Question

Using the parametric form of convexity and the example of a two-year coupon bond, coupons being paid at the end of each year, show that the convexity of a bond is a decreasing function of coupon rate.

Answer 1

Solution =

Convexity of a Bond is a measure that shows the relationship between bond price and Bond yield, i.e., the change in the duration of the bond due to a change in the rate of interest, which helps a risk management tool to measure and manage the portfolio’s exposure to interest rate risk and risk of loss of expectation

Example,

For a Bond of Face Value USD1,000 with a semi-annual coupon of 8.0% and a yield of 10% and 6 years to maturity and a present price of 911.37, the duration is 4.82 years, the modified duration is 4.59 and the calculation for Convexity would be:

Annual Convexity : Semi-Annual Convexity/ 4= 26.2643Semi Annual Convexity : 105.0573

In the above example, a convexity of 26.2643 can be used to predict the price change for a 1% change in yield would be:

If the only modified duration is used:

Change in price =   – Modified Duration *Change in yield

Change in price for 1% increase in yield = ( – 4.59*1%) = -4.59%

So the price would decrease by 41.83

To accommodate the convex shape of the graph the change in price formula changes to:

Change in price = [–Modified Duration *Change in yield] +[1/2 * Convexity*(change in yield)2]

Change in price for 1% increase in yield = [-4.59*1 %] + [1/2 *26.2643* 1%] = -4.46%  

So the price would decrease by only 40.64 instead of 41.83