Duration and Convexity: Reproducing a Bond Price Shock

State the instrument assumptions and sensitivity units

Assume an option-free fixed-cash-flow position with starting full value USD 1,000,000, modified duration 6, and annual convexity 45. The duration and convexity use the same yield convention and valuation point. These are synthetic parameters rather than values extracted from a security. The exercise applies an instantaneous yield shift while freezing contractual cash flows, credit conditions, and all other model inputs.

CFA Institute's duration reading distinguishes modified duration from cash-flow timing measures. Its convexity reading explains the curvature correction. Those sources support the method, not the invented position. Specifying modified duration matters because substituting Macaulay duration without conversion would change the result. Likewise, a benchmark-curve sensitivity based on changing cash flows would answer a different question from this fixed-cash-flow yield approximation.

Convert the shock and calculate both terms

Convert 100 basis points to 0.01 in decimal yield units. The first-order term is negative modified duration times the yield change: negative 6 times 0.01 equals negative 0.06, or negative 6%. The curvature term is one-half times convexity times the squared yield change: 0.5 times 45 times 0.01 squared equals 0.00225, or positive 0.225%.

Adding the terms gives negative 0.05775, equivalent to negative 5.775%. For USD 1,000,000, multiply by that decimal change to obtain negative USD 57,750. The remaining estimated price value is USD 942,250. The 0.225 figure is a percentage-point adjustment to the duration-only return estimate, not a 22.5% return and not 0.225 dollars per unit of face value.

Use symmetric shocks to expose curvature

Reverse the hypothetical shock to a 100-basis-point decline. Duration now contributes positive 6%, while the squared convexity term remains positive 0.225%. The combined estimate becomes positive 6.225%. Equal-size upward and downward yield changes therefore do not produce equal-size opposite price changes under this second-order model. Positive curvature improves both results relative to the linear approximation.

The table also includes 50-basis-point moves. Halving the shock halves the duration term but quarters the convexity adjustment. This scaling is a useful implementation check. None of the rows is a probability-weighted scenario or forecast. The sensitivities remain fixed at their assumed initial values, so the grid illustrates a local approximation rather than a sequence of re-estimated positions after successive market moves.

Keep the calculation separate from total return

The result is a price effect at a frozen point in time. A holding-period return additionally depends on coupon income, reinvestment, movement along the curve as maturity approaches, and endpoint valuation. Starting yield cannot simply be added without specifying the interval and cash-flow conventions. A 5.775% estimated price decline is therefore not a claim that the investment loses exactly that amount over a year.

The starting USD value is a position valuation, not assumed face amount or margin posted. Translating the same percentage into currency loss requires the appropriate price-value base. For a derivative exposure, cash deposited and economic rate exposure can differ substantially. An unqualified return on margin would not be comparable with this price change on the stated USD 1,000,000 position.

Recognize when full repricing is necessary

Duration and convexity are local derivatives. Their approximation can deteriorate as shocks grow or cash-flow assumptions change. Callable bonds and mortgage securities can require a model for exercise or prepayment behavior, rather than the fixed-cash-flow treatment used here. An observed credit-spread move or liquidity discount also cannot be explained solely by inserting a government-yield shock into this formula.

A portfolio adds another assumption if one duration and one shock are used for all maturities. Curve twists can affect exposures differently even when the average yield move looks small. Identify key-rate sensitivities or reprice against a changed curve when that is the question. The present calculation deliberately isolates a common yield change and does not establish that all market-rate disturbances are parallel.

Preserve the arithmetic and its limited claim

An audit record should retain starting price value, sensitivity definitions, yield units, shock sign, formulas, and rounding. Compute the decimal result before formatting it as a percentage, and check the currency effect independently. Comparing duration-only and curvature-adjusted estimates helps reveal unit errors, but agreement between implementations validates arithmetic only. It does not prove that the assumed sensitivities match an actual security.

The publication date belongs to this hypothetical method, not to a purported rate event. Its defensible conclusion is that the specified approximation produces negative 5.775% excluding income for a positive 100-basis-point yield shock. The result neither predicts a yield move nor identifies a suitable bond allocation, and its numerical precision should not be mistaken for evidence that real-world repricing would match every displayed decimal.

What to retain

• Use a decimal yield change of 0.01 for a 100-basis-point shock.
• The stated parameters imply -6% plus 0.225 percentage points, or -5.775%.
• The result excludes income and remains a local price approximation.

Source notes

1. CFA Institute: Yield-Based Bond Duration Measures and Properties
2. CFA Institute: Yield-Based Bond Convexity and Portfolio Properties

Related research