Where duration and convexity stop explaining the loss

Identify the rate behind the duration

A duration figure without its definition is not a complete exposure measure. Modified duration commonly describes price sensitivity to the bond's yield, while effective duration depends on repricing under changes to a benchmark curve and can incorporate changing cash flows. Key-rate sensitivities isolate particular curve locations. Establish which measure appears in the report and which variable the scenario actually shocks.

Do not multiply a benchmark-rate shock by a spread duration simply because both numbers use the word duration. Inspect the pricing setup, including coupon frequency, day-count convention, accrued interest, and the chosen base price. A price-change percentage should use a consistent denominator. Differences in conventions can produce a reconciliation gap before any economically meaningful disagreement about the scenario has occurred.

Use curvature as an approximation improvement

CFA Institute's bond-convexity reading explains why the price-yield relationship for an option-free fixed-rate bond is curved and why duration alone misses part of a larger move. With matching annualized conventions, a common approximation is minus modified duration times the yield change, plus one half times convexity times the squared yield change. Enter yield changes as decimals in that expression.

Convexity does not make the approximation exact. The starting sensitivities themselves can change substantially as yields move. A report that displays many decimal places may still rely on a coarse local expansion. Compare calculations across shock sizes and retain the exact pricing assumptions used for any benchmark valuation, rather than judging quality solely from whether the convexity term has been included.

Hypothetical calculation: check basis points carefully

Take an invented USD 2,000,000 option-free bond position with modified duration 6 and convexity 50. Assume an instantaneous parallel yield increase of 150 basis points, equal to 0.015. The duration term is minus 0.09. The convexity term is 0.5 times 50 times 0.015 squared, equal to 0.005625. Together they imply an approximate price change of minus 8.4375%.

Multiplying by the starting position value produces an estimated USD 168,750 loss. For an equally sized yield decrease, the duration contribution changes sign while the convexity contribution stays positive, producing an estimated gain of USD 191,250. These hypothetical calculations exclude coupon accrual, credit-spread changes, defaults, currency movements, and transaction costs. They are deliberately simple sensitivity checks, not historical bond returns.

Look beyond a parallel curve move

Two portfolios can share aggregate duration while carrying very different maturity exposures. A concentration in intermediate maturities is not equivalent to an offsetting combination of short and long bonds. Ask for sensitivity by curve segment, then inspect steepening, flattening, and localized changes. The same headline rate increase can leave one maturity nearly unchanged while moving another substantially, with different valuation consequences.

The Basel Committee's interest-rate-risk framework discusses measurement and shock scenarios in the banking-book context. Its relevance here is conceptual: different rate configurations and modeling assumptions deserve explicit attention. This portfolio guide does not import bank capital thresholds or claim regulatory compliance. Use the framework to sharpen the question being asked, while retaining the actual instrument terms and portfolio measurement boundary.

Separate credit and embedded options

A corporate bond can lose value through a widening spread even when the benchmark curve falls. If the model shocks all-in yield and separately applies a credit-spread loss, inspect whether the spread move has been counted twice. Conversely, a benchmark-only calculation can omit material credit exposure. Require a clear mapping from each shocked input to the valuation effect attributed to it.

Callable and prepayable instruments add another complication because expected cash flows may change with rates. Positive convexity from an option-free example should not be assumed for these holdings. Ask which exercise or prepayment assumptions are embedded in the stressed price. A duration number calculated before the shock can be particularly misleading if the scenario extends or shortens the effective life of the instrument.

Reconcile portfolio and liability views

For a portfolio supporting liabilities, an asset-only loss does not describe the change in the funded position. Revalue liabilities under compatible assumptions and show both sides separately. An apparent offset may depend on a particular discount curve, timing convention, or cash-flow estimate. Avoid subtracting two sensitivity totals unless their units and bases are genuinely comparable and the offset is economically relevant.

Also distinguish economic-value changes from cash earnings over a reporting period. A mark-to-market decline need not create an immediate payment, while refinancing and collateral terms can generate cash pressure even when longer-term value offsets appear favorable. The report should explain which question it answers. Combining economic value, income, and funding needs in one unlabeled number makes a precise calculation practically unusable.

Escalate the approximation gap

A useful review compares duration-only, duration-plus-convexity, and appropriate full-revaluation results for the same holdings and inputs. Explain the difference by instrument group and materiality. Do not average the methods to manufacture a compromise estimate. The purpose is to identify where a simplified method remains informative and where more detailed treatment is needed before drawing a portfolio-level conclusion.

Close the review with a bounded action: verify a curve mapping, obtain missing option terms, or recalculate a material maturity bucket. State whether the reported loss includes accrual and spread effects. This produces an interpretable record of rate sensitivity without turning a mathematical approximation into a prediction, an investment recommendation, or evidence that an illustrative tool reproduces a bank's supervisory stress engine.

Review checkpoints

• Match the duration definition to the shocked rate.
• Convert basis points to decimals before applying the formula.
• Curve shape, credit, and embedded options require separate scrutiny.

Source notes

1. CFA Institute: Yield-Based Bond Convexity and Portfolio Properties
2. Basel Committee: Interest rate risk in the banking book, April 2016

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