Why Convexity Calculations Go Wrong
A duration-and-convexity estimate is compact, but it combines several conventions in one expression: the yield quotation, the frequency of payments, the units of the yield change, and the units of the price change. A calculation can look plausible while being wrong by a factor of 100, by a factor tied to payment frequency, or simply by a sign.
The estimate is a local approximation, not an exact repricing formula. It assumes that the cash flows remain fixed, that the yield convention does not change, and that the yield shift is applied consistently to all payments. These assumptions matter when deciding whether a calculation is set up correctly.
For a conventional bond with positive fixed cash flows, modified duration is positive and convexity is positive. Thus, a yield increase produces a negative first-order contribution, while the convexity contribution is positive for either direction of yield change. With signed cash flows, these sign properties are not guaranteed; the measures must be calculated from the actual price function.
Four Checks Before Substituting
Determine whether the yield is an effective annual rate or a nominal annual rate convertible a specified number of times per year. Use measures defined with respect to that same yield variable.
A rise of 0.6 percentage points is \(\Delta y=0.006\), not \(0.6\). A rise of 60 basis points is also \(0.006\).
For a nominal annual yield convertible \(m\) times per year, the periodic rate is \(y/m\). If the bond has \(k\) periods remaining, use \(k\), not automatically the number of years, in period-based sums.
The formula estimates a relative change, \(\Delta P/P\). Multiply by the current price \(P\) to obtain an estimated dollar change, then add that dollar change to \(P\) to estimate the new price.
Why the Approximation Has These Signs
Taylor’s theorem gives a local explanation. For a twice-differentiable price function around the current yield \(y\), the change in price is approximately its first derivative times the yield change, plus one-half its second derivative times the squared yield change. Dividing by the current price and using the definitions above gives the duration-and-convexity expression.
The squared term cannot be negative, so its sign in the approximation is determined by \(C_y\), not by whether yields rise or fall. For positive convexity, it raises the estimate relative to duration alone in both cases. This does not mean that the bond price rises for both yield directions: the duration term ordinarily dominates for a sufficiently small shock.
Worked Example: Annual Yield and a One-Percentage-Point Shock
Worked Example: Annual Yield and a One-Percentage-Point Shock
A two-year bond pays $10 at the end of year 1 and $110 at the end of year 2. Its effective annual yield is 10%, and its current price is $100. Estimate the relative and dollar price change if the yield rises by one percentage point. Use annual compounding and the duration-and-convexity approximation.
Method. First calculate the price-weighted duration and convexity with respect to the effective annual yield. For annual cash flows \(CF_t\) at times \(t\), the price function is \(P(y)=\sum_t CF_t(1+y)^{-t}\). Differentiating this function gives the measures used here:
Step 1: Verify the current price. At \(y=0.10\), the present values of the two payments are
Step 2: Calculate duration. Equivalently, weight each payment’s present value by \(t/(1+y)\), then divide by price:
Step 3: Calculate convexity. Weight each present value by \(t(t+1)/(1+y)^2\):
Step 4: Use a decimal yield change. A one-percentage-point increase is \(\Delta y=0.01\). The estimated relative price change is
This is approximately a 1.712246% decrease. The estimated dollar change is \(100(-0.01712246)\approx-\$1.71\), so the estimated new price is approximately $98.29. The check is consistent: with positive duration, a yield increase produces a price decrease, and the positive convexity term makes the estimated decrease slightly smaller than the duration-only decrease.
Common mistake avoided: The convexity contribution is \(\tfrac12 C_y(\Delta y)^2\), not \(C_y\Delta y\). The yield change must also be \(0.01\), not \(1\), because the shock is one percentage point.
Payment Periods and Yield Units Must Match
When a nominal annual yield \(y\) is convertible \(m\) times per year, the periodic rate is \(y/m\). If \(CF_k\) is paid after period \(k\), then \(P(y)=\sum_k CF_k(1+y/m)^{-k}\). Differentiating with respect to the quoted annual nominal yield \(y\) introduces a factor of \(1/m\) in duration and \(1/m^2\) in convexity:
These are measures with respect to the annual nominal yield \(y\). They are not the same as measures taken with respect to the periodic rate \(y/m\). A frequent error is to use the number of years in place of the number of payment periods, or to use measures with respect to one yield variable and a shock measured in another.
Worked Example: Semiannual Payments and a Nominal Yield
A zero-coupon bond pays $100 after four semiannual periods. Its nominal annual yield convertible semiannually is 6%. Using duration and convexity with respect to that annual nominal yield, estimate the relative price change when the yield falls by one percentage point.
Step 1: Identify the period rate and price. There are four periods and the periodic yield is \(0.06/2=0.03\). Thus,
Step 2: Calculate measures with respect to the quoted annual yield. For a single payment at period \(k=4\), the price weights cancel. With \(m=2\),
Step 3: Apply the shock with the correct sign and units. A fall of one percentage point means \(\Delta y=-0.01\). Therefore,
The estimated relative increase is about 1.9653%. The estimated dollar increase is \(88.8487(0.01965313)\approx\$1.7467\), or $1.75 to the nearest cent. The resulting estimated price is about $90.60. The sign check is positive, as expected for a yield decrease on this positive-cash-flow bond.
Common mistake avoided: The four semiannual payments count as four periods, not two years’ worth of annual periods in the formula. Since the measures are with respect to the annual nominal yield, use \(\Delta y=-0.01\); do not substitute a periodic yield change into those annual-yield measures.
Relative Changes Are Not Dollar Changes
Duration and convexity in the displayed approximation produce \(\Delta P/P\), a dimensionless relative change. If that value is \(-0.02\), the estimated change is a 2% decrease, not a decrease of $0.02. The corresponding dollar change depends on the current price. Keeping these quantities labelled during the calculation prevents a common interpretation error.
Worked Example: Converting the Estimate to Dollars
A bond is currently priced at $250. Its modified duration and convexity, both measured with respect to the same effective annual yield, are \(D_y=3.2\) and \(C_y=14\). Estimate the relative and dollar price changes for a 50-basis-point yield decrease. State the approximate new price.
Step 1: Convert the shock. Fifty basis points is \(0.50\) percentage points, so \(\Delta y=-0.005\).
Step 2: Calculate the relative change.
The estimated relative increase is 1.6175%. The positive convexity contribution is \(0.000175\), or 0.0175 percentage points of price.
Step 3: Convert to dollars and check.
To the nearest cent, the estimated dollar gain is $4.04 and the estimated new price is $254.04. The change is positive because the yield fell. The relative result is 1.6175%; the dollar result is approximately $4.04.
Common mistake avoided: Reporting \(0.016175\) as a dollar change confuses a proportion with money. Multiplying by $250 is necessary to obtain the dollar estimate.
Common Error Patterns and Corrections
| Error | Why it fails | Correction |
|---|---|---|
| Using 1 for a one-percentage-point yield change | The formula uses a rate change in decimal form, not a number of percentage points. | Use \(\Delta y=0.01\). |
| Counting years rather than payment periods for a semiannual bond | The cash-flow exponents count periods when the periodic rate is \(y/2\). | Use the number of semiannual periods, and convert yield derivatives consistently. |
| Adding the duration term for a yield increase | For positive modified duration, the first-order price response has the opposite sign to the yield change. | Use \(-D_y\Delta y\). |
| Giving the convexity term the opposite sign when yields fall | The yield change is squared, so the sign of the shock does not change this term. | Use \(+\tfrac12 C_y(\Delta y)^2\), retaining the sign of \(C_y\). |
| Calling a relative change a dollar change | The approximation first estimates \(\Delta P/P\). | Multiply by the current price to estimate \(\Delta P\). |
A useful audit is to calculate the duration term and convexity term separately before adding them. Check the magnitude of \(\Delta y\), confirm that the measures use the same yield variable, and write “relative” or “dollars” beside the result. For an ordinary positive-cash-flow bond with positive convexity, the convexity contribution should be positive whether yields rise or fall. If an answer contradicts these checks, revisit the setup before rounding.
The approximation is most informative for modest yield changes. It omits third- and higher-order terms, so it is not a substitute for exact repricing when a precise price is required or when the shock is large. Its purpose is to estimate the local response and make the roles of duration and convexity explicit. Accurate use of the conventions is aligned with the SOA Exam FM (Financial Mathematics) syllabus.
Check Your Understanding
For each problem, keep the yield convention and the units of the final result explicit. Treat the stated duration and convexity as measured with respect to the yield variable named in the problem.
- A three-year zero-coupon bond has an effective annual yield of 5%. Estimate its relative price change for a 25-basis-point yield increase using duration and convexity. (Answer: approximately \(-0.7109\%\))
- A zero-coupon bond pays after six semiannual periods and has a nominal annual yield of 8% convertible semiannually. Using measures with respect to that annual nominal yield, estimate the relative price change for a 0.5-percentage-point yield decrease. (Answer: approximately \(+1.4544\%\))
- A bond priced at $480 has \(D_y=2.5\) and \(C_y=9\), measured with respect to its effective annual yield. Estimate the dollar change and new price for a 40-basis-point yield increase. (Answer: approximately a $4.77 decrease; new price $475.23)
- A bond is priced at $1,250 and has \(D_y=1.8\) and \(C_y=5.4\), measured with respect to its effective annual yield. Estimate the dollar gain and new price for a 0.75-percentage-point yield decrease. (Answer: approximately a $17.06 gain; new price $1,267.06)
- A bond priced at $1,000 has duration 3.6 and convexity 18, both with respect to its nominal annual yield convertible semiannually. An analyst estimates the change for a 1-percentage-point yield increase by substituting \(0.005\) into the formula, claiming this is the per-period yield change. Identify the unit error and give the corrected relative and dollar estimates. (Answer: use \(\Delta y=0.01\); approximately \(-3.510\%\), or a $35.10 decrease)