Bond Risk

Interest rate risk, duration, default risk

1 Introduction

When you invest in bonds, you expect to receive a stream of coupon payments and eventually get your principal back at maturity. But between the day you buy a bond and the day it matures, many things can go wrong. Understanding these risks is essential for any investor who wants to make informed decisions about fixed-income securities.

Bonds face two primary categories of risk that can affect your realized return. The first is interest rate risk, which encompasses the ways that changes in market interest rates can hurt your investment. The second is default risk, the possibility that the bond issuer fails to make promised payments. These risks operate through different mechanisms and require different analytical tools, but both are fundamental to understanding how bonds behave in the real world.

This lecture develops your understanding of both risk types. We begin with interest rate risk, exploring how it manifests through price changes and reinvestment opportunities, and then introduce duration as the key metric for measuring and managing this risk. We conclude with default risk, examining how credit markets assess and price the probability that issuers will fail to meet their obligations.

2 Interest Rate Risk: Two Opposing Forces

Interest rate risk actually consists of two distinct components that work in opposite directions, which creates an interesting dynamic for bond investors.

Price risk is the risk that rising interest rates will decrease the market value of your bond. When market rates increase, newly issued bonds offer higher yields, making your existing bond (with its now-below-market coupon) less attractive. To compensate, its price must fall. If you need to sell before maturity, you could suffer a capital loss.

Reinvestment rate risk works in the opposite direction. When you receive coupon payments, you typically reinvest them. If interest rates have fallen since you bought the bond, you’ll reinvest those coupons at lower rates than you anticipated, reducing the future value of your investment. Conversely, when rates rise, you can reinvest coupons at higher rates—a silver lining to offset the price decline.

This offsetting relationship is crucial. Consider what happens when interest rates suddenly increase: the market price of your bond falls (bad news), but your coupon payments can now be reinvested at higher rates (good news). Whether you end up better or worse off depends on your investment horizon—how long you planned to hold the bond—and a property called duration.

Here is the key insight: if your investment horizon exactly matches the bond’s duration, these two effects perfectly cancel out. The capital loss from the price decline is exactly offset by the higher reinvestment income. This principle underlies a strategy called immunization, where portfolio managers deliberately match the duration of their bond holdings to their investment horizon to neutralize interest rate risk. We’ll develop the mathematics of duration next.

3 Bond Duration

3.1 Macaulay Duration: The Concept

A bond’s Macaulay duration measures the weighted-average time until you receive the bond’s cash flows, where the weights reflect the present value of each payment relative to the bond’s total price. Think of it as the “center of gravity” of the bond’s payment stream measured in time.

Why does this matter? Duration captures how sensitive a bond is to interest rate changes. A bond with cash flows concentrated far in the future (high duration) is more sensitive to rate changes than one with cash flows arriving soon (low duration). This makes intuitive sense: if you’re going to receive most of your money quickly, changes in discount rates don’t affect you much. If you’re waiting decades for your principal, those discount rates compound over many periods, making the present value highly sensitive to rate changes.

The formula for Macaulay duration is:

\[D = \sum_{t=1}^{n} \frac{t/F \cdot PV(CF_t)}{P_0}\]

where each term weights the time of payment by the present value of that payment as a fraction of the bond’s price. In expanded form for a standard coupon bond:

\[D = \frac{1}{F} \cdot \frac{C/F}{P_0(1+Y/F)^1} + \frac{2}{F} \cdot \frac{C/F}{P_0(1+Y/F)^2} + \cdots + \frac{n}{F} \cdot \frac{C/F + Par}{P_0(1+Y/F)^n}\]

Variable	Definition
\(D\)	Macaulay duration (in years)
\(P_0\)	Current bond price
\(Y\)	Annual yield to maturity (YTM)
\(C\)	Annual coupon payment
\(F\)	Coupon frequency per year (e.g., 2 for semiannual)
\(n\)	Total number of coupon periods remaining (\(T \times F\), where \(T\) is years to maturity)
\(Par\)	Par value (face value) of the bond

In Excel, you can calculate Macaulay duration using the DURATION function with the standard bond parameters: settlement date, maturity date, coupon rate, yield, frequency, and basis.

3.2 Modified Duration

While Macaulay duration tells us about the timing of cash flows, modified duration directly measures price sensitivity. It equals Macaulay duration divided by the gross yield per period:

\[D^* = \frac{D}{1 + Y/F}\]

where \(D^*\) denotes modified duration. The division by \((1 + Y/F)\) converts from a time measure to a pure price-sensitivity measure. Modified duration tells you approximately how much the bond’s price will change, in percentage terms, for a one-unit change in yield.

In Excel, use the MDURATION function with the same parameters as DURATION.

4 What Determines Duration?

Three factors primarily determine a bond’s duration, and understanding each helps you predict which bonds carry more interest rate risk.

4.1 Duration Increases with Maturity

Longer-maturity bonds have higher duration because more of their cash flows occur in the distant future. A 30-year bond has much more of its present value coming from payments decades away compared to a 1-year bond, making it more sensitive to discount rate changes.

Example 1: Duration and Maturity

Consider three bonds, all with a 6% coupon rate and 6% YTM (semiannual payments, par value = $1,000). They differ only in maturity: Bond A matures in 1 year, Bond B in 10 years, and Bond C in 30 years. Calculate their Macaulay duration.

Solution to Example 1

Since coupon rate equals YTM, all bonds trade at par ($1,000). Using Excel’s DURATION function or the formula:

Bond	Maturity	Macaulay Duration
A	1 year	0.985 years
B	10 years	7.662 years
C	30 years	14.253 years

The pattern is clear: as maturity increases from 1 to 10 to 30 years, duration increases from about 1 year to 7.7 years to 14.3 years. Note that duration increases less than proportionally with maturity—the 30-year bond doesn’t have 30 times the duration of the 1-year bond because the coupon payments received along the way pull the average time forward.

4.2 Duration Increases as YTM Decreases

When yields are lower, distant cash flows are discounted less heavily, so they contribute more to the bond’s present value and weight the duration calculation more heavily toward the future.

Example 2: Duration and Yield to Maturity

Consider three bonds, all with a 4% coupon rate and 10 years to maturity (semiannual payments, par value = $1,000). They differ in YTM: Bond A has a YTM of 2%, Bond B has 4%, and Bond C has 6%. Calculate their Macaulay duration.

Solution to Example 2

Using Excel’s DURATION function:

Bond	YTM	Price	Macaulay Duration
A	2%	$1,180.46	8.497 years
B	4%	$1,000.00	8.339 years
C	6%	$851.23	8.169 years

As YTM increases, duration decreases. The effect is more subtle than maturity but still meaningful: lower yields mean higher duration and more interest rate sensitivity.

4.3 Duration Increases as Coupon Rate Decreases

Lower coupon rates mean you receive less of your return through periodic payments and more through the final principal payment. Since that principal arrives at maturity, lower coupons push the weighted-average timing later.

Example 3: Duration and Coupon Rate

Consider three bonds, all with 6% YTM and 10 years to maturity (semiannual payments, par value = $1,000). They differ in coupon rate: Bond A pays 3%, Bond B pays 6%, and Bond C pays 9%. Calculate their Macaulay duration.

Solution to Example 3

Using Excel’s DURATION function:

Bond	Coupon Rate	Price	Macaulay Duration
A	3%	$776.84	8.495 years
B	6%	$1,000.00	7.662 years
C	9%	$1,223.16	7.133 years

As the coupon rate increases, duration decreases. Lower coupon bonds have higher duration because more of their value comes from the distant principal repayment rather than near-term coupon payments.

The following table summarizes these relationships:

Factor	Change	Effect on Duration	Intuition
Maturity	Increase	Duration increases	Cash flows pushed further into the future
YTM	Increase	Duration decreases	Distant cash flows discounted more heavily
Coupon rate	Increase	Duration decreases	More value received through near-term coupons

5 Using Duration to Estimate Price Changes

Duration’s practical value comes from its ability to estimate how much a bond’s price will change when interest rates move. This is essential for risk management and hedging.

5.1 The Macaulay Duration Approach

For a bond with Macaulay duration \(D\), the percentage price change resulting from a yield change of \(x\) (in decimal form) is approximately:

\[\frac{P_1 - P_0}{P_0} \approx -D \cdot \frac{x}{1 + Y/F}\]

The negative sign reflects the inverse relationship between prices and yields: when yields rise (\(x > 0\)), prices fall.

5.2 The Modified Duration Approach

Modified duration provides a more direct calculation:

\[\frac{P_1 - P_0}{P_0} \approx -D^* \cdot x\]

This is simpler because modified duration already incorporates the yield adjustment.

Example 4: Estimating Price Changes with Duration

Return to the bonds from Examples 1, 2, and 3. For each bond:

1. Use Macaulay duration to estimate the percentage price change if YTM increases by 0.50%.

2. Use modified duration to estimate the percentage price change if YTM decreases by 0.75%.

Solution to Example 4

Part (a): Using Macaulay Duration for a 0.50% YTM Increase

The formula is: \(\%\Delta P \approx -D \cdot \frac{x}{1 + Y/F}\)

Example 1 Bonds (all have 6% YTM, semiannual):

Bond	Duration	Calculation	% Price Change
A	0.985	\(-0.985 \times \frac{0.005}{1.03}\)	−0.478%
B	7.662	\(-7.662 \times \frac{0.005}{1.03}\)	−3.720%
C	14.254	\(-14.253 \times \frac{0.005}{1.03}\)	−6.919%

Example 2 Bonds (different YTMs):

Bond	Duration	YTM	Calculation	% Price Change
A	8.497	2%	\(-8.497 \times \frac{0.005}{1.01}\)	−4.207%
B	8.339	4%	\(-8.339 \times \frac{0.005}{1.02}\)	−4.088%
C	8.169	6%	\(-8.169 \times \frac{0.005}{1.03}\)	−3.965%

Example 3 Bonds (all have 6% YTM, semiannual):

Bond	Duration	Calculation	% Price Change
A	8.495	\(-8.495 \times \frac{0.005}{1.03}\)	−4.124%
B	7.662	\(-7.662 \times \frac{0.005}{1.03}\)	−3.719%
C	7.133	\(-7.133 \times \frac{0.005}{1.03}\)	−3.462%

Part (b): Using Modified Duration for a 0.75% YTM Decrease

First, calculate modified duration: \(D^* = D / (1 + Y/F)\)

Then apply: \(\%\Delta P \approx -D^* \times x = -D^* \times (-0.0075)\)

Example 1 Bonds:

Bond	Macaulay D	Modified D*	% Price Change
A	0.985	0.957	+0.718%
B	7.662	7.439	+5.579%
C	14.253	13.839	+10.379%

Example 2 Bonds:

Bond	Macaulay D	Modified D*	% Price Change
A	8.497	8.413	+6.310%
B	8.339	8.179	+6.132%
C	8.169	7.931	+5.948%

Example 3 Bonds:

Bond	Macaulay D	Modified D*	% Price Change
A	8.495	8.248	+6.186%
B	7.662	7.439	+5.579%
C	7.133	6.925	+5.194%

Notice that longer-duration bonds experience larger price swings in both directions. The 30-year bond (C in Example 1) moves more than 10% for a 0.75% rate decrease, while the 1-year bond barely moves at all.

6 Default Risk

While interest rate risk affects all bonds, default risk applies specifically to bonds where the issuer might fail to make promised payments. U.S. Treasury securities are considered free of default risk because the federal government can always raise taxes or print money to meet its obligations. Corporate bonds, municipal bonds, and sovereign debt from other countries all carry varying degrees of default risk.

Default risk matters because it directly affects the expected return on your investment. If a company files for bankruptcy, bondholders may receive only a fraction of their principal back—or nothing at all. Even if default doesn’t occur, the mere possibility of default depresses bond prices and elevates required yields.

6.1 How Default Risk Is Measured

Investors and analysts assess default risk through several approaches. The most widely used is credit ratings assigned by agencies such as Moody’s, Standard & Poor’s, and Fitch. These ratings evaluate an issuer’s ability and willingness to meet its debt obligations based on factors like financial strength, cash flow stability, industry conditions, and management quality.

The following table shows the major rating categories and what they signify:

Rating (S&P/Fitch)	Rating (Moody’s)	Category	Interpretation
AAA	Aaa	Investment Grade	Highest quality, minimal risk
AA	Aa	Investment Grade	High quality, very low risk
A	A	Investment Grade	Upper-medium quality, low risk
BBB	Baa	Investment Grade	Medium quality, moderate risk
BB	Ba	Speculative Grade	Speculative, substantial risk
B	B	Speculative Grade	Highly speculative
CCC	Caa	Speculative Grade	Very high risk
CC	Ca	Speculative Grade	Near default
C	C	Speculative Grade	Lowest rated, default imminent
D	—	Default	In default

The dividing line between investment grade (BBB−/Baa3 and above) and speculative grade (BB+/Ba1 and below, often called “junk” or “high-yield” bonds) is particularly important. Many institutional investors, such as pension funds and insurance companies, are restricted by regulation or internal policy from holding speculative-grade bonds. This creates a significant price impact when bonds cross this threshold.

6.2 The Yield Spread as a Default Risk Premium

Markets price default risk through yield spreads—the difference between a bond’s yield and the yield on a comparable Treasury security. This spread compensates investors for bearing the risk that the issuer might default.

\[\text{Yield Spread} = Y_{corporate} - Y_{Treasury}\]

For example, if a 10-year corporate bond yields 5.5% and the 10-year Treasury yields 4.0%, the spread is 150 basis points (1.50%). This spread reflects both the expected loss from default and a risk premium for the uncertainty involved.

Yield spreads vary systematically with credit ratings. Higher-rated bonds have narrower spreads because default is less likely, while lower-rated bonds demand wider spreads to compensate for greater risk. Spreads also widen during economic downturns when default becomes more likely across the board, a phenomenon that reflects the systematic component of credit risk.

6.3 Other Indicators of Default Risk

Beyond credit ratings, several market-based measures help assess default risk. Credit default swaps (CDS) are derivative contracts that provide insurance against default; the CDS spread indicates how much investors pay for this protection and serves as a market-based measure of perceived default risk. Bond prices relative to par also signal credit concerns—a bond trading at 70 cents on the dollar implies the market expects significant losses.

Financial ratios provide fundamental indicators of an issuer’s ability to service debt. Common metrics include the interest coverage ratio (EBIT divided by interest expense), which measures how easily a firm can pay interest from operating income, and the debt-to-equity ratio, which indicates financial leverage. Deteriorating ratios often precede rating downgrades and spread widening.

7 Key Takeaways

Bond investing requires understanding two fundamental risks that operate through different channels. Interest rate risk affects bond prices and reinvestment opportunities in opposing ways: rising rates hurt prices but help reinvestment, while falling rates help prices but hurt reinvestment. The degree of exposure to interest rate risk is captured by duration, which measures both the weighted-average timing of cash flows (Macaulay duration) and direct price sensitivity (modified duration). Bonds with longer maturities, lower yields, or lower coupon rates have higher duration and thus greater interest rate risk.

Duration provides a practical tool for estimating price changes when yields move. Using either the Macaulay or modified duration formula, investors can quickly approximate the percentage price impact of rate changes—essential for risk management, hedging, and immunization strategies. When a portfolio’s duration matches the investment horizon, interest rate risk is neutralized because price effects and reinvestment effects offset.

Default risk adds another layer of concern for non-Treasury bonds. Credit ratings from major agencies provide standardized assessments of default probability, with the investment-grade versus speculative-grade distinction carrying particular significance for institutional investors. The market prices default risk through yield spreads above Treasury rates, and these spreads fluctuate with economic conditions and issuer-specific developments. Together, understanding interest rate risk and default risk equips you to evaluate bond investments comprehensively and construct portfolios aligned with your risk tolerance and investment objectives.

8 Key Formulas Summary

Concept	Formula	When to Use
Macaulay Duration	\(D = \sum_{t=1}^{n} \frac{t/F \cdot PV(CF_t)}{P_0}\)	Finding weighted-average time of cash flows; immunization matching
Modified Duration	\(D^* = \frac{D}{1 + Y/F}\)	Converting Macaulay duration to price sensitivity measure
Price Change (Macaulay)	\(\frac{\Delta P}{P_0} \approx -D \cdot \frac{x}{1 + Y/F}\)	Estimating % price change using Macaulay duration
Price Change (Modified)	\(\frac{\Delta P}{P_0} \approx -D^* \cdot x\)	Estimating % price change using modified duration (simpler)
Yield Spread	\(\text{Spread} = Y_{corporate} - Y_{Treasury}\)	Measuring default risk premium in market prices

9 Practice Problems

Practice Problem 1: Duration and Maturity

Bonds X, Y, and Z all have a 5% coupon rate and 5% YTM (semiannual payments, par value = $1,000). Bond X matures in 2 years, Bond Y matures in 15 years, and Bond Z matures in 25 years. Calculate the Macaulay duration for each bond.

Solution to Practice Problem 1

Since coupon rate equals YTM, all bonds trade at par ($1,000). Using Excel’s DURATION function:

Bond	Maturity	Macaulay Duration
X	2 years	1.928 years
Y	15 years	10.727 years
Z	25 years	14.536 years

Practice Problem 2: Duration and Yield to Maturity

Bonds D, E, and F all have a 5% coupon rate and 8 years to maturity (semiannual payments, par value = $1,000). Bond D has a YTM of 3%, Bond E has a YTM of 5%, and Bond F has a YTM of 7%. Calculate the Macaulay duration for each bond.

Solution to Practice Problem 2

Using Excel’s DURATION function:

Bond	YTM	Price	Macaulay Duration
D	3%	$1,141.31	6.79 years
E	5%	$1,000.00	6.691 years
F	7%	$879.06	6.585 years

Practice Problem 3: Duration and Coupon Rate

Bonds G, H, and I all have a 5% YTM and 12 years to maturity (semiannual payments, par value = $1,000). Bond G has a 2% coupon rate, Bond H has a 5% coupon rate, and Bond I has an 8% coupon rate. Calculate the Macaulay duration for each bond.

Solution to Practice Problem 3

Using Excel’s DURATION function:

Bond	Coupon Rate	Macaulay Duration
G	2%	10.451 years
H	5%	9.166 years
I	8%	8.425 years

Practice Problem 4: Estimating Price Changes

Using the bonds from Practice Problems 1, 2, and 3:

1. Use Macaulay duration to estimate the percentage price change if YTM increases by 0.25%.

2. Use modified duration to estimate the percentage price change if YTM decreases by 1.00%.

Solution to Practice Problem 4

Part (a): Using Macaulay Duration for a 0.25% YTM Increase

Formula: \(\%\Delta P \approx -D \cdot \frac{x}{1 + Y/F}\)

Practice Problem 1 Bonds (5% YTM):

Bond	Duration	Calculation	% Price Change
X	1.9278	\(-1.928 \times \frac{0.0025}{1.025}\)	−0.470%
Y	10.727	\(-10.727 \times \frac{0.0025}{1.025}\)	−2.62%
Z	14.536	\(-14.536 \times \frac{0.0025}{1.025}\)	−3.545%

Practice Problem 2 Bonds (different YTMs):

Bond	Duration	YTM	Calculation	% Price Change
D	6.790	3%	\(-6.790 \times \frac{0.0025}{1.015}\)	-1.672%
E	6.691	5%	\(-6.691 \times \frac{0.0025}{1.025}\)	-1.632%
F	6.585	7%	\(-6.585 \times \frac{0.0025}{1.035}\)	-1.591%

Practice Problem 3 Bonds (5% YTM):

Bond	Duration	Calculation	% Price Change
G	10.451	\(-10.451 \times \frac{0.0025}{1.025}\)	-2.549%
H	9.166	\(-9.166 \times \frac{0.0025}{1.025}\)	-2.236%
I	8.425	\(-8.425 \times \frac{0.0025}{1.025}\)	-2.055%

Part (b): Using Modified Duration for a 1.00% YTM Decrease

First calculate modified duration, then apply: \(\%\Delta P \approx -D^* \times (-0.01)\)

Practice Problem 1 Bonds:

Bond	Modified D*	% Price Change
X	1.881	+1.881%
Y	10.465	+10.465%
Z	14.181	+14.181%

Practice Problem 2 Bonds:

Bond	Modified D*	% Price Change
D	6.690	+6.690%
E	6.528	+6.528%
F	6.362	+6.362%

Practice Problem 3 Bonds:

Bond	Modified D*	% Price Change
G	10.196	+10.196%
H	8.942	+8.942%
I	8.219	+8.219%

10 Ask an LLM

Here are three questions you might ask an AI assistant to deepen your understanding of these concepts:

• How does convexity improve upon duration as a measure of interest rate risk, and when does the linear approximation from duration break down?
• If I’m building a bond portfolio with a 7-year investment horizon, walk me through how I would implement an immunization strategy step by step.
• What happened to yield spreads during the 2008 financial crisis and the 2020 COVID shock, and what does that tell us about how markets price default risk during periods of stress?
• Why might a bond’s yield spread exceed what credit ratings alone would suggest, and what other factors do sophisticated investors consider when assessing default risk?
• How do callable bonds and other embedded options affect duration, and why does “effective duration” become necessary for these securities?