Bond Duration and Bond Convexity Explained
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- Опубліковано 21 тра 2024
- Ryan O'Connell, CFA, FRM explains bond duration and bond convexity.
Chapters:
0:00 - Introduction to Bond Duration and Bond Convexity
0:14 - Bond Duration Definition
0:41 - Key Factors Affecting Duration
2:15 - Calculating Macaulay Duration in Excel
4:38 - Plotting Bond Prices based on Duration in Excel
6:11 - Why Bond Convexity is Important
6:43 - Graphing Bond Duration + Convexity
7:50 - Approximate Convexity Formula
8:29 - Change in Bond Price Formula
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*Disclosure: This is not financial advice and should not be taken as such. The information contained in this video is an opinion. Some of the information could be wrong. This channel is owned and operated by Portfolio Constructs LLC. Some of the links above are affiliate links, meaning, at no additional cost to you, I will earn a commission if you click through and make a purchase.
💾 Download Free Excel File:
► Grab the file from this video here: ryanoconnellfinance.com/product/bond-duration-and-convexity-explainer-toolkit/
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► Join me for personalized finance tutoring tailored to your goals: ryanoconnellfinance.com/finance-tutoring/
Excellent video! Interested in the excel file but that link just downloads an image in png format.
@@BlessedGTG Thank you for letting me know, this is fixed now!
I am a chartered accountancy's final level student.i forgot the whole concept of convexity and bond duration,you just brush up my whole concepts in 10 mins! thanks alot,you are great teacher!
Love from India
I really appreciate it! You've given me motivation to make my next video 💪
Hi Ryan, just put my comment here to say thank you for this channel. Outstanding in simplicity of explanation. Great job.
You're very welcome! I really appreciate the positive feedback as it helps me stay motivated 👍
Thanks for this video! I periodically get back here since these concepts are very abstract and you explained it the better way I could find on the Internet.
Thank you so much for that feedback, it means a lot!
Doing Level 3 at the moment. This was the best explanation and visualisation of duration I have seen during the whole time I've been doing CFA. Seriously concise and helpful video. Thank you!
I really appreciate that feedback Cam! Best of luck with Level 3
Guys, Macaulay Duration is not a price sensitivity measure. You cannot use Macaulay duration to calculate price change. It is only same as modified duration when the interest rate is compounded continuously.
Needed a quick revision of the terminologies and this video exactly did the job. Kudos!
Glad it helped Dushyant!
Best explanation of convexity I've ever seen. Well done sir!
Very well explained. Clear and to the point. Well done!
Thank you Robert!
Best video I watched on this topic so far, this chapter of CFA is pretty daunting but this video made it easy. Thanks.
It is a tough topic! I'm glad it helped and really appreciate the feedback
Thank you - this helped me clear up a bit on duration/Convexity on the surface. Will check out more of your videos!
Really appreciate it Alec!
Thank you. This is a brilliant explanation of the bond convexity! 🙏
You're very welcome!
Thank you you’re an amazing teacher
Really appreciate it Madara!
Great video, clear and concise explanation, thanks for this.
Thank you for the positive feedback Rob!
This explanation is very easy to understand 👍
So easy to understand! Thank you for the explanation - better than my uni lectures
You're very welcome!
Absolutely brilliant explanation! 👍 very helpful, Thanks!
Thank you for the kind words!
My dude this video was awesome made a complex subject so simple!
Thank you Rafael, I appreciate the feedback!
I must said this material is really nice, fixed income is an complicated product, but sir you make it easy to comprehend, respect
Thank you for the feedback and I'm really happy to see you are finding this useful
studying for level1 of CFA, and this was super helpful!
Glad it helped Alyssa! Goodluck with the test
Very well explained in simple manner
Thank you!
Thanks for sharing!! you are professor by nature
Wow, thank you!
Great Job, thank you for the explanation !
You are welcome! And thank you
I love this fixed income playlist! Thanks for the videos
My pleasure! Glad you are enjoying it
Helped a lot. Thanks man.
My pleasure Ahmet
Easiest explanation on whole UA-cam. 👍🏻 Could you please also explain the possibility and reason of MBS having negative duration.
Thank you Monika! I will look into that in the future
Thank you! That was helpful
Appreciate the feedback Manar!
Amazing really helped me for the FI part
Really glad it helped you! Best of luck
Awesome videos, Thank you!
Thank you Katarina!
Very understandable! Thanks!
My pleasure!
One important thing to add is one of terminology: in a practical setting people can use duration to mean DV01 which is the change in dollar value of your position for a bp move
Thank you for your job!
It is my pleasure!
thank you. better than all professors😘
Wow, thank you!
Thanks Ryan!
My pleasure!
thank you Ryan!
It's my pleasure!
comment for the support ! great content !
Thank you for the support!
Thanks for sharing ❤
My pleasure!
Phenomenal content
Thank you!
Thank you!
You're welcome Berat!
U look like Sheldon 😂 also the way you are teaching loved it 😅🎉😂😂
Haha this is my first time hearing that! Not sure that is a good thing for me😂 Glad you enjoyed the video
Awesome!
Outstanding video, do you have a video where you leverage Excel to calculate and chart out convexity?
I don't yet but that is a great idea for a future video!
Sir, you are the boss....
Thank you my friend!
Good one 😊
Thanks Amit
Omg, you are very smart ❤
Haha only in this one specific area Manuel!
"If interest rate goes up by i% then bonds value go down by = duration*i%.
Thanks for the video, really helpful!
My pleasure!
this is an awesome video, but I am new to this, may I know is bond duration and macaulay duration are the same?
Liked this video and subscribed to your channel.
Could you cover CDS in detail? From CFA Level 3 perspective
Glad to have you on board! I'll add it to my long list of future videos haha
ty prof
This is superb - thank you. I have a newb question - how does this formula work for bonds bought/sold in the secondary market? Is the market value used instead of par value?
That is correct! You would use the market value instead of the par value
Thank man. Just wondering how do you find a bond with positive convex in the market, from a FI portfolio manager perspective
"A bond is said to have positive convexity if duration rises as the yield declines". Most option free bonds have positive convexity and satisfy this condition to my understanding
Ryan, could you please draft shortly what is the difference between PV01 and Modified Duration? Both are to explain how the price will change due to IR movement.
Hello, I will have a detailed video on modified duration coming up in the next few weeks here.
The modified duration signifies the estimated percentage variation in price for a 1% alteration in yield. On the other hand, DV01 indicates the expected monetary shift in response to a 0.01% change in yield. Hence, DV01 can be represented as (D)(Price)/10,000. It's primarily a matter of units!
To phrase it differently, dollar duration equates to the product of duration and price, whereas DV01 is this dollar duration adjusted by dividing by 10,000.
Hi! I want to ask. Since we are trying to show the sensitivity of a bond's price, why are we showing it in a bond price to ytm graph, and not a bond price to change in interest rate graph? does that mean that the market interest rate change and the ytm change are interchangeable? thanks!
Yes, the market interest rate, the yield to maturity, the required rate of return, the discount rate... all these terms are generally pretty interchangeable in bond pricing
Is the change in YTM in the convexity formula calculated as the difference between V- interest rate and V+ interest rate or is it the difference between V0 interest rate and V+ or V- interest rate?
Hello Dominik, it should be the 2nd option you listed, the "difference between V0 interest rate and V+ or V- interest rate"
studying for actuarial exam fm 🙏
Good luck with the test!
Just curious if it is right to use macauley duration to measure price fluctuations with changing interest rate. Is it not better to use modified duration and if not what is the difference? Thanks.
This is a very good breakdown that will answer all of these questions for you: www.investopedia.com/ask/answers/051415/what-difference-between-macaulay-duration-and-modified-duration.asp
Hello Hope all is well. In the Macaulay duration is usually in years can you explain why in this video you mentioned the 3 duration indicate that the price will decrease by 3% for 1% change in yield. Shouldn't the 3 indicated the weighted average time to maturity? thank you in advance
Hello, it is true of both things and can be interpreted both ways! I have a more comprehensive video on this if you are curious: ua-cam.com/video/2tXjJR1W0YU/v-deo.html
This is clarity
Thanks Julia!
At 4:37, shouldn't you use ModDur to calculate bond price % change? Here the diff is v minor, but just curious
Hello, modified duration is more appropriate than Macaulay duration for this purpose. I have a video breaking out the differences in the two here: ua-cam.com/video/2tXjJR1W0YU/v-deo.html
What's the keyboard shortcut you use to lock in cell values?
F4
tysmmm@@RyanOConnellCFA
@@rahilshah712 My pleasure
Thanks. Shouldn't the Modified duration be the one that determines the relationship between interest rate and bond price.
So modified duration = Macaulay duration / ( 1+ yield to maturity).
So, the relationship is with modified duration and not Macaulay duration.
I was thinking abou the same when watching the video. Although the linear relationship holds in that case as well.
As Balazs said, both formulas for Macaulay Duration and Modified Duration show a linear relationship in which bond price is dependent on interest rates
Hey Ryan, im really struggling with understanding duration completely. If market interest falls, by 1%, and the bonds goes up in value, the yield will become smaller. Does this smaller yield result with the duration increasing? Because if you end up buying these bonds at the current price, the average weighted time to get your investment back is longer compared to before the change in market interest? Or is the duration fixed?
Hey there! Yes, you've got the right idea. Duration measures a bond's sensitivity to interest rate changes, so it's a fixed characteristic that doesn't change with market interest rates. However, when market interest rates fall, causing bond prices to rise, the yield (current income as a percentage of the bond price) decreases. This doesn't directly affect the bond's duration, but it does mean that if you buy the bond at this new higher price, your rate of return until maturity is lower, not because the duration has changed, but because the bond's price and yield have shifted.
How do you calculate duration for bonds purchased between two coupons dates please ?
It would be similar to the calculation that I did in this video expect the weighted average for the time period of each cashflow would be reduced by the amount of the year that has passed. So if it was an annual coupon rate bond, and we are halfway to the next coupon, we would use 0.5 instead of 1
My study of the relation between percentage price change, duration and convexity turns out that this is only the other format of two order Taylor approximation.
If you look at the left side of a positive convex curve, it is more convex than its right side. This means that when the interest rate increases, the price will drop less than it would increase if the interest rate were to fall. So, the left side of the curve has a higher duration than its right side? Is the greater the convexity, the greater the duration?
Hi @luismxavier, that's an insightful observation! Yes, greater convexity can indeed reflect higher duration on the left side of the curve, as it indicates the bond's price is more sensitive to decreases in interest rates than to increases. However, it's crucial to note that while convexity adds to the sensitivity measured by duration, they are distinct concepts; greater convexity doesn't always imply greater duration, but it does mean the bond will exhibit less price volatility for interest rate changes.
@@RyanOConnellCFA thanks.
I agree with you. Greater convexity means that a more convex bond will exhibit higher price than a less convex bond for the same interest rate level.
However, I’ve read on Investopedia, I guess, that for the purpose of risk diversification, in the context of portfolio management, less convexity is preferred to more convexity. I think this is contradictory to the less price volatility of higher convexity. Is it correct?
Hi ryan, everything cool ?? This concept you call as macauly duration for me is called as modified durantion, no ? In my mind Macauly duration is a time measure and based of this you can calculate the modified duration and than using modified duration you have a sensity of the behavior of the price facing a interests changes
Hey Matheus, you are correct that you calculate modified duration using Macauly duration! But what I calculated in the video is Macauly duration. To get the modified duration you would take the value I calculated in the video and then divide it by (1 + (YTM/n))
Question: I'm teaching duration and there's basically two interpretations of Macaulay Duration:
1) The "expected length" of the bond with respect to the proportion of the bond's price that corresponds to the each year (in analogous to the discrete expected value formula from probability)
2) A slightly modified relative rate of change of a bond's price with respect to the yield rate (or spot rates).
Which is better in practice!? Both are correct and make sense to ME but no one addresses which is most important in practice...
You're right that they both are true. I think the first one would make more intuitive sense to a new student but I'm not sure. I'd probably go with whichever version you think your students will find easier to grasp!
@@RyanOConnellCFA Ok thanks for that insight. I'm teaching an Exam FM for Actuaries class so I guess I'll have to ask around.
First of all, many thanks. Your video was excellent, it is greatly appreciated.
In my opinion, even though the Mac duration is what is firstly taught in Finance, I think it always makes sense to make estimations based on the Mod duration instead, since it is literally the derivative of the Price with respect to the yield.
On the other hand, I also understand why you did it for sake of simplicity, since for values very small the factor (1/(1+y)) approximates to 1.
Nevertheless, I think it would have been better to mention your assumptions behind in order to avoid any confusion.
Please do not get me wrong, it is just a constructive comment. Many thanks again, please keep doing some more great content.
this is an excellent question, btw
Thanks! What I ended up doing is making a video deriving Macaulay Duration as an expected time for anyone interested. See ua-cam.com/video/ysVoEgtY2oA/v-deo.html
In my Exam FM prep course I just teach the definition and emphasize that the slight difference between Macaulay Duration and Modified Duration makes formulas for Macaulay Duration a bit simpler (and refer them to the above video).@@frerejacques
At 4:41, I think u were referring to the modified duration rather than Marcaulay
Are you referring to where I said that the duration calculation assumes a linear relationship between interest rates and the bond price change? That should apply to both Macaulay duration and modified duration
Why is bond convexity a thing though? Is it a behavioral phenomenon (after all investors are the ones setting the prices)? Like I still don't understand why the price change is greater if interest rates decrease than if interest rates increase. Thanks Mr. O'Connell.
Hey Ethan, I took this out of an investopedia article so I hope these examples help:
"Higher coupon bonds, for example, tend to have higher convexity than lower coupon bonds because they are more sensitive to changes in interest rates."
Convexity is a measure of the curvature of duration. So why would a bond with higher coupons have higher convexity? Because when calculating the present value of a bond with coupons, its price will be more greatly affected by changes in interest rates.
As for "why the price change is greater if interest rates decrease than if interest rates increase", this is only the case for bonds with positive convexity. There are negative convexity bonds as well which would be the opposite
@@RyanOConnellCFA Thanks Mr. O'Connell!
With the rapidly increasing inflation, what's your prediction on how much bond prices will decrease in the future?
It is a hard question to answer William. There are a lot of different factors involved, including the time to maturity, the coupon rate, and the credit risk of the bond in question
Where can I get the data from?
I made up the data in this video. Are you talking about data for bonds in the market?
Why do we add convexity?? Is it because bond's downside is risk is capped?
Hello! Great question. Convexity is an important concept to consider in bond pricing because it accounts for the curvature in the price-yield relationship. While duration measures the linear relationship between bond price and yield changes, it doesn't fully capture the price sensitivity when there are larger fluctuations in interest rates.
Convexity helps to refine the estimate of price change, especially when interest rates change significantly. The curvature effect results in bonds with higher convexity having less downside risk when interest rates rise, and greater upside potential when interest rates fall. By incorporating convexity, we get a more accurate assessment of a bond's price sensitivity to interest rate changes. I hope this helps clarify the concept!
@@RyanOConnellCFA thanks a lot!
5:12
Could you kindky write ✍️ down the mathematical formula of how did you get here?
What are those $ symbols?😮
Hello! This just means that I am looking in the cell references. Google "lock cell references in Excel" and you will be able to make sense of the formula once you understand that
@@RyanOConnellCFA
Ahh now it makes sense
Only one thing I'd like to add here
0:28
You should have specified that the Bonds duration meant Modified duration here and not Macaulay duration cause people get confused between the two
Btw, the way you explained Macaulay duration as weighted average in another video was astounding 👏
@@gokuvegeta9500 Very good point! I have another video here where I get really granular on the differences between Macaulay Duration and Modified Duration if you are curious here: ua-cam.com/video/MzJihqG2DEA/v-deo.html
There is an error in your convexity formula - denominator factor is: 2*DeltaY^2*V0 instead of DeltaY^2*V0 . Otherwise very clear explanation of the topic. Also why do you use Macaulay duration instead of modified duration?
I used the convexity formula shown in the CFA Level 1 textbooks which is accurate! We could use Modified instead of Macaulay, I wanted to make the video simpler.
He divides it later on by 2 in the second term of the Taylor approximation expression at minute 8:30.
I have also seen in other youtube videos where they divide it by 2 within the same Convexity formula as you mentioned, but later on in the Taylor approximation they just simply multiply the convexity by (delta YTM)^2 and do not divide it by 2 again.
The reason they do this slightly differently, it's that they would like to show the Taylor expression (formula at minute 8:30) as the Modified duration (first expression) plus a correction factor (second expression) Convexity x delta YTM^2, since they already divided by 2 the convexity so they don't need to do it again.
I think many of the ways these exercises are being taught for the "sake of simplicity" are really based on the assumption that the students cannot understand fundamental calculus easily, so they just talk about these sub-components of the formulas in a way students can "copy-paste" them and get easily the results without having to understand too much of the formulas behind, also usually done in the name of "practical application".
However, if you do understand a bit of calculus, think of it simply as the average time to maturity weighted on cash flows (Macaulay duration), the 1st derivative of the Price with respect to the yield (Modified duration), 2nd derivative of Price with respect to the yield (Convexity), and a Taylor series for a more accurate bond price approximation through the curve, where they only use the first 2 terms of the expression for simplicity.
Blessings.
other things equal, why an increase in bond's YTM will decrease its interest rate risk??
Higher coupon rate and time to maturity will decrease the duration and hence the interest rate risk. Think about the weighted average present values in the cash flows that I calculated in excel. The higher the coupon rate, the greater percentage of the overall present value will be pushed into earlier periods, lowering the duration. Does that make sense?
@@RyanOConnellCFAIt makes sense for Coupon Rate. But how about the YTM of the bond, YTM is the discount factor of the cash flow, YTM increase meaning less present value of the coupon can be received....how can the interest rate risk decrease as well.
@@DAVIDJC565 YTM is in the denominator of the Macaulay Duration formula, so changing YTM will change duration. Try downloading the Excel file in the description. Then change the YTM up and down and watch how Macaulay Duration changes. It may be easiier to understand that way
@@RyanOConnellCFA *yield to maturity
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Why do you say coupon two different ways? You say q-pawn rate but zero coupon.
Lol never thought about it until now. I'm from Wisconsin and we pronounce our "A"s and "O"s weird
When the rate is 5%, the bond price should be 1000, not 1001,12.
qpon
I need help understanding WHY bond prices are convex. What makes a bond more convex than another? I can understand the example of a callable bond leading to negative convexity, but what’s the rationale behind positive convexity? Is it cash flow related?
The convexity of a bond price is due to the way its duration changes with interest rate movements; when rates change, the present value of future cash flows adjusts non-linearly, resulting in a convex price-yield relationship. A bond is more convex when its cash flows are distributed further in the future, as these distant payments are more sensitive to interest rate changes, amplifying the non-linear price impact. Positive convexity, seen in most standard bonds, indicates that the bond's price increases by a greater rate as interest rates fall, and decreases at a slower rate when rates rise, reflecting this asymmetric response to rate changes.
Does that help?
We call that as Volatility of a Bond, Price sensitivity
So, if macaulay duration is 6.74, does that mean it’ll take 6.74 years to receive cashflows equivalent to the initial purchase price of the bond?
Hello Erick, I'd say that is about right. Here is the Investopedia definition: "The Macaulay duration is the weighted average term to maturity of the cash flows from a bond. The weight of each cash flow is determined by dividing the present value of the cash flow by the price."