Bond Risk Measures: Duration, Convexity, and Immunization

A bond portfolio can gain value when rates fall and still leave a pension fund less able to meet its obligations. The liabilities may rise by more than the assets. Managing that mismatch requires a clear choice of what to hedge: a bond's price, the dollar surplus over liabilities, or the funding ratio. These are different quantities, even when all use duration.

This chapter develops duration and convexity as first- and second-order price sensitivities, then uses them to construct and test a portfolio against a future liability. We begin with fixed, nonnegative cash flows and a common yield convention. The approximations describe nearby yield changes; they are not exact repricing rules for arbitrary rate moves.

Immunization matches asset present value and cash-flow timing to a liability under stated assumptions. In the single-liability example, matching Macaulay duration to the payment horizon removes first-order exposure to an immediate persistent change in one flat yield. It does not guarantee payment under every future rate path or remove default and liquidity risk.

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.

For the fixed-cash-flow bonds considered here, price is the present value of promised payments. The first formula uses one yield per payment period, with positive discount base and nonnegative cash flows:

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

For these fixed positive payments, a higher yield lowers price. A lower-coupon bond generally trades below an otherwise comparable higher-coupon bond when both are priced at the same yield. The coupon itself is contractual: an issuer does not automatically change an existing bond's payments when market yields move.

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: A fixed payment due later has greater percentage yield sensitivity at the same yield convention. Longer coupon-bond maturity also increases duration in the examples below, but maturity alone is not a universal ordering rule across all coupon and yield combinations.
• Coupon rate: Holding maturity, yield and payment frequency fixed, lower coupons increase duration when there is a coupon date strictly before maturity. A zero-coupon bond has all its present-value weight at maturity, giving the greatest duration for those terms. Higher coupons shift present-value weight to earlier receipts when those dates exist. If all remaining payments arrive at maturity, changing the coupon does not change duration or create an earlier receipt.
• 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.

For the fixed nonnegative cash flows used here, the price-yield curve bends upward. From the same starting yield, a nonzero yield decrease gives a larger price gain than the loss caused by an equal increase, provided the discount base remains positive. Embedded options can change this convexity pattern.

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 payment periods in this formula
• $t$: the integer payment-period index; divide by the annual payment frequency to express duration in years
• $C_t$: 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, so the result is a proper weighted average.

Macaulay duration summarizes the timing of present-value weights; it is not a payback date or a promise that accumulated coupons have recovered the purchase price. A zero-coupon bond has one receipt time, so its duration equals maturity. With positive earlier coupons, some present-value weight arrives sooner and the weighted mean is shorter than maturity.

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. With a positive coupon payment strictly before maturity, Macaulay duration is less than maturity because that earlier payment receives positive present-value weight. If all remaining cash flow arrives at maturity, duration equals maturity.

While Macaulay duration measures weighted-average time, we need a measure that directly quantifies price sensitivity. Practitioners and risk managers care about when cash flows arrive and 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 is the dollar price sensitivity per unit change in decimal yield; multiply it by a small yield change to approximate the dollar price change. 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.

The derivation above differentiates with respect to a per-period yield, so its duration is in periods. For nominal annual yield $y$ compounded $k$ times per year, differentiating $(1+y/k)^{-t}$ introduces $t/k$. With $D_{Mac}$ now in years, this gives modified duration with respect to the quoted annual decimal yield:

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 overstates the bond's loss. For a 2% yield decrease, it understates the bond's price gain. 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 first derivative describes the slope of the price-yield curve; the second derivative describes how that slope changes. Modified duration is the negative first derivative divided by price, while convexity is the second derivative divided by price. Consequently, $dD_{Mod}/dy=D_{Mod}^2-C$, rather than simply $C$.

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

Taking the derivative again,

With nonnegative cash flows and a positive discount base, the second-derivative terms are nonnegative; at least the principal term is positive. Thus the fixed-cash-flow price-yield curve is convex: its slope becomes less negative as yield increases.

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)

For positive convexity, the second-order term is nonnegative because $(\Delta y)^2\geq0$. It is positive for a nonzero yield change and zero at the expansion point. Fixed nonnegative cash flows with positive discount bases have positive convexity under the yield convention used here; embedded options can change that conclusion.

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 approximates this difference to second order; higher-order terms remain.

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.

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.

Money duration (also called dollar duration) is modified duration multiplied by the position's market value. It is the magnitude of the first-order dollar sensitivity to a unit change in decimal yield. Multiply it by 0.01 for a one-percentage-point move:

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 is a one-basis-point yield move expressed in decimal units: 1 bp = 0.01 percentage points = 0.0001.

Use the same decimal-yield unit as the derivative: a one-basis-point increase is $\Delta y=0.0001$, so the first-order dollar loss is $D_{Mod}P\times0.0001$. Traders use DV01 to compare dollar sensitivities to small yield movements.

These measures are particularly useful for hedging, as they express risk in dollar terms that can be directly compared across positions. If one unit has a DV01 of $50 and another has a DV01 of $25 to the same yield factor, two short units of the second offset the first-order exposure of one long unit of the first. This does not also match their convexity, curve-shape or credit exposures.

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 interest rate exposure by changing the mix of bonds. With positive portfolio value and a positive-value addition, holding the other positions fixed, a bond whose modified duration exceeds the current portfolio duration raises it; one below the current portfolio duration lowers it. These comparisons use the same yield-change coordinate.

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 a horizon equal to Macaulay duration, the price and reinvestment effects cancel to first order in a small immediate persistent parallel yield change, assuming fixed cash flows and reinvestment at the same shocked yield. Finite shocks leave higher-order exposure. Matching one duration does not protect against arbitrary subsequent rate paths.

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 portfolio Macaulay duration in years equal to the liability horizon $T$. Equivalently, match modified duration to $T/(1+y/k)$ under the same flat nominal annual yield and frequency.
2. Present value matching. Ensure the portfolio's current market value equals the present value of the liability.

In the common-flat-yield fixed-cash-flow model, these conditions remove first-order terminal-value exposure to an immediate persistent parallel shift. Positive asset convexity relative to the single-payment liability controls the local second-order surplus response. These conditions do not establish immunity to all finite shocks or future rate paths.

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

Liability-Driven Investing extends immunization concepts to manage portfolios against complex liability streams. Rather than targeting a single future payment, LDI addresses situations where institutions must meet a series of obligations over time. The framework shifts the investment objective from maximizing returns to ensuring that assets will be sufficient to meet liabilities under various market conditions.

Pension funds provide a classic example. A defined-benefit pension fund must pay retirees monthly benefits potentially spanning decades. The present value of these liabilities depends on interest rates. When rates fall, the present value of fixed liability payments rises. Funding can weaken if liabilities rise faster than assets in percentage terms; whether a deficit develops also depends on the initial funding position and the size of the rate move. A fund that appeared adequately funded at high interest rates may suddenly face a deficit when rates decline, even if the actual benefit obligations have not changed.

The duration gap measures the mismatch between asset duration and liability duration weighted by the inverse funding ratio:

where:

• $D_A$: duration of the asset portfolio
• $D_L$: duration of the liabilities
• $PV_A$: present value of assets
• $PV_L$: present value of liabilities
• $PV_L / PV_A$: the inverse of the funding ratio

The duration gap measures first-order exposure of dollar surplus $S=A-L$. Under a common small decimal yield change, $\Delta S\approx(-D_AA+D_LL)\Delta y=-A\,\text{Duration Gap}\,\Delta y$. A positive gap means rising yields reduce dollar surplus; a negative gap means rising yields increase it. This concerns dollar surplus, not the percentage changes in asset and liability values.

A zero gap equates asset and liability money durations, so dollar surplus has zero first-order yield sensitivity. The funding ratio $F=A/L$ has a different derivative: $dF/dy=F(D_L-D_A)$. Its first-order sensitivity vanishes when the two modified durations are equal. For an overfunded fund, that is generally not the same condition as a zero surplus duration gap.

The negative duration gap (-4.9) indicates that liabilities are more rate-sensitive than assets. A 1% rate increase benefits the fund because liabilities fall more than assets, improving the funding ratio. Conversely, falling rates would damage the fund's position. This analysis highlights the interest rate risk that pension funds face when their asset duration does not match their liability duration.

Pension funds and insurers employ several LDI approaches to manage the duration gap and protect their funding status:

• Cash flow matching: Structure bond portfolios so that coupon and principal payments exactly match liability payment dates in both timing and amount. This approach eliminates reinvestment risk entirely because each cash inflow is immediately used to fund an outflow. However, cash flow matching may be expensive or impossible to implement for very long-dated liabilities, as appropriate bonds may not exist or may trade at premium prices.

• Duration matching: Match portfolio duration to liability duration, accepting some reinvestment risk in exchange for flexibility. This approach allows the use of a broader range of bonds and can be implemented at lower cost than perfect cash flow matching. The tradeoff is that the fund remains exposed to non-parallel yield curve shifts.

• Key rate duration matching: Match sensitivity to specific points on the yield curve, and protects against non-parallel curve movements. Instead of a single duration number, key rate durations measure sensitivity to changes at specific maturities (such as the 2-year, 5-year, 10-year, and 30-year points). Matching key rate durations across assets and liabilities gives more stable protection but requires more sophisticated analysis and implementation.

• Liability-driven benchmarking: Compare the portfolio's cash-flow-adjusted percentage return with the percentage change in a consistently valued liability benchmark. Asset returns above that benchmark indicate relative outperformance, even when both percentage changes are negative. Contributions and benefit payments must be handled consistently, and liability remeasurement must be distinguished from investment performance. This comparison is one performance measure, not a complete judgment of a plan's success.

The current portfolio's funding ratio improves when rates rise because its liability percentage sensitivity exceeds its asset sensitivity. The surplus-matched portfolio holds dollar surplus constant in this linear scenario model, but its funding ratio still changes. The added equal-duration series instead holds the funding ratio constant within the model. Neither construction removes convexity, credit, liquidity or general curve-shape risk, and the finite-shock curves use approximate rather than exactly repriced values.

Duration and convexity are powerful tools, but they rest on assumptions that may not hold in practice. Understanding these limitations helps practitioners apply these measures appropriately and recognize when more sophisticated analysis is required.

Common-factor scenario limitation: An individual bond's own-YTM modified duration is a local price sensitivity; it does not assume that all benchmark-curve yields move together. The portfolio and immunization scenarios in this chapter use a common yield change, so a single matched scalar exposure does not describe arbitrary curve-shape changes. In reality, short rates and long rates often move differently. The curve can steepen, flatten, or twist in ways that a single scalar exposure cannot capture. For example, if short-term yields rise while long-term yields stay fixed on an initially upward-sloping curve, the long-minus-short spread narrows: the curve flattens over those maturities. A single common-factor duration approximation would not fully describe portfolio performance in such a scenario. Key rate durations address this limitation by measuring sensitivity to changes at specific maturities, such as the 2-year, 5-year, 10-year, and 30-year points. However, they add complexity to the analysis.

Instantaneous shift assumption: Duration is a local price derivative at a fixed valuation time. Calendar passage changes remaining cash-flow dates, and coupon payments can cause jumps in the measured duration. A previous asset-liability match can therefore drift and require rebalancing; duration need not decrease smoothly or monotonically along every coupon-bond path.

Yield level dependency: Both duration and convexity change as yields change. A portfolio immunized at 5% yields will have different risk characteristics at 3% yields. Large rate movements can materially alter a portfolio's risk profile. The duration and convexity calculated before a rate move may not accurately describe the portfolio's behavior after the move.

Credit and liquidity risks: Duration addresses interest rate risk only. Corporate bonds face additional risks from credit spread changes, defaults, and liquidity conditions that duration does not capture. Spread duration measures sensitivity to credit spread changes and should be considered alongside interest rate duration for corporate bond portfolios. Default and liquidity risks require separate analysis entirely.

Embedded options: Callable bonds and mortgage-backed securities contain embedded options that create negative convexity in certain yield ranges. When rates fall, callable bonds may be redeemed by the issuer, limiting the investor's upside. Standard duration measures can be misleading for these instruments because the cash flow pattern depends on the path of interest rates. Effective duration, which uses scenario-based revaluation to account for how embedded-option cash flows change with rates, is an appropriate sensitivity measure for these securities, alongside full scenario repricing.

Reinvestment assumptions: Immunization assumes coupons can be reinvested at prevailing rates. In practice, transaction costs, minimum investment sizes, and market liquidity affect actual reinvestment rates. An investor receiving a small coupon payment may not be able to reinvest at the quoted market rate due to bid-ask spreads or minimum trade sizes.

Despite these limitations, duration and convexity remain foundational risk measures. They provide a common language for discussing interest rate exposure, enable rapid approximate calculations for portfolio analysis, and form the building blocks for more sophisticated risk management systems. Practitioners who understand both the power and the limitations of these tools can apply them effectively while recognizing situations that require additional analysis.

This chapter developed the core quantitative tools for measuring and managing bond interest rate risk:

Duration measures a bond's sensitivity to interest rate changes. Macaulay duration represents the weighted average time to receive cash flows. Modified duration directly quantifies the percentage price change for a given yield change. For example, a bond with a modified duration of 7 will lose approximately 7% of its value for each 1% increase in yields.

Convexity adds the second-order price term omitted by duration. It improves the approximation in the worked examples, but finite-shock errors remain. For positive convexity the term is positive for a nonzero yield change and zero at the expansion point.

Portfolio measures aggregate individual bond durations and convexities using market-value weights. DV01 and dollar duration express risk in dollar terms useful for hedging.

Immunization against a single payment matches present value and Macaulay duration to the payment horizon under a consistent yield convention. It removes first-order exposure to the stated immediate persistent parallel shock in the fixed-cash-flow model, not every possible rate path or source of risk.

Liability-driven investing extends immunization concepts to manage complex streams of future obligations. The duration gap measures the mismatch between asset and liability sensitivities, guiding portfolio construction for pension funds and insurers.

Own-YTM duration and convexity are local price sensitivities, not assumptions that a benchmark yield curve moves in parallel. The common-yield portfolio and immunization scenarios here do not cover arbitrary curve-shape changes. These measures have other limitations, but they remain essential tools that every fixed income practitioner must master. More advanced techniques build upon this foundation by adding sensitivity to curve shape changes, credit spreads, and embedded options.

The key parameters for bond risk measures and immunization are:

• Face Value: The contractual principal due at maturity. The examples use $1,000 per bond; denominations depend on the instrument.
• Coupon Rate: Annual coupon payment as a percentage of face value. At fixed maturity, yield and payment frequency, higher coupons shorten Macaulay duration when there is a coupon date strictly before maturity. If all remaining payments arrive at maturity, duration is unchanged.
• Years to Maturity: Time until principal is due. Moving cash flows later changes duration and convexity; compare the complete cash-flow schedules and yield conventions rather than relying on maturity alone.
• Yield to Maturity (y): The discount rate that equates the bond's price to the present value of its cash flows. For fixed nonnegative cash flows with at least one positive future payment and a positive discount base, higher yields reduce modified duration. Macaulay duration does not increase; it falls strictly when positive cash flows occur at more than one date.
• Frequency: Number of coupon payments per year. These examples use two, so both pricing and reinvestment use the corresponding semiannual periodic yield.
• Modified Duration: Measures percentage price sensitivity to yield changes. Used for hedging and risk management.
• Convexity: Measures the curvature of the price-yield relationship. Positive convexity benefits bondholders as it creates asymmetric price responses.
• Target Duration: For the single-liability flat-yield example, the horizon matched by portfolio Macaulay duration to remove first-order terminal-value yield exposure.

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