Set the model before the measure
An educational bond pays 80 after one year, 80 after two years and 1,080 after three years. Use one annual yield of 8% and annual discounting. The present value of the payments is 1,000. Assume known dates, no default, amortisation, indexation or embedded right to change the cash flow. SEC notes that bond contracts can have other properties,[14](#source-cm-corporate) so this one model is not universal. In particular, it cannot be applied unchanged to unknown future coupons or an instrument that may terminate early.
In this model, modified duration calculated through the derivative of price with respect to yield is approximately 2.5771. For a small yield change, relative price change is approximated by minus duration times the rate change in decimal form. When the yield rises from 8% to 9%, the change is 0.01, not the number 1. The linear estimate gives a price of approximately 974.23. A units error here would increase the calculated effect one hundredfold and completely distort the conclusion.
Why a second adjustment appears
Directly discounting the same three payments at 9% gives approximately 974.69. The difference from the linear estimate arises because price is not a straight-line function of yield. Convexity describes that curvature. In the selected model, normalised second derivative is approximately 9.3002. Adding “one half times convexity times the squared yield change” gives about 974.6940. This is a mathematical adjustment to the model, not an additional independent payment to the bondholder.
If the yield falls to 7%, the exact model price is approximately 1,026.24. The increase need not exactly equal the decrease under a symmetric rise in yield. This shows why nonlinearity matters. Even second order remains an approximation, not an exact calculation for every movement. With a large change or changed payments, revalue the model. Do not extend a tangent indefinitely and call the line a law of future market quotations.
There is no single sensitivity to everything
The duration described here concerns a parallel change in one selected yield. The real curve can change slope and shape; credit risk, liquidity and embedded rights can also change. The measure therefore does not automatically explain all observed security return. Identify whether the number is modified duration, another variant or a figure produced by another method. Similar names in two services do not prove identical assumptions, units or calculation dates.
To check another estimate, ask about cash flow, calendar, accrual convention, base rate and change scenario. Then compare the approximation with direct recalculation for at least one scenario. The [interest-risk guide](/guides/bond-rate-risk/) explains direction, [coupon and yield](/guides/coupon-versus-yield/) separates payment rate from required yield, and [offers](/guides/maturity-and-offer/) describes events that can change the cash flow itself. Even an exact model sensitivity does not establish liquidity, debtor solvency or suitability for a particular person.
Control the derivative model
The calculated measure belongs to the chosen price function. Changing accrual convention or calendar can change the numerical value. Therefore, comparing durations from different sources starts with method, not with the conclusion that a larger figure must be wrong. Our task uses whole-year periods and one yield; that makes the calculation reproducible but not universal.
For self-checking, reduce the size of the scenario shift and compare the linear approximation with direct recalculation. Under the original assumptions, the gap should behave like approximation error, not a separate source of profit. If a party’s right changes the cash flow, recalculating the old payments does not test the new contractual scenario. First redefine amounts and dates. Do not transfer the convexity obtained here to every callable or otherwise optional bond: fixed cash flow was essential. Sensitivity is tested together with the function’s boundaries, not separately from the obligation text.