Why Bond Risk Comparisons Need Controlled Conditions
A statement such as “Bond A is riskier than Bond B when rates change” is incomplete unless the meaning of risk and the comparison conditions are clear. A longer maturity often means greater price sensitivity, for example, but differences in coupon, yield, or cash-flow structure can also affect the result. A useful comparison holds other features constant where possible and identifies the measure being ranked.
This tutorial focuses on price sensitivity to yield changes. It uses modified duration for a first-order comparison and convexity to improve estimates when the yield change is not infinitesimal. The calculations assume fixed, noncallable cash flows and a common change in yield. They do not account for defaults, embedded options, or changes in the shape of the yield curve.
A Framework for Ranking Sensitivity
Modified duration ranks bonds by their first-order percentage sensitivity: a bond with larger modified duration has a larger approximate percentage price response to the same small yield change. For a finite yield change, the duration-convexity approximation is usually more informative:
Here \(\Delta y\) is the change in annual yield expressed as a decimal: one percentage point is \(0.01\), not \(1\). If the shift is upward, duration contributes a negative term and convexity contributes a positive term. For a downward shift, both terms are positive for a conventional option-free bond. Convexity therefore makes the estimated gain from a yield decrease larger than the estimated loss from an equal yield increase.
The percentage change and dollar change are related but are not interchangeable. The estimated dollar change is \(P\) times the estimated percentage change. Two bonds can have the same percentage sensitivity but different dollar changes because their prices differ. A risk ranking should therefore state whether it concerns percentage price sensitivity or dollar exposure.
Why Duration and Convexity Give These Comparisons
For fixed cash flows \(CF_t\) paid at integer times \(t\), discounted at an annual effective yield \(y\), the price is \(P=\sum_t CF_t/(1+y)^t\). Differentiating this expression explains why duration is a cash-flow-weighted measure of sensitivity.
The middle weighted sum is Macaulay duration: each payment time is weighted by that payment’s share of the bond’s present value. Differentiating a second time gives the annual-yield convexity formula:
The duration-convexity estimate follows from the second-order Taylor approximation to price around the current yield: retain the first derivative and half the second derivative, then divide by the current price. It is an approximation, not an exact price calculation; it is most useful for moderate changes when the cash flows and yield convention remain unchanged.
Controlled Comparisons: Maturity, Coupon, and Yield
For conventional fixed-cash-flow bonds, the following comparisons are useful when the named feature changes and the others are held constant. They are directional rules, not substitutes for calculating duration when several features differ.
| Feature changed | Typical comparison for an option-free bond | Condition for a fair comparison |
|---|---|---|
| Maturity | Longer maturity generally means greater price sensitivity. | Keep coupon, yield, payment frequency, and other cash-flow features the same. |
| Coupon | Lower coupon generally means greater price sensitivity. | Keep maturity, yield, payment frequency, and other features the same. |
| Yield level | For the same fixed cash flows, a lower yield generally produces greater duration and convexity. | Compare the same bond cash flows at different yields using the same compounding convention. |
| Duration and convexity | Greater modified duration means greater first-order percentage sensitivity; convexity improves finite-change estimates. | Specify the direction and size of the yield change, and compare the same kind of measure. |
The maturity rule reflects the timing of the cash flows: extending payments generally shifts more present-value weight toward later dates. A lower coupon likewise leaves a greater share of value in the later principal payment. For a given set of positive fixed cash flows, a lower yield increases the relative present-value weight of later payments, which tends to increase duration. These effects can be offset if the bonds also differ in other features, so a label such as “longer” or “lower coupon” alone is not enough to establish a ranking.
Worked Example: Comparing Maturity With Coupon and Yield Held Constant
Two hypothetical noncallable bonds each have a $1,000 face value, a 5% annual coupon, annual payments, and a 5% annual effective yield. Bond A matures in 2 years; Bond B matures in 5 years. Rank their modified durations and estimate each percentage price change for a 1-percentage-point yield increase.
Method. Since coupon, yield, payment frequency, and face value are the same, this is a controlled maturity comparison. At a coupon rate equal to the yield, each bond sells at par. For these par bonds, the present-value weights can be used to obtain Macaulay duration. Using \(v=1/1.05=0.952381\), the par-bond expression is \(D_{\mathrm{Mac}}=((1.05)/0.05)(1-v^n)=21(1-v^n)\), where \(n\) is the number of years.
For Bond A, \(v^2=0.907029\), so \(D_{\mathrm{Mac}}=21(1-0.907029)=21(0.092971)=1.952391\) years, or 1.952381 years using unrounded \(v^2\). Therefore \(D_{\mathrm{mod}}=1.952381/1.05=1.859410\) years.
For Bond B, \(v^5=0.783526\), so \(D_{\mathrm{Mac}}=21(1-0.783526)=21(0.216474)=4.545954\) years, or 4.545951 years using unrounded \(v^5\). Therefore \(D_{\mathrm{mod}}=4.545951/1.05=4.329477\) years.
For a 1-percentage-point increase, \(\Delta y=+0.01\). The first-order estimates are \(-1.859410(0.01)=-0.0185941\), or \(-1.8594\%\), for Bond A and \(-4.329477(0.01)=-0.0432948\), or \(-4.3295\%\), for Bond B. Bond B is more sensitive by modified duration and by this first-order estimate.
Check and common mistake. The 5-year bond’s greater duration is consistent with its later principal payment. A common error is to compare maturity while overlooking different coupons or yields; that would no longer isolate the maturity effect. Another is to enter \(1\) for a 1-percentage-point change, which would incorrectly treat it as a 100-percentage-point change.
Worked Example: Adding Convexity to a Bond Comparison
Compare two hypothetical 2-year bonds, each with $1,000 face value, annual payments, and an annual effective yield of 10%. Bond Z is a zero-coupon bond. Bond C pays a 10% annual coupon. Calculate modified duration and convexity for each, then estimate their percentage price changes for a 1-percentage-point yield increase.
Method. Calculate present-value weights, then use \(D_{\mathrm{mod}}=\sum_t w_t t/(1+y)\) and \(C=\sum_t w_t t(t+1)/(1+y)^2\), where \(w_t\) is the present value of the cash flow at time \(t\), divided by the bond price.
For Bond Z, the price is \(P=1000/(1.10)^2=1000/1.21=\$826.45\). Its only cash flow is at year 2, so its entire present-value weight is \(w_2=1\). Thus \(D_{\mathrm{mod}}=2/1.10=1.818182\), and \(C=2(3)/(1.10)^2=6/1.21=4.958678\).
For Bond C, the cash flows are $100 at year 1 and $1,100 at year 2. Their present values are \(100/1.10=\$90.9091\) and \(1100/1.21=\$909.0909\), giving price \(P=\$1,000\). The weights are \(w_1=90.9091/1000=0.0909091\) and \(w_2=909.0909/1000=0.9090909\). Therefore \(D_{\mathrm{mod}}=(0.0909091(1)+0.9090909(2))/1.10=1.9090909/1.10=1.735537\). Its convexity is \(C=(0.0909091(1)(2)+0.9090909(2)(3))/(1.10)^2=(0.1818182+5.4545454)/1.21=4.658151\).
With \(\Delta y=+0.01\), Bond Z’s estimate is \(-1.818182(0.01)+\tfrac12(4.958678)(0.01)^2=-0.01818182+0.000247934=-0.01793389\), or \(-1.7934\%\). Bond C’s estimate is \(-1.735537(0.01)+\tfrac12(4.658151)(0.01)^2=-0.01735537+0.000232908=-0.01712246\), or \(-1.7122\%\). Bond Z has the larger estimated percentage decline for this specified yield increase.
Check and common mistake. Both estimates are negative, as expected for a yield increase, and convexity reduces the size of each estimated decline. Ranking by coupon alone would be a poor substitute for calculating the measures when the question asks for a numerical comparison. Also, the convexity term uses \((\Delta y)^2\), so it is positive for either direction of the yield shift.
Worked Example: Isolating the Effect of Yield Level
Consider the same hypothetical 2-year zero-coupon bond with $1,000 face value, first valued at a 4% annual effective yield and then at a 10% annual effective yield. Compare modified duration, convexity, and the duration-convexity estimate for a 1-percentage-point yield increase at each yield.
Method. The cash flows and maturity are unchanged, so the yield level is the feature being compared. For a zero-coupon bond maturing at year 2, \(D_{\mathrm{mod}}=2/(1+y)\) and \(C=6/(1+y)^2\).
At \(y=0.04\), the price is \(1000/(1.04)^2=1000/1.0816=\$924.56\). Modified duration is \(2/1.04=1.923077\), and convexity is \(6/1.0816=5.547337\). For \(\Delta y=+0.01\), the estimate is \(-1.923077(0.01)+\tfrac12(5.547337)(0.01)^2=-0.01923077+0.000277367=-0.01895340\), or \(-1.8953\%\).
At \(y=0.10\), the price is \(1000/(1.10)^2=1000/1.21=\$826.45\). Modified duration is \(2/1.10=1.818182\), and convexity is \(6/1.21=4.958678\). For the same \(+0.01\) shift, the estimate is \(-1.818182(0.01)+\tfrac12(4.958678)(0.01)^2=-0.01818182+0.000247934=-0.01793389\), or \(-1.7934\%\).
At the lower yield, both modified duration and convexity are higher, and the estimated percentage decline is larger. This is a percentage comparison: the prices differ, so a dollar comparison would require multiplying each percentage estimate by its own starting price.
Check and common mistake. The bond’s cash flows are identical at both yields; only the discount rate changes. Do not describe the comparison as two different bonds with different maturities or cash flows. Also, the direction of the shift is explicitly an increase; changing it to a decrease changes the sign of the duration term.
Interpreting Rankings and Avoiding Traps
When two bonds have different duration and convexity, the ranking can depend on the size and direction of the yield change. Modified duration alone ranks the first-order response. For a specified finite change, calculate both terms of the duration-convexity estimate before concluding which has the larger estimated change in absolute value.
For example, suppose Bond X has \(D_{\mathrm{mod}}=3.1\) and \(C=14\), while Bond Y has \(D_{\mathrm{mod}}=2.8\) and \(C=30\). X has the larger first-order percentage sensitivity. For a 1-percentage-point increase, X’s estimate is \(-3.1(0.01)+\tfrac12(14)(0.01)^2=-0.031+0.0007=-0.0303\), or \(-3.03\%\). Y’s is \(-2.8(0.01)+\tfrac12(30)(0.01)^2=-0.028+0.0015=-0.0265\), or \(-2.65\%\). X still has the larger absolute estimated change. For a 1-percentage-point decrease, the estimates instead are \(+3.1(0.01)+0.0007=3.17\%\) and \(+2.8(0.01)+0.0015=2.95\%\). X remains larger, but the estimates differ from those for an increase.
- Do not mix percentage and dollar rankings. Multiply the percentage estimate by the bond’s initial price to obtain an estimated dollar change.
- Do not leave the yield shift ambiguous. State both the magnitude and direction; the duration term changes sign when the direction changes.
- Do not treat the approximation as exact. For a large shift or a bond with cash flows that change when rates change, a fixed-cash-flow duration-convexity estimate may be inadequate.
- Do not assume every pairwise ranking is controlled. Maturity and coupon rules isolate one feature only when relevant competing features are held constant.
These comparisons matter when an investor or financial analyst needs to understand how a bond allocation may respond to yield movements. For an actuary, the same discipline helps distinguish a sensitivity measure from a forecast: duration and convexity describe local price behavior under stated assumptions, not the return that an investment will necessarily realize. The figures are aligned with the SOA VEE Accounting and Finance syllabus.
Check Your Understanding
For each problem, state the relevant comparison measure and keep yield changes in decimal form.
- Two hypothetical zero-coupon bonds have 3-year maturities and annual effective yields of 5% and 9%. Calculate each modified duration and convexity, then identify which has the larger values. (Answer: At 5%, \(D_{\mathrm{mod}}=3/1.05=2.8571\) and \(C=12/1.05^2=10.8844\). At 9%, \(D_{\mathrm{mod}}=3/1.09=2.7523\) and \(C=12/1.09^2=10.1001\). The 5% yield bond has both larger.)
- Bond A has modified duration 4.0 and convexity 20; Bond B has modified duration 3.7 and convexity 36. For a 1-percentage-point yield decrease, calculate both duration-convexity estimates and rank their absolute values. (Answer: A: \(4.0(0.01)+\tfrac12(20)(0.01)^2=0.0410=4.10\%\). B: \(3.7(0.01)+\tfrac12(36)(0.01)^2=0.0388=3.88\%\). A has the larger absolute estimate.)
- A 4-year zero-coupon bond is valued at a 6% annual effective yield. Calculate its price for $1,000 face value and its modified duration; estimate the percentage price change for a 0.5-percentage-point yield increase using duration only. (Answer: \(P=1000/(1.06)^4=\$792.09\); \(D_{\mathrm{mod}}=4/1.06=3.7736\); estimate \(=-3.7736(0.005)=-1.8868\%\).)
- Two noncallable bonds have the same maturity, yield, payment frequency, and face value, but Bond L has a lower coupon than Bond H. Which is typically more price-sensitive, and what must be checked before applying this rule? (Answer: Bond L is typically more sensitive; confirm the cash-flow comparison is otherwise controlled, with no differing embedded options or other features that change cash flows.)
- A bond has a price of $980, modified duration 5.2, and convexity 32. Estimate the dollar price change for a 0.75-percentage-point yield increase using duration and convexity, then distinguish the percentage estimate from the dollar estimate. (Answer: \(\Delta y=0.0075\). Percentage estimate \(=-5.2(0.0075)+\tfrac12(32)(0.0075)^2=-0.0381=-3.81\%\). Dollar estimate \(=-0.0381(\$980)=-\$37.34\), to the cent.)