Fixed-Income Analysis and Valuation

Introduction

Fixed-income securities represent contractual claims on future cash flows. Governments, financial institutions and corporations issue them to obtain funding, while investors use them to generate income, preserve capital, diversify portfolios and match future liabilities. Despite the term “fixed income,” bond returns are not necessarily fixed: interest-rate movements, reinvestment opportunities, inflation, credit events, liquidity and embedded options can materially change realised outcomes.

Fixed-income analysis therefore combines contractual analysis, time-value-of-money valuation, yield-curve modelling, interest-rate risk measurement and credit assessment. These capabilities are central to treasury management, insurance asset–liability management, banking, pension investment and portfolio management.

Learning Objectives

After studying this chapter, students should be able to:

• explain the contractual and structural features of bonds;
• calculate bond prices, accrued interest and major yield measures;
• interpret yield curves, spot rates and forward rates;
• calculate and apply duration, DV01 and convexity;
• design a basic immunisation strategy and explain reinvestment risk;
• assess credit quality, default probability, recovery and credit spreads;
• distinguish major securitised instruments and their risks; and
• conduct interest-rate and credit-spread scenario analysis.

1. Anatomy of a Bond

A bond is a debt instrument under which the issuer promises specified payments to investors. The legal document defining these obligations is usually called the indenture or trust deed.

1.1 Principal features

For a fixed-rate bond with annual coupon rate c, face value F and m payments per year, each regular coupon is:

Here, C is the coupon payment in currency units. The formula assumes equal periodic coupons.

1.2 Common bond structures

• Fixed-rate bond: pays a predetermined coupon.
• Floating-rate note: pays a reference rate plus a quoted margin, subject in some cases to caps or floors.
• Zero-coupon bond: pays no periodic coupon and is issued below its redemption value.
• Callable bond: gives the issuer the right to redeem early.
• Putable bond: allows the investor to sell the bond back under specified conditions.
• Convertible bond: may be exchanged for issuer equity according to stated terms.
• Inflation-linked bond: adjusts principal, coupons or both using an inflation index.
• Perpetual bond: has no contractual maturity, although it may contain call provisions.

Embedded options alter cash-flow timing. Consequently, a callable or mortgage-backed security should not be analysed as though its cash flows were fixed.

2. Bond Pricing

2.1 Present-value principle

The value of a bond equals the present value of its expected contractual cash flows discounted at rates appropriate to their timing and risk. If a single yield to maturity is used for a conventional fixed-rate bond, its price is:

where:

• P = bond price in currency units;
• C = coupon per payment period;
• F = face value;
• y = nominal annual yield compounded m times per year;
• N = remaining number of coupon periods.

This formula assumes fixed promised cash flows, regular coupon intervals and the same periodic discount rate for every cash flow. Using one yield is convenient but does not reproduce the full term structure.

2.2 Worked example: pricing a coupon bond

A five-year bond has face value 1,000, an annual coupon rate of 6%, annual payments and a yield to maturity of 7%.

Using the five-year annuity factor of approximately 4.10020:

The bond trades below par because its 6% coupon rate is lower than the required 7% yield. If coupon rate equals yield, a standard bond prices at par on a coupon date; if coupon rate exceeds yield, it trades at a premium.

2.3 Clean price, dirty price and accrued interest

The dirty price is the full settlement value, including accrued interest. The clean price excludes accrued interest and is commonly the quoted market price.

Define β as the fraction of the current coupon period that has elapsed since the previous coupon date. Under simple proportional accrual:

Thus:

Let τ = 1 − β be the fraction of a coupon period remaining until the next coupon. If the next payment is indexed by k = 1, the dirty price under a single periodic yield is:

This notation is internally consistent: β measures elapsed time, while τ measures time to the next coupon. Actual accrued interest depends on the bond’s day-count convention, such as Actual/Actual or 30/360.

Example. A semiannual coupon is 30 and 40% of the coupon period has elapsed. Accrued interest is 0.40 × 30 = 12. If the clean price is 982, the dirty price is 994.

2.4 Pricing with spot rates

Arbitrage-free valuation discounts each payment using the spot rate corresponding to its maturity:

Here, s_t is the annual effective spot rate for maturity t, and CF_t is the cash flow at that date. The equation assumes annual periods; compounding conventions must be adjusted when market practice differs.

3. Yield Measures

A yield summarises return under particular assumptions. Different yield measures answer different questions.

3.1 Current yield

Current yield ignores capital gains or losses, reinvestment and the time value of principal.

3.2 Yield to maturity

Yield to maturity (YTM) is the single discount rate that equates the bond’s dirty price with the present value of its promised cash flows. It is an internal rate of return, not a guaranteed realised return. Realising the quoted YTM requires, among other conditions, holding the bond to maturity, avoiding default and reinvesting interim coupons at the same yield.

3.3 Yield to call and yield to worst

Yield to call assumes redemption on a specified call date at the contractual call price. Yield to worst is generally the lowest yield among relevant permitted redemption scenarios, excluding issuer default. These measures are especially relevant when market yields have fallen and an issuer has an economic incentive to refinance.

3.4 Realised compound return

Realised return depends on the sale price or redemption proceeds and the actual reinvestment rates earned on coupons. It can therefore differ considerably from initial YTM.

4. Term Structure, Spot Rates and Forward Rates

The term structure of interest rates describes the relationship between yields and time to maturity for securities of comparable credit quality and liquidity.

Figure: Common yield-curve shapes; actual curves can also be humped or irregular.

4.1 Interpretations of the yield curve

• Pure expectations theory: long rates reflect expected future short rates, abstracting from term premiums.
• Liquidity preference theory: investors usually require compensation for holding longer maturities, creating a positive term premium.
• Market segmentation theory: supply and demand within maturity sectors influence yields because investors do not regard all maturities as close substitutes.
• Preferred habitat theory: investors prefer certain maturity ranges but may move if adequately compensated.

Observed curves may reflect all these forces as well as monetary policy, inflation expectations, fiscal conditions and risk aversion.

4.2 Spot-rate bootstrapping

Bootstrapping extracts spot rates sequentially from prices of default-free instruments. Suppose a one-year zero-coupon rate is 4% and a two-year 5% annual-coupon bond trades at par. For face value 1,000:

Therefore:

Hence s₂ ≈ 5.025%. Bootstrapping requires consistent prices, settlement dates, day-count rules and compounding conventions.

4.3 Forward rates

A forward rate is the interest rate implied today for borrowing or investing between two future dates. For annual effective spot rates, the one-year forward rate from year n−1 to year n is:

For example, if s₁ = 4% and s₂ = 5%:

An implied forward rate is not necessarily an unbiased forecast. It may include term, liquidity and risk premiums.

5. Interest-Rate Risk: Duration and DV01

Bond prices generally move inversely to required yields. The relationship is nonlinear and is affected by maturity, coupon, yield level and embedded options.

5.1 Macaulay duration

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

Here, t_i is measured in years, PV(CF_i) is the present value of cash flow i, and P is the full price. Macaulay duration is measured in years.

5.2 Modified duration

For a fixed-cash-flow bond valued using a nominal annual YTM compounded m times:

For a small change in yield:

The yield change must be expressed as a decimal. The approximation assumes a small, uniform change in the bond’s yield and unchanged cash flows.

5.3 Worked example: duration

Consider a three-year bond with face value 1,000, an 8% annual coupon and a 10% annual yield.

For a 1 percentage-point increase in yield, the duration-only estimate is a price decline of approximately 2.525%.

5.4 DV01

Dollar value of one basis point (DV01) estimates the currency change in value for a one-basis-point yield movement:

For the preceding bond:

Thus, a one-basis-point yield increase reduces the bond’s value by approximately 0.240 currency units. Portfolio DV01 is computed using total portfolio market value and portfolio duration, subject to the same small-shift assumptions.

6. Convexity

Duration is a first-order, straight-line approximation. Convexity captures curvature in the price–yield relationship.

For an annual-pay, fixed-cash-flow bond using an annually compounded yield, a discrete formula is:

This particular expression should not be applied unchanged to other compounding frequencies. With convexity, the price-change approximation becomes:

6.1 Worked example: convexity

For the three-year bond in the duration example:

For a yield increase of ∆y = 0.01:

The estimated new price is therefore approximately:

Direct repricing at 11% gives approximately 926.69, demonstrating the improvement obtained by including convexity.

Most option-free bonds have positive convexity. Callable bonds and mortgage-backed securities can exhibit negative convexity when falling rates increase the probability of early redemption or prepayment.

7. Reinvestment Risk and Immunisation

7.1 Reinvestment risk

Reinvestment risk is the possibility that interim cash flows will be reinvested at rates lower than initially anticipated. High-coupon bonds have greater coupon reinvestment exposure than otherwise similar low-coupon bonds. Zero-coupon bonds held to maturity have no interim coupon reinvestment risk, although they retain price risk if sold early and default risk where applicable.

Price risk and reinvestment risk move in opposite directions. Rising rates reduce current bond prices but improve the reinvestment rate on coupons; falling rates increase prices but reduce reinvestment income.

7.2 Immunisation principles

Immunisation seeks to make asset value sufficient to meet a liability despite small interest-rate changes. A classical single-liability strategy generally requires:

1. asset market value equal to the present value of liabilities;
2. asset duration equal to liability duration;
3. adequate asset convexity relative to liability convexity; and
4. periodic rebalancing as time passes and yields change.

7.3 Worked example: immunising a liability

An institution must pay 100,000 in four years. The annual effective interest rate is 5%. It can invest in two-year and six-year zero-coupon bonds.

The liability present value is:

Because a zero-coupon bond’s Macaulay duration equals its maturity, let A₂ and A₆ be the current market values invested in the two-year and six-year zeros:

Duration matching requires:

Solving gives:

The corresponding redemption amounts are approximately 45,351.47 for the two-year zero and 55,125.00 for the six-year zero. The two-year proceeds must be reinvested after maturity, so the strategy is not a cash-flow match. It is protected only approximately against small, parallel yield-curve changes and must be rebalanced.

Cash-flow matching is more restrictive: asset cash flows are scheduled to meet liability payments directly. It reduces reinvestment dependence but may be more costly or difficult to implement.

8. Credit Analysis and Credit Spreads

8.1 Sources of credit risk

Credit risk includes more than outright default. Investors must consider:

• failure to pay interest or principal;
• rating migration and spread widening;
• reduced recovery value after default;
• covenant deterioration;
• liquidity loss; and
• correlated exposure to industries, regions or economic factors.

Analysts evaluate business risk, profitability, cash-flow stability, leverage, interest coverage, liquidity, collateral, seniority and governance. Sovereign analysis additionally considers fiscal capacity, external balances, monetary credibility, political stability and currency regime.

8.2 Expected loss

A basic one-period credit-loss model is:

where:

• PD = probability of default over the horizon;
• LGD = loss given default as a proportion of exposure;
• EAD = exposure at default in currency units.

If PD is 2%, expected recovery is 40%, and EAD is 5 million, then LGD is 60% and:

Expected loss is not the maximum possible loss and does not fully capture uncertainty, concentration or tail dependence.

8.3 Ratings and transition risk

Credit-rating agencies express relative creditworthiness using rating categories. Ratings are opinions rather than guarantees. Investors should supplement them with independent analysis because ratings may lag changing conditions and can be affected by model, information and conflict-of-interest limitations.

A transition matrix estimates the probabilities that an issuer moves from one rating category to another over a stated horizon. Migration to a lower rating can produce losses even if no default occurs.

8.4 Credit spreads

A credit spread is the yield compensation above a benchmark rate. It may compensate for expected default loss, unexpected credit risk, liquidity, taxation, risk aversion and market technical factors.

A spread of 180 basis points equals 1.80%. On a 10 million exposure, 1.80% represents 180,000 per year—not 18,000—when applied as a simple annual rate difference:

Common spread measures include benchmark spread, interpolated spread, zero-volatility spread and option-adjusted spread. Option-adjusted spread is particularly useful when embedded options make promised cash flows interest-rate dependent, although its estimate depends on the chosen interest-rate and option model.

9. Securitised Instruments

Securitisation pools financial assets and issues securities backed by their cash flows. A special-purpose vehicle is commonly used to separate the pool from the originator, subject to legal requirements.

Cash flows are often allocated through a waterfall. Senior tranches receive priority and usually have lower expected loss, while junior tranches absorb losses first. Credit enhancement may include subordination, overcollateralisation, reserve accounts or guarantees.

Prepayment creates distinctive interest-rate behaviour. When rates fall, mortgage borrowers may refinance, returning principal earlier and limiting price appreciation—contraction risk. When rates rise, prepayments may slow, lengthening duration as values decline—extension risk.

Securitisation can improve funding and risk distribution, but complex structures may obscure underwriting quality. Investors must investigate loan-level data, servicing standards, legal enforceability, model assumptions and incentive alignment.

10. Interest-Rate and Spread Scenario Analysis

Duration and convexity summarise local sensitivity but cannot represent every possible yield-curve movement. Scenario analysis revalues securities under explicitly defined changes.

Important scenarios include:

• parallel shift: all maturities move by the same number of basis points;
• steepening: long-term rates rise relative to short-term rates;
• flattening: the difference between long- and short-term rates narrows;
• curvature change: intermediate maturities move differently from the ends;
• spread shock: credit or liquidity spreads widen or tighten;
• volatility shock: changes in rate volatility alter the value of embedded options;
• default or downgrade: cash flows, recovery assumptions and liquidity change.

Key-rate duration measures sensitivity to selected maturity points and is more informative than one aggregate duration for non-parallel curve changes. Effective duration is generally preferred for option-bearing instruments because it allows projected cash flows to change when rates change.

11. Applied Case: Insurance Liability Portfolio

Background

This is an illustrative case. Meridian Life must meet a known four-year claim payment while also investing part of its surplus in corporate bonds. Management is concerned about rising rates, credit-spread widening and governance scrutiny.

Problem

A liability-only strategy based on short-term instruments creates reinvestment risk. A long-duration corporate portfolio may provide yield but exposes the insurer to price volatility and credit deterioration. Management therefore needs an integrated asset–liability and credit approach.

Analysis and approach

The treasury team first discounts the known liability using an appropriate high-quality yield curve. It matches present value and duration using high-quality zero-coupon exposures, following the immunisation calculation presented earlier. Because duration matching assumes small parallel shifts, the team also examines key-rate durations and tests steepening and flattening scenarios.

For the surplus portfolio, a proposed 10 million corporate position offers a spread of 180 basis points. The gross annual spread compensation is 180,000. The team does not treat this amount as profit automatically: expected default loss, liquidity cost, capital consumption, transaction cost and option risk must be deducted or evaluated separately.

The insurer performs issuer analysis, limits sector concentration, estimates PD and LGD, and stress-tests a spread widening of 100 basis points. Callable holdings are valued using an option-adjusted framework rather than relying solely on YTM.

Outcome

Management adopts the immunised high-quality portfolio for the contractual liability and a diversified, limit-controlled corporate allocation for surplus funds. Quarterly rebalancing thresholds are established because durations drift as time passes and yields change.

Learning points

• Liability matching and return maximisation are different objectives.
• Duration matching is approximate and requires monitoring.
• Credit spreads compensate for several risks, not merely expected default.
• Embedded options and non-parallel curve changes require scenario-based valuation.
• Governance limits and independent model validation are integral to portfolio construction.

12. Governance, Regulation and Technology

Banks and insurers must manage interest-rate and credit risks within applicable capital, liquidity, accounting and supervisory frameworks. Sound governance includes approved risk limits, independent valuation, model validation, stress testing, accurate data lineage and escalation of breaches.

Ethical concerns arise when issuers obscure leverage, originators weaken underwriting, rating processes face conflicts, or portfolio managers chase yield without communicating downside risk. Fiduciaries should disclose valuation uncertainty and avoid representing model estimates as certainties.

Modern fixed-income systems bootstrap curves, calculate key-rate exposures, simulate defaults and process large quantities of transaction and loan-level data. Machine-learning models can assist credit monitoring and anomaly detection, but unstable relationships, biased training data and limited explainability can create new risks. Human oversight and validation remain essential.

13. Analytical Limitations and Best Practices

Fixed-income models simplify reality. YTM assumes a common discount rate; duration relies on local rate changes; credit models depend on estimated probabilities and recoveries; and option-adjusted valuation depends on volatility and behavioural assumptions. Market prices may also reflect liquidity, regulation, collateral eligibility and forced trading.

Professional best practice is to:

1. verify cash flows, settlement rules and embedded options;
2. use consistent compounding and day-count conventions;
3. value cash flows using an appropriate term structure;
4. supplement duration with convexity, key-rate measures and full revaluation;
5. separate expected credit loss from unexpected and liquidity risks;
6. stress-test rates, spreads, defaults and prepayments jointly;
7. monitor concentration, collateral and covenant quality; and
8. document assumptions and validate models independently.

Chapter Summary

Bond value is the present value of future cash flows, discounted at rates consistent with timing, credit quality, liquidity and optionality. Clean and dirty prices differ by accrued interest, while yield measures provide simplified return indicators subject to important assumptions. Spot rates support arbitrage-free valuation, and forward rates express rates implied between future dates.

Duration measures first-order interest-rate sensitivity, convexity improves the estimate for larger yield changes, and DV01 expresses sensitivity in currency terms. Immunisation combines present-value and duration matching but requires convexity consideration and rebalancing. Credit analysis evaluates default probability, loss severity, migration, liquidity and governance. Securitised instruments add structural, prepayment and tranche risks. Effective professional analysis integrates valuation, scenario testing, independent credit work and disciplined risk governance.

Key Terms

Accrued Interest: Coupon interest earned since the previous coupon date but not yet paid.

Clean Price: Quoted bond price excluding accrued interest.

Convexity: Measure of curvature in the bond price–yield relationship.

Credit Spread: Yield difference between a credit-risky instrument and a benchmark.

Dirty Price: Full settlement price including accrued interest.

Duration: Measure of the timing of cash flows or sensitivity to yield changes.

DV01: Approximate currency change in value for a one-basis-point yield movement.

Expected Loss: Probability-weighted credit loss, commonly calculated as PD × LGD × EAD.

Forward Rate: Interest rate implied today for a future period.

Immunisation: Strategy designed to protect the ability of assets to meet liabilities against specified rate changes.

Loss Given Default: Proportion of exposure lost if default occurs.

Option-Adjusted Spread: Spread estimated after accounting for embedded-option effects.

Reinvestment Risk: Risk that interim cash flows will be reinvested at unfavourable rates.

Securitisation: Process of issuing securities backed by pooled financial assets.

Spot Rate: Current discount rate applicable to a single cash flow at a specified maturity.

Term Structure: Relationship between interest rates and maturity for comparable instruments.

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

Zero-Coupon Bond: Bond that makes no periodic coupon payments and pays its face value at maturity.

Review Questions

Short Answer Questions

1. Distinguish between a bond’s clean price and dirty price.
2. Why can realised return differ from yield to maturity?
3. What information does DV01 provide?
4. Explain the difference between contraction risk and extension risk.
5. Identify the three components of the expected-loss formula.

Long Answer Questions

1. Explain the features that determine a bond’s contractual cash flows and risk profile.
2. Compare Macaulay duration, modified duration, effective duration and key-rate duration.
3. Discuss the main theories used to interpret the shape of the yield curve.
4. Evaluate the benefits and limitations of credit ratings in portfolio management.
5. Explain how securitisation reallocates credit and prepayment risks among investors.

Analytical / Application Questions

1. A six-year annual-pay bond has a 7% coupon, face value 1,000 and YTM of 8%. Calculate its price and explain whether it trades at a premium or discount.
2. Construct a duration-matched portfolio for a five-year single liability using three-year and seven-year zero-coupon bonds. State the assumptions and limitations of your solution.
3. Analyse how a simultaneous 100-basis-point rate increase and 150-basis-point credit-spread widening would affect an option-free corporate-bond portfolio. Explain which measures and scenarios should be used.

Multiple Choice Questions with Answers

Question 1

A five-year annual-pay bond has face value 1,000, a 6% coupon and a 7% YTM. Its approximate price is:

A. 930.00

B. 959.00

C. 1,000.00

D. 1,041.00

Correct Answer: B. 959.00

Question 2

A semiannual coupon is 30 and 40% of the coupon period has elapsed. Under proportional accrual, accrued interest is:

A. 12

B. 18

C. 30

D. 42

Correct Answer: A. 12

Question 3

If the one-year spot rate is 4% and the two-year spot rate is 5%, both annual effective rates, the implied one-year forward rate beginning in one year is approximately:

A. 4.00%

B. 5.00%

C. 6.01%

D. 9.20%

Correct Answer: C. 6.01%

Question 4

A bond has modified duration of 4.5. Ignoring convexity, a 50-basis-point increase in yield produces an approximate price change of:

A. −2.25%

B. −0.225%

C. +2.25%

D. −4.50%

Correct Answer: A. −2.25%

Question 5

What is the principal benefit of including convexity in a bond price-change estimate?

A. It eliminates default risk

B. It captures curvature in the price–yield relationship

C. It guarantees reinvestment at the original YTM

D. It converts a dirty price into a clean price

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

Question 6

A credit exposure has PD of 2%, recovery rate of 40% and EAD of 5 million. Expected loss is:

A. 40,000

B. 60,000

C. 100,000

D. 200,000

Correct Answer: B. 60,000

Question 7

A spread of 180 basis points applied to 10 million for one year represents:

A. 18,000

B. 80,000

C. 180,000

D. 1,800,000

Correct Answer: C. 180,000

Question 8

To duration-match a four-year liability using only two-year and six-year zero-coupon bonds at the same yield, what proportions of present market value should be invested in each?

A. 25% in the two-year and 75% in the six-year

B. 50% in each

C. 75% in the two-year and 25% in the six-year

D. 100% in the four-year liability fund

Correct Answer: B. 50% in each

Question 9

Which risk occurs when falling interest rates encourage mortgage borrowers to repay principal earlier than expected?

A. Extension risk

B. Contraction risk

C. Currency risk

D. Settlement risk

Correct Answer: B. Contraction risk

Question 10

Which measure is generally most appropriate for estimating interest-rate sensitivity when a bond's projected cash flows change as rates change?

A. Current yield

B. Macaulay duration based on fixed cash flows

C. Effective duration

D. Coupon rate

Correct Answer: C. Effective duration