Yield-Based Bond Convexity and Portfolio Properties

While duration is a linear approximation of the sensitivity of a bond’s price to changes in yield, the true relationship between a bond’s price and its yield-to-maturity is a curved (convex) line. We introduce convexity as a complementary risk measure to improve bond price change estimates based on modified duration alone to account for this non-linear relationship. The convexity adjustment becomes more important when considering larger moves in yield-to-maturity and longer-maturity bonds. These lessons will also show how to estimate duration and convexity for a portfolio of bonds, as well as highlight limitations due to underlying assumptions.

Most of the examples and exhibits used throughout the reading can be downloaded as a Microsoft Excel workbook. Each worksheet in the workbook is labeled with the corresponding example or exhibit number in the text.
 

• Convexity is a complementary risk metric that measures the second-order (non-linear) effect of yield changes on price for an option-free fixed-rate bond. The convexity adjustment adds to the linear price estimate provided by modified duration.Convexity is always positive for an option-free fixed-rate bond, such that estimated price increases from a decline in yields are higher than duration alone would suggest and estimated price decreases from an increase in yields are lower than duration alone would suggest. T herefore, convexity is valuable to investors.
• Convexity has the same relationship with bond features as duration: a f ixed-rate bond will have greater convexity the longer its time-to-maturity, the lower its coupon rate, and the lower its yield-to-maturity.
• Money convexity expresses convexity in terms of currency units or percent of par for a position in a bond because it is the product of a bond’s annual convexity and its full price.
• Portfolio duration and convexity can be calculated (1) as the weighted average of time to receipt of the aggregate cash flows or (2) by using the weighted averages of the durations and convexities of the individual bonds that make up the portfolio.
• While the first method is theoretically correct, it is difficult to use in practice. The second method is commonly used by portfolio managers but implicitly assumes parallel shifts in the yield curve, which are rare.