:

A. A 4% price increase for a 1% yield increase

B. A 4% price decrease for a 1% yield increase

C. A 0.25% price decrease for a 1% yield increase

D. No price change

Correct Answer: B. A 4% price decrease for a 1% yield increase

Question 5

Convexity is useful because it:

A. Measures issuer recovery directly

B. Captures curvature in the price–yield relationship

C. Replaces the need for a bond price

D. Equals the coupon rate

Correct Answer: B. Captures curvature in the price–yield relationship

Question 6

In the simplified expected-loss formula, LGD means:

A. Liquidity growth duration

B. Loss given default

C. Long-term government discount

D. Legal guarantee deduction

Correct Answer: B. Loss given default

Question 7

Classical immunisation generally seeks to match asset and liability:

A. Coupon rates only

B. Present values and durations

C. Credit ratings only

D. Trading volumes

Correct Answer: B. Present values and durations

Question 8

A dirty bond price equals:

A. Clean price minus accrued interest

B. Clean price plus accrued interest

C. Face value plus coupon rate

D. Yield plus duration

Correct Answer: B. Clean price plus accrued interest

Question 9

If PD is 3%, LGD is 45% and EAD is 10 million, expected loss is:

A. 13,500

B. 135,000

C. 450,000

D. 3,000,000

Correct Answer: B. 135,000

Question 10

Which strategy most directly aligns assets with the timing and characteristics of institutional liabilities?

A. Liability-driven investing

B. Unrestricted yield chasing

C. Random maturity selection

D. Currency speculation

Correct Answer: A. Liability-driven investing