Shape of Yield Curves Under Instrument‐Based Parallel Shifts

ABSTRACT

This paper studies the shape of zero and par yield curves through the carry and convexity of zero‐bond and par–swap butterflies under instrument‐based parallel shifts : the component instruments are held fixed, and each instrument's own quote receives the same additive shock. Although this framework is a conditional scenario experiment rather than a model‐independent restriction on admissible yield‐curve shapes, it is simple and widely used in fixed‐income risk management. For zero‐coupon bonds, maturity is the natural horizontal coordinate; for par–swaps, the nonlinear bootstrap relation makes the fixed‐leg annuity, or PV01, the relevant coordinate. Under the stated instrument‐based evolution rule, an appropriately weighted zero‐cost butterfly has a nonnegative instantaneous revaluation under either direction of an admissible common shift, while convexity of the initial curve in the relevant coordinate also generates positive short‐horizon carry. These results provide a partial economic rationale for the commonly observed concavity of yield curves: convex segments can support positive‐carry, level‐neutral butterflies under the benchmark shift convention. We then examine daily US dollar par–swap rates from July 3, 2000 to October 28, 2016. The data are obtained from the Federal Reserve Bank of St. Louis FRED database. The empirical results show that convexity is more frequently detected in annuity coordinates than under maturity weighting and is concentrated primarily at the front end of the curve. Realized short‐horizon movements are seldom instrument‐parallel, and the gross advantage of convex butterflies is substantially reduced by nonlevel curve movements and transaction costs. The evidence, therefore, supports the use of annuity coordinates for constructing and evaluating swap butterflies, while also showing that the benchmark parallel‐shift scenario should be interpreted as a risk‐management device rather than as a description of typical realized curve dynamics.