 # Convexity Perplexity

What is the combined Duration and Convexity effect in the following situation: Duration = 4.0 Convexity = -1.6 What is the percentage price change for a 150 bp increase in rates? I think it is the following: = (4.0 X (-150/100)) + (-1.6(-.015)^2 )*100 = -6.0 + -0.036 = -6.036

Edit: Your duration part is incorrect. It should be (-0.015*Duration)

depends on your definition of “convexity effect”. If you think it is the same as “convexity”, then your solution would be right (except the duration mistake pointed out above). However, if you think “convexity effect” is Convexity*(percentage change in rates)^2, then you wouldn’t be right.

if the “convexity effect” given a 150 bps increase in interest rates is given as -120bps, and the the the duration is given as as 300bps, i believe the calculation would be -120bps + -(300*1.5) = -570bps. someone please tell me if this is incorrect. thanks

jackoliver Wrote: ------------------------------------------------------- > What is the combined Duration and Convexity effect > in the following situation: > > Duration = 4.0 > Convexity = -1.6 > > What is the percentage price change for a 150 bp > increase in rates? > > I think it is the following: > > = (4.0 X (-150/100)) + (-1.6(-.015)^2 )*100 > = -6.0 + -0.036 > = -6.036 Answer came to -5.7

It is [duration + convexity] and then you flip the sign of the result to reflect the opposite direction of movement of rates. So that would give you =(.04 * 1.5) + [(-1.6)(.015^2)(100)] =.06 + (-.036) =.024 or 2.4% Since rates are going up, price is going down, which yields a -2.4% change in price. Also note that the duration is the percentage change in price for a 100bp movement. When it says 4, that means 4% movement for 100bp movement.

I think I confused having to calculatate the convexity effect versus being given the convexity effect. Until recently I had never seen a convention where Convexity effect of -1.2 means the % change. Thanks all for the clarification.

I’m glad I guessed right, I checked that question multiple times and never got either of the negative answers (-5.7 and -3.3 I think)! I took the Dec 1 version.

i always thought about conv as a positive number… but when it is negative it think you should do like this: % price change ( duration) + ( - “convexity adjustment” )

zero, my understanding is that if convexity is negative and there is a positive change in rates (increase in rates), the effect from the change in convexity should be negative and thus increase the absolute value of the change (not decreasing, as you have indicated) in the price of the bond to a value lower than if convexity was zero. furthermore, in the event of a decrease in interest rates, the negative convexity would mute the absolute value of the positive price change of the duration effect and thus lead to a lower bond price than if convexity was zero. in both cases (increase and decrease in interest rates), negative convexity leads to a lower price level than a bond with zero convexity. someone please let me know if i am incorrect in these assumptions. thanks

I didnt use any calculations. I simply assumed that due to a bond price having positive convexity, than as the yield increased, the bond price would decrease…at a decreasing rate…and that the convexity would adjust the price upward. However, given the negative convexity factor, does this prove that finance03^ is correct? Could this have been a callable bond at a low yield hence, the convexity would have been negative?

Finance03 is right. I was looking too much at the equation jackoliver wrote and not enough at the problem. If convexity is negative, it is bending the linear duration approximation down to meet the curve, so to speak. This would lead to a lower actual price than what duration alone would give you regardless of the direction of yield movement. … and the opposite is true for positive convexity (i.e. bending up). Thanks for noticing that, finance03.

finance03 Wrote: ------------------------------------------------------- > if the “convexity effect” given a 150 bps increase > in interest rates is given as -120bps, and the the > the duration is given as as 300bps, i believe the > calculation would be -120bps + > -(300*1.5) = -570bps. someone please tell me if > this is incorrect. thanks This is correct, right?

Yeh, I’m still confused? I marked -5.7 too. What is the correct answer then? 2.4 or -5.7? (btw, what exam code was this from 1010/1111? Bcs I think 1010/1111 had a convexity adjustment of -1.2)

I think the -1.2 means that the total part of the convexity effect was -1.2. I came to this conclusion because the the 4 possible choices where 4.5% -1.2% =3.3 4.5%+1.2% =5.7 -4.5%-1.2%=-5.7 -4.5%+1.2%=-3.3% For a option free bond, convexity should always be +. In this case, it was negative which means it was a callable bond. Anyhow, I got -5.7%

i thought callable bonds only exibit negatice convexity at low interest rates

“i thought callable bonds only exibit negatice convexity at low interest rates” yes u are right but when you are moving in negative callable convexity area in both sides (rates increasing, rates decreasing), then convexity adjustment is still negative. The conxity is positive from a special breakeven point, which is similar to option free bond

never seen a q w/ negative convexity adjustment like that b4… gave me trouble…

completly agree with LongOnCFA. Negative convexity arises when market yields fall under the critical yield y`and bond prices become increase at lower pace. So, it´s a callable bond and convexity effect must be substract after duration effect. the correct one is -4.5%-1.2%=-5.7% for me too.

It would not be a call feature in this bond that would cause it to exhibit negative convexity given rising interest rates. As stated above, a callable bond would exhibit negative convexity when interest rates fall. With this problem, interest rates are rising and the calculation above depicts a price change greater than with effective duration alone. This is typical of a MBS. As interest rates rise, the prepayments on the securities slow down and extend the cash flows of the asset which results in larger than expected decreases in the price than from ED alone.