Estimating price changes with duration and convexity

CFA Level I Fixed Income
#1

Change in full bond price = -annual modified duration (change in YTM) + 0.5(annual conexity)(change in YTM squared)

8% bond with a full price of $908 and a YTM of 9%.

Bonds duration is 9.42.

Convexity is 68.33.

Does anyone know if I want to calculate the change in price for:

a) 1% increase in YTM to 10%

b) 1% decrease in YTM to 8%

-9.42 x 0.01 = -0.0942 (duration effect)

0.5 x 68.33 x 0.01(squared) = 0.0034165 (convexity effect)

-0.0942 + 0.0034165 = -9.07835% (expected change in bond price)

so a 1% increase in YTM will lead to a fall in price of -9.07835%?

Thus, the bond priced at $908 will fall to $825.57 (decrease of 82.43)

so a 1% decrease in YTM will lead to a rise in price of +9.76165%? (subtracting the convexity effect instead of adding? - is that correct)?

Thus, the bond priced at $908 will rise to $996.64 (increase of 88.64)

I feel like I am going wrong some where here…

Does anyone know what the correct answers are? and can elaborate on what I am doing wrong please?

#2

You’re doing it right. If you’re confused because the 1% decrease in YTM creates a larger percentage change in price than the 1% increase, it’s because of the convexity effect.

An equal decrease in YTM produces a greater price effect than an equal increase because the curveature of the price to YTM relationship.

#3

You add the convexity effect in both cases.

#4

Hi guys, thanks for getting back so quick!

@S2000magician - With regard to the 1% decrease in YTM (to 8%) - when I add the convexity effect, I get an estimate of $990.43 which is not as accurate as the estimate from just ModDur?

In the schweser book, p.96:

Actual Figures:

1% increase in YTM (to 10%): $822.47

1% decrease in YTM (to 8%): $1,000

Modified Duration:

1% increase in YTM (to 10%): $828.41

1% decrease in YTM (to 8%): $993.53

Modified Duration with convexity adjustment (my calculations)

1% increase in YTM (to 10%): $825.57 (CLOSER than ModDur)

1% decrease in YTM (to 8%): $990.43 (FURTHER AWAY than ModDur) - that can’t be right? - that’s why I subtracted the convexity effect because it gave a closer answer?

#5

You’ve done something wrong in your duration/convexity calculation when Δy = -1%; it should be $996.64. Recheck your work.

#6

Hi Onestar,

You add the convexity effect in both cases.

Where the yield decreases the calc is

-9.42 x -0.01 = 0.0942 the duration effect

Convexity effect is the same as the rise in yield example as the squaring of the yield term negates the negative yield growth figure. ie 0.5 x 68.33 x -0.01(squared) = 0.0034165 (convexity effect)

Therefore change in price = 0.0942 + 0.0034165 = 0.0976165 0r 9.76165%

If you increase the 908 by this you get the 996.64 you are looking for.

I think what has happened is you used the yield change as 0.01 and not -0.01 in the duration calculation which will give you exactly the same % change as for the increase in yield. However because you know conceptually that the decrease in yield will add to a price rise you have added this percentage rather than taken it away which is why you have got the price of 990.42. This is also why subtracting the convexity effect when the yield decreases has got you a closer answer than adding, because the sign for the duration effect is incorrect. I know this because I used to do the same.

If you have anything else you can get me at freecfa@outlook.com or follow me on twitter on @freecfa

#7

Thank you very helpful!