Fixed-Income, Credit and Liability Management

Introduction and Context

Fixed-income securities are contractual claims under which an issuer promises specified cash flows to an investor. They include government bonds, corporate bonds, certificates of deposit, commercial paper, mortgage-backed securities and other debt instruments. In investment banking and wealth management, fixed income serves three related purposes: generating income, preserving capital and matching future liabilities.

The apparent simplicity of a bond—a loan exchanged for interest and principal—conceals several risks. Market yields change, altering present value. Inflation reduces real purchasing power. Issuers may default or be downgraded. Embedded options may change cash-flow timing. Liquidity conditions can make a theoretically attractive bond difficult to trade. Effective portfolio management therefore requires more than selecting securities with the highest coupon: it requires analysis of cash flows, interest-rate sensitivity, credit quality, liquidity, tax and liability timing.

This chapter develops a framework for pricing bonds, measuring duration and convexity, interpreting the term structure, analysing credit spreads and securitisation, and constructing portfolios for income, capital preservation and liability matching.

Learning Objectives

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

1. Price fixed-income securities using discounted cash flow methods.
2. Explain the relationship among coupon, yield, price, maturity and interest-rate risk.
3. Calculate and interpret Macaulay duration, modified duration and convexity.
4. Analyse the term structure and major theories of the yield curve.
5. Distinguish default, credit-spread, downgrade, liquidity and recovery risks.
6. Explain securitisation, credit enhancement and the agency problem.
7. Construct portfolios for income, capital preservation and liability matching.
8. Apply immunisation principles while recognising their assumptions and limitations.

1. The Economics of a Bond

A bond’s promised cash flows normally consist of periodic coupons and repayment of principal at maturity. A zero-coupon bond pays no interim coupon and makes a single payment at maturity. A coupon bond with face value F, annual coupon rate c and n annual periods pays C = cF each period and F at maturity.

The price is the present value of promised cash flows discounted at the yield per period y:

Here, P is price, C is the coupon per period, F is face value, y is the yield per period and n is the number of periods. The formula assumes known promised cash flows, a constant yield for all periods and payment periods consistent with the yield convention. If coupons are semi-annual, C is the annual coupon divided by two, y is the annual quoted yield divided by two, and n is twice the number of years.

For a zero-coupon bond:

The bond trades at par when coupon rate equals the required yield, at a discount when coupon rate is below the required yield and at a premium when coupon rate exceeds the required yield. For a conventional option-free bond, price and yield move inversely. The relationship is nonlinear: price rises by more when yields fall than it falls for an equal-sized yield increase, a property associated with convexity.

Worked Example 1: Pricing a Coupon Bond

A three-year bond has face value 1,000, an annual coupon rate of 6%, and a yield to maturity of 8%, with annual coupons. Its cash flows are 60 in years 1 and 2 and 1,060 in year 3.

The bond therefore trades below par because its 6% coupon is less than the market yield of 8%. Yield to maturity is an internal rate of return: it is the discount rate that equates the present value of promised cash flows with the observed price, assuming coupons are paid as promised and reinvested at that yield. It is not necessarily the investor’s realised return.

Accrued Interest and Quoted Prices

Between coupon dates, the seller has earned part of the next coupon. The clean price excludes accrued interest; the dirty price includes it:

Under a simple day-count convention, accrued interest equals coupon payment × elapsed days ÷ days in the coupon period. Actual market conventions differ by instrument and jurisdiction. Settlement, ex-coupon periods, taxes and day-count rules must therefore be checked before executing a transaction.

2. Interest-Rate Risk: Duration and Convexity

Macaulay Duration

Macaulay duration is the cash-flow-weighted average time at which an investor receives the bond’s promised cash flows:

CFₜ is the cash flow at period t, P is bond price, y is yield per period and t is measured in periods. If periods are semi-annual, duration in periods must be divided by two to express years. Macaulay duration is particularly useful for immunisation under parallel yield-curve shifts.

Modified Duration

Modified duration measures approximate percentage price sensitivity to a change in yield:

For a small yield change ∆y:

The negative sign reflects the inverse price-yield relationship. Yield and duration must use the same period frequency. The approximation is most reliable for small changes and option-free bonds whose cash flows do not change when yields move.

Dollar Duration and DV01

Dollar duration converts percentage sensitivity into currency:

DV01, or price value of a basis point, is the approximate price change for a one-basis-point change in yield:

For a portfolio, DV01 is summed across positions, using position market values and appropriate effective durations. It is widely used by dealers and portfolio managers to manage small parallel yield movements.

Convexity

Duration is a first-order approximation. Convexity captures the curvature of the price-yield relationship:

For discrete cash flows, a commonly used approximation is:

For a yield change ∆y, the duration-convexity approximation is:

Convexity is expressed in squared periods or squared yield units, depending on convention. A standard option-free bond generally has positive convexity: for equal yield changes, the price gain from a fall in yields exceeds the price loss from a rise. Callable bonds may exhibit negative convexity over some yield ranges because falling yields increase the probability of early redemption.

Worked Example 2: Duration Estimate

Suppose a bond portfolio is worth 10,000,000 and has modified duration of 5.2. If yields rise by 25 basis points, the approximate price impact is:

The estimated loss is 130,000. This is an estimate, not a guaranteed result: curve twists, spread changes, convexity, liquidity and cash-flow changes may produce a different outcome.

3. Term Structure and Yield-Curve Analysis

The term structure describes the relationship between yields and maturities for otherwise comparable instruments. A yield curve may be upward sloping, flat, inverted or humped. It is important to distinguish a government benchmark curve from a corporate curve, because the latter includes credit and liquidity components.

Spot Rates and Forward Rates

A spot rate is the yield applicable to a single cash flow at a particular maturity. For a bond with cash flows CFₜ, valuation using spot rates sₜ is:

A one-period forward rate fₜ,ₜ₊₁ implied by annual spot rates satisfies:

Forward rates are implied prices, not certain forecasts of future short-term rates. They may contain term premia, liquidity premia and risk compensation.

Yield-Curve Theories

The expectations hypothesis argues that long-term yields reflect expected future short-term rates, possibly adjusted for compounding. The liquidity-preference theory adds a positive term premium because investors generally require compensation for holding longer maturities. The preferred-habitat approach argues that investors have maturity preferences but may move between segments when compensation is sufficient. Market segmentation theory emphasises distinct supply and demand conditions in different maturity sectors.

In practice, no single theory explains every curve movement. A steepening curve may reflect expected monetary tightening, increased term premium, fiscal supply or deteriorating inflation expectations. Portfolio decisions should therefore analyse both macroeconomic expectations and market risk premia.

Key Rate and Spread Duration

Total duration can conceal exposure to particular maturities. Key-rate duration measures sensitivity to a change at one maturity while holding other curve points approximately constant. It is useful for managing non-parallel shifts, such as a 2-year yield falling while a 10-year yield rises.

Spread duration measures price sensitivity to a change in credit spread, holding the risk-free curve constant. For a credit portfolio, separating interest-rate duration from spread duration helps identify whether risk arises from government yields or issuer compensation.

4. Credit Risk and Credit Spreads

Credit risk is the possibility that an obligor fails to pay principal or interest as promised, or that the market value of its debt falls because perceived credit quality deteriorates. Major components include:

• Default risk: failure to meet contractual obligations.
• Migration risk: rating upgrade or downgrade that changes required yield.
• Recovery risk: uncertainty about value recovered after default.
• Spread risk: widening of the risk premium without immediate default.
• Liquidity risk: inability to transact at a reasonable price.
• Counterparty risk: failure of a trading or derivative counterparty.
• Concentration risk: excessive exposure to one issuer, sector, geography or factor.

A credit spread is the yield difference between a risky bond and a comparable risk-free or benchmark bond:

The spread compensates for expected loss, unexpected loss, liquidity, risk aversion, tax and other factors. It is not identical to the expected default loss.

A simplified expected-loss relationship is:

If recovery rate is R, then loss given default is approximately 1 − R. Thus:

PD is probability of default over the relevant horizon. This relationship is a simplification: timing, discounting, changing exposure, correlated defaults and risk premia must be incorporated in a full valuation model.

Credit Analysis

Fundamental analysis examines an issuer’s business model, competitive position, profitability, leverage, liquidity, cash-flow coverage, refinancing needs, asset quality and legal structure. Quantitative ratios may include debt-to-EBITDA, interest coverage, funds from operations-to-debt and current liquidity. Qualitative analysis considers governance, strategy, regulation and management credibility.

Credit ratings provide a common language but are not substitutes for independent analysis. A professional investor should review financial statements, covenants, maturity walls, secured versus unsecured ranking, guarantees and event-risk provisions. Internal credit limits should consider both probability of default and loss severity.

5. Securitisation and Structured Credit

Securitisation pools contractual receivables—such as mortgages, auto loans, credit-card balances or student loans—and transfers them to a special-purpose vehicle (SPV). The SPV issues securities whose payments depend on the pool’s cash flows.

Figure: Simplified securitisation structure and tranche cash-flow waterfall.

Credit enhancement may be internal—subordination, overcollateralisation, excess spread or reserve accounts—or external, such as guarantees and insurance. Senior tranches receive payment before junior tranches and therefore generally have lower expected loss, although they remain exposed to modelling, correlation, servicing, liquidity and tail risks.

Securitisation can diversify funding and improve balance-sheet efficiency, but it may create opacity and incentives for weak underwriting. Investors must assess asset quality, historical performance, servicing arrangements, trigger mechanisms, waterfall rules, prepayment assumptions and stress scenarios. Post-crisis regulation has placed greater emphasis on risk retention, disclosure, capital requirements and model governance.

6. Liability Management and Immunisation

Liability management begins with identifying the amount, timing, currency, tax status and certainty of future obligations. A pension fund, insurer, bank or wealthy individual may have liabilities such as benefit payments, tuition, mortgage obligations or planned charitable distributions.

Cash-Flow Matching

Cash-flow matching constructs an asset portfolio whose contractual cash flows cover liabilities as they fall due. It provides strong protection against interest-rate movements when cash flows are certain and assets can be held to maturity. Its disadvantages include reinvestment constraints, imperfect availability of maturities, credit risk and reduced liquidity.

Duration Matching and Classical Immunisation

Classical immunisation seeks to protect a present value surplus against small parallel yield changes by matching asset and liability duration:

and

where PV_A and PV_L are present values of assets and liabilities, and D_A and D_L are their durations. The portfolio should also have at least as much convexity as the liabilities for a stronger local protection condition.

The reasoning is that a yield increase reduces both asset and liability present values, while a yield decrease increases both. If duration is matched, first-order effects on the surplus approximately offset. Immunisation is not permanent: time passes, cash flows are received, yields change non-parallelly and credit spreads move.

Rebalancing

An immunised portfolio requires monitoring and rebalancing. Rebalancing may be triggered by duration drift, changes in liability estimates, curve twists, rating migration, cash-flow deviations or funding-ratio thresholds. Transaction costs, taxes and liquidity should be included in the decision. Derivatives such as interest-rate futures and swaps can adjust duration efficiently, but introduce basis, collateral and counterparty risks.

7. Portfolio Construction for Different Objectives

Income Generation

An income-oriented portfolio emphasises sustainable coupon and distribution cash flows. It may use laddered maturities, diversified investment-grade credit and selected higher-yield exposures. The investor must distinguish nominal income from total return: a high coupon may compensate for substantial default or duration risk. Reinvestment risk is especially important when bonds mature or coupons are paid during falling-rate environments.

Capital Preservation

Capital preservation prioritises credit quality, liquidity, short or intermediate duration and diversification. Treasury bills, high-quality sovereign debt, deposits and short-term investment-grade instruments may be appropriate, subject to jurisdictional and deposit-insurance limits. Preservation is not equivalent to eliminating risk: inflation, taxes, issuer concentration and market-value volatility remain relevant.

Liability Matching

A liability-driven portfolio first maps obligations, then determines whether exact cash-flow matching, duration matching, or a combination is feasible. A core matching portfolio can be supplemented by a return-seeking portfolio, provided that liquidity and funding-ratio limits are respected. Currency hedging is necessary when assets and liabilities are denominated in different currencies unless the mismatch is deliberate and risk-budgeted.

A practical construction process is:

1. Specify objectives, constraints and risk tolerance.
2. Build a liability schedule and identify uncertain cash flows.
3. Define permissible issuers, ratings, maturities and concentration limits.
4. Estimate interest-rate, spread, liquidity, inflation and currency exposures.
5. Select securities using scenario analysis and relative-value analysis.
6. Execute with attention to transaction costs and market liquidity.
7. Monitor performance, exposures, covenant events and liability changes.
8. Rebalance under documented governance rules.

8. Illustrative Case: Pension Fund Immunisation

Background. The illustrative Meridian Pension Fund must pay 4 million in one year, 5 million in three years and 6 million in five years. Its investment committee wants to reduce sensitivity to interest-rate movements while retaining liquidity for unexpected benefit payments.

Problem. The fund currently holds long corporate bonds with a duration of 7.4 years, while the present-value-weighted liability duration is 4.8 years. A parallel rise in yields would reduce asset value more sharply than liability value, weakening the funding ratio.

Analysis. The committee separates the portfolio into a liability-matching sleeve and a return-seeking sleeve. It uses high-quality government and investment-grade bonds for the matching sleeve, with maturities near the payment dates. The return-seeking sleeve is capped by a risk budget and may contain diversified corporate credit.

Suppose the matching sleeve has market value 60 million and modified duration 4.5. Its approximate DV01 is:

A 10-basis-point rise in the relevant yield curve would therefore imply an approximate loss of 270,000 in the sleeve before convexity and other effects. If the liability portfolio has materially different DV01, the fund can alter maturities or use a swap overlay. The overlay must be calibrated to the liability curve, not merely to a broad market index.

Approach and solution. The fund establishes limits for issuer exposure, minimum liquidity, rating migration and derivative collateral. It performs parallel and non-parallel curve shocks, spread widening, inflation shocks and severe but plausible default scenarios. Quarterly reviews compare actual and projected cash flows, duration, key-rate exposure and funding ratio.

Outcome. The revised structure reduces the portfolio’s mismatch and makes the source of risk transparent. It does not eliminate risk: liabilities may change, corporate spreads may widen, bonds may become illiquid and the assumed curve may not describe actual funding costs.

Learning points. Liability management is an integrated process rather than a single duration calculation. Cash-flow certainty, credit quality, liquidity, governance and stress testing are as important as mathematical matching.

9. Regulation, Ethics and Governance

Fixed-income professionals have duties to represent valuation assumptions honestly, disclose material risks and avoid misleading yield comparisons. Suitability requires matching duration, credit quality, liquidity and complexity to the client’s objectives and capacity for loss. Complex securitised products should not be presented as equivalent to government bonds merely because they carry a high rating.

Valuation governance is essential when instruments are illiquid. Independent price verification, documented models, observable market inputs and valuation reserves reduce the risk of overstated performance. Conflicts may arise when an investment bank structures, underwrites, distributes and researches the same product. Information barriers, disclosure and supervisory controls are therefore central.

Regulatory frameworks differ by jurisdiction, but commonly address capital, liquidity, disclosure, risk retention, suitability, market abuse, stress testing and resolution planning. Professionals should consult applicable rules rather than relying on general textbook principles.

10. Technology, Data and Contemporary Developments

Electronic trading, portfolio analytics and machine-learning tools can improve pricing, liquidity monitoring and scenario analysis. AI may help classify issuer disclosures, detect covenant changes or identify unusual spread movements. However, models can reproduce data bias, mistake correlation for causation or fail during structural breaks. Human review, explainability, data lineage, model validation and limits on automated execution remain necessary.

Contemporary fixed-income markets are also affected by inflation uncertainty, central-bank balance-sheet policy, private credit growth, climate-transition risk and increased use of derivatives. Climate and transition factors may affect default probability, recovery value, refinancing access and asset stranding. They should be incorporated through financially material scenarios rather than unsupported labels.

Challenges and Limitations

Bond valuation depends on assumptions about discount rates, cash flows and timing. Yield-to-maturity can obscure reinvestment and default risk. Duration assumes small changes and stable cash flows. Convexity approximations may be inaccurate for large shocks or callable and mortgage-backed securities. Credit ratings can lag market information. Historical default data may not represent future regimes. Securitisation models depend on prepayment, loss, correlation and recovery assumptions. Immunisation works best for small, parallel shifts and known liabilities, whereas real portfolios face curve twists, optionality, liquidity constraints and changing obligations.

A robust process therefore combines analytical measures with scenario analysis, stress testing, judgement, diversification and governance. No single metric captures fixed-income risk completely.

Chapter Summary

Fixed-income valuation discounts contractual cash flows using market-consistent yields. Bond prices generally move inversely to yields, while duration measures first-order interest-rate sensitivity and convexity improves the estimate for larger movements. Spot rates and forward rates support term-structure analysis, but forward rates are not certain forecasts.

Credit risk includes default, migration, recovery, spread, liquidity and concentration risk. Credit spreads compensate for expected loss and several risk premia. Securitisation transforms pools of receivables into tranched securities, using waterfalls and credit enhancement while introducing model, structural and agency risks.

Portfolio construction must begin with the investor’s objective. Income portfolios focus on sustainable cash flow, capital-preservation portfolios emphasise quality and liquidity, and liability-driven portfolios match asset cash flows or duration to obligations. Immunisation can reduce sensitivity to small parallel rate changes, but requires monitoring and rebalancing. Ethical valuation, suitability, disclosure, model governance and stress testing are indispensable professional practices.

Key Terms

Bond: A debt security promising contractual payments of interest and principal.

Coupon: Periodic interest payment on a bond.

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

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

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

Duration: Measure of the timing or interest-rate sensitivity of bond cash flows.

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

Modified duration: Approximate percentage price sensitivity to a yield change.

Convexity: Curvature of the price-yield relationship.

DV01: Approximate price change for a one-basis-point yield movement.

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

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

Recovery rate: Proportion of exposure recovered after default.

Securitisation: Transformation of pooled receivables into marketable securities.

Special-purpose vehicle: Legal entity that holds securitised assets and issues notes.

Subordination: Priority structure placing junior claims below senior claims.

Cash-flow matching: Aligning asset payments with liability dates and amounts.

Immunisation: Portfolio technique designed to offset asset and liability interest-rate sensitivity.

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

Spread duration: Sensitivity to changes in credit spreads.

Review Questions

Short Answer Questions

1. Why does a bond trade at a discount when its coupon rate is below its yield to maturity?
2. Define Macaulay duration and modified duration.
3. What is the difference between a spot rate and a forward rate?
4. Identify four components of credit risk.
5. What conditions are required for classical duration immunisation?

Long Answer Questions

1. Explain bond pricing and discuss the assumptions underlying yield to maturity.
2. Derive the duration-convexity approximation and explain when it may be unreliable.
3. Compare expectations, liquidity-preference, preferred-habitat and market-segmentation theories of the yield curve.
4. Explain the securitisation process, including tranching, credit enhancement and key risks.
5. Discuss how an investment manager should construct a portfolio for capital preservation and liability matching.

Analytical / Application Questions

1. A bond has a market value of 8 million and modified duration of 6. If yields rise by 40 basis points, estimate the percentage and currency price change using duration alone. State the limitations of your estimate.
2. A pension fund’s assets have duration 6.8 years and liabilities have duration 4.2 years. Explain the likely effect of a parallel rate increase and recommend two methods of reducing the mismatch.
3. A corporate bond offers a spread of 250 basis points over a government bond. Discuss why this spread cannot be interpreted solely as expected default loss, and identify the additional analysis required before investment.

Multiple Choice Questions with Answers

Question 1

A three-year bond pays annual coupons of 60 and has face value 1,000. If its yield is 8%, which price is closest?

A. 900.00

B. 948.95

C. 1,000.00

D. 1,060.00

Correct Answer: B. 948.95

Question 2

For a conventional option-free bond, what is the usual effect of a rise in yield?

A. Price rises linearly

B. Price falls

C. Coupon rate rises automatically

D. Maturity shortens automatically

Correct Answer: B. Price falls

Question 3

If Macaulay duration is 5 years and yield per period is 4%, modified duration is closest to:

A. 4.00

B. 4.81

C. 5.20

D. 9.00

Correct Answer: B. 4.81

Question 4

A portfolio worth 10,000,000 has modified duration 5. A 20-basis-point yield increase implies an approximate price change of:

A. +100,000

B. −10,000

C. −100,000

D. −1,000,000

Correct Answer: C. −100,000

Question 5

Which measure captures the curvature of the bond price-yield relationship?

A. Coupon rate

B. Convexity

C. Recovery rate

D. Current yield

Correct Answer: B. Convexity

Question 6

A credit spread is best defined as:

A. Coupon minus face value

B. Risky-bond yield minus benchmark yield

C. Face value minus market value

D. Default probability minus recovery rate

Correct Answer: B. Risky-bond yield minus benchmark yield

Question 7

If probability of default is 3% and loss given default is 40%, the simplified expected loss is:

A. 0.12%

B. 1.20%

C. 7.50%

D. 43%

Correct Answer: B. 1.20%

Question 8

In a securitisation waterfall, which tranche generally receives payments first?

A. Equity tranche

B. Mezzanine tranche

C. Senior tranche

D. Residual tax tranche

Correct Answer: C. Senior tranche

Question 9

Classical immunisation generally requires matching:

A. Coupon rates only

B. Asset and liability present values and durations

C. Issuer names and maturities exactly

D. Credit ratings and currencies only

Correct Answer: B. Asset and liability present values and durations

Question 10

Key-rate duration is especially useful for analysing:

A. A single coupon payment

B. Non-parallel yield-curve movements

C. Accounting depreciation

D. Recovery proceeds after liquidation only

Correct Answer: B. Non-parallel yield-curve movements