Bond Convexity

in this video we will talk about convexity of a bond we will make use of some of the concepts covered in videos on duration the price you curve shown in this graph shows a nonlinear relationship between the price of a bond and its yield the gradient of this straight line that's tangent to the price yield curve at yield y bar is called duration and it's used to approximate the change in bond price in response to change at interest rate take for example the change in yield from y bar to Y 1 since duration is a linear measure it will use this straight line that's changing to the price youth curve to estimate the new price of the bond of P 1 at the new yield y 1 but because bond price and yield do not have a linear relationship there's some measurement error in the price estimated using duration and the actual price at the new yield this measurement error occurs because Jewish it is unable to capture the nonlinear relationship between the price of a bond and its yield so to better approximate change in bond price due to changing yield we need to know the convexity measure that will help us capture that part of price change that duration is unable to capture let convexity be denoted by C then mathematically convexity can be calculated as follows the price of a bond if you'v goes up plus the price of a bond if yield goes down minus 2 times the initial price of a bond divided by 2 times the initial price of a bond times changing yield square let's take an example consider a two-year 8% coupon bond where coupons are paid semi-annually and face value of this bond equals $1,000 and yield to maturity this bond is 10% so the current price of this bond equals nine sixty four point five four dollars and the modified duration of this bond denoted by D with a subscript M or D equals one point seven nine five five we know that modified duration measures bonds approximate price sensitivity to changes in interest rates so if yield increases by one percent bond price is expected to come down by one point seven nine five five percent and if you decreases by one percent price is expected to increase by approximately one point seven nine five five percent let's check how accurately duration measures this percentage change in price if yield goes up from 10% to 11% the new price of the bond goes down to nine forty seven point four two dollars this implies a percentage change in price equal to the new price of nine forty seven point four two dollars minus the original price of nine sixty four point five four dollars divided by the original price of nine sixty four point five four dollars and this equals negative one point seven five percent we're showing this to three decimal places because we want to highlight the impact of convexity adjustment later on on the other hand if yield decreases from 10 percent to 9 percent the price of a bond goes up to nine eighty two point zero six dollars this corresponds to a percentage change in price equal to the new price of nine eighty two point zero six dollars minus the original price of nine sixty four point five four dollars divided by the original price of nine sixty four point five four dollars which equals one point eight one seven percent so while duration does a reasonable job at estimating the percentage change in price there's still some measurement error for example duration predicted that if you goes up from 10% to 11% the price of the bond will come down by one point seven nine five five percent whereas in reality it came down by one point seven seventy five percent on the other hand duration predicted that if you goes down from 10 percent to nine percent price of the bond will go up by one point seven nine five five percent whereas in reality the bond price went up by one point eight one seven percent it is this measurement error that can be reduced by convexity measure and in our example convexity equals the price of a bond if yield goes up from 10 percent to 11 percent which is nine forty seven point four two dollars plus the price of a bond if yield goes down from 10 percent to nine percent which is 980 2.0 $6 minus 2 times the initial price of 960 4.5 $4 divided by 2 times the initial price of 960 4.5 $4 times the change in yield which is 1% in our example square and this equals two point zero nine so in addition to duration estimated change the convexity adjustment that we have to make equals convexity times the change in yield square which in our example equals convexity of two point zero nine times the change in yield which is one percent square which equals zero point zero two zero nine percent so let's check how this convexity adjustment improves our estimation of percentage change in price when heel goes up from 10% to 11% generation estimates that bond price will change by negative one point seven nine five five percent but there's a convexity adjustment to that equal to zero point zero two zero nine percent which brings the total price change equal to negative one point seven seven five percent this is exactly equal to the actual change of negative one point seven seven five percent similarly when yield decreases from ten percent to 9 percent duration would predict that one price would go up by one point seven nine five 5% and adding the convexity adjustment to that we get the total percentage change in price equal to one point eight one six percent this change is very close to the actual percentage price change of one point eight one seven percent so we can see how convexity adjustment has improved our estimation of percentage change in price notice how convexity adjustment is positive in both cases whether yields go up or they go down which is consistent with the fact that a non callable bond has positive convexity in other words we know that any bond price calculated using duration alone would be estimated using this straight Cajun line since this line lies below the price yield curve if you changes from Y bar any price estimated using this line will be less than the actual price of the bond at that you yield so the benefit of positive convexity adjustment is that it adds to the new price and reduces the measurement error that we get using duration alone so we have learned how to calculate convexity of a bond and how convexity adjustment is applied to capture that part of bonds price change that duration is unable to capture if there are any questions or comments please feel free to post

Channel: finCampus Lecture Hall

Views: 95,160

Rating: 4.8296838 out of 5

Keywords: Bond Convexity, Convexity, Duration, Yield, Yield Curve, Interest Rates, Fixed Income

Id: pOpaTmLO-lE

Length: 10min 22sec (622 seconds)

Published: Sat May 25 2013