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# Duration of a Floating Rate Note

#### girishkhare

##### New Member
Hi David,

In FRM handbook, it is given that the duration of the Floating Rate Note immediately after the rate adjust is zero and the duration in the intermediate period is time left till next rate readjustment. In other words, suppose the note readjusts the coupon based on LIBOR every six months, say on January 1 and July 1. On January 1, the duration of the Floating Rate Note would be zero while the duration on February 1 would be equal to five months.

Duration is the average time one has to wait till the payment is received. If the duration is zero, it would mean that the whole payment should be received immediately. However, this is obviously not the case. I am getting confused on the meaning of zero-duration of a floating rate note. Similarly, if it is February 1 and the duration of the note is five months. Even though the time till maturity is long, say 10 years, how can the duration be equal to 5 months?

Any help on this?

Girish

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
HI Girish,

This fascinated me, also, as I was preparing for Saturday's webinar b/c we reviewed an interest rate swap pricing problem (see #3 at http://www.bionicturtle.com/forum/viewreply/6344/)
… and my sub-question included: what is the swap's duration.
Where I tried to be careful to say "in practice, we round down the duration of the FRN to zero" (i.e., the IRS = long a fixed plus short a FRM)

Starting with your final point: my preferred way of saying Macaulay duration, that I got from Sanjay Nawalkha @ http://www.fixedincomerisk.com/ is "w,eighted average maturity of bond, where weights are PV of cash flows." In which case, the FRN has Mac duration ~ time to next coupon because all of the subsequent cash flows have a weight of zero (as they do not contribute to price). So this is closely related to the idea that a FRN must price at par immediately upon coupon settlement. Some further discussion on this phenomenon and a very brief XLS "proof" here

So, FRN prices at par at the moment of settlement, such that FRN duration approximately (~=) time to next coupon. In the webinar, I said "round down to zero" b/c an FRM question (for example) would typically assume FRN duration = 0, but unless it is immediately before settlement that is strictly incorrect (although my example shows how minimal the error is).

So, I disagree, technically, with "the duration of the Floating Rate Note immediately after the rate adjust is zero" because at that point in time, the next coupon is essentially fixed as the rate has already been determined -- zero duration cannot be true as there is a bit of price risk until the next reset. (it is just a wrong as saying a six month zero fixed coupon bond has 0 duration). The Mac duration is nearer to 0.4xx years (< 0.5 years).
... but this is why I was worried about presenting it: in my example, the FRN duration is nearer to 0.5 but i "rounded down" to zero (and the impact on swap DV01 was only $2). hope that helps, David #### girishkhare ##### New Member Thanks a lot David. BT rocks. #### bluekaktus ##### New Member Hi David, FRNs can also have negative duration too? #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi bluekaktus, In my opinion, no, unless you stretch the definition of FRN. By FRM, i assume a principal repayment such that the instrument prices to par at the next coupon. On the other hand, an interest only (IO) tranche generally does have negative duration: the underlying pool has non negative duration, such that structuring can create inverse floating tranches (of the sort recently in the news about Freddie Mac http://www.propublica.org/article/freddy-mac-mortgage-eisinger-arnold ) and floating tranches. If a two tranche structure can create a tranche with duration greater than underlying average maturity (e.g., the inverse floaters) which is like leveraging duration, then, by definition, the other tranche must have negative duration. Thanks, #### jcb05 ##### New Member I am curious how a margin on top of the floating rate coupon might affect the instrument's duration. For example, if the instrument is priced at a deep discount because of a wide spread. If I had a floating rate instrument with a coupon of 1M libor +350 bps resetting monthly, I have been told I could think about this as a 1M libor floater, which has a duration of .083 years, (1month), in addition to a bond with a fixed 3.5% coupon with the same term. Is that right? My next question is a little more specific, though in the same vein. If i were to have a credit card account with a floating rate coupon and a large spread to the index, and it pays off entirely each month, aka pays no interest, how should I think about the duration? As it is a floater, I would expect the duration to be no longer than the 1 month reset period, but the large spread to the index would add duration as it can be treated as a fixed rate portion. But, since none of the cashflows are being driven by the coupon, because the current balance is paid in full each month, it seems like this should also be ignored. I know this is a pretty specific question, but any help, or resources toward which you could point me, would be greatly appreciated. Thanks! J #### Praveen_India ##### Member HI Girish, This fascinated me, also, as I was preparing for Saturday's webinar b/c we reviewed an interest rate swap pricing problem (see #3 at http://www.bionicturtle.com/forum/viewreply/6344/) … and my sub-question included: what is the swap's duration. Where I tried to be careful to say "in practice, we round down the duration of the FRN to zero" (i.e., the IRS = long a fixed plus short a FRM) Starting with your final point: my preferred way of saying Macaulay duration, that I got from Sanjay Nawalkha @ http://www.fixedincomerisk.com/ is "w,eighted average maturity of bond, where weights are PV of cash flows." In which case, the FRN has Mac duration ~ time to next coupon because all of the subsequent cash flows have a weight of zero (as they do not contribute to price). So this is closely related to the idea that a FRN must price at par immediately upon coupon settlement. Some further discussion on this phenomenon and a very brief XLS "proof" here So, FRN prices at par at the moment of settlement, such that FRN duration approximately (~=) time to next coupon. In the webinar, I said "round down to zero" b/c an FRM question (for example) would typically assume FRN duration = 0, but unless it is immediately before settlement that is strictly incorrect (although my example shows how minimal the error is). So, I disagree, technically, with "the duration of the Floating Rate Note immediately after the rate adjust is zero" because at that point in time, the next coupon is essentially fixed as the rate has already been determined -- zero duration cannot be true as there is a bit of price risk until the next reset. (it is just a wrong as saying a six month zero fixed coupon bond has 0 duration). The Mac duration is nearer to 0.4xx years (< 0.5 years). ... but this is why I was worried about presenting it: in my example, the FRN duration is nearer to 0.5 but i "rounded down" to zero (and the impact on swap DV01 was only$2).

hope that helps, David
Hi David,

As mentioned by Girish in the beginning:

"In FRM handbook, it is given that the duration of the Floating Rate Note immediately after the rate adjust is zero and the duration in the intermediate period is time left till next rate readjustment."

However you seem to have disagreed.

Its true right? because when coupon is reset based on some reference, say LIBOR, there is no interest rate risk, as coupon is matching perfectly with market rates at that particular time. But when market rates start moving after coupon has been set then the interest rate risk sets in and duration is until the next coupon reset.

Thanks,
Praveen

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Praveen_India yes, strictly i disagree with that: if it were true, then at what instant (what moment in time) would the duration shift from zero to "time left till next rate readjustment?" This thought experiment, IMO, shows the fallacy. The duration of the FRN is always time to next reset; so it is converging on zero as the settlement approaches. But as soon as the coupon pays (or really, what we mean is: as soon as the next coupon is determined), then at that moment when the next cash flow is already decided, then interest rate risk is created. So, to me, if (say) it's a semi-annual FRN, the duration is highest immediately after settlement (or at settlement, if you like), when duration is ~ six months, then declining toward zero. Then "snapping back up" to six months at the next coupon, etc. I hope that helps,

#### Praveen_India

##### Member
Yes, true. Thanks for the explanation David.

#### sharman.jamie

##### Member
I'm looking at floating rate bonds and I think you also need to take into account the spread. For instance the duration of a floating rate bond with a spread of 1% over ie libor is equivalent to the duration of any fixed cashflows (ie ones on or before the refix date like you have mentioned) plus the duration of a fixed coupon bond with coupon 1% going out to maturity. It sounds trivial but given rates are so low, this can certainly makes a difference.

#### lRRAngle

##### Member
Hi David,

I know it's been stale for a while but can you answer the question Sharman.Jamie raised?

Thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @lRRAngle I don't think it's a question, but rather a comment. (I can't actually respond necessarily to every comment. The forum is meant to be a collaborative resource). I agree with it, in part, although I have not taken the time to model or research it. The part I disagree with is where sharman appears to count the principal twice. I'm sure this is covered in some text somewhere, but my quick thought is that, for example, If you have a 10-year floater with index (e.g., LIBOR) plus 1.0% margin, then this is equivalent to a floater plus a stream of 1.0% coupons but not again the final principal (which after all is already included to price the floater to par).

I do agree that this margin, to the extent it is additive to the discount rate, impacts the duration and renders it non-negative, but I'm thinking the impact is small (because it's only the incremental coupons). For a 10-year stream at +1.0% on a $100.0 notional, when discounted at 2.0%, I get a PV of$8.98; i.e., a floater + 1.0% might be worth $108.98. Then if i reprice just the +1% stream at 3.0% (i.e. a +100 basis point shock), the value of the stream drops to$8.53. That's only a \$0.45 drop, admittedly which is 0.45 years on a par priced bond. So, in theory, I think I do agree. At the same time, the impact seems to be much less than if we mistakenly treated the spread as its own bond (with principal). I think that's why I believe that I've read in fabozzi somewhere (don't quote me please) that the key assumption, which permits ignoring the index--or really I think it's really rounding down--is a constant spread added to the index.

My quick back of the envelope would tend to justify nullifying this, or perhaps running the calculations to show how this adds maybe less than one year of duration, if the spread is constant. But again, I think the key secondary assumption is that the discount rate is different than (less than) the sum of the index and spread. If we wanted to assume the appropriate discount rate is approximately the index plus the spread (e.g., for risk), then I think we can be back to pricing at par: certainly it is easy to show that, if we discount the variable flows at the same rate used to determine the case flows, we price to par at each coupon. But all in, without doing the research, my instinct is that theoretically this does add something to the duration, under the assumptions. On the other hand, realistically, there is a measure called spread duration (to account for varying spread). Ultimately I would want to model this to be comfortable in how it is treated because I can think of arguments on both sides actually. Thanks,

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#### lRRAngle

##### Member
Great & wow you are fast to respond This makes sense and very interesting. Your answer raised another question for me, what king of instrument would generate to a negative duration? I am sure there are a number of instruments but I am curious what characteristics would make it such.

Many thanks

Staff member
Subscriber

#### sharman.jamie

##### Member
Yes I did intend it as just a comment. Simply because I had been doing some duration calculations and found that the spread should be taken into account (atleast when comparing my own calculations with blombergs calculated oas1 duration).

I'm glad you brought up spread01 (ie risk a credit spread like z spread will change) because that is relevant but not the spread I was referring to here (the fixed coupon in addition to a floating coupon that corporate bonds add on to compensate investors), although I think you knew that! In anycase not my intention to be pedantic or add confusion; because this is not asked in the frm, just something i noticed when dealing with these 'in the wild'.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@sharman.jamie Yes thank you! Good to know that you did take into account the spread on the floating rate note. I'd be very interested in the calculations (in general) that you used, if it's possible to share. I was thinking, as mentioned above, to parse the floater into a (i) a par floater (i.e., cash flows at discount rate plus principal = par) plus (ii) the stream of "spread coupons" without the principal. So on a quick back-of-the-envelope, I was getting about +0.5 years duration on a 1.0% spread over ten years, but i'd be keen to know if there is a better approach. Thank you!

#### sharman.jamie

##### Member
The way I did it was to model the whole bond, including the floating legs and the spread. Then use the numerical approach to blip both yield and libor up and down, adjusting the floating coupons, then reprice to produce the effective duration. I'd be really interested if you had some way to use the numerical mcauley/mod duration formula to produce something similar.

I don't advocate the Bloomberg OAS1 approach at all. It looks like they just put the fixed+floating coupons into the mcauley duration equation as is, and so as such produce a vastly too large duration figure. Basically if all the coupons were fixed. http://www.treasurer.ca.gov/cdiac/webinars/2012/20120215/presentation.pdf.

#### sharman.jamie

##### Member
The way I did it was to model the whole bond, including the floating legs and the spread. Then use the numerical approach to blip both yield and libor up and down, adjusting the floating coupons, then reprice to produce the effective duration. I'd be really interested if you had some way to use the numerical mcauley/mod duration formula to produce something similar.

I don't advocate the Bloomberg OAS1 approach at all. It looks like they just put the fixed+floating coupons into the mcauley duration equation as is, and so as such produce a vastly too large duration figure. Basically if all the coupons were fixed. http://www.treasurer.ca.gov/cdiac/webinars/2012/20120215/presentation.pdf.
Also meant to add, you can see this effect on US172967KC44 (If you have access to a terminal)
Number of years to refix date = 0.13
Numerical Duration = 0.14 (ie Slightly larger because of the 1.31% spread)
BBG OAS1/Mod Duration =2.48

#### jjking

##### New Member
Hi @David Harper CFA FRM

Would you please be able to redo this example with inverse floaters?

Such as: a 3-year inverse floater with semi-annual coupons. So in the instance of a direct floater, the duration resets to 6 months at each coupon.

What would the likely duration be for each of these over time? Would it increase as the settlement approached?

Best Wishes

Hi @Praveen_India yes, strictly i disagree with that: if it were true, then at what instant (what moment in time) would the duration shift from zero to "time left till next rate readjustment?" This thought experiment, IMO, shows the fallacy. The duration of the FRN is always time to next reset; so it is converging on zero as the settlement approaches. But as soon as the coupon pays (or really, what we mean is: as soon as the next coupon is determined), then at that moment when the next cash flow is already decided, then interest rate risk is created. So, to me, if (say) it's a semi-annual FRN, the duration is highest immediately after settlement (or at settlement, if you like), when duration is ~ six months, then declining toward zero. Then "snapping back up" to six months at the next coupon, etc. I hope that helps,

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#### Nicole Seaman

Staff member
Subscriber
Hi @David Harper CFA FRM

Would you please be able to redo this example with inverse floaters?

Such as: a 3-year inverse floater with semi-annual coupons. So in the instance of a direct floater, the duration resets to 6 months at each coupon.

What would the likely duration be for each of these over time? Would it increase as the settlement approached?

Best Wishes
Hello @jjking

As the forum is getting extremely busy before the exam, I just wanted to recommend using the search function here in the forum. A search of "inverse floaters" brings up a great deal of information that has already been discussed in the forum. If you do not find answers, I'm sure someone will be able to help you here. David's time becomes stretched VERY thin right before the exam, so I want to make sure that everyone is utilizing the search function before asking questions.

Thank you,

Nicole