Convexity, duration, price change for a bond: Question

For a 25bps change in yield, what is the approx price change for a bond with the following characteristics: Market price= 95% of par ED=MD=8.5 Effective CV=130 Doubt: 25bps looks like a very small change and can be assumed to be an almost parallel shift. Then, why do we have to take CV into account also. Why can’t we just ignore it?

25 bps is a huge change, 1 bps is called a small change. In Eurodollar 25bps change for 1 lot is $625. And in your example it will be $0.03 (for $100 Par) change and will run into thousands as one bond usually cost $1mn above and not $100 :slight_smile:

So, there’s that! What do I have to do to become like you?

P.S. on a serious note.

I’m not sure what you mean by “an almost parallel shift”. You have a 25bp shift in the yield for this maturity; they’ve told you nothing about the rest of the yield curve.

I think you’re confusing the idea of a parallel vs. nonparallel shift with the idea of a small vs. large yield change. Two completely different ideas.

In any case, if they give you both duration and convexity, they expect you to use both duration and convexity. So . . . throw these numbers into your formula for ΔP. Bob’s your uncle.

A small change in the yield will lead to a small movement on the pruice yield curve. This small movement can be approximated as movement along a straight line both to the left and to the right for an increase and a decrease in yield respectively. So, it is “an almost parallel shift” in the yield.

Probably yes. It is difficult to internalize this.

just wanted to make sure, I understand the concept well.

So, you are a Briton? Just curious to know who I had been talking to for a while on AF :stuck_out_tongue: I gotta go, get some sleep :slight_smile:

The term “parallel shift” applies to the entire yield curve: all yields change by the same amount. It could be 2bp, or 20bp, or -200bp. The magnitude doesn’t matter; what matters is that it’s the same everywhere.

The only thing we know about this 25bp shift is that it happens to the YTM for this bond (whose maturity we don’t know, though it’s probably about 10 years). We don’t know the change for a 2-year YTM, the 5-year YTM, the 20-year YTM, and so on. It would be a parallel shift if all of them changed 25bp, but we don’t know that that’s the case. Fortunately, we don’t care; we care only about the yield change for our bond.

The small movement left and right you’re discussing is a movement along the price vs. yield curve for this bond. That’s not the curve under consideration when someone refers to a parallel shift; they’re talking about the yield vs. maturity curve (more specifically: the par yield curve). Please make sure that you have these ideas straight in your mind.

When people calculate modified or effective duration (and convexity), they start by assuming a certain change in yield (Δy) to get P– and P+. Common values for Δy are 25bp, 50bp, and 100bp.

When you are using the calculated values for modified or effective duration to determine a price change for a given change in yield, you don’t need to know whether your change in yield is small or large; what you need to know is whether or not it is close to the value used to compute the duration. If it’s close, you likely don’t need to use the convexity adjustment; if it’s not, you like will need to use the convexity adjustment. So, if they used 25bp and you use 20bp, you probably don’t need convexity; if they use 50bp and you use 20bp, you probably do. As I said, when CFA Institute gives you both the duration and the convexity, they expect you to use both.

Nope: born and reared in Southern California. Scottish, English, Czech, and Cherokee (at least) by blood; I probably identify more with Scottish than anything.

As for the expression, I pick them up all the time; pronunciations as well. I’ve been accused of being a Briton, a Canuck, and an Aussie (and maybe more).

Just a simple investment analyst, teacher, magician, warhead designer, and equestrian.


haha. Thanks :smiley:

Hi S2000magician,

After reading your good article about KRD (snapshot is as follows):

I have a question that why you emphasize we use _ par _ rate?

I know that when calculating macauley or modified duration, we need to use YTM. But par curve is only the YTM for _ par bonds _, and the bonds we are calculating duration are not necessarily par bonds.

Can you explain it further? Thanks in advance.

I emphasize it for two reasons:

  • It’s true
  • Many people get it wrong

The yield shift used to compute a key-rate duration is a shift at one specific maturity on the par curve, keeping all other par rates unchanged. This results in a change in the spot curve at that maturity and all longer maturities (and, for whatever it’s worth, a change in the forward curve at that maturity and all longer maturities), so it has implications for the prices of bonds at maturities other than (specifically, longer than) the maturity at which the par yield is changed, and some of those implications are unusual (e.g., discount bonds having negative key-rate durations for key rates shorter than their maturity).

Many people incorrectly teach that a key-rate duration is computed by changing the spot rate at the chosen maturity; they’re wrong, but they do it anyway. Someone has to clean up their mess; I’ve decided to do my part to help in the clean-up.

Can you share the link to that article about KRD.

Here you go:

(Full disclosure: as of 4/25/16, there is a charge to read the articles on my website. You can get an idea of the quality of the articles by looking at the free samples here:

Thanks for your reply.

Now, I have 2 questions:

  1. Macauley duration & modified duration use YTM in the denominator △y;

approximate modified duration & effective duration use benchmark curve — i.e. par rate in the denominator △y.

What do these relate to parallel or non-parallel shift ?

  1. Approximate modified duration is a type of ’ yield duration’;

while effective duration is a type of ’ curve duration’.

How to explain the difference between them?


Par rate and YTM are the same thing.

If you’re talking about a single bond, it doesn’t matter whether the shift is parallel or not; all that matters is the change at that bond’s maturity.

If you’re talking about a portfolio of bonds with several different maturities, then Macaulay, modified, and effective duration each assume a parallel shift of the par curve.

Modified duration assumes that cash flows _ do not _ change when YTM changes.

Effective duration allows that cash flows may change when YTM changes.

I have no idea what the distinction between “yield” duration and “curve” duration is supposed to mean.

You’re welcome.

I got it.

Question: Do we assume that we hold the bond to maturity here in the context of Macaulay, modified, and effective duration?

Good to hear.

For Macaulay duration – which is a weighted-average time to receipt of cash flows – yes; we’re assuming that we receive the cash flows as promised by the bond issuer, so we’re assuming that we don’t sell it early.

For modified and effective duration – which measure the price sensitivity of the bond to a change in its YTM – no; those price changes occur in the market, and the market doesn’t know (or care) whether we intend to hold the bond to maturity or not.

I got it.

Whether bond will be called back / put back depends on market.

Further question:

According to your statement about modified and effective duration above, is the following opinion of my own understanding right?

Because option-embedded bonds’ future CFs may change -->

YTM of bond itself is no more applicable -->

so we turn to use YTM of par bond, i.e. par rate -->

then par rate of maturities other than the bond’s maturity might also impact the bond’s CFs (other than YTM of specific bond — only the YTM of the same maturity as the bond will have impact on the bond’s value) -->

thus impact the bond’s value -->

thus if we designate △y = 1%, it just indicates that we assume a parallel upward shift of 1%, which is not practical in real world?

Thanks :slightly_smiling_face:

Hi All

Would really appreciate if someone could please explain why we divide convexity by (period)2 to get annualised convexity instead of just convexity/periods. Thanks !

The units on convexity are years2.

Sounds about right.

You’re now talking about what’s known as key rate duration (i.e., how does the price of a bond change when one par rate changes and all other par rates remain unchanged), which is a Level II topic.

If you cannot control your curiosity, check out the article I wrote on key rate duration: However, when you’ve finished it, get back to studying your Level I material. :wink:

(Full disclosure: as of 4/25/16, there is a charge to read the articles on my website. You can get an idea of the quality of the articles by looking at the free samples here: