Sep 24

Bond convexity

by David Harper, CFA, FRM, CIPM

FRM |

convexConcave2

Learning Outcome (2007 FRM)

  • LO 23.4: Calculate and interpret the convexity of a security, given a change in yield and the resulting change in price.

Bond convexity is the bend in the price/yield curve

A plain-vanilla bond is convex (or upwardly concave): its price/yield curve bends up and away from any tangent line. As below, the purple line is a tangent line to the green price/yield curve:

convexityIllustrated

In risk (in the FRM), we care deeply about price sensitivity. In this case, how sensitive is bond price to changes in yield? If the price/yield curve were linear, this would be an easy question to answer. But the price/yield curve is nonlinear, as illustrated by the green line above. Duration is the slope of the (purple) tangent line. Duration is a first-order derivative. As such, it is a linear approximation to the nonlinear reality.

So we want to know, if yield changes by, say one percent, how much will the bond price change? Graphically, if we move right or left on the x-axis (above), what is the up/down price impact on the y-axis? But the gap between the green price/yield curve (i.e. the true relationship) and the purple duration-based linear approximation reveals duration's problem: it only works well if you zoom in real close and perform tiny changes to the yield. Duration is only locally accurate.

The convexity adjustment "plugs the gap" between the linear duration and the nonlinear price/yield curve. Notice the actual (green) price/yield curve is always above the duration-based (purple) line: the convexity adjustment will always be positive. This is interesting because greater convexity benefits the bond holder under both rising and falling rates!

 

The convexity formula(s)

For FRM candidates, it is good to connect the convexity adjustment to the Taylor series. The Taylor Series is critically useful in general. It allows us to "expand" a simple function (based on derivatives) in order to approximate a complex function. Here is the Taylor Series, copied from the Wikipedia entry:wikiconvexity2

The second term (in yellow) contains a first derivative; in the case of a bond price, this corresponds to duration. The next term (in green) contains a second derivative. This corresponds to convexity. In fact, this term is the convexity adjustment, where f''(a) is the convexity measure.

The helps to explain why convexity has two steps:

  1. Calculate the convexity measure; i.e., the second-order derivative approximation
  2. Plug the convexity measure into the convexity adjustment

 

The convexity measure is given by either of the following (the difference is the '2' in the denominator):

convexityregular

 

The reason it does not matter is, the convexity measure does not mean much on its own. It is plugged into the convexity adjustment. So, the adjustment either scales the 1/2 or does not. So you get to the same place.

 

Jorion's convexity

To make things even more confusing, FRM candidates who use the FRM Handbook get to the convexity a third way. With this formula:

convexityregularjorion

Where D- is the dollar duration, if we shock the yield down; and D+ is the dollar duration if we shock the yield up. But this, too, is another road to the same destination. The example below illustrates all of this.

 

An EditGrid example

Here are the assumptions. Note we need the same four assumptions we need to price a bond (par value, coupon, YTM, and years to maturity). Beyond that, we only need to say how much we are shocking the bond (in this case, we assume 50 basis points).

  • Par: $100
  • Coupon: 3% (1.5% every six months)
  • Yield: 5%
  • Years to maturity: 10 years
  • Shock @ 50 basis points

The left-hand column in this EditGrid spreadsheet contains the assumptions; the right-hand column computes the convexity adjustment. Given a 50 bps shock value, the bond is re-priced at 5.5% (shock up) and 4.5% (shock down). After that, convexity is estimated two ways:

  • A dollar duration is calculated at both yields. Given the dollar durations, the Jorion convexity is a simple matter: 81.26 = (8.57 - 8.16)/(0.005).
  • The other approach applies the full formula above. So, in this case, 40.63 = (80.97 + 88.03 - [2][84.41])/([2][84.41][.005^2])

The second is one-half the first, and since these are convexity measures, they are the same answer when plugged into the convexity adjustment. For example, given a 1% yield change, the convexity adjustment equals 0.41% = (40.63)(1%^2). That is an estimate of the gap between the (linear) tangent line and the (nonlinear) price/yield curve.

EditGrid Spreadsheet by bt/admin.

Comments

  1. Urvija 03 Oct 2007

    Hi David,
    in the formula to calculate duration(grid H7), should we not put a 2 in the denominator, as in the original duration calculation?
    Thanks a lot
    Urvija

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