Fixed-Income Valuation and Market Risk

Introduction

Fixed-income securities are central to bank liquidity portfolios, insurance investments, pension assets and treasury funding. Their contractual cash flows may appear predictable, but their market values respond to interest rates, credit conditions, embedded options, liquidity and volatility. Treasury professionals must therefore understand both valuation and risk measurement.

This chapter develops cash-flow valuation, yield-curve construction, credit-spread interpretation, duration and convexity, and portfolio exposure measurement. Unless otherwise stated, examples assume no taxes, transaction costs or default between valuation and settlement dates.

Learning Objectives

After studying this chapter, students should be able to:

• value fixed-rate and zero-coupon debt securities;
• distinguish spot, par, forward and yield-to-maturity rates;
• bootstrap discount factors and spot rates from market instruments;
• interpret credit spreads without confusing rate-based and currency-based measures;
• calculate duration, DV01 and convexity;
• estimate portfolio exposure to parallel, non-parallel and spread movements;
• evaluate the limitations and governance requirements of market-risk models.

1. Fixed-Income Cash Flows and Prices

A conventional bond promises periodic coupons and repayment of principal. A zero-coupon bond pays only principal at maturity. Floating-rate notes reset coupons using a reference rate plus a contractual margin, while callable, puttable and mortgage-backed securities contain options that alter future cash flows.

1.1 Present-value principle

Let a security produce cash flows CF_i at times t_i. If D(t_i) is the appropriate discount factor, its theoretical price is:

Here, P and CF_i are measured in currency units, while D(t_i) is dimensionless. The discount factor represents the present value of one currency unit received at time t_i.

If annual effective spot rates z_i apply to annual cash-flow dates, then:

This form assumes t_i is measured in years and each spot rate is effective annually. Other compounding conventions require corresponding adjustments.

1.2 Valuation using yield to maturity

For a fixed-rate bond with face value F, coupon rate c, m coupon payments per year and N remaining coupon periods, the coupon per period is C=Fc/m. If all cash flows are discounted at one nominal annual yield to maturity y, compounded m times per year:

Yield to maturity is the internal rate of return implied by the price. It is not generally an expected return because it assumes no default, reinvestment of coupons at the same yield, and holding to maturity. Using one yield also suppresses the term structure embedded in spot rates.

A bond sells:

• at par when its coupon rate equals its yield under consistent conventions;
• at a premium when its coupon rate exceeds its yield;
• at a discount when its coupon rate is below its yield.

1.3 Clean price, accrued interest and dirty price

Quoted bond prices are commonly clean prices, excluding accrued coupon interest. Settlement uses the dirty price:

Under a simple proportional convention:

Here, d is the number of accrued days since the previous coupon, D is the number of days in the coupon period under the applicable day-count convention, and C is the coupon payment. Actual market conventions differ across securities.

2. The Term Structure of Interest Rates

The term structure describes how interest rates differ by maturity for instruments of comparable credit quality and contractual characteristics.

2.1 Spot, par and forward rates

These rates are related but not interchangeable.

Figure: Illustrative relationships among upward-sloping spot, par and forward curves; actual ordering and shape depend on market conditions.

2.2 Par-yield formula

Consider a bond priced at face value 1, paying an annualized par coupon rate c_par on dates t₁,…,t_n. Let α_i be the year fraction for coupon period i and D(t_i) the corresponding discount factor. Its par condition is:

Therefore:

The numerator is dimensionless, while the denominator is measured in years, so c_par is an annualized rate. For equally spaced coupons, α_i=1/m. The formula applies when the bond is valued at par and principal is repaid at the final date.

2.3 Bootstrapping a zero-coupon curve

Bootstrapping recursively extracts discount factors from instruments whose earlier cash flows can already be valued. For instrument n with market price P_n, known earlier cash flows and one final cash flow at t_n:

This requires consistent prices, settlement dates, day-count rules and cash-flow assumptions.

#### Worked example: annual par bonds

Suppose three default-free bonds with face value 100 pay annual coupons and trade at par:

For the one-year bond:

For the two-year bond:

For the three-year bond:

The annual effective spot rates are obtained from z_t=D(t)^-1/t-1:

In practice, instruments do not mature on every required date. Treasuries therefore use interpolation, smoothing splines or parametric curve models. The method must avoid implausible discontinuities and should be independently validated.

2.4 Forward rates

For annual effective rates, the one-year forward rate from year 1 to year 2 satisfies:

Using the bootstrapped factors:

An implied forward rate is primarily a no-arbitrage rate. It is not necessarily an unbiased forecast of the future spot rate because risk and term premiums may be present.

2.5 Explanations of curve shape

• Expectations hypothesis: long rates reflect expected future short rates, in its pure form without term premiums.
• Liquidity-premium theory: investors generally require compensation for holding longer maturities.
• Market-segmentation theory: supply and demand within maturity sectors affect rates.
• Preferred-habitat theory: investors prefer particular maturities but may move when adequately compensated.

Observed curves can be upward-sloping, flat, inverted or humped. No single theory explains all market conditions.

3. Credit Risk and Credit Spreads

A credit spread is the yield difference between a credit-risky instrument and an appropriately matched benchmark. Comparisons should control for maturity, currency, seniority, liquidity, coupon structure and embedded options.

3.1 Dimensionally consistent expected loss

For a horizon h, expected credit loss in currency units is:

Here:

• PD_h is the probability of default over horizon h, dimensionless;
• LGD is loss given default as a fraction of exposure, dimensionless;
• EAD is exposure at default in currency units.

Thus, EL_currency is a currency amount. It cannot be directly added to, or compared with, a credit spread expressed as a rate.

Expected loss normalized by exposure is:

This is a cumulative loss rate over horizon h. Under a simple annualization for short horizons, its annualized approximation is PD_h×LGD/h, where h is measured in years. Annualization becomes unreliable for long horizons or non-uniform default timing.

Under a reduced-form model with constant annual default intensity λ and fractional recovery R, a common continuous-time approximation is:

The spread s and intensity λ are rates per year. This approximation assumes simplified recovery, liquidity and risk-premium conditions; it is not a universal spread identity.

#### Numerical interpretation

Suppose one-year PD=2\%, LGD=60\% and EAD=10 million currency units.

The normalized one-year expected-loss rate is:

If the observed spread is 190 basis points, the additional 70 basis points may reflect risk aversion, liquidity, uncertainty, taxes, market technical factors and model differences. It should not automatically be labelled a pure risk premium.

3.2 Components of observed spreads

Conceptually, an observed spread may compensate for:

• normalized expected default loss;
• systematic credit-risk premium;
• liquidity and transaction costs;
• embedded options;
• tax and regulatory treatment;
• uncertainty and market supply–demand effects.

These components interact and are not always separately observable.

3.3 Z-spread and option-adjusted spread

A Z-spread is a constant spread added to each benchmark spot rate so that discounted promised cash flows equal the market price. An option-adjusted spread (OAS) attempts to remove the value of embedded options using an interest-rate model. OAS is model-dependent and is particularly important for callable bonds and mortgage-backed securities.

4. Interest-Rate Sensitivity

4.1 Macaulay duration

Macaulay duration is the present-value-weighted average time to receive a security's cash flows:

Time t_i and D_Mac are measured in years. The formula describes cash-flow timing under the selected discount factors.

4.2 Modified duration

For a bond valued using one nominal yield y compounded m times annually:

Therefore, for a small parallel yield change:

The yield change must be expressed in decimal form; 25 basis points equals 0.0025. Modified duration assumes fixed cash flows and a small parallel movement in the yield used for valuation.

4.3 Worked duration example

Consider a three-year bond with face value 100, annual coupon rate 6% and annual yield of 5%.

Its Macaulay duration is:

Thus:

For a 50-basis-point yield increase:

The duration-only estimated price is approximately 101.34. Direct repricing at 5.5% gives approximately 101.35; the small difference reflects curvature.

4.4 Dollar duration and DV01

Dollar duration measures value sensitivity to a unit change in yield:

DV01, or value change for a one-basis-point movement, is:

For the preceding bond, DV01 is approximately 102.723×2.701×0.0001=0.0277 currency units per 100 face value. By convention, DV01 is often quoted as the positive magnitude of the loss from a one-basis-point yield increase.

4.5 Convexity

Duration is a first-order approximation. For a bond discounted by a nominal annual yield y with m compounding periods, convexity is:

Its units are years squared under this yield convention. The second-order approximation is:

Option-free bonds normally have positive convexity. Callable bonds and mortgage-backed securities may exhibit negative convexity because falling rates increase the likelihood of early redemption.

4.6 Effective and key-rate duration

When cash flows change as rates change, analytical modified duration is inappropriate. Effective duration uses full repricing:

Here, P_- is the price after a downward shift, P_+ after an equal upward shift, and P₀ the initial price. Embedded options must be revalued in both scenarios.

Key-rate duration measures sensitivity to movements at selected curve maturities while holding other key rates approximately fixed. It identifies curve-shape exposure that a single duration conceals.

5. Spread and Portfolio Risk

5.1 Spread duration

Spread duration estimates price sensitivity to a small change in credit spread, holding the benchmark curve constant:

For instruments with options, spread duration should be based on model-consistent OAS changes. Spread widening produces losses for a long credit position.

5.2 Portfolio aggregation

For positions i with market values V_i and modified durations D_i, approximate portfolio DV01 is:

This signed aggregation is valid for a common small parallel shock and consistent valuation conventions. Offset may be misleading when instruments respond to different curve points, currencies, spreads or optionality.

A first-order factor representation is:

Here, DD_k is dollar sensitivity to benchmark rate factor k, and SD_j is dollar spread sensitivity to spread factor j. Changes are expressed as decimal rates.

5.3 Volatility, correlation and VaR

If linear factor sensitivities are collected in vector b and the covariance matrix of rate and spread changes is Σ, approximate profit-and-loss variance is:

Under a zero-mean normal approximation, one-tailed value at risk is:

For example, z_0.99≈2.326. VaR is a loss threshold at a stated confidence level and horizon, not the maximum possible loss. Historical simulation, Monte Carlo simulation, expected shortfall and stress testing are essential complements, especially where distributions are nonlinear or heavy-tailed.

6. Illustrative Treasury Case: Managing a Bond Portfolio

Background

This fictional case concerns Meridian Bank, which holds the following liquidity and investment portfolio:

Problem

The risk committee requests an estimate of losses if benchmark rates rise by 75 basis points, corporate spreads widen by 40 basis points and mortgage spreads widen by 25 basis points.

Analysis

The approximate benchmark-rate loss is:

The spread loss is:

Total first-order loss is therefore:

The portfolio's benchmark-rate DV01 is:

Thus, its approximate loss per one-basis-point rate increase is 28,550 currency units.

Approach and outcome

A pay-fixed, receive-floating interest-rate swap can provide negative duration and reduce benchmark-rate exposure. If a 10-million-notional swap has a DV01 magnitude of 7,800, the approximate hedge notional is:

The hedge addresses parallel benchmark-rate risk but does not eliminate credit-spread, mortgage prepayment, curve, basis or liquidity risk. The treasury should therefore combine the swap with key-rate limits, spread limits and stress testing.

Learning points

• Rate duration and spread duration measure different risk factors.
• Aggregated duration can hide curve and optionality risks.
• Hedges create basis, counterparty and liquidity exposures.
• Full repricing is preferable for large shocks and option-sensitive assets.

7. Limitations, Governance and Contemporary Practice

7.1 Model limitations

Yield-based duration assumes a parallel change, stable cash flows and small shocks. Credit spreads are not pure default measures. OAS results depend on volatility, prepayment and interest-rate models. Market prices may also be unreliable in illiquid periods.

Best practice includes independent price verification, model validation, sensitivity analysis, backtesting, stress testing and transparent treatment of valuation adjustments.

7.2 Regulation and accounting

Banks must distinguish trading-book market risk from banking-book interest-rate risk and maintain approved limits, escalation procedures and capital processes. Insurers must consider asset–liability duration mismatches and policyholder options. Accounting classifications affect whether valuation changes enter profit, other comprehensive income or amortized-cost calculations. Expected-credit-loss accounting and fair-value measurement serve different purposes and should not be conflated.

7.3 Ethics and controls

Curve selection, liquidity adjustments and model assumptions can materially affect reported profit and capital. Governance should prevent selective use of models to conceal losses. Independent valuation control, documented data sources, segregation of duties and audit trails are essential.

7.4 Technology and emerging developments

Modern treasury systems combine market data, cash-flow engines, scenario simulation and real-time limits. Machine learning may assist with liquidity estimation, prepayment forecasting and anomaly detection, but opaque models can amplify governance risk. Benchmark transitions, electronic trading, climate-related repricing and episodes of rapid monetary tightening have increased the importance of basis risk, liquidity stress and intraday monitoring.

Chapter Summary

Fixed-income valuation discounts each contractual or model-dependent cash flow using an appropriate discount factor. Spot rates value individual cash flows, par yields determine coupons that produce par value, and forward rates are implied by no-arbitrage relationships. Bootstrapping recursively derives discount factors from traded instruments.

Credit spread analysis must preserve dimensional consistency: expected loss in currency equals PD × LGD × EAD, whereas the normalized loss rate equals PD × LGD over the stated horizon. Observed spreads additionally reflect risk premiums, liquidity, options and market conditions.

Duration measures first-order interest-rate sensitivity, convexity captures curvature, and effective duration accommodates rate-dependent cash flows. DV01 converts duration into currency sensitivity. Key-rate and spread durations reveal exposures hidden by aggregate duration. Portfolio management requires factor-based measurement, full repricing, stress testing, disciplined hedging and strong model governance.

Key Terms

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

Bootstrapping: Recursive extraction of discount factors from market prices.

Clean Price: Bond price excluding accrued interest.

Convexity: Second-order measure of price sensitivity to yield changes.

Credit Spread: Yield difference relative to an appropriately matched benchmark.

Dirty Price: Clean price plus accrued interest.

Discount Factor: Present value of one currency unit paid at a future date.

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

Effective Duration: Repricing-based duration for instruments with rate-dependent cash flows.

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

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

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

Macaulay Duration: Present-value-weighted average cash-flow time.

Modified Duration: First-order percentage price sensitivity to yield.

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

Par Yield: Coupon rate that causes a bond to trade at par.

Probability of Default: Likelihood of default over a specified horizon.

Spot Rate: Rate used to discount a single cash flow at a specific maturity.

Spread Duration: Price sensitivity to a change in credit spread.

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

Review Questions

Short Answer Questions

1. Distinguish clean price from dirty price.
2. Why is yield to maturity not necessarily an investor's realized return?
3. Define a discount factor and explain its units.
4. Why cannot PD × LGD × EAD be directly compared with a credit spread?
5. What additional risk does key-rate duration reveal?

Long Answer Questions

1. Explain spot, par, forward and yield-to-maturity rates and their relationships.
2. Derive the par-yield formula from the present-value condition for a par bond.
3. Describe a complete bootstrapping procedure, including data and interpolation requirements.
4. Compare Macaulay, modified, effective and spread duration.
5. Critically evaluate VaR as a measure of fixed-income portfolio risk.

Analytical / Application Questions

1. Bootstrap two- and three-year discount factors from a set of annual par bonds and interpret the resulting forward rates.
2. Estimate the rate and spread losses on a mixed government and corporate bond portfolio under a specified stress scenario.
3. Design a duration hedge for an option-sensitive bond portfolio and identify the risks that remain after hedging.

Multiple Choice Questions with Answers

Question 1

A one-year zero-coupon bond with face value 100 is priced at 95.24. Assuming annual compounding, its spot rate is closest to:

A. 4.00%

B. 5.00%

C. 5.24%

D. 6.00%

Correct Answer: B. 5.00%

Question 2

A two-year annual-coupon par bond pays a 5% coupon. If D(1)=0.96, what is D(2)?

A. 0.8571

B. 0.9067

C. 0.9520

D. 1.0500

Correct Answer: B. 0.9067

Question 3

Which formula gives the annualized par coupon rate for accrual fractions α_i and discount factors D(t_i)?

A. c_par=(1-D(t_n))/∑α_iD(t_i)

B. c_par=D(t_n)/∑D(t_i)

C. c_par=∑α_iD(t_i)/(1-D(t_n))

D. c_par=1/D(t_n)-1

Correct Answer: A. c_par=(1-D(t_n))/∑α_iD(t_i)

Question 4

A loan has one-year PD of 1.5%, LGD of 40% and EAD of 5 million. What is its normalized one-year expected-loss rate?

A. 0.06%

B. 0.60%

C. 1.50%

D. 3.75%

Correct Answer: B. 0.60%

Question 5

Why is PD×LGD×EAD not directly comparable with a quoted credit spread?

A. It excludes coupon income.

B. It is always measured after tax.

C. It is a currency amount, whereas a spread is a rate over time.

D. It applies only to government securities.

Correct Answer: C. It is a currency amount, whereas a spread is a rate over time.

Question 6

A bond has modified duration 4.0. For a 25-basis-point yield increase, its duration-based percentage price change is approximately:

A. +1.00%

B. +0.10%

C. −0.10%

D. −1.00%

Correct Answer: D. −1.00%

Question 7

A portfolio worth 2 million has modified duration 3.0. Its approximate DV01 is:

A. 60 currency units

B. 600 currency units

C. 6,000 currency units

D. 60,000 currency units

Correct Answer: B. 600 currency units

Question 8

If D(1)=0.97 and D(2)=0.92, the annual effective one-year forward rate from year 1 to year 2 is closest to:

A. 2.17%

B. 5.00%

C. 5.43%

D. 8.70%

Correct Answer: C. 5.43%

Question 9

Which measure is most appropriate for a callable bond whose expected cash flows change when interest rates change?

A. Current yield

B. Macaulay duration based on unchanged cash flows

C. Effective duration based on full repricing

D. Accounting maturity

Correct Answer: C. Effective duration based on full repricing

Question 10

A pay-fixed, receive-floating interest-rate swap is used to hedge a long fixed-rate bond portfolio primarily because it:

A. adds positive benchmark-rate duration

B. provides negative benchmark-rate duration

C. eliminates all credit-spread risk

D. guarantees positive convexity

Correct Answer: B. provides negative benchmark-rate duration