Interest Rates, Inflation and Fixed-Income Valuation

Introduction

Interest rates are the price of using money over time. They influence household saving, corporate investment, government borrowing, exchange rates, asset prices, and financial stability. Fixed-income securities—especially bonds—are contracts that promise specified or formula-based cash flows. Their values depend primarily on the level and structure of interest rates, expected inflation, and the probability that promised payments will not be made.

This chapter develops a framework for understanding interest-rate determination and for valuing bonds. It then examines yield measures, the price–yield relationship, duration, interest-rate risk, credit ratings, and default risk. The emphasis is on interpretation as well as calculation: a finance professional must understand not only what a bond is worth, but why its value changes and what risks that valuation contains.

Learning Objectives

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

1. Explain the economic determinants of nominal and real interest rates.
2. Describe the term structure of interest rates and distinguish spot, forward, and yield-curve concepts.
3. Explain inflation expectations, liquidity premiums, and credit spreads.
4. Value coupon bonds, zero-coupon bonds, and perpetuities.
5. Calculate current yield, yield to maturity, and approximate yield to call.
6. Analyze the inverse price–yield relationship and reinvestment risk.
7. Calculate and interpret Macaulay duration, modified duration, and convexity.
8. Assess credit ratings, credit spreads, and default risk.
9. Apply fixed-income analysis to investment and corporate-finance decisions.

1. The Economics of Interest Rates

1.1 Nominal and real interest rates

A nominal interest rate is the stated rate before adjusting for inflation. A real interest rate measures the change in purchasing power. The exact Fisher relationship is:

Here, i is the nominal interest rate per period, r is the real interest rate, and π^e is expected inflation over the same period. Therefore:

For modest rates, the approximation i ≈ r + π^e is often useful. Rates must refer to the same period and compounding convention. If a nominal annual rate is compounded monthly, it must first be converted appropriately before comparison with an annual real rate.

The real rate compensates lenders for postponing consumption and bearing risk. Expected inflation compensates them for the anticipated loss of purchasing power. If inflation is unexpectedly high, borrowers generally benefit and lenders lose, unless contracts adjust payments for inflation.

1.2 Supply and demand for loanable funds

Interest rates are influenced by the supply of savings and the demand for borrowing. Household saving, corporate retained earnings, government surpluses, and foreign capital contribute to supply. Business investment, household credit demand, government deficits, and working-capital needs contribute to demand.

An increase in desired borrowing, holding other factors constant, tends to raise rates. Greater saving tends to lower rates. In practice, central banks influence short-term rates through monetary policy, while market expectations determine much of the longer-term rate structure.

1.3 Risk and liquidity premiums

Investors require compensation for risks beyond the time value of money. A risky bond’s yield can be represented conceptually as:

The risk-free benchmark is usually associated with highly creditworthy government securities in the relevant currency, although even sovereign securities can contain inflation, liquidity, and sovereign-risk components. Important premiums include:

• Default premium: compensation for possible failure to pay interest or principal.
• Liquidity premium: compensation for difficulty selling quickly at a fair price.
• Maturity premium: compensation for exposure to changing rates over a longer horizon.
• Tax premium or discount: adjustment for differences in tax treatment.
• Option premium: compensation associated with embedded call, put, or conversion features.

2. Inflation and Interest Rates

Inflation is a sustained increase in the general price level. It affects fixed-income securities because bond payments are usually fixed in nominal currency. A bond paying $1,000 in ten years is less valuable today if investors expect substantial inflation.

Inflation affects rates through two channels. First, lenders demand compensation for expected loss of purchasing power. Second, uncertainty about future inflation increases the risk premium. Unexpected inflation can also redistribute wealth between borrowers and lenders.

Inflation-linked bonds adjust principal or coupons according to an inflation index. Their yields are often interpreted as real yields, while the difference between comparable nominal and inflation-linked yields is commonly called the breakeven inflation rate. This measure is informative but not a pure forecast because it may include liquidity differences, risk premiums, and indexation features.

3. The Term Structure of Interest Rates

The term structure describes how interest rates vary with maturity for securities of comparable credit quality, liquidity, and contractual features. A graph of yields against maturities is the yield curve.

A spot rate is the annualized return on a zero-coupon investment for a particular maturity. A forward rate is the rate implied today for borrowing or lending during a future period. For annual compounding, the one-year forward rate beginning in year 1 is derived from two spot rates:

Thus:

Here, s₁ and s₂ are one- and two-year spot rates, and f_1,2 is the one-year forward rate for the second year. This calculation assumes annual compounding and no arbitrage.

3.1 Yield-curve theories

Three traditional theories explain the term structure:

1. Expectations theory: long-term rates reflect expected future short-term rates. An upward curve may indicate expectations of rising short-term rates.
2. Liquidity-preference theory: investors prefer shorter maturities and require a positive maturity premium to hold longer securities. The curve may therefore slope upward even when future short-term rates are expected to remain stable.
3. Market-segmentation theory: investors and borrowers operate in preferred maturity segments, so supply and demand differ across maturities.

Modern interpretations generally combine expectations with time-varying risk premiums. Consequently, a yield curve is not a mechanical forecast of future rates.

3.2 Shapes and interpretations

A normal curve slopes upward, a flat curve shows similar yields across maturities, and an inverted curve has lower long-term than short-term yields. A humped curve may reflect changing expectations and maturity-specific demand. Analysts should examine the curve together with inflation expectations, central-bank policy, economic growth, fiscal conditions, and market liquidity.

4. Bond Characteristics and Valuation

A bond’s key contractual terms include face value, coupon rate, coupon frequency, maturity, seniority, security, covenants, and embedded options. The face value is the amount repaid at maturity. The coupon payment is:

where F is face value, c is the annual coupon rate, and m is the number of coupon payments per year. The periodic required return is y/m, where y is the annual yield consistent with the compounding convention.

The general bond-pricing formula is:

Here, P is the full (dirty) price, n is the total number of coupon periods, and all rates and periods must use the same frequency. The formula assumes promised cash flows, constant discounting at yield y, and no default or transaction costs.

A zero-coupon bond has no interim coupons:

A perpetuity with level coupon C and no maturity has value:

This perpetuity formula assumes a constant periodic yield and an infinite sequence of payments.

Worked Example 1: Coupon-bond value

A three-year bond has face value $1,000, an annual coupon rate of 6%, annual payments, and a required yield of 8%. The annual coupon is $60. Therefore:

P = 60/1.08 + 60/(1.08)² + 1,060/(1.08)³

The components are approximately $55.56, $51.44, and $841.52. Thus, P ≈ $948.52. The bond sells below par because its 6% coupon is less than the market yield of 8%.

If the required yield were 4%, the price would exceed $1,000. If it equaled 6%, the price would equal par, assuming identical payment frequency and no special features.

Accrued interest and quoted price

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

For a simple bond with equal coupon periods, accrued interest is approximately the coupon payment multiplied by the fraction of the coupon period elapsed. Market conventions differ, so professional valuation must follow the relevant day-count convention.

5. Yield Measures

5.1 Current yield

Current yield measures annual coupon income relative to the bond’s market price:

It ignores capital gain or loss and therefore is not a complete measure of return.

5.2 Yield to maturity

Yield to maturity (YTM) is the discount rate that equates the present value of promised cash flows with the bond’s market price. It is the internal rate of return under the assumptions that the bond is held to maturity, all promised payments occur, and coupons are reinvested at the YTM.

For a coupon bond, YTM usually must be found by trial and error, a financial calculator, or numerical software. It is not necessarily the investor’s realized return because default, sale before maturity, changing reinvestment rates, and taxes can alter outcomes.

5.3 Yield to call and yield to worst

A callable bond may be redeemed before maturity. Yield to call is calculated using the call date and call price instead of maturity date and face value. Investors commonly examine yield to maturity, yield to call, and other relevant redemption scenarios; the lowest applicable yield is often called yield to worst.

Worked Example 2: Current yield

A bond pays a $70 annual coupon and trades for $950. Its current yield is:

70/950 = 0.073684 ≈ 7.37%

This does not include the potential $50 gain if the bond eventually pays $1,000 at maturity.

6. Price–Yield Relationships and Interest-Rate Risk

Bond prices and yields move inversely. When market yields rise, existing fixed coupons become less attractive, so their prices fall. When yields fall, existing coupons become more attractive, so prices rise. The relationship is nonlinear: the price–yield curve is convex, meaning the price increase from a yield decline is generally larger than the price decrease from an equal yield increase.

Two major risks affect bond investors:

• Price risk: rising yields reduce the market value of an existing bond.
• Reinvestment risk: falling rates reduce the return available when coupons are reinvested.

Longer-maturity and lower-coupon bonds generally have greater price sensitivity to yield changes. A zero-coupon bond with maturity equal to the investor’s horizon has no interim coupon-reinvestment risk, but it can have substantial price risk before maturity.

7. Duration and Convexity

7.1 Macaulay duration

Macaulay duration is the weighted average time until a bond’s promised cash flows are received. The weights are the present values of individual cash flows divided by bond price:

CF_t is the cash flow at period t, P is price, and D_M is measured in coupon periods. Convert to years by dividing by m.

7.2 Modified duration

Modified duration estimates percentage price sensitivity to a small change in yield:

When ∆y is the change in annual yield expressed as a decimal, the first-order approximation is:

This assumes a parallel shift in the relevant yield and a small change. It is less accurate for large changes or securities with embedded options.

7.3 Convexity adjustment

Convexity improves the estimate for larger yield changes:

Convexity is positive for ordinary option-free bonds. Callable bonds may exhibit negative or changing convexity when the call option becomes valuable.

Worked Example 3: Duration estimate

Suppose a bond has modified duration of 5.2 years. If its yield rises by 0.50 percentage points, ∆y = 0.005. The estimated percentage price change is:

∆P/P ≈ −5.2 × 0.005 = −0.026

The price is expected to decline by approximately 2.6%, before considering convexity. If the bond is priced at $1,020, the estimated dollar decline is about $26.52, giving an estimated price of $993.48.

8. Credit Risk, Ratings and Spreads

Credit risk is the possibility that a borrower will fail to pay interest or principal as promised. It includes default risk, downgrade risk, recovery risk, and spread risk. A credit spread is the difference between the yield on a risky bond and the yield on a suitable benchmark of similar maturity and currency:

The spread compensates for expected loss and risk premiums. It is not identical to the expected default loss because it also reflects liquidity, risk aversion, taxes, market segmentation, and other factors.

Credit-rating agencies classify issuers and issues according to relative credit quality. Investment-grade ratings generally indicate lower assessed default risk; below-investment-grade securities carry greater credit risk and are often called high-yield or speculative-grade bonds. Ratings are opinions, not guarantees, and may lag rapidly changing conditions.

A simplified expected-loss framework is:

Loss given default is approximately one minus the recovery rate:

For example, if annual probability of default is 2% and expected recovery is 40%, expected loss is 0.02 × 0.60 = 0.012, or 1.2% of exposure, before considering timing and uncertainty. Real credit models use transition probabilities, exposure at default, collateral, seniority, and scenario analysis.

Credit analysis

Professional analysis considers:

• profitability, leverage, liquidity, and cash-flow coverage;
• industry structure and business-cycle sensitivity;
• management quality and governance;
• collateral, guarantees, seniority, and covenants;
• refinancing needs and maturity concentration;
• macroeconomic and foreign-exchange exposure.

Credit spreads can widen even without an actual default. A downgrade, weaker economic outlook, reduced market liquidity, or increased investor risk aversion can lower bond prices.

9. Illustrative Case: Financing Northstar Manufacturing

Background

Northstar Manufacturing, a fictional mid-sized producer, plans to issue five-year bonds with $1,000 face value and a 7% annual coupon. Comparable government securities yield 4.5%, while bonds of firms with similar credit quality trade at a 2.0% credit spread.

Problem

Management must estimate the issue price, understand the effect of a credit deterioration, and assess whether the financing creates significant interest-rate exposure.

Analysis

The required yield is approximately:

4.5% + 2.0% = 6.5%

Because the 7% coupon exceeds the 6.5% required yield, the bond should issue above par. Its approximate value is:

P = 70/1.065 + 70/(1.065)² + 70/(1.065)³ + 70/(1.065)⁴ + 1,070/(1.065)⁵

This is approximately $1,020. The precise price depends on market conventions and the exact benchmark curve.

Suppose recession concerns increase the credit spread by 1.5 percentage points while the government yield remains unchanged. The required yield becomes 8.0%, and the bond’s price falls below par. Existing investors experience a mark-to-market loss even though Northstar has not yet defaulted.

Approach and outcome

Northstar should stress-test rates and spreads, avoid concentrating maturities in one year, consider fixed versus floating-rate debt, and maintain liquidity reserves. Investors should evaluate leverage, cash-flow coverage, covenant protection, and recovery prospects rather than relying solely on the rating.

Learning points

The case demonstrates that bond values respond to both general interest rates and issuer-specific spreads. A high coupon does not eliminate price risk, and a rating does not guarantee payment. Financing decisions should integrate valuation, risk management, and governance.

10. Limitations, Regulation and Professional Practice

Bond valuation is model-dependent. A single YTM can conceal a non-flat yield curve, and assuming a constant yield may be inappropriate when spot rates differ by maturity. Duration approximations can be inaccurate for large yield movements and bonds with embedded options. Credit spreads can be distorted by illiquidity, stale prices, or unusual market conditions.

Best practice includes using observable market curves, documenting assumptions, applying consistent day-count conventions, independently checking cash flows, and conducting sensitivity and scenario analysis. Valuation adjustments should be governed by clear controls, segregation of duties, and audit trails.

Ratings should not be treated as the sole basis for investment decisions. Ethical practice requires transparent disclosure of model risk, conflicts of interest, valuation uncertainty, and material credit deterioration. Technology can improve portfolio monitoring by automating cash-flow schedules, calculating duration, and identifying unusual spread movements. However, artificial-intelligence systems require validated data, explainable outputs, human oversight, and controls against model drift and biased training data.

Contemporary Developments

Fixed-income markets increasingly use electronic trading, algorithmic execution, real-time risk dashboards, and climate-related disclosure. Central-bank balance-sheet policies and large institutional demand can materially affect the yield curve. Sustainability-linked and green bonds introduce additional disclosure and verification considerations, although their financial valuation still depends on cash flows, credit quality, liquidity, and contractual terms. Investors must distinguish measurable contractual benefits from marketing claims and assess reporting credibility.

Chapter Summary

Interest rates combine compensation for time value, expected inflation, liquidity, credit, maturity, tax, and option-related risks. The Fisher relationship links nominal rates, real rates, and expected inflation. The term structure describes rates across maturities; spot rates discount individual cash flows, while forward rates are implied by spot rates. Yield curves reflect expectations and risk premiums rather than providing perfect forecasts.

Bond value equals the present value of promised cash flows. Prices fall when required yields rise and rise when yields fall. Current yield measures coupon income only, whereas YTM is an internal rate of return subject to reinvestment and payment assumptions. Duration estimates interest-rate sensitivity, and convexity improves that estimate for larger movements. Credit spreads compensate for default and other risks, while ratings provide opinions rather than guarantees. Sound fixed-income practice combines quantitative valuation with credit analysis, scenario testing, governance, and ethical disclosure.

Key Terms

Nominal interest rate: Stated monetary interest rate before inflation adjustment.

Real interest rate: Interest rate adjusted for purchasing-power changes.

Expected inflation: Anticipated rate of increase in the general price level.

Spot rate: Yield for a zero-coupon investment of a particular maturity.

Forward rate: Rate implied today for a future borrowing or lending period.

Yield curve: Relationship between yields and maturities for comparable securities.

Face value: Amount repaid by a bond issuer at maturity.

Coupon: Contractual interest payment on a bond.

Yield to maturity: Rate equating a bond’s price with the present value of promised payments.

Current yield: Annual coupon divided by the bond’s market price.

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

Macaulay duration: Present-value-weighted average time to receive cash flows.

Modified duration: Approximate percentage price sensitivity to yield changes.

Convexity: Curvature of the bond price–yield relationship.

Credit risk: Risk of loss arising from failure to meet contractual obligations.

Credit spread: Risky-bond yield minus a comparable benchmark yield.

Probability of default: Likelihood that a borrower fails to meet obligations.

Recovery rate: Proportion recovered after default.

Liquidity premium: Compensation for difficulty selling an asset quickly at fair value.

Yield to call: Return measure based on a bond’s possible early redemption.

Review Questions

Short Answer Questions

1. Distinguish between nominal and real interest rates.
2. What is the term structure of interest rates?
3. Why do bond prices and yields move in opposite directions?
4. Define credit spread and identify two components it may reflect.
5. What is the difference between current yield and yield to maturity?

Long Answer Questions

1. Explain the expectations, liquidity-preference, and market-segmentation theories of the yield curve.
2. Derive and interpret the general bond-pricing equation, including its assumptions.
3. Discuss price risk and reinvestment risk for fixed-income investors.
4. Explain Macaulay duration, modified duration, and convexity and their limitations.
5. Describe how credit ratings, financial analysis, covenants, and recovery affect bond valuation.

Analytical / Application Questions

1. A five-year, $1,000 bond pays an 8% annual coupon and investors require 10%. Calculate its price and explain why it trades at a discount.
2. A bond has modified duration of 6.4 years. Estimate the percentage price change if its yield falls by 0.75 percentage points. Explain why the actual change may differ.
3. A company’s benchmark yield is 5.2% and its bond yield is 8.1%. Analyze the credit spread and explain three reasons it could widen without an immediate default.

Multiple Choice Questions with Answers

Question 1

Which relationship is the exact Fisher relationship?

A. 1 + i = (1 + r)(1 + π^e)

B. i = r − π^e

C. 1 + r = (1 + i)(1 + π^e)

D. i = r × π^e

Correct Answer: A. 1 + i = (1 + r)(1 + π^e)

Question 2

A bond’s annual coupon is $60 and its market price is $800. What is its current yield?

A. 6.00%

B. 7.50%

C. 8.00%

D. 13.33%

Correct Answer: B. 7.50%

Question 3

If the required yield rises, the price of an option-free fixed-coupon bond generally:

A. Rises

B. Remains unchanged

C. Falls

D. Becomes equal to face value

Correct Answer: C. Falls

Question 4

Which measure is the present-value-weighted average time of a bond’s cash flows?

A. Current yield

B. Macaulay duration

C. Credit spread

D. Recovery rate

Correct Answer: B. Macaulay duration

Question 5

A risky bond yields 8.4% and a comparable benchmark yields 5.9%. What is the credit spread?

A. 1.5 percentage points

B. 2.5 percentage points

C. 3.5 percentage points

D. 14.3 percentage points

Correct Answer: B. 2.5 percentage points

Question 6

If expected recovery is 30%, loss given default is:

A. 30%

B. 50%

C. 70%

D. 130%

Correct Answer: C. 70%

Question 7

For a small increase in yield, modified duration estimates percentage price change as:

A. D_mod × Δy

B. −D_mod × Δy

C. D_mod ÷ Δy

D. −Δy ÷ D_mod

Correct Answer: B. −D_mod × Δy

Question 8

A bond selling below face value most directly indicates that its coupon rate is:

A. Above the required yield

B. Equal to the required yield

C. Below the required yield

D. Unrelated to the required yield

Correct Answer: C. Below the required yield

Question 9

Which risk increases when coupon payments must be reinvested at lower market rates?

A. Reinvestment risk

B. Default risk

C. Settlement risk

D. Recovery risk

Correct Answer: A. Reinvestment risk

Question 10

A credit rating is best understood as:

A. A guarantee of payment

B. A legally fixed market price

C. An opinion about relative credit quality

D. A calculation of current yield

Correct Answer: C. An opinion about relative credit quality