Duration and Convexity

Convexity and Duration 

When interests rate rise, it hurts fixed income, but does not punish bonds. We shorten duration if interest rates go up and vice versa. Duration helps to figure out bond price. We use duration to increase gain and reduce loss. Duration is a better description than maturity, because it let’s us know how fast we can get our money back.

How does price respond to a change in interest rates? When yield goes up, price goes down. Convexity helps us to manage this change better. Convexity is always positive.

For any given bond a graph of the relationship between price and yield is convex. This means that the graph forms a curve rather than a straight-line (linear). The degree to which the graph is curved shows how much a bond's yield changes in response to a change in price. In this class we will take a look at what affects convexity and how we can use it to compare bonds. 

If we graph a tangent at a particular price of the bond (touching a point on the curved price-yield curve), the linear tangent is the bond's duration, which is shown in graph below. The exact point where the two lines touch represents Macaulay duration. Modified duration, as we saw in the previous class, must be used to measure how duration is affected by changes in interest rates. But modified duration does not account for large changes in price. If we were to use duration to estimate the price resulting from a significant change in yield, the estimation would be inaccurate. The dark portions of the graph show the ranges in which using duration for estimating price would be inappropriate. 

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Furthermore, as yield moves further from Y*, the dark space between the actual bond price and the prices estimated by duration (tangent line) increases. 

The convexity calculation, therefore, accounts for the inaccuracies of...