Bond Risk Measures: Duration, Convexity, and Immunization

Fixed income securities form the backbone of global financial markets, with the bond market's size dwarfing equity markets. Yet bonds, despite their reputation as safe investments, carry significant risks. The most fundamental of these is interest rate risk, the exposure of bond prices to changes in prevailing interest rates. When rates rise, bond prices fall, and vice versa. This inverse relationship creates challenges for portfolio managers, pension funds, and insurance companies who must manage assets against future liabilities.

This chapter develops the quantitative tools that practitioners use to measure and manage interest rate risk. Duration tells us how sensitive a bond's price is to rate changes and summarizes the bond's interest rate exposure in a single number. Convexity refines this measure by capturing the curvature in the price-yield relationship, which duration alone misses. Together, these measures enable us to approximate price changes for any given shift in interest rates.

Beyond measurement, we explore immunization: the practice of structuring bond portfolios to be insensitive to interest rate movements. This technique is essential for institutions with fixed future liabilities. Pension funds, for example, must pay retirees decades from now regardless of where rates move in the interim.

Before developing formal risk measures, we establish the fundamental relationship between interest rates and bond prices. Understanding this relationship provides the intuition that underlies all duration and convexity calculations.

A bond represents a contractual promise to pay a stream of future cash flows. These cash flows consist of periodic coupon payments plus the return of principal at maturity. The question every bond investor faces is: what is this stream of future payments worth today? The answer depends critically on the discount rate applied to those payments.

The bond's price is the present value of its future cash flows, discounted at the prevailing yield:

where:

• $P$: the bond price
• $C_t$: the cash flow at time $t$ (coupon payments and principal)
• $y$: the yield to maturity (expressed as a decimal)
• $T$: the total number of periods until maturity

This formula immediately reveals why bond prices move inversely with yields: higher discount rates produce lower present values. The logic is straightforward. When market interest rates rise, newly issued bonds offer higher coupon payments than existing bonds. Investors will only purchase older, lower-coupon bonds at a discount that compensates them for the reduced income. Conversely, when rates fall, existing bonds with higher coupons become more valuable because they offer income streams that exceed what new bonds provide.

The magnitude of this price sensitivity depends on several bond characteristics, and understanding these relationships helps investors anticipate how their holdings will respond to rate changes:

• Maturity: Longer-maturity bonds are more sensitive to rate changes because their cash flows are discounted over longer periods. A change in the discount rate applied over twenty years has a far greater impact than the same change applied over two years. This occurs because the compounding effect of the discount rate grows exponentially with time.
• Coupon rate: Lower-coupon bonds are more sensitive because a larger proportion of their value comes from the distant principal payment. A zero-coupon bond derives all its value from the final maturity payment, making it maximally sensitive to rate changes. Higher-coupon bonds receive more of their value from near-term cash flows, which are less affected by rate changes.
• Current yield level: Price sensitivity is higher at lower yield levels due to the convex nature of the price-yield curve. Moving from a 2% yield to a 3% yield produces a larger percentage price change than moving from 8% to 9%, even though both represent a 100 basis point increase.

Let's examine how a bond's price changes across different yield levels:

The curve exhibits convexity, meaning it is not a straight line but curves upward. This convexity means that for equal-sized rate increases and decreases, the price gain from a rate decrease exceeds the price loss from a rate increase. This asymmetry becomes important when we develop more sophisticated risk measures, as it represents a fundamental property of fixed income securities that benefits bondholders.

Frederick Macaulay introduced duration in 1938 as a measure of the effective maturity of a bond. His insight was that a bond's stated maturity date tells only part of the story. A 10-year bond that pays substantial coupons behaves quite differently from a 10-year zero-coupon bond, even though both mature on the same date. Macaulay sought a single number that would capture the economic timing of all a bond's cash flows.

Rather than using only the final maturity date, duration accounts for all cash flows and their timing. The concept is analogous to finding the center of gravity of the cash flow stream. Just as a physical object balances at its center of mass, a bond's duration represents the "balance point" of its payments over time. For a bond paying cash flows $C_t$ at times $t = 1, 2, \ldots, n$:

where:

• $D_{Mac}$: Macaulay duration in years
• $t$: the time period when cash flow is received (in years or periods, depending on convention)
• $C_t$: the cash flow at time $t$
• $y$: the yield to maturity per period (expressed as a decimal)
• $P$: the bond price (sum of all present values)
• $PV(C_t)$: the present value of cash flow $C_t$, equal to $C_t/(1+y)^t$
• $n$: the total number of cash flow periods

Each term $t \cdot PV(C_t)$ weights the time period by the present value of the cash flow received at that time. Dividing by the total price $P$ normalizes these weights to sum to one, ensuring that the result is a proper weighted average.

Intuitively, Macaulay duration answers the question, "When, on average, do I get my money back?" Consider this question from the perspective of an investor who needs to know when they will recover their investment. A zero-coupon bond returns all money at maturity, so its duration equals its maturity. There is no ambiguity because there is only one cash flow. A coupon bond returns money earlier through periodic payments, pulling the average receipt time forward. This explains why higher-coupon bonds have shorter durations: more cash arrives sooner, reducing the weighted average time until the investor recovers their investment.

For a zero-coupon bond, the calculation is trivial. The only cash flow is the face value at maturity, so Macaulay duration equals the time to maturity. For coupon bonds, duration is always less than maturity because earlier coupon payments pull the weighted average forward. The difference between maturity and duration depends on the coupon rate, with higher coupons creating a larger gap.

Let's calculate duration for several bonds to see how bond characteristics affect it:

The results confirm our intuition: longer maturity increases duration, and lower coupons increase duration because less of the bond's value comes from near-term cash flows. Notice that the 10-year, 2% coupon bond has a duration of nearly 9 years, while the 10-year, 8% coupon bond has a duration closer to 7 years. The higher coupon payments shift the center of gravity of the cash flows earlier in time.

While Macaulay duration measures weighted-average time, we need a measure that directly quantifies price sensitivity. Practitioners and risk managers care not just about when cash flows arrive, but about how much a bond's price will change when interest rates move. Modified duration provides this essential link between the time-weighted concept of Macaulay duration and the practical concern of price volatility.

The derivation of modified duration comes from calculus, specifically from examining how the bond price function responds to small changes in yield. This mathematical derivation reveals that the connection between time-weighted cash flows and price sensitivity is not accidental. Rather, it emerges naturally from the structure of the present value formula.

Starting with the bond price formula,

Taking the derivative with respect to yield $y$,

This derivative tells us the dollar change in price for a small change in yield. To express this in percentage terms, we divide both sides by the price $P$:

where the last step recognizes that $\frac{1}{P}\sum_{t=1}^{n} \frac{t \cdot C_t}{(1+y)^{t}}$ is exactly the definition of Macaulay duration.

This result shows that price sensitivity and time-weighted cash flows are directly connected. The derivative naturally produces the Macaulay duration formula, adjusted by a factor of $(1+y)$. This gives us modified duration:

where:

• $D_{Mod}$: modified duration
• $D_{Mac}$: Macaulay duration
• $y$: the annual yield to maturity
• $k$: the number of compounding periods per year (e.g., $k=2$ for semi-annual)

The negative sign in the derivative confirms the inverse relationship between price and yield. Modified duration is expressed as a positive number with the understanding that price and yield move in opposite directions. When we use modified duration to estimate price changes, we explicitly include the negative sign.

The price change approximation using modified duration is:

or equivalently:

where:

• $\Delta P$: the change in bond price
• $P$: the current bond price
• $D_{Mod}$: modified duration
• $\Delta y$: the change in yield (expressed as a decimal, e.g., 0.01 for a 1% change)

The negative sign confirms that price and yield move in opposite directions: when yields rise ($\Delta y > 0$), prices fall ($\Delta P < 0$). This approximation is essentially a first-order Taylor expansion of the price function around the current yield. It uses the slope of the price-yield curve at the current point to estimate prices at nearby yields.

The modified duration of 7.79 tells us that for each 1% (100 basis point) increase in yield, we expect the bond price to decrease by approximately 7.79%. Since the bond is priced at par ($1,000), this translates to roughly a$77.90 price change per 100 basis points of yield movement. This gives portfolio managers a quick way to assess interest rate exposure: multiply the modified duration by the expected yield change to estimate the percentage price impact.

We can verify that modified duration provides a reasonable approximation for small yield changes:

The approximation works well for small changes but introduces noticeable error for larger movements. This error arises because duration is a linear approximation of a curved (convex) relationship. When we use duration alone, we are essentially drawing a tangent line to the price-yield curve at the current point and using that straight line to estimate prices at other yields.

For a 2% yield increase, duration underestimates the new price and suggests the bond loses more value than it actually does. For a 2% yield decrease, it underestimates the price gain and suggests the bond gains less than it actually does. This systematic pattern motivates the need for convexity. The actual price is always higher than the linear approximation, regardless of whether rates rise or fall.

Duration provides a first-order approximation to price changes, analogous to using the slope of a curve to estimate nearby points. However, the price-yield relationship is not a straight line but a curve that bends upward. Convexity measures this curvature and allows us to improve our price estimates significantly.

The concept can be understood through an analogy with physics. If duration represents velocity (the rate of price change), then convexity represents acceleration, measuring how quickly that rate of change itself changes. Just as knowing both velocity and acceleration allows better prediction of future position, knowing both duration and convexity allows better prediction of future prices.

Convexity is derived from the second derivative of the price function with respect to yield. Starting from the first derivative:

Taking the derivative again,

Notice that the second derivative is positive because all terms in the sum are positive. This mathematical fact confirms what we observe graphically: the price-yield curve is convex (curves upward), meaning its slope becomes less negative as yields increase.

Dividing by price to get the convexity measure,

where:

• $C$: convexity
• $P$: bond price
• $t$: time period
• $C_t$: cash flow at time $t$
• $y$: yield to maturity
• $n$: total number of periods

For semi-annual compounding, the formula adjusts to:

where the factor of $1/4$ (equivalently $1/k^2$ for frequency $k$) arises from converting the period-based calculation to annual terms. This adjustment accounts for the fact that both the time units and the yield compounding are expressed in periods rather than years. Without this adjustment, the convexity value would be scaled incorrectly when used in the price approximation formula with yield changes expressed in annual terms.

With both duration and convexity, we can form a second-order Taylor expansion for price changes. The Taylor series provides a systematic way to approximate a function near a point using its derivatives. By including the second derivative (convexity), we capture the curvature that the first-order approximation using duration alone misses.

The Taylor series expansion of price around the current yield $y_0$ is

Dividing by $P$ and substituting our definitions of duration and convexity,

where:

• $\Delta P / P$: the percentage change in bond price
• $D_{Mod}$: modified duration (first-order sensitivity)
• $C$: convexity (second-order sensitivity)
• $\Delta y$: change in yield (as a decimal)

The convexity term is always positive regardless of the direction of the yield change, because it involves $(\Delta y)^2$. Squaring any number, whether positive or negative, produces a positive result. This means convexity always adds to the price estimate for bonds with positive convexity. Most standard bonds without embedded options have positive convexity.

Intuitively, this reflects the asymmetric benefit of convexity. Bondholders gain more from falling rates than they lose from rising rates of the same magnitude. This asymmetry arises because the price-yield curve bends upward, being convex to the origin. Any movement away from the current yield produces a price that lies above the tangent line approximation. The convexity term captures exactly this difference between the curved reality and the linear approximation.

Adding convexity dramatically improves the approximation. The duration-convexity model captures nearly all price movements even for substantial yield changes of 200 basis points. The remaining small errors arise from higher-order terms in the Taylor expansion that we have not included.

The visualization shows the duration approximation as a tangent line that diverges from the actual curve as we move away from the current yield. The straight red dashed line represents what duration alone predicts, touching the actual curve only at the current yield of 5%. The duration-convexity approximation, being a parabola (a quadratic function), tracks the actual curve much more closely across the entire yield range. The green dashed-dotted curve hugs the blue actual price curve, demonstrating the value of including the second-order term.

Practitioners often work with dollar-denominated risk measures rather than percentage changes. This preference arises for practical reasons. When hedging a portfolio or calculating profit and loss, dollar amounts matter directly. Two common metrics translate duration into dollar terms.

Dollar Duration measures the dollar change in price for a one percentage point (100 basis point) change in yield.

where $D_{Mod}$ is modified duration and $P$ is the bond price.

This measure tells us the absolute dollar exposure to interest rate changes. A position with higher dollar duration has greater dollar risk, regardless of whether that position represents a large investment in low-duration bonds or a small investment in high-duration bonds.

DV01 (Dollar Value of a 01, also called "price value of a basis point" or PVBP) measures the dollar change for a one basis point (0.01%) change in yield:

The factor 0.0001 converts the yield change from percentage points to basis points (1 bp = 0.01% = 0.0001 in decimal form).

To understand this conversion, consider that modified duration tells us the percentage price change per 1% (100 basis points) yield change. Dividing by 100 gives us the change per basis point. Multiplying by the price converts to dollar terms. DV01 has become the standard risk measure in fixed income trading because basis points are the natural unit for discussing yield changes in the market.

These measures are particularly useful for hedging, as they express risk in dollar terms that can be directly compared across positions. If one bond has a DV01 of $50 and another has a DV01 of$25, we need two units of the second bond to hedge one unit of the first.

Individual bond risk measures become even more powerful when extended to portfolios. A portfolio manager holding dozens or hundreds of bonds needs a way to summarize the aggregate interest rate exposure. Fortunately, duration and convexity combine in a straightforward way: they are computed as weighted averages, where the weights are each bond's market value as a proportion of total portfolio value.

For a portfolio with $n$ bonds, each with market value $V_i$ and modified duration $D_i$:

where:

• $D_{Portfolio}$: the portfolio's modified duration
• $w_i$: the weight of bond $i$ in the portfolio, calculated as $w_i = V_i / \sum_j V_j$
• $V_i$: the market value of bond $i$
• $D_i$: the modified duration of bond $i$
• $n$: the number of bonds in the portfolio

This formula follows from the linearity of the price change approximation. If each bond's percentage price change is given by $-D_i \cdot \Delta y$, then the portfolio's percentage change is the value-weighted average of the individual changes. The same weighting applies to convexity:

where $C_i$ is the convexity of bond $i$.

This weighted-average approach allows portfolio managers to adjust overall interest rate exposure by changing the mix of bonds. Adding long-duration bonds increases portfolio duration, while adding short-duration bonds decreases it.

The portfolio duration of approximately 8.4 years reflects the weighted average of individual bond durations. Even though the 30-year bond has the longest duration at over 14 years, its relatively small weight (approximately 12% of portfolio value) moderates its impact on the overall duration. The portfolio convexity of approximately 115 indicates substantial curvature in the portfolio's price-yield relationship, which will provide protection against large rate movements in either direction. Higher convexity is generally desirable because it means the portfolio benefits more from the asymmetry between price gains and losses.

Immunization is a strategy that structures a bond portfolio to protect against interest rate risk by matching the portfolio's duration to a target investment horizon. The technique addresses a fundamental challenge facing investors with future obligations: how can one ensure that today's investments will meet tomorrow's needs, regardless of how interest rates move in the interim?

The core insight is that interest rate changes affect bond investors through two offsetting channels:

1. Price effect. When rates rise, bond prices fall, creating a negative impact. An investor who needs to sell bonds before maturity will realize less than expected.
2. Reinvestment effect. When rates rise, coupon payments can be reinvested at higher rates, creating a positive impact. An investor who reinvests coupons will earn more than expected on those reinvestments.

These two effects work in opposite directions. Rising rates hurt through lower prices but help through higher reinvestment income. Falling rates help through higher prices but hurt through lower reinvestment income. The question becomes: is there a holding period at which these effects exactly cancel?

At the duration point, these two effects exactly offset each other. A portfolio held for exactly its duration period will achieve approximately the same terminal value regardless of immediate interest rate changes. If rates rise, the price loss is offset by enhanced reinvestment returns. If rates fall, the reinvestment shortfall is offset by the higher price received when selling remaining bonds.

Consider an investor with a liability due in exactly $T$ years. To immunize against interest rate risk, two conditions must be satisfied:

1. Duration matching. Set the portfolio's modified duration equal to the liability horizon $T$.
2. Present value matching. Ensure the portfolio's current market value equals the present value of the liability.

When these conditions are met, small parallel shifts in the yield curve will not affect the investor's ability to meet the liability. The portfolio is said to be immunized against interest rate risk.

We can verify that our immunized portfolio performs well under different interest rate scenarios: