# Convexity problem

Hello,

1. For an option-free bond, what are the effects of the convexity adjustment on the magnitude (absolute value) of the approximate bond price change in response to an increase in yield and to a decrease in yield ?

A. (decrease in yield leads to increase in magnitude) / (increase in yield leads to decrease in magnitude)

B. (decrease in yield leads to increase in magnitude) / (increase in yield leads to increase in magnitude)

B. (decrease in yield leads to decrease in magnitude) / (increase in yield leads to increase in magnitude)

A. Option free bonds have positive convexity and the effect of positive convexity is to increase the magnitude of the price increase when yields fall and to decrease the magnitude of the price decrease when yields rise.

* What i get up to this point is for option free bond, the convexity’s adjustment is to increase the bond price from the approximate duration to the actual value based on the convexity of the bond (inline with the formula of 1/2 * annual convexity * delta(ytm) ^ 2 )

Hi,

I’m not sure whether this will help in understanding the answer explanation:

If you view it mathematically, by incorporating 1st and 2nd order effect we have:

%change in bond price = -AnnDur (ΔYTM) + ½(Convexity)(ΔYTM)2

Case 1: YTM decreases

%change in bond price = - AnnDur ( -ve ) + ½(Convexity)( -ve )2

1st part of the equation return a positive value.

2nd part of the equation return a positive value.

Overall, % change in bond price = \$ A

Case 2: YTM increases

%change in bond price = - AnnDur ( +ve ) + ½(Convexity)( +ve )2

1st part of the equation return a negative value.

2nd part of the equation return a positive value.

Overall, % change in bond price = \$ B

Comparing both case 1 and 2, we will arrive at the following:

A>B

Positive Convexity = bond prices increase faster than they decrease (this creates the curvature of the typical price yield curve) Negative Convexity = bond prices increase at a decreasing rate (for a call option this is because as the price increases and approaches call price, the chances of getting called are greater so price flattens)

I’d encourage you to draw a picture, such as the one I have in this article I wrote on convexity: http://financialexamhelp123.com/convexity/. If you compare the price change using duration alone (red line) to the price change using both duration and convexity (blue line), you’ll see that:

• when the yield decreases, the blue line is above the red line: the change using both duration and convexity is greater than the change using duration alone
• when the yield increases, the blue line is above the red line: the change using borh duration and convexity is less than the change using duration alone

Rates go up, bond prices go down…but postive convexity decreases the magnitude of the bad price move.

Rates go down, bond prices go up…and positive convexity increases the magnitude of the good price move.

thank you all for the responses. i will double check it.

And thanks for this, i really like the background color and font color choosen, its very convenient to read.

i will surely check your web in the future before posting a question in this forum

Thanks! I’m glad that it’s easy on the eyes.

I hope that it’s easy on the brain as well!