Convexity and Approximation

Read curvature as a local property

CFA Institute's convexity reading describes the second-order effect omitted by a duration-only estimate. In a regular option-free bond with positive fixed receipts, price is a decreasing but convex function of yield wherever the compounding base is positive. The slope changes as yield changes. A single duration number therefore cannot exactly describe equal-sized upward and downward yield moves away from the starting point.

The useful question is not whether duration is wrong, but how much of the curved relationship its local line captures over the chosen scenario. Convexity supplies the next term of that local expansion. It should be presented with a starting yield and shock definition, because a statistic calculated at one point does not describe curvature everywhere along the price-yield curve with equal accuracy.

Use one definition throughout the formula

Here C denotes normalized yield convexity, defined as the second derivative of price with respect to annual nominal yield, divided by price. It has the conventional time-squared dimension. With periodic index k and frequency m, compute C as the sum of CF_k*k*(k+1)/[m^2*(1+y/m)^(k+2)], divided by P. This definition already includes the frequency conversion and does not include the one-half Taylor coefficient.

The second-order relative-price estimate is -Dmod*dy + 0.5*C*dy^2, where dy is a decimal annual nominal yield change. Some reporting conventions package factors differently, so a number called convexity should not be inserted blindly. Applying the one-half coefficient twice, or omitting the frequency-squared denominator, changes the result even though the formula may still produce a plausible-looking positive adjustment.

Choose a single-payment benchmark

Consider a hypothetical USD 1,000 zero-coupon bullet due in exactly two years, valued with a 5% annually compounded yield. Price is 1000/(1.05)^2 = USD 907.029478. All cash arrives at maturity, with no calls, default, accrued interest, taxes or transaction costs. The zero-coupon case is chosen because both derivatives can be verified directly from a simple power function.

Modified duration is 2/1.05 = approximately 1.904762 years. Convexity is 2*3/(1.05)^2 = approximately 5.442177 years squared. These are baseline quantities calculated before a shock. Keep their unrounded values in the approximation. Rounding the derivatives before applying the scenario can obscure whether a small discrepancy comes from the approximation itself or merely from displayed input precision.

Compare three prices for an upward shock

Raise the annual yield from 5% to 6%, so dy = +0.01. Duration alone gives USD 889.752726. Adding the convexity term gives USD 889.999537. Exact repricing gives 1000/(1.06)^2 = USD 889.996440. All three calculations hold the maturity date and promised USD 1,000 payment fixed. The scenario contains no passage of time or intervening coupon distribution.

The convexity-adjusted estimate is much closer in this example, but it still exceeds the exact value by about USD 0.003097. That residual is not evidence of a missing fee or default assumption. It reflects higher-order terms excluded from the approximation. Keeping the residual visible is more informative than rounding both answers to cents and declaring them mathematically identical.

Apply the same convention to a downward shock

For a decline from 5% to 4%, dy = -0.01. Duration alone gives USD 924.306230, duration plus convexity gives USD 924.553041, and exact repricing gives 1000/(1.04)^2 = USD 924.556213. The convexity contribution is positive for both shock directions because the yield change is squared. The first-order duration contribution changes sign; the second-order contribution does not.

CFA Institute's convexity primer emphasizes that the price-yield relationship is curved rather than straight. In this fixed-payment example, equal absolute yield changes therefore do not create equal absolute price changes. The upward-yield loss is smaller than the downward-yield gain. This is a statement about the stipulated mathematical structure, not a forecast that either scenario will occur or a claim that convexity eliminates investment losses.

Do not extend fixed-flow convexity to optional payments

The positive-curvature argument relies on holding positive cash flows and their dates fixed. A callable bond can change its expected remaining payments when rates change, so the same derivative interpretation does not settle its price response. A mortgage-like claim can have analogous payment-timing changes. Those instruments require models of the contingent cash flows and are outside the parent tool's regular noncallable bullet scope.

Nor does a scalar yield convexity capture every cross-effect of a multi-point curve scenario. Different maturities can move independently, creating several first and second derivatives with respect to curve inputs. A one-variable expansion cannot reproduce that complete surface. Explain the one-yield experiment honestly rather than treating a more elaborate-looking formula as evidence that the valuation now includes arbitrary term-structure changes.

Choose an error comparison that answers the question

Report approximation error in both the relevant price unit and, where helpful, relative to the starting price. An absolute error that appears small per 100 face can become material after position scaling. Conversely, a large displayed count of decimal places is not a reason to imply practical transaction precision. The example's detailed decimals exist to make the arithmetic comparison inspectable, not to suggest executable pricing.

The parent framework supports one, two or four regular periods per year and whole-year maturities from one through thirty. Within that boundary, full repricing is the direct comparator for a stated yield shock. Duration and convexity remain useful explanations of why price moved and how a local approximation behaves. They should accompany the exact calculation where needed, not replace a readily available exact calculation with an unmeasured approximation.

Primary references

1. CFA Institute: Yield-Based Bond Convexity and Portfolio Properties
2. CFA Institute: A Bond Convexity Primer