Hypothetical worked example and primary educational references; not observed market data.
Start with present value
A bond price links future payments to a discount rate. FINRA's yield explanation provides the present-value foundation for this relationship. For a hypothetical zero-coupon bond promising USD 1,000 in five years, price equals 1,000 divided by one plus yield raised to the fifth power. The equation is curved when plotted against yield. That curvature explains why equal yield increases and decreases generally produce unequal price changes for a conventional option-free bond. The following analysis is an original mathematical illustration, not a reported observation or a claim about the sensitivity of a particular traded security.
Supporting references: 1. FINRA: Understanding Bond Yield and Return
Choose a clear baseline
Assume an annual yield of 5%, giving a baseline modeled price of 783.53. Keeping the payment and five-year timing fixed, a 4% yield produces a price of 821.93, while a 6% yield produces 747.26. These invented yield inputs let the reader inspect the same contract under symmetric one-percentage-point changes. Relative to the exact unrounded baseline, the decrease in yield raises price by approximately 4.90%, and the increase lowers it by approximately 4.63%. Rounding the displayed prices is convenient, but calculations should retain full precision until the final reported percentage.
Hypothetical worked example
Exact zero-coupon repricing around a 5% baseline
Model setup
Read these assumptions before comparing the scenarios.
- All monetary amounts in this example are hypothetical U.S. dollars (USD), not current prices, quoted account terms or observed market data. Percentages, counts, time periods and index levels retain their stated units.
- Face value is hypothetically 1,000, paid in five years.
- Annual compounding; no default, options, fees or time passage.
Modeled price (USD)
- 4.00821.93
- 5.00783.53
- 6.00747.26
Expanded scale, not zero-based; dots show small differences.
- Lowest displayed value
- 747.26
- Highest displayed value
- 821.93
Hypothetical cases from the table below, in the stated units. Not market observations, a forecast or a recommended choice. Read the scenario assumptions before comparing.
| Assumed yield (%) | Modeled price (USD) |
|---|---|
| 4.00 | 821.93 |
| 5.00 | 783.53 |
| 6.00 | 747.26 |
Calculation details
- Price at 4%
- 821.93 USD
(1000 / (1.04 ^ 5)) - Price at 5%
- 783.53 USD
(1000 / (1.05 ^ 5)) - Price at 6%
- 747.26 USD
(1000 / (1.0600000000000001 ^ 5))
Why the curved response appears
The discount denominator changes multiplicatively across five periods. Lowering the yield makes every period's discount smaller, and those reductions compound in the valuation formula. Raising the yield works in the opposite direction, but the two results are not mirror images around the initial price. A tangent line at the baseline can approximate both moves, yet it cannot follow the entire curve. Duration describes that local slope; convexity describes curvature. Understanding both terms as features of a model is more helpful than interpreting them as independent sources of guaranteed performance.
A second-order estimate remains an estimate
An analytical price-change approximation can add one half of convexity multiplied by the squared yield change to the duration term, using compatible conventions. That adjustment can improve a small-shock estimate for stable cash flows. However, directly revaluing this simple zero-coupon promise is clearer than introducing an additional assumed convexity statistic. For a real portfolio, the convention, instrument features, and yield scenario matter. The numerical comparison here teaches the reason a linear estimate has residual error. It does not establish that every security has the same curvature or that a second-order formula produces an executable market price.
Options can change the payment path
An issuer's call right can alter the bond's expected life when yields decline. The investor may not receive the full upside implied by discounting an unchanged schedule because the issuer can replace that schedule through permitted redemption. Some securities can therefore behave differently from the positive-curvature zero-coupon illustration. FINRA discusses call and reinvestment risks that help explain this boundary. Before applying an option-free model, identify whether future payments themselves are sensitive to market conditions. A sophisticated formula cannot compensate for assuming a fixed maturity when the relevant contractual path is conditional.
Supporting references: 2. FINRA: Bonds
Interpret curvature within the portfolio
Convexity is useful when stress testing interest-rate exposure, particularly for larger shocks that make duration alone less informative. It is not a reason to buy a bond without analyzing price, creditworthiness, liquidity, and the investor's horizon. The hypothetical gain and loss above arise from assumed repricing with every other input fixed. A real outcome could include spread changes, coupon income, trading costs, and a passage of time. Record the baseline and the changed assumptions when presenting any sensitivity table. That preserves the distinction between a model of valuation and an expectation about future realized returns.
Test another situationRevalue a short payment exactly before using an approximation
A separate hypothetical case. Its inputs differ from the main worked example above.
Revalue a short payment exactly before using an approximation
Imagine an option-free hypothetical promise of USD 1,210 paid in two years. At an assumed annually compounded discount yield of 10%, its present value is USD 1,000. Revalue the unchanged promise immediately at a hypothetical zero yield, with no time passing. The modeled increase is exactly USD 210. This is a deliberately large sensitivity, not a plausible-range claim or expected gain. The example makes curvature visible by returning directly to discounted cash flows. Paying for exposure to that sensitivity also accepts losses when yields rise; the favorable revaluation alone cannot establish the attractiveness of the purchase or the reliability of payment.
Use curvature only after identifying conditional cash flows
FINRA's yield discussion ties price to the payment schedule. An exact discounting calculation is useful only if that schedule is the right one. Before applying an option-free convexity interpretation, ask whether redemption, prepayment or another contractual feature could change amounts or dates under the new yield. If so, show the changed schedule rather than treating the previous promised endpoint as unconditional. Numerical precision cannot resolve an incorrect cash-flow path. For a portfolio, also distinguish yield shocks affecting all payments equally from changes concentrated at particular maturities. Curvature describes a valuation relationship, not an independent source of guaranteed wealth.
Supporting references: 1. FINRA: Understanding Bond Yield and Return
Check this case’s calculations
- Baseline discounted promise
- 1,000.00 USD
(1210 / (1.1000000000000001 ^ 2)) - Zero-yield repricing gain
- 210.00 USD
(1210 - (1210 / (1.1000000000000001 ^ 2)))
Limits of this additional case
- The unusually large hypothetical shock is used to expose nonlinearity and excludes options, credit and market execution.
Limits of this analysis
- The model applies to a fixed option-free payment schedule.
- Callable instruments can have different curvature.
- Modeled present values exclude credit, liquidity and transaction-price effects.
Sources and scope
- FINRA: Understanding Bond Yield and Return
Coupon, current yield and yield to maturity measure different things.
Scope: U.S. investor education; supports the stated mechanism, not the hypothetical scenario or a forecast.. Reference accessed 2026-10-06.
- FINRA: Bonds
Bondholders face interest-rate, credit, call, inflation and liquidity risks.
Scope: U.S. investor education; supports the stated mechanism, not the hypothetical scenario or a forecast.. Reference accessed 2026-10-06.
Illustrative calculations are not forecasts or personal recommendations. Editorial standards
