 # Why is this statement incorrect? (Convexity/Duration)

From CFAi volume 5, page 516, reading 69, problem number 24, statement 2: “Incorporating convexity into the analysis of a non-callable bond’s price changes as interest rates change always results in higher bond price estimates than derived by using only the bond’s duration. This is true whether interest rates increase or decrease.” My assumption here is that when we’re referring to a non-callable bond, we’re implying the positive convexity that we see with an option free bond. Thus, when dealing with positive convexity, the duration being the first derivative (yes, I actually like that definition) will run tangent to the price/yield schedule, and all points above and below the point of intersection (tangent point) are below the price/yield schedule. The answer key on page A-82 says “…Because of convexity, actual prices (i.e. those on the actual pricing curve) will always be above the tangent line. However, duration and convexity adjustments for callable bonds are more complex…” Can someone clarify? Is this another mistake, or am I missing something here?

What’s your question? Bullets/non-callables/option free bonds (whichever term you prefer) exhibit positive convexity. Bonds with embedded options exhibit negative convexity…ie Callables and MBS.

CFAI’s definition is correct… I don’t understand what you’re trying to ask here… link: http://www.schaeffersresearch.com/schaeffersu/advanced/advancedbonds/advancedbonds6.aspx Just something I googled real quick, but notice that using the bond’s duration is like using the straight line and when you incorporate convexity, it accounts for the curving factor. The curve is the actual price of the bond while the line is an estimate derived from duration analysis alone. Note that the true price of a option-free bond is always above price derived from the duration. Edit: Oh wow, how did I end up in the LI forum… I thought I was in General Discussion. I knew it was kinda weird for someone to be asking this in General D…

I stated my question in the title of the thread. The appendix had the answer D, which indicated that the statement above is false. What I’m asking, is the statement really false, or is this an error in the answer key? That’s what I’m asking.

If that’s the question, then I do not see any problem with the first statement. Therefore this might be something you want to email the CFAI about (and perhaps tell them to add it to their errata list).

Yes, the statement “interest rates change always results in higher bond price estimates than derived by using only the bond’s duration. This is true whether interest rates increase or decrease.” is false for the following reason: Just remember that the higher the interest rate goes, the less sensitive the bond price will be when the interest rate changes, and that duration does not take this into account. Therefore, for a given change in interest rate, a decrease in the rate will result in a bigger increase in bond price than a corresponding increase in the rate would result in a decrease in bond price. Consequently, because duration estimate provided by the book does not take into account whether interest rates changes were the result of increase or decrease in rate, just using duration would result in a higher than actual bond price estimate for an increase in interest rate, but a lower than actual bond price estimate for a decrease in interest rate.

doubled Wrote: ------------------------------------------------------- > Therefore, for a given change in interest rate, a > decrease in the rate will result in a bigger > increase in bond price than a corresponding > increase in the rate would result in a decrease in > bond price. Call me crazy, but it sounds to me like you’re actually validating the CFAI quote that you’re attempting to refute, which like others have said, is consistent with my understanding of how the convexity adjustment refines the price estimated via duration. To rephrase your quote in caveman terms I can understand: “Upon a decline in interest rates, bond price will increase by more than duration alone would predict. Upon an increase in interest rates, bond price will decline by less than duration alone would predict.” Therefore, duration alone always underestimates the price of a bullet bond for a given increase/decrease in interest rates. The convexity adjustment should serve to increase the price estimated by duration, delivering us a superior estimate of actual price change. That sound about right? However, I do agree with your other statement that duration is only useful for small changes in yield and that duration changes at various points in the price/yield chart. Anyway, I haven’t touched FI for months, I’ll revisit my books over the weekend. Forgive me if I’ve misinterpreted your post.

ah you are right hiredguns1. The statement should be true.

The statement is false. % price change = - duration effect + convexity effect The opposite signs for the duration and convexity effects should tell you why using duration alone will result in a higher bond price estimate than using the both together.

Nah. The “duration effect” means that bonds with higher duration are more impacted by interest rate changes than lower duration bonds. It’s not really a math statement (and virtually all fixed income instruments drop in value when interest rates rise except stuff like inverse floaters)

Joey, you are right but this is the statement we are disputing or agreeing to: “Incorporating convexity into the analysis of a non-callable bond’s price changes as interest rates change always results in higher bond price estimates than derived by using only the bond’s duration. This is true whether interest rates increase or decrease.” Hence, my reasoning to use the mathematical equation which combines duration and convexity effects on price change. As per my understanding, the equation simply shows that incorporating convexity into the analysis results in lower bond price estimates with changing interest rates, rather than higher price estimates. Oh well, this gives me an added reason to hammer down duration and convexity. There was this one question on the PM sessions which I was totally stumped by.

sweft Wrote: ------------------------------------------------------- > The statement is false. > > % price change = - duration effect + convexity > effect > > The opposite signs for the duration and convexity > effects should tell you why using duration alone > will result in a higher bond price estimate than > using the both together. Sweft, you equation is right. But your conclusion is not. For example, if interest rate increase, duration effect would be negative and convexity effect is positive which means bond price will decline less than considering duration effect only. So considering the convexity effect will lead to higher bond price. I also think the statement is correct unless there are more information to consider. Maybe you can post the other choices as well. Sometimes, it is matter of choosing the best or most suitable answer.

Disptra, Your explanation is excellent. Can it be that the reason the statement is false, is because of the last sentence in the line. “THIS IS TRUE WHETHER INTEREST RATES INCREASE OR DECREASE.” If I read it right, only if interest rates increase would the phrase be correct. However, because the phrase also incorporates interest rate decreases, the phrase is false?

The statement is TRUE unless the posted statement is incomplete.

I think that we’re barking up the worng tree here. The CFA exams aren’t going to trick you with ambiguous wording like “duration effect” which may be positive or negative. The idea is to understand concepts like increased duration means increased interest rate risk, option free bonds always have positive convexity that always accrues to the bondholder, options can cause negative convexity at some interest rates, etc…

Thank you for the responses. To avoid any further confusion, let me post the entire question: 24. Consider the following two statements. Are the two statements correct? Statement 1: “For non-callable bonds, duration provides only a linear approximation of a bond’s price changes as interest rates change.” Statement 2: “Incorporating convexity into the analysis of a non-callable bond’s price changes as interest rates change always results in higher bond price estimates than derived by using only the bond’s duration. This is true whether interest rates increase or decrease.” …Statement 1…Statement 2 A…No…No… B…No…Yes… C…Yes…Yes… D…Yes…No… Back of the book says the answer is D Normally on these binary, yes/no questions, answers A and D are either Yes/Yes or No/No. Maybe the publisher/author accidentally swapped answers C and D and that may have caused the mixup. -g

I seem to remember reading a post about MBS can have negative convexity adjustment. I think call option is not the only thing that changes cash flow. I just googled this. I read part of the paper, it says that prepayment can cause negative convexity. It totally makes sense. When rate declines, prepayment will increase, duration will decline causing negative convexity. It basically has the similar effect as call option.

But an MBS with prepayment option is a callable bond (a prepayment is just a bond call).