Fixed-Income Securities and Bond Portfolio Management

Introduction

Fixed-income securities promise contractually specified payments, usually interest and principal, over defined periods. They are central to bank liquidity portfolios, insurance liability funding, pension investment, government borrowing and securitized credit markets. Although their cash flows often appear predictable, their market values remain exposed to interest rates, yield-curve movements, inflation, credit deterioration, liquidity conditions, embedded options and reinvestment risk.

Bond portfolio management therefore combines valuation with risk control. Managers must distinguish coupon rates from yields, interpret the term structure, quantify duration and convexity, assess default and recovery risk, and select strategies consistent with liabilities, benchmarks and regulatory constraints.

Learning Objectives

After studying this chapter, students should be able to:

1. Explain major bond features and classifications.
2. Price fixed-rate, zero-coupon and floating-rate instruments.
3. calculate and interpret yield measures and holding-period return.
4. Explain spot rates, forward rates and yield-curve theories.
5. Measure interest-rate exposure using duration, convexity and key-rate duration.
6. Evaluate credit risk and securitized products.
7. Apply immunization, laddering, indexing and active bond strategies.
8. Recognize governance, ethical and technological issues in fixed-income management.

1. Bond Features and Classification

A bond is a debt instrument under which an issuer promises payments to investors. Its legal terms are specified in an indenture or offering document.

1.1 Major Types

Government bonds generally have low default risk in domestic currency, although they retain interest-rate and inflation risk. Corporate bonds add issuer credit and liquidity risk. Municipal or sub-sovereign bonds are issued by regional authorities. Zero-coupon bonds make no periodic coupon payments and are issued below face value. Inflation-linked bonds adjust principal or coupons using an inflation index.

A callable bond permits the issuer to redeem before maturity, usually when rates decline. A putable bond allows the investor to sell the bond back under specified conditions. Convertible bonds may be exchanged for equity and therefore combine debt and equity-option characteristics.

2. Valuation of Fixed-Rate Bonds

A bond's value equals the present value of its promised cash flows, discounted at rates appropriate to timing and risk.

For a conventional fixed-rate bond valued using one yield to maturity:

where:

• P = full or dirty price in currency units;
• C = annual coupon cash amount;
• F = face value;
• y = annual nominal yield to maturity compounded m times per year;
• m = coupon payments per year;
• N=mT = total payment periods for maturity T years.

This single-yield formula assumes that all promised cash flows are discounted at the same yield. Arbitrage-consistent valuation instead discounts each cash flow using its maturity-specific spot rate.

Worked Example: Correct Pricing of a Three-Year Bond

A three-year bond has face value $1,000, an annual coupon rate of 6%, annual payments and an 8% yield. Thus, the annual coupon is $60.

The bond's value is therefore $948.46. It sells at a discount because its 6% coupon rate is below the required 8% yield.

2.1 Price–Yield Relationship

For an option-free bond with positive cash flows, price falls when yield rises. The relationship is convex rather than linear because:

Consequently, for equal-sized yield changes, the percentage price gain from a yield decrease is generally larger than the price loss from a yield increase.

Figure: The decreasing curve is globally convex in price–yield space; its absolute slope becomes smaller as yield rises.

2.2 Clean Price, Dirty Price and Accrued Interest

Between coupon dates, the buyer compensates the seller for earned coupon interest:

Under a simple proportional convention:

Actual calculation depends on the market's day-count convention, such as Actual/Actual or 30/360. Market quotations commonly report clean prices, while settlement uses dirty prices.

3. Yield and Realized Return Measures

3.1 Current Yield and Yield to Maturity

Current yield measures annual coupon income relative to clean market price:

It ignores capital gains, losses and time value. Yield to maturity, or YTM, is the discount rate equating the full price with promised cash flows. For a conventional bond with positive cash flows, it is found numerically.

YTM is not a guaranteed realized return. Realization requires holding the bond to maturity, no default and reinvesting interim coupons at the calculated YTM.

Yield to call replaces maturity cash flows with the call price and call date. For bonds with embedded options, analysts often examine yield to worst and option-adjusted spread rather than relying solely on YTM.

3.2 Holding-Period Return

For a beginning price P₀, ending price P₁, coupon income I_C and interest earned by reinvesting coupons I_R, all measured in currency units, the holding-period return is:

Every monetary component is divided by beginning market value, so the result is dimensionless. If coupons are not reinvested, I_R=0. Taxes and transaction costs should be deducted when calculating an investor's net return.

4. Term Structure of Interest Rates

The term structure relates yields to time to maturity for securities of comparable credit quality and liquidity.

• Spot rate: annualized return on a zero-coupon payment at a specified maturity.
• Forward rate: rate implied today for borrowing or investing during a future interval.
• Par yield: coupon rate that makes a bond's price equal to par.
• Yield curve: graphical relationship between yield and maturity.

Using annual spot rates z_t, an option-free bond is valued as:

This formula assumes annual periods; compounding must be adjusted for other conventions.

4.1 Bootstrapping Example

Suppose a one-year zero-coupon rate is 4%, and a two-year annual-pay 5% coupon bond trades at par. With face value $100:

Solving gives z₂ \approx 5.025\%. The one-year forward rate beginning one year from now satisfies:

Thus f_1,2\approx6.061\%. This is an implied no-arbitrage rate, not necessarily an unbiased forecast of the future spot rate.

4.2 Explanations of Yield-Curve Shape

Expectations theory associates long rates with expected future short rates, under restrictive assumptions excluding term premiums. Liquidity-premium theory argues that investors normally require additional compensation for longer maturities. Market-segmentation theory emphasizes maturity-specific supply and demand. Preferred-habitat theory allows investors to leave preferred maturity sectors when sufficiently compensated.

Observed curves reflect expected monetary policy, inflation, risk premiums, government issuance, regulation, collateral demand and market liquidity. Therefore, no single theory explains every curve movement.

5. Floating-Rate Instruments

A floating-rate note, or FRN, pays a coupon linked to a reference rate plus a quoted margin. A coupon for period i is commonly:

where R_i-1 is the reference rate fixed at the preceding reset date, s_q is the contractual quoted margin, and \delta_i is the year fraction under the applicable day-count convention.

Given discount factors DF_i, its general projected-cash-flow value is:

Here \widehat{R}_i-1 denotes the known or projected reference rate. Projection and discount curves may differ in modern multi-curve valuation. Credit and liquidity spreads must also be incorporated into appropriate discount rates.

Immediately after a reset, an FRN tends toward par if the quoted margin equals the market-required margin and issuer credit quality is unchanged. It need not trade at par when credit spreads, liquidity or contractual caps and floors change.

Worked Example: FRN Spread Valuation

Consider a two-year quarterly FRN with face value $1,000. Assume a flat projected benchmark of 5.0%, a quoted margin of 0.6%, and a market-required annual rate of 5.2%, all using simple quarterly compounding. The quarterly coupon is:

The required quarterly return is 0.052/4=0.013. Under the stated flat-rate assumptions:

The note trades above par because its $14 quarterly coupon exceeds the $13 payment required by the market rate. FRNs have low benchmark-rate duration near reset dates, but they retain credit-spread, liquidity, basis, cap and floor risk.

6. Duration and Interest-Rate Exposure

6.1 Macaulay and Modified Duration

Macaulay duration is the present-value-weighted average time to cash flow. For m payments per year:

Its unit is years. Modified duration measures first-order price sensitivity to a change in nominal annual yield:

For a small yield change \Delta y:

The approximation assumes a small, parallel yield change and unchanged cash flows. Dollar value of one basis point is approximately:

Worked Example: Duration

For the three-year 6% bond priced earlier, the present values are $55.5556, $51.4403 and $841.4622. Therefore:

A one-percentage-point yield increase therefore produces a first-order estimated price decline of approximately 2.619%.

6.2 Convexity

Duration is linear, whereas the actual price–yield relationship is curved. For a conventional bond with periodic yield compounding, convexity in years squared is:

The duration–convexity approximation is:

For the example bond, annual payments give convexity of approximately 9.507\text{ years}². For a yield rise from 8% to 9%:

The exact repriced value is approximately $924.06, also a decline of about 2.572%. The close result reflects the relatively small yield movement.

Callable and mortgage-backed securities can exhibit negative convexity because falling rates encourage calls or prepayments, limiting price appreciation. Their effective duration and convexity should be estimated using option-adjusted valuation models.

6.3 Key-Rate Duration

Parallel-shift duration does not capture twists or changes in curve curvature. Key-rate duration measures sensitivity to a localized movement at a selected maturity:

Here P_i^- and P_i^+ are prices after decreasing and increasing the selected key rate by \Delta z_i, while other key rates are held constant. A set of key-rate durations supports yield-curve hedging.

7. Credit Risk and Credit Spreads

Credit risk includes default risk, downgrade risk, recovery uncertainty and spread volatility. A simplified expected-loss model is:

where PD is probability of default, LGD is loss given default, and EAD is exposure at default. All inputs must refer to a consistent time horizon.

Credit spreads compensate investors not only for expected default loss but also for unexpected loss, liquidity, risk aversion, taxes and market technical factors. Important analytical inputs include leverage, interest coverage, cash-flow stability, collateral, covenant protection, industry conditions and sovereign exposure.

Credit ratings provide useful standardized opinions but are not guarantees. Managers should conduct independent analysis and monitor migration risk, concentration and correlated defaults. Credit default swaps may transfer part of the default exposure but introduce counterparty, basis and documentation risk.

8. Securitized Fixed-Income Products

Securitization pools financial assets and issues claims against their cash flows. Major products include mortgage-backed securities, asset-backed securities and collateralized loan obligations.

Pass-through securities distribute principal and interest from an asset pool after fees. Collateralized mortgage obligations divide cash flows into tranches with different priorities and prepayment sensitivities. Credit tranching reallocates losses: junior tranches absorb losses before senior tranches, subject to transaction rules.

Mortgage-backed securities expose investors to:

• Prepayment risk: principal returns sooner when refinancing increases.
• Extension risk: principal returns more slowly when rates rise.
• Negative convexity: price appreciation is constrained when rates fall.
• Model risk: valuation depends on uncertain borrower behavior.
• Servicing and legal risk: documentation and collection quality affect cash flows.

Asset-backed securities may be supported by auto loans, receivables, student loans or leases. Analysis must evaluate pool quality, excess spread, overcollateralization, reserve accounts, waterfall rules and stress performance rather than relying only on tranche ratings.

9. Bond Portfolio Strategies

9.1 Passive Strategies

Buy-and-hold emphasizes income and low turnover but does not eliminate default or reinvestment risk. Laddering distributes maturities across time, providing recurring liquidity and reducing dependence on a single reinvestment date. A bullet portfolio concentrates maturity near one target date, while a barbell combines short and long maturities.

Bond indexing seeks to replicate a benchmark's return and risk characteristics. Because bond indices may contain thousands of issues, managers often use stratified sampling or optimization based on duration, sector, rating and key-rate exposures. Tracking error, transaction costs and index turnover remain important.

9.2 Immunization

Classical single-period immunization protects a liability-funded portfolio against small parallel yield shifts by satisfying:

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

Immunization is not permanent. Duration changes with time, cash flows and yields, requiring rebalancing. It is also imperfect under nonparallel curve changes, defaults, transaction costs and embedded options.

9.3 Active Strategies

Active managers seek returns through:

• duration positioning based on rate views;
• yield-curve steepener or flattener trades;
• sector rotation;
• credit selection;
• liquidity-premium capture;
• relative-value and spread trades;
• option-adjusted analysis of callable and securitized securities.

Futures, swaps and options can modify duration efficiently. Their use requires controls over leverage, collateral, counterparty exposure and basis risk. Forecasting skill must exceed fees, trading costs and model error for active management to add value consistently.

10. Illustrative Case: Immunizing an Insurance Liability

Background

This fictional case concerns Meridian Life, which must pay a guaranteed $5 million benefit in five years. The annual effective market yield is 5%, and the investment team can buy three-year and seven-year zero-coupon government bonds.

Problem

The insurer wants to protect the funded position against a small parallel shift in interest rates while avoiding credit risk.

Analysis and Approach

The present value of the liability is:

Because zero-coupon bond Macaulay duration equals maturity, let w be the proportion invested in the three-year zero. Duration matching requires:

Thus w=0.5. Meridian allocates half of the liability's present value, $1,958,815.42, to each bond.

The face amounts are:

The asset portfolio's Macaulay duration is five years, matching the liability. Its convexity is:

The liability's convexity is:

Outcome

The portfolio matches present value and duration and has greater convexity than the liability. It is therefore locally immunized against small parallel rate movements. The three-year proceeds must later be reinvested, and the seven-year bond must be sold at the five-year horizon. Meridian must rebalance as time and rates change.

Learning Points

Immunization is a disciplined risk-control method rather than a guarantee. Its effectiveness depends on curve movements, transaction costs, liquidity, model assumptions and timely rebalancing.

11. Governance, Regulation and Contemporary Practice

Banks and insurers must integrate bond investment with liquidity requirements, capital rules, asset–liability management and stress testing. Governance should establish limits for duration, credit ratings, sectors, issuers, currencies, derivatives and illiquid assets. Independent valuation and model validation are particularly important for structured products.

Ethical risks include misleading yield presentation, unsuitable product sales, concealed leverage, inflated securitization quality and greenwashing in sustainable bonds. Portfolio managers should disclose methodology, conflicts, fees, option risks and material assumptions.

Electronic trading, alternative data and machine learning increasingly support liquidity estimation, credit monitoring and trade execution. However, historical data may underrepresent crises, while opaque models can amplify correlated trading. Human oversight, explainability, cybersecurity and data governance remain essential.

Recent developments include greater use of secured overnight reference rates, growth in green and sustainability-linked bonds, increased attention to sovereign debt vulnerability, and more frequent climate-related stress testing. These developments change analytical inputs but do not replace cash-flow valuation and disciplined risk measurement.

Chapter Summary

Fixed-income valuation begins with discounting expected cash flows at rates appropriate to their timing, credit quality, liquidity and optionality. Bond prices move inversely and convexly with yields. Yield to maturity summarizes promised cash flows but does not guarantee realized return, while holding-period return includes income, reinvestment income and price change relative to beginning value.

Spot and forward rates describe the term structure and support arbitrage-consistent pricing. Floating-rate notes reset coupons with reference rates but remain exposed to spread, credit and liquidity changes. Duration estimates first-order interest-rate sensitivity; convexity improves estimates for larger changes, while key-rate duration captures nonparallel curve exposure.

Credit analysis combines expected-loss measurement with qualitative assessment. Securitized products require analysis of collateral, payment waterfalls, prepayments and embedded options. Portfolio managers may use laddering, indexing, immunization or active strategies according to liabilities, benchmarks and risk tolerance. Effective governance requires independent analysis, transparent reporting, stress testing and ongoing rebalancing.

Key Terms

Accrued Interest: Coupon interest earned since the previous payment date.

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

Clean Price: Quoted bond price excluding accrued interest.

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

Credit Spread: Yield premium over a reference rate for credit and related risks.

Dirty Price: Settlement price including accrued interest.

Duration: Measure of cash-flow timing or price sensitivity to yield changes.

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

Floating-Rate Note: Debt instrument whose coupon resets using a reference rate.

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

Holding-Period Return: Total investment return over a specified ownership period.

Key-Rate Duration: Sensitivity to a yield change at a selected curve maturity.

Laddering: Strategy of distributing bond maturities across several dates.

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

Negative Convexity: Condition in which price gains are constrained when yields fall.

Option-Adjusted Spread: Spread after accounting for embedded-option value.

Securitization: Pooling assets and issuing securities backed by their cash flows.

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

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

Review Questions

Short Answer Questions

1. Why does a bond sell at a discount when its coupon rate is below its required yield?
2. Distinguish clean price from dirty price.
3. Why is yield to maturity not necessarily the realized return?
4. What risks remain in a floating-rate note after its coupon resets?
5. What causes negative convexity in mortgage-backed securities?

Long Answer Questions

1. Explain the valuation and risk characteristics of fixed-rate, zero-coupon and floating-rate bonds.
2. Compare expectations, liquidity-premium, market-segmentation and preferred-habitat theories.
3. Explain Macaulay duration, modified duration, convexity and key-rate duration.
4. Discuss the credit and structural risks of securitized products.
5. Compare laddering, indexing, immunization and active bond management.

Analytical / Application Questions

1. Price a five-year 7% annual coupon bond with $1,000 face value at a 6% annual yield, and explain whether it trades at a premium or discount.
2. Construct an immunized portfolio for a six-year liability using four-year and eight-year zero-coupon bonds, assuming equal credit quality.
3. Evaluate how a simultaneous rise in benchmark rates and issuer credit spreads would affect fixed-rate and floating-rate corporate bonds.

Multiple Choice Questions with Answers

Question 1

A three-year bond has a $1,000 face value, a 6% annual coupon and an 8% annual yield. What is its price?

A. $946.28

B. $948.46

C. $1,000.00

D. $1,060.00

Correct Answer: B. $948.46

Question 2

A one-year zero rate is 4%. A two-year 5% annual coupon bond with $100 face value trades at par. What is the two-year spot rate?

A. 4.000%

B. 4.762%

C. 5.025%

D. 6.061%

Correct Answer: C. 5.025%

Question 3

A bond has modified duration of 4 and convexity of 20. Using the duration–convexity approximation, what is its percentage price change when yield falls by 0.50 percentage points?

A. −2.025%

B. −1.975%

C. +1.975%

D. +2.025%

Correct Answer: D. +2.025%

Question 4

A bond is purchased for $980 and sold for $990. During ownership it pays $40 in coupons, which generate $1 of reinvestment income. What is the holding-period return?

A. 4.18%

B. 5.10%

C. 5.20%

D. 6.12%

Correct Answer: C. 5.20%

Question 5

When will a floating-rate note generally trade near par immediately after a coupon reset?

A. When its quoted margin equals the market-required margin and credit quality is unchanged

B. Whenever its maturity exceeds one year

C. Only when the reference rate is zero

D. Whenever its coupon is paid quarterly

Correct Answer: A. When its quoted margin equals the market-required margin and credit quality is unchanged

Question 6

A loan portfolio has exposure at default of $10 million, one-year default probability of 2% and loss given default of 40%. What is expected loss?

A. $8,000

B. $40,000

C. $80,000

D. $200,000

Correct Answer: C. $80,000

Question 7

Which conditions are central to classical first-order bond immunization?

A. Equal coupon rates and equal face values

B. Equal present values and matched durations

C. Equal maturities and equal credit ratings only

D. Maximum yield and minimum market value

Correct Answer: B. Equal present values and matched durations

Question 8

Why may a mortgage-backed security exhibit negative convexity?

A. Borrowers tend to prepay more rapidly when interest rates fall

B. Default probability always falls when interest rates rise

C. Its coupon necessarily increases when market rates decline

D. Its maturity is legally fixed and cannot change

Correct Answer: A. Borrowers tend to prepay more rapidly when interest rates fall

Question 9

A bond has a clean price of $1,012 and accrued interest of $8. What is its dirty price?

A. $1,004

B. $1,012

C. $1,016

D. $1,020

Correct Answer: D. $1,020

Question 10

Which strategy distributes bond maturities across several dates to provide recurring liquidity and reduce concentration in one reinvestment date?

A. Laddering

B. Credit tranching

C. Yield-curve inversion

D. Short selling

Correct Answer: A. Laddering