Bond and Credit Markets

Introduction

Bond markets transfer funds from investors to governments, municipalities, financial institutions and corporations. Unlike equity, a conventional bond establishes contractual payments: periodic interest and repayment of principal. Credit markets extend beyond publicly traded bonds to loans, private placements, commercial paper, securitized credit and credit derivatives.

Bond analysis integrates the time value of money, interest-rate risk, default risk, liquidity, taxation and contractual design. A bond’s quoted yield is therefore not simply a return measure; it reflects the benchmark yield curve, credit compensation, embedded options and market conditions.

Learning Objectives

After studying this chapter, students should be able to:

• distinguish government, municipal and corporate bonds;
• calculate bond prices, accrued interest and major yield measures;
• use spot rates to value contractual cash flows;
• calculate and interpret duration, DV01 and convexity;
• evaluate credit ratings, default probability, recovery and credit spreads;
• explain covenants, issuance procedures and secondary-market trading;
• assess the effects of interest-rate and credit-spread changes; and
• recognize relevant regulatory, ethical and technological developments.

1. Bond Contracts and Market Structure

A bond is a debt instrument under which an issuer promises specified payments to investors. The face value, or par value, is the principal amount used to calculate coupons and normally repaid at maturity. The coupon rate determines contractual interest, while the maturity date identifies when principal becomes due.

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

Here, C and F are currency amounts, c is an annual decimal rate, and m is the number of coupon periods per year. The formula assumes equal regular coupon periods.

1.1 Major bond categories

Government securities are often treated as reference instruments in their domestic currency, but they are not universally risk-free. Sovereigns can default, restructure debt, impose capital controls or repay nominal amounts whose purchasing power has been reduced by inflation.

Municipal debt commonly includes general obligation bonds, supported by broad taxing resources, and revenue bonds, supported by specified projects or revenue streams. Tax treatment is jurisdiction-specific. Credit analysis must not assume that all municipal debt carries an identical public guarantee.

Corporate bonds may be secured by collateral, senior unsecured, subordinated or structurally subordinated to debt issued by operating subsidiaries. Priority affects expected recovery if default occurs.

1.2 Primary and secondary markets

In the primary market, issuers raise new funds through auctions, syndicated offerings, public issues or private placements. Key participants include the issuer, legal advisers, trustees, rating agencies and investment banks. Underwriters may purchase an issue and resell it or act on a best-efforts basis.

In secondary markets, existing bonds trade through exchanges, electronic platforms or dealer-based over-the-counter networks. Institutional bonds frequently trade by request for quote rather than through a continuous central order book. Liquidity differs substantially by issue size, age, credit quality and market conditions.

A bond’s bid price is the price at which a dealer will buy; the ask price is the price at which the dealer will sell. Their difference compensates for inventory, search and adverse-selection risks.

2. Bond Pricing

2.1 Present-value model

A bond’s value equals the present value of its promised cash flows at discount rates appropriate to their timing and risk. If a single yield to maturity is used, the full or dirty price is:

where:

• P = dirty price in currency units;
• CF_t = cash flow paid in coupon period t;
• y = nominal annual yield compounded m times per year;
• m = coupon and compounding frequency per year; and
• N = remaining number of payment periods.

The model assumes the promised cash flows are known and the same periodic yield discounts every payment. For credit-risky or option-embedded bonds, this is a valuation convention rather than a complete economic model.

For a level-coupon bond, CF_t=C before maturity and CF_N=C+F at maturity. Price and yield move inversely because a higher discount rate reduces present value.

Worked Example 1: Price of a coupon bond

A three-year bond has face value $1,000, an annual coupon rate of 5%, annual payments and a yield to maturity of 6%.

Because the 5% coupon rate is below the 6% required yield, the bond trades below par. The calculation assumes no default and settlement exactly on a coupon date.

2.2 Clean price, dirty price and accrued interest

Quoted bond prices are often clean prices, excluding accrued interest. The settlement amount is the dirty price:

For a regular coupon period, a common approximation is:

Here, AI is accrued interest, d is the number of days elapsed since the previous coupon, and D is the number of days in the coupon period under the applicable day-count convention. Actual/Actual, 30/360 and Actual/360 conventions can produce different amounts.

If a semiannual coupon is $30 and 70 of 182 convention days have elapsed, accrued interest is 30(70/182)=\$11.54. If the clean price is $990.00, the dirty price is $1,001.54.

2.3 Spot-rate valuation

Using one yield for every cash flow can conceal the term structure of interest rates. If s_t is the annual spot rate for maturity t years with annual compounding, value is:

Each cash flow is discounted at its matching zero-coupon rate. For other compounding conventions, discount factors must be adjusted consistently. Spot-rate valuation is central to arbitrage-free pricing and yield-curve construction.

Suppose a two-year annual-pay bond provides $50 after one year and $1,050 after two years. If one- and two-year spot rates are 4% and 5%, respectively:

2.4 Yield measures

Current yield relates annual coupon income to clean market price:

It ignores capital gains, losses and timing beyond one year.

Yield to maturity (YTM) is the discount rate that equates the dirty price with promised cash flows. It is an internal rate of return, conditional on holding to maturity, receiving all promised payments and reinvesting interim coupons at the YTM. The reinvestment assumption is especially important for high-coupon bonds.

Yield to call substitutes the call date and call price for contractual maturity. Investors in callable bonds commonly examine yield to worst, the lowest yield among relevant permitted redemption scenarios, subject to market convention.

For a zero-coupon bond with n years to maturity and annual compounding:

This special-case formula does not apply directly to coupon-paying bonds.

2.5 Tax-equivalent municipal yield

Where municipal interest is exempt from an investor’s relevant income tax, the taxable-equivalent yield is:

Here, y_M is the tax-exempt municipal yield and \tau is the investor’s applicable marginal tax rate. At a 4% municipal yield and a 25% tax rate, y_TE=4\%/0.75=5.33\%. This comparison is valid only after considering differences in credit, liquidity, options, taxation and maturity.

3. Interest-Rate Sensitivity

3.1 Why maturity alone is inadequate

Two bonds with the same maturity can react differently to yield changes because coupons alter the timing of value received. Duration measures the weighted timing and first-order price sensitivity of a bond.

3.2 Macaulay and modified duration

For a bond discounted at a nominal annual yield y compounded m times per year, Macaulay duration in years is:

Modified duration is:

For a small parallel yield change \Delta y expressed as a decimal annual rate:

Macaulay duration has units of years. Modified duration is conventionally interpreted as percentage price sensitivity per unit change in annual yield. The approximation assumes small yield changes, unchanged cash flows and a parallel movement in the relevant yield.

For the three-year bond in Worked Example 1:

A 1 percentage-point increase in yield therefore produces a first-order estimated price decline of approximately 2.6957%.

3.3 DV01

Dollar value of one basis point, or DV01, estimates the currency change in price for a one-basis-point yield movement:

One basis point equals 0.01 percentage point, or 0.0001 in decimal form. For a portfolio worth $2,000,000 with modified duration 4.0, DV01 is approximately $800. A one-basis-point yield increase reduces value by about $800, assuming the stated duration approximation applies.

3.4 Convexity

The price–yield relationship is curved rather than linear. For an option-free bond under the same discrete-compounding convention, convexity is:

Convexity has units of years squared when yield is an annual decimal rate. The second-order approximation is:

For the three-year bond, convexity is approximately 10.00 years². If yield rises from 6% to 7%:

The estimated price is approximately $947.52. Direct valuation at 7% gives approximately $947.51, illustrating the value of the convexity adjustment.

Figure: The dashed duration linearization passes through and is tangent to the price–yield curve at the initial valuation point. Convexity captures the curve’s departure from that tangent.

The curve lies above its tangent when price is measured on the vertical axis, reflecting positive convexity for a conventional option-free bond. Callable bonds can exhibit negative convexity when falling yields make redemption increasingly likely.

3.5 Effective and key-rate duration

When cash flows change with rates, as with callable or mortgage-backed securities, modified duration based on fixed promised cash flows is inadequate. Model-based effective duration is:

Here, P_- is model value after a downward curve shift, P_+ is value after an equal upward shift, P₀ is initial value, and \Delta y is the absolute decimal shift. The model must re-estimate option exercise and cash flows under each scenario.

Key-rate duration measures exposure to changes at selected yield-curve maturities. It is preferable when the curve may steepen, flatten or twist rather than move in parallel.

4. Credit Risk and Credit Spreads

4.1 Components of credit risk

Default risk is the possibility that an issuer fails to make a required payment or breaches another condition constituting default. Downgrade risk is the risk that deteriorating credit quality reduces market value even without default. Recovery risk concerns the amount and timing recovered after default.

A basic expected-loss measure is:

where:

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

If one-year PD is 2%, expected recovery is 40% of exposure, and EAD is $5 million, then LGD is 60% and expected loss is 0.02\times0.60\times\$5{,}000{,}000=\$60{,}000. Expected loss is not the maximum loss and does not capture unexpected-loss volatility.

4.2 Credit ratings

Rating agencies classify relative creditworthiness using ordinal grades. Investment-grade categories generally indicate lower assessed credit risk than speculative-grade categories, although exact symbols vary by agency.

Ratings summarize analysis of leverage, cash-flow stability, industry conditions, governance, liquidity, sovereign constraints and debt structure. They are opinions rather than guarantees. Investors should conduct independent analysis because ratings can lag information, differ across agencies and change abruptly.

A rating transition matrix estimates probabilities of migration among rating categories over a specified horizon. Transition estimates are sensitive to economic cycles, sample composition and methodology.

4.3 Credit spreads

A credit spread is a yield difference between a credit-risky bond and a benchmark with comparable maturity or duration:

Observed spreads compensate for more than expected default loss. They can include unexpected credit loss, liquidity, risk aversion, taxation, regulatory capital costs, embedded options and market segmentation. Consequently, spread should not be interpreted as PD alone.

The option-adjusted spread (OAS) is the constant spread added to benchmark spot rates in a valuation model after explicitly allowing for embedded options. Its reliability depends on the interest-rate and option-exercise models employed.

Under a highly simplified constant-intensity model with default intensity \lambda and fractional recovery R, a short-horizon credit spread may be approximated by:

This relation assumes simplified recovery, liquidity and risk-premium conditions. It is not a general formula for observed bond spreads.

4.4 Structural and reduced-form perspectives

Structural credit models view corporate debt as contingent on the value of the issuer’s assets. Default occurs when asset value becomes insufficient relative to debt obligations. This perspective connects leverage and asset volatility to credit risk.

Reduced-form models treat default as a stochastic event governed by an estimated intensity or hazard rate. They are often convenient for pricing credit derivatives but require assumptions about intensity dynamics and recovery.

5. Covenants, Security and Recovery

A bond indenture or trust deed defines the legal rights and obligations of issuer and investors. Covenants reduce agency conflicts but may restrict managerial flexibility.

Affirmative covenants require actions such as providing financial statements, maintaining insurance or paying taxes. Negative covenants prohibit or limit actions such as issuing additional secured debt, selling important assets, paying excessive dividends or entering related-party transactions.

Financial covenants may specify minimum interest coverage or maximum leverage. Maintenance covenants must be satisfied regularly, whereas incurrence covenants are tested only when the issuer undertakes a specified action.

Recovery depends on collateral quality, priority, guarantees, insolvency law, jurisdiction and time to resolution. Seniority does not guarantee full repayment. Structural subordination occurs when creditors of an operating subsidiary have priority over claims against cash flows remitted to a parent holding company.

6. Issuance, Trading and Market Practice

6.1 Issuance process

A typical corporate issue involves:

1. determining funding amount, maturity, currency and structure;
2. appointing underwriters and legal advisers;
3. conducting due diligence and preparing disclosure;
4. obtaining ratings where appropriate;
5. marketing the issue to investors;
6. setting the coupon and issue spread;
7. allocating securities and settling the transaction; and
8. complying with continuing disclosure and covenant obligations.

A book-building process collects investor orders at indicative spreads. Strong demand can permit tighter pricing, while weak demand may require a higher spread or smaller issue.

Governments often use competitive auctions. Auction design differs by jurisdiction and can include multiple-price or uniform-price formats.

6.2 Secondary-market analysis

Professional traders compare a bond with benchmark government securities, swaps and peer issuers. Relative-value analysis should account for duration, seniority, liquidity, currency, call features and tax status.

Liquidity can disappear during stress precisely when investors most need to trade. Matrix prices or evaluated prices may then differ from executable prices. Bid–ask spreads, dealer inventories and transaction frequency provide useful but incomplete liquidity evidence.

7. Applied Case: Managing a Corporate Bond Position

Background

The following is an illustrative case. Meridian Bank owns $10 million face value of the three-year, 5% annual-pay corporate bond valued earlier at a 6% yield. Its market value is approximately $9,732,698. Modified duration is 2.6957 and convexity is approximately 10.00 years².

Problem

The treasury manager expects benchmark rates may rise by 100 basis points. Credit analysts also warn that the issuer’s leverage is increasing, creating a possible spread-widening risk. Management needs to distinguish interest-rate exposure from credit exposure.

Analysis

The position’s DV01 is:

The duration–convexity estimate for a 100-basis-point rate increase is a 2.6457% loss, or approximately $257,000. Direct repricing produces a similar result.

Suppose spread duration is separately estimated at 2.6. An 80-basis-point spread increase would imply, to first order:

That is an additional approximate loss of $202,000 if evaluated separately. Rate and spread effects should not be added mechanically when they interact, when the yield curve moves non-parallel, or when default probability and liquidity change materially.

Approach

Meridian can reduce benchmark-rate DV01 with government-bond futures or a pay-fixed interest-rate swap. Hedge ratios must use the DV01 of the chosen hedge instrument and, for futures, appropriate conversion-factor and cheapest-to-deliver analysis. Such a hedge does not eliminate corporate spread risk.

The bank could reduce spread exposure by selling part of the position, purchasing credit protection where a suitable contract exists, or replacing the bond with a shorter-spread-duration security. Each choice introduces transaction costs, basis risk and counterparty or liquidity considerations.

Outcome

Management executes a partial interest-rate hedge but reduces the bond holding because credit deterioration cannot be neutralized adequately with the available instruments. Limits are established separately for benchmark DV01, spread duration and issuer concentration.

Learning Points

• Interest-rate risk and credit-spread risk are economically distinct.
• Duration is a local approximation, not a complete stress test.
• Hedging one risk factor can leave other exposures unchanged.
• Liquidity, basis risk and counterparty risk affect hedge effectiveness.
• Credit analysis must supplement external ratings.

8. Regulation, Governance and Ethics

Bond regulation typically addresses disclosure, market conduct, suitability, capital adequacy and trading transparency. Exact requirements vary by jurisdiction. Banks are also subject to prudential standards governing credit concentration, market risk, liquidity and capital.

Ethical concerns include selective disclosure, misleading bond marketing, manipulation of evaluated prices, conflicts in underwriting and rating, and inaccurate sustainability claims. Green or sustainability-linked bonds require credible use-of-proceeds rules, measurable targets and transparent reporting. Labelling alone does not establish superior credit quality or environmental impact.

Issuers should disclose material risks and covenant terms clearly. Investors should avoid blind reliance on ratings, perform suitability assessments and document valuation assumptions. Governance committees should separate trading incentives from independent risk control.

9. Technology and Contemporary Developments

Electronic trading has improved price discovery in liquid segments, while regulatory trade-reporting systems have increased post-trade transparency in several markets. Nevertheless, many corporate and municipal bonds remain comparatively illiquid.

Machine learning can support default prediction, document analysis, news classification and liquidity estimation. Its limitations include unstable historical relationships, biased data, low interpretability and rare default observations. Human review remains necessary, especially when models affect lending, investment or regulatory decisions.

Other developments include climate-related credit analysis, growth in private credit, transition away from legacy interbank offered rates, central-bank balance-sheet changes and increasing use of portfolio trading. These developments affect benchmark curves, liquidity and risk transmission.

10. Limitations and Professional Best Practices

Bond models are simplifications. YTM assumes a single discount rate; duration assumes small changes; convexity is only a second-order correction; ratings can lag; and observed spreads combine several risk premia. Model prices may be unreliable when trading is sparse or embedded options are difficult to estimate.

Sound professional practice includes:

• valuing cash flows with internally consistent curves and conventions;
• reconciling clean and dirty prices before settlement;
• reporting rate, spread, option and liquidity risks separately;
• supplementing duration with key-rate and stress scenarios;
• verifying covenant, priority and collateral provisions;
• conducting independent credit analysis;
• documenting model assumptions and data sources; and
• monitoring concentration, counterparty and wrong-way risk.

Chapter Summary

Bonds are contractual claims whose values depend on the timing and risk of promised cash flows. Government, municipal and corporate bonds differ in taxation, security, liquidity, legal structure and default exposure. Prices can be calculated using a single yield or maturity-specific spot rates, while clean prices must be distinguished from dirty settlement values.

Macaulay duration measures weighted cash-flow timing, modified duration estimates first-order yield sensitivity, DV01 expresses that sensitivity in currency terms, and convexity improves estimates for larger rate changes. Effective and key-rate duration are needed when cash flows change with rates or the yield curve moves non-parallel.

Credit analysis considers PD, LGD, exposure, ratings, migration, recovery, seniority and covenants. Credit spreads contain expected loss as well as liquidity and risk premia. Effective management therefore separates benchmark interest-rate risk from spread, default and liquidity risks and combines quantitative models with legal and fundamental analysis.

Key Terms

Accrued Interest: Coupon interest earned by the seller since the previous coupon date.

Clean Price: Quoted bond price excluding accrued interest.

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

Credit Rating: Ordinal opinion about an issuer’s or instrument’s relative creditworthiness.

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

Covenant: Contractual requirement or restriction protecting creditors or governing issuer conduct.

Dirty Price: Bond settlement value including accrued interest.

Duration: Measure of cash-flow timing and local interest-rate sensitivity.

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

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

Face Value: Principal amount normally repaid at maturity.

Investment Grade: Rating classification associated with relatively lower assessed credit risk.

Loss Given Default: Proportion of exposure lost after accounting for recovery.

Option-Adjusted Spread: Model-based spread after allowing explicitly for embedded options.

Recovery Rate: Proportion of exposure recovered following default.

Spot Rate: Yield on a zero-coupon cash flow at a specified maturity.

Structural Subordination: Lower effective priority resulting from debt being issued at different levels of a corporate group.

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

Review Questions

Short Answer Questions

1. Distinguish a clean bond price from a dirty bond price.
2. Why does a conventional bond’s price normally fall when its yield rises?
3. What is the difference between Macaulay duration and modified duration?
4. Identify three components that may be embedded in an observed credit spread.
5. Distinguish maintenance covenants from incurrence covenants.

Long Answer Questions

1. Compare government, municipal and corporate bonds with respect to risk, taxation, liquidity and repayment sources.
2. Explain the assumptions and limitations of yield to maturity as a return measure.
3. Derive the economic interpretation of duration and explain why convexity improves price-change estimates.
4. Evaluate the roles and limitations of credit ratings in institutional investment decisions.
5. Discuss how security, seniority, guarantees and covenants influence expected recovery and bond pricing.

Analytical / Application Questions

1. A four-year annual-pay bond has face value $1,000, a 6% coupon and a 7% yield. Calculate its price and explain whether it trades at a premium or discount.
2. A $5 million portfolio has modified duration 3.8 and convexity 18. Estimate the percentage and currency change for a 75-basis-point increase in yield.
3. Analyze how an investor should compare a tax-exempt municipal bond yielding 3.9% with a taxable corporate bond yielding 5.2%, given a 30% marginal tax rate and differing credit and liquidity risks.

Multiple Choice Questions with Answers

Question 1

A one-year zero-coupon bond pays $1,000 at maturity. What is its price at an annual yield of 5%?

A. $950.00

B. $952.38

C. $1,000.00

D. $1,050.00

Correct Answer: B. $952.38

Question 2

A bond has a clean price of $101.25 per $100 face value and accrued interest of $1.40. What is its dirty price?

A. $99.85

B. $101.25

C. $102.65

D. $142.65

Correct Answer: C. $102.65

Question 3

A tax-exempt municipal bond yields 4%. What is its taxable-equivalent yield for an investor facing a 25% marginal tax rate?

A. 3.00%

B. 4.00%

C. 5.00%

D. 5.33%

Correct Answer: D. 5.33%

Question 4

A bond has modified duration of 2.7. Ignoring convexity, what is the approximate price effect of a 100-basis-point yield increase?

A. A 2.7% decrease

B. A 1.0% decrease

C. A 2.7% increase

D. A 27.0% decrease

Correct Answer: A. A 2.7% decrease

Question 5

For a conventional option-free bond with positive convexity, what does the convexity term contribute to the estimated percentage price change for either a rise or fall in yield?

A. A negative adjustment in both cases

B. A positive adjustment in both cases

C. No adjustment unless the bond defaults

D. A positive adjustment only when yields rise

Correct Answer: B. A positive adjustment in both cases

Question 6

Other factors equal, which debt claim normally has the strongest direct collateral position?

A. Subordinated unsecured bond

B. Senior unsecured bond

C. Senior secured bond

D. Common equity

Correct Answer: C. Senior secured bond

Question 7

Which statement about an observed corporate bond credit spread is most accurate?

A. It equals default probability in all circumstances.

B. It contains only expected default loss.

C. It is unaffected by market liquidity.

D. It may reflect credit loss, liquidity and risk premia.

Correct Answer: D. It may reflect credit loss, liquidity and risk premia.

Question 8

A $2,000,000 bond portfolio has modified duration of 4.0. What is its approximate DV01?

A. $80

B. $400

C. $800

D. $8,000

Correct Answer: C. $800

Question 9

Which covenant generally restricts actions such as issuing additional secured debt or selling key assets?

A. Negative covenant

B. Affirmative covenant

C. Coupon convention

D. Settlement convention

Correct Answer: A. Negative covenant

Question 10

What does a credit-rating transition matrix describe?

A. Coupon payments at each future date

B. Probabilities of migration among rating categories

C. The relationship between clean and dirty prices

D. Daily bid–ask spreads for government bonds

Correct Answer: B. Probabilities of migration among rating categories