Bond fund Maturity vs Duration

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bertilak
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Bond fund Maturity vs Duration

I think I know what maturity and duration are, but just to be sure:
• Maturity: The time between a bond's issue date and the date its principle is paid back.
• Duration: A complex calculation representing a bond's sensitivity to chnges in interest rates. I think it can be approximated by the remaining time until the maturity date.
Other than clarifying or correcting the above, my question is, when looking at a bond fund's vital stats, what is the significance of the difference between the two?

For example here is what Morningstar reports about VBTLX (Vanguard's Total Bond fund)

Code: Select all

``````Avg Eff Duration	5.99 Yrs
Avg Eff Maturity	8.20 Yrs``````
What would it mean if, for example, maturity was as reported (8.20 Yrs) but Duration was 4.99 Yrs?
May neither drought nor rain nor blizzard disturb the joy juice in your gizzard. -- Squire Omar Barker, the Cowboy Poet

dkturner
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Re: Bond fund Maturity vs Duration

bertilak wrote:I think I know what maturity and duration are, but just to be sure:
• Maturity: The time between a bond's issue date and the date its principle is paid back.
• Duration: A complex calculation representing a bond's sensitivity to chnges in interest rates. I think it can be approximated by the remaining time until the maturity date.
Other than clarifying or correcting the above, my question is, when looking at a bond fund's vital stats, what is the significance of the difference between the two?

For example here is what Morningstar reports about VBTLX (Vanguard's Total Bond fund)

Code: Select all

``````Avg Eff Duration	5.99 Yrs
Avg Eff Maturity	8.20 Yrs``````
What would it mean if, for example, maturity was as reported (8.20 Yrs) but Duration was 4.99 Yrs?
Once you get your head around the concepts of maturity and duration you have to factor in the impact of principal prepayments and call features. Problem is different sources of maturity/duration statistics calculate differently.

bertilak
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Re: Bond fund Maturity vs Duration

dkturner wrote:
bertilak wrote:I think I know what maturity and duration are, but just to be sure:
• Maturity: The time between a bond's issue date and the date its principle is paid back.
• Duration: A complex calculation representing a bond's sensitivity to chnges in interest rates. I think it can be approximated by the remaining time until the maturity date.
Other than clarifying or correcting the above, my question is, when looking at a bond fund's vital stats, what is the significance of the difference between the two?

For example here is what Morningstar reports about VBTLX (Vanguard's Total Bond fund)

Code: Select all

``````Avg Eff Duration	5.99 Yrs
Avg Eff Maturity	8.20 Yrs``````
What would it mean if, for example, maturity was as reported (8.20 Yrs) but Duration was 4.99 Yrs?
Once you get your head around the concepts of maturity and duration you have to factor in the impact of principal prepayments and call features. Problem is different sources of maturity/duration statistics calculate differently.
Ah, yes. I can see how a callable bond has extra risk which decreases as the maturity date approaches. I never heard of princilpe prepayment but can guess that it is a kind of partial call.
May neither drought nor rain nor blizzard disturb the joy juice in your gizzard. -- Squire Omar Barker, the Cowboy Poet

alex_686
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Re: Bond fund Maturity vs Duration

There are different types of duration out there.

There is "Macaulay Duration" which is what you see on a fund's fact sheet. It is the time weighted cash flows of a bond. For example a 10 year zero coupon bond would have a duration of 10. A 10 year coupon bond would have a duration closer to 9 to account for the coupons. The formula for this one is dead simple.

There is "Modified Duration" which measures price sensitivity. The formula is a little bit more complex and requires you to input certain assumptions. The SEC does not like funds using imputed assumptions so you will never see this number even though it is the more important number in my opinion.

Assuming a flat yield curve, for option free bonds (no calls, puts, etc.) the duration for these 2 would be the same. Most of the time they are close.
bertilak wrote:I never heard of princilpe prepayment but can guess that it is a kind of partial call.
These are most commonly associated with mortgage and other asset back bonds. Each payment has a portion of interest and principle.

alex_686
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Re: Bond fund Maturity vs Duration

bertilak wrote:For example here is what Morningstar reports about VBTLX (Vanguard's Total Bond fund)

Code: Select all

``````Avg Eff Duration	5.99 Yrs
Avg Eff Maturity	8.20 Yrs``````
What would it mean if, for example, maturity was as reported (8.20 Yrs) but Duration was 4.99 Yrs?
Strictly speaking, it would mean the cash flows would come earlier to you. Bigger coupons, implying premium bonds. Maybe more MBS, bonds with sinking funds, etc. Honestly, I don't pay much attention to Maturity.

Doc
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Re: Bond fund Maturity vs Duration

Looking at what Alex said from a different prospective:

"There is "Modified Duration" which measures price sensitivity."

"There is "Macaulay Duration" which is what you see on a fund's fact sheet. It is the time weighted cash flows of a bond." Sometimes referred to as the time it takes you to get your money back or the time to indifference to interest rate changes.

As far as the retail investor is concerned it probably makes no difference. Both of them will be about the same. But often both with be different from the maturity. The greater the coupon the greater that difference will be.

If you are choosing between bond funds use the duration not the maturity.
A scientist looks for THE answer to a problem, an engineer looks for AN answer and lawyers ONLY have opinions. Investing is not a science.

bertilak
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Re: Bond fund Maturity vs Duration

Can I summarize things as follows?
• The duration is intended to represent what maturity (if issued today) the bond in question would need to make it equivalent to other newly-issued bonds with the same coupon.
• The calculation to reach that number can be made in several ways, with varying assumptions getting various answers.
• However calculated, it can be used to compare bonds. You might want to do this to see which bond is the better deal -- (risk-adjusted?) payout vs. price. Some risks (default?, call?, inflation?) are left as an exercise to the buyer.
Am I close?

But to answer my original question, the difference between maturity and duration is of little significance to most buyers. The duration is intended to be the number of interest.
Last edited by bertilak on Wed Apr 19, 2017 2:02 pm, edited 1 time in total.
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dm200
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Re: Bond fund Maturity vs Duration

Especially with mortgage related bonds, prepayment can be a significant contributor to the lower duration vs maturity.

bertilak
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Re: Bond fund Maturity vs Duration

dm200 wrote:Especially with mortgage related bonds, prepayment can be a significant contributor to the lower duration vs maturity.
So that risk is already baked into the duration?
May neither drought nor rain nor blizzard disturb the joy juice in your gizzard. -- Squire Omar Barker, the Cowboy Poet

alex_686
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Re: Bond fund Maturity vs Duration

bertilak wrote:The duration is intended to represent what maturity (if issued today) the bond in question would need to make it equivalent to other newly-issued bonds with the same coupon.
No. It is mainly used as a risk measure. Duration and maturity are the same only for zero coupon bonds and it is next to impossible to find replacement bonds (same maturity and coupons). It is really more usefully when comparing bonds with different maturities and different coupons and figuring out which of the bonds is riskier. If bonds have different maturities and different coupons but have the same duration then these different bonds probably have the same risk.

Consider this for a second: A 10 year floating coupon bond which resets every year will have a duration of about .5. Barbecue 1 = 1 year and we assume we are 1/2 through the year. Long maturity but almost no risk.
bertilak wrote:The calculation to reach that number can be made in several ways, with varying assumptions getting various answers.
meh. The formulas are pretty much standardized. However different formuals are used to figure out different things. What are you trying to figure out? That will more or less drive which formula you use. Heck, even what assumptions you use won't affect the end result that much.
bertilak wrote:However calculated, it can be used to compare bonds. You might want to do this to see which bond is the better deal -- (risk-adjusted?) payout vs. price. Some risks (default?, call?, inflation?) are left as an exercise to the buyer.
The general formula for yield = Risk Free Rate (Treasury) + Credit Spread + Option Adjusted

Modified Duration measures the risk to parallel changes in the yield curve. Mainly this is from the change in the risk free rate. It would also capture if the credit spread changed. It would not capture if the bond moved from one ratings to another, say from AA to A.

dkturner
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Re: Bond fund Maturity vs Duration

bertilak wrote:
dm200 wrote:Especially with mortgage related bonds, prepayment can be a significant contributor to the lower duration vs maturity.
So that risk is already baked into the duration?
Principal prepayments are baked into the duration number at current interest rates. When rates rise prepayments tend to slow down, resulting in an increase in duration. With falling rates prepayments tend to increase, resulting in a decrease in duration. This is a good thing for debtors, but not what the owners of mortgage loans want. It's called negative convexity. The point being is that even a "static" portfolio can have increasing or decreasing duration and average maturity.

alex_686
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Re: Bond fund Maturity vs Duration

bertilak wrote:
dm200 wrote:Especially with mortgage related bonds, prepayment can be a significant contributor to the lower duration vs maturity.
So that risk is already baked into the duration?
Kind of the other way around. Modified Duration is a measure of risk. So not so much as "baked in" but a "representative of". remember where I said that different assumptions did not matter too much because those assumptions tended to bunch together? This is the exception to that rule. MBS bonds will always have a prepayment assumption when calculating the duration. There is even a industry standard on what that prepayment speed is. Conditional Prepayment Rate or CPR. Everybody agrees that the values are bunk but it is the industry standard. Different firms will come up with their own internal model for internal evaluation. When the SEC comes around they always pull out the CPR.

Doc
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Re: Bond fund Maturity vs Duration

bertilak wrote:Can I summarize things as follows?
• The duration is intended to represent what maturity (if issued today) the bond in question would need to make it equivalent to other newly-issued bonds with the same coupon.
• The calculation to reach that number can be made in several ways, with varying assumptions getting various answers.
• However calculated, it can be used to compare bonds. You might want to do this to see which bond is the better deal -- (risk-adjusted?) payout vs. price. Some risks (default?, call?, inflation?) are left as an exercise to the buyer.
Am I close?

But to answer my original question, the difference between maturity and duration is of little significance to most buyers. The duration is intended to be the number of interest.
What alex said.

"But to answer my original question, the difference between maturity and duration is of little significance to most buyers"

No. High coupon bonds means you get your money back faster so if you compare two bonds with the nearly the same maturity the one with the higher coupon and therfore shorter duration has less term risk and risk is important to all investors.

(On the other hand, the difference between the Macaulay and Modified duration calculations may not be very important to the retail investor.)
A scientist looks for THE answer to a problem, an engineer looks for AN answer and lawyers ONLY have opinions. Investing is not a science.

baw703916
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Re: Bond fund Maturity vs Duration

bertilak wrote:
dm200 wrote:Especially with mortgage related bonds, prepayment can be a significant contributor to the lower duration vs maturity.
So that risk is already baked into the duration?
Well, not exactly. If interest rates drop then prepayment and refinancing is likely, but if interest rates rise it isn't. So it's an asymmetric risk, which can't be captured by a single number. More generally, the duration only gives the sensitivity of the bond price to a very small change in interest rates (if you want to think in terms of calculus it's the first derivative). But it isn't appropriate for large changes in interest rates: imagine a 30 year zero coupon Treasury (if the coupon is zero and there's no possibility of prepayment then the maturity and duration are identical). Then say interest rates go up by 4%. According to the formula

change in price = -1*duration*change in interest rate,

the bond should drop in value by -120% and be worth less than zero!

To have a more meaningful picture of how bond prices respond to more substantial changes in interest rates, and also reflect the asymmetry of callable bonds, a quantity called convexity comes into play (mathy people can think of it as the second derivative).

Shorter answer: duration is not the whole story.
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bertilak
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Re: Bond fund Maturity vs Duration

So, this reminds me of the poor Dude in The Big Lebowski:
• DUDE
It's a complicated case. Lotta ins.
Lotta outs. And a lotta strands to
keep in my head, man. Lotta strands ...
I think it would take much more study to understand bonds than I am ready to invest.

I will simply keep in mind that it is duration that is important and shorter duration means less risk, all else being equal. (I am right here, aren't I?)

While thinking about this I figured I'd look at another TBM fund beside VBTLX so I checked out Fidelity's (FTBFX). Morningstar doesn't even quote an average maturity for FTBFX. Its duration (5.63) shorter than VBTLX's (5.99). I was therefore surprised by a comparison chart between the two. FTBFX took a big hit in 2008 whereas VBTLX sailed right through with hardly a ripple: FTBFX vs VBTLX so there is more to this than duration. I guess all else isn't equal between Fidelity's and Vabguard's TBM funds.
baw703916 wrote:Shorter answer: duration is not the whole story.
Amen!
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grabiner
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Re: Bond fund Maturity vs Duration

bertilak wrote:While thinking about this I figured I'd look at another TBM fund beside VBTLX so I checked out Fidelity's (FTBFX). Morningstar doesn't even quote an average maturity for FTBFX. Its duration (5.63) shorter than VBTLX's (5.99). I was therefore surprised by a comparison chart between the two. FTBFX took a big hit in 2008 whereas VBTLX sailed right through with hardly a ripple
FTBFX is not an index fund; it is a broad-based bond fund. It took a hit in 2008 because it holds lower-quality bonds than the index, including about 15% junk.

Fidelity does have an equivalent to Vanguard's index; the fund is Fidelity US Bond Index, and the Premium class has 0.07% expenses. These two track almost perfectly, including 2008.
David Grabiner

bertilak
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Re: Bond fund Maturity vs Duration

grabiner wrote:
bertilak wrote:While thinking about this I figured I'd look at another TBM fund beside VBTLX so I checked out Fidelity's (FTBFX). Morningstar doesn't even quote an average maturity for FTBFX. Its duration (5.63) shorter than VBTLX's (5.99). I was therefore surprised by a comparison chart between the two. FTBFX took a big hit in 2008 whereas VBTLX sailed right through with hardly a ripple
FTBFX is not an index fund; it is a broad-based bond fund. It took a hit in 2008 because it holds lower-quality bonds than the index, including about 15% junk.

Fidelity does have an equivalent to Vanguard's index; the fund is Fidelity US Bond Index, and the Premium class has 0.07% expenses. These two track almost perfectly, including 2008.
Yes, I picked the wrong Fidelity fund to properly match VBTLX but I stumbled upon one that demonstrates duration is not the whole risk story, perhaps not even the biggest part of the story.
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alex_686
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Re: Bond fund Maturity vs Duration

bertilak wrote:Yes, I picked the wrong Fidelity fund to properly match VBTLX but I stumbled upon one that demonstrates duration is not the whole risk story, perhaps not even the biggest part of the story.
If I recall correctly, duration explains about 80% of returns for government bonds and 70% for corporate.

Another point. I think the difference between the 2 funds is not down to duration exactly. If Fidelity is following a different index - or no index at all - then the duration of the 2 funds may be very different over the past 10 years. Fidelity may have always had a longer duration. I know that VBTLX's duration has increased over the last 10 years. So while they are currently apples to apples I don't know if there were 10 years ago. Strategies always shift.

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Re: Bond fund Maturity vs Duration

Textbook definition of duration :" A measure of the average life of a bond, defined as the weighted average of the times until each payment is made, with the weights proportional to the present value of the payment." ( Investments, Bodie Kane Marcus, ninth edition.)

You could call it the average time until you get your money. Because cash flows are weighted by their present values, earlier payments have higher weights. Holding maturity constant, a bond's duration is lower when the coupon rate is higher. For a zero coupon bond, duration equals time to maturity. For a fund, duration will be an average of whatever bonds are in that fund.

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Re: Bond fund Maturity vs Duration

grabiner wrote:
bertilak wrote:While thinking about this I figured I'd look at another TBM fund beside VBTLX so I checked out Fidelity's (FTBFX). Morningstar doesn't even quote an average maturity for FTBFX. Its duration (5.63) shorter than VBTLX's (5.99). I was therefore surprised by a comparison chart between the two. FTBFX took a big hit in 2008 whereas VBTLX sailed right through with hardly a ripple
FTBFX is not an index fund; it is a broad-based bond fund. It took a hit in 2008 because it holds lower-quality bonds than the index, including about 15% junk.

Fidelity does have an equivalent to Vanguard's index; the fund is Fidelity US Bond Index, and the Premium class has 0.07% expenses. These two track almost perfectly, including 2008.
Thanks for posting that. I was wondering what was going on with the different graphs.

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Re: Bond fund Maturity vs Duration

baw703916 wrote: If interest rates drop then prepayment and refinancing is likely, but if interest rates rise it isn't. So it's an asymmetric risk, which can't be captured by a single number. More generally, the duration only gives the sensitivity of the bond price to a very small change in interest rates (if you want to think in terms of calculus it's the first derivative).
When interest rates rise home owners are less likely to trade up and thus their current mortgages​ get extended so it is not completely asymmetric. Either way is bad for the lenders - us.

Yes on the calculus.
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alex_686
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Re: Bond fund Maturity vs Duration

baw703916 wrote:Well, not exactly. If interest rates drop then prepayment and refinancing is likely, but if interest rates rise it isn't. So it's an asymmetric risk, which can't be captured by a single number. More generally, the duration only gives the sensitivity of the bond price to a very small change in interest rates (if you want to think in terms of calculus it's the first derivative). But it isn't appropriate for large changes in interest rates: imagine a 30 year zero coupon Treasury (if the coupon is zero and there's no possibility of prepayment then the maturity and duration are identical). Then say interest rates go up by 4%. According to the formula

change in price = -1*duration*change in interest rate,

the bond should drop in value by -120% and be worth less than zero!

To have a more meaningful picture of how bond prices respond to more substantial changes in interest rates, and also reflect the asymmetry of callable bonds, a quantity called convexity comes into play (mathy people can think of it as the second derivative).
You point on convexity is spot on. You analysis of MBS needs a little work.

A 30 year whole mortgage if held to maturity has a duration of about 15 years. After all, duration is the time weighted steam of payments.

When you pool a couple of thousands of mortgages together you get a duration of between 7 and 10. You expect some prepayments, either people putting a little extra towards their mortgage, refi, move, or whatever.

A MBS will have a even tighter duration than that. A mortgage pool will often be split up into 20 to 30 different bits, or tranches. Some of these tranches are supporting tranches. If prepayments are faster than expected a specific tranches will take the hit, if slower than another. Thus the majority of the MBS issued from the mortgage pool are insulated from rate changes.

So while the pool as a whole may be convex the individual bonds issued are not. Historically this has worked quite well. Expect in 2008 where the tranches designated to take the hit from defaults were burned through and losses flowed into the main tranches.

baw703916
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Re: Bond fund Maturity vs Duration

I wrote that post while thinking about callable munis, while the actual question was about MBS. My bad!

For an issuer of munis there is an asymmetric risk. If say a bond has a 30 year maturity but is callable after 20 years, the issuer will only refinance if the market interest rates are lower than the interest rate of the bond. So if interest rates are dropping it's a 20 year bond, but if they're rising it's a 30 year.

Yes, for MBSs it's more complicated, as the bond always gets repaid when the property is sold (unless it's assumable). Thanks to both of you for explaining the details.
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Re: Bond fund Maturity vs Duration

bertilak wrote:I think I know what maturity and duration are, but just to be sure:
• Maturity: The time between a bond's issue date and the date its principle is paid back.
No. Maturity is the time between today and the date its principal is paid back. What you said is the bond's term. Maturity is extremely important because a 30-year term bond with a maturity of 5 years and a 7-year term bond with a maturity of 5 years will both have the same YTM. It is an amazing thing about bonds, that regardless of term and coupon, the yield will be the same for the same maturity. This is the whole point of having a yield curve (which you will note is only based on maturity, not term). If you don't believe this, go look at an open market listing of bonds for sell. You will see bonds with all kinds of terms and coupons, but for identical maturities, they will all have the same yield (YTM). That does not mean they are equivalent, because coupon will make a difference if it is sold before maturity.
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Doc
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Re: Bond fund Maturity vs Duration

kolea wrote:It is an amazing thing about bonds, that regardless of term and coupon, the yield will be the same for the same maturity.
Duration.
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kolea
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Re: Bond fund Maturity vs Duration

Doc wrote:
kolea wrote:It is an amazing thing about bonds, that regardless of term and coupon, the yield will be the same for the same maturity.
Duration.
I think maturity is right. Here is an example from today's open market on US Treasuries.

CUSPID: 912828ND8, Term: 10 years. Maturity date: 5/15/2020 (3 years). Coupon: 3.50%. YTM: 1.387%

CUSPID: 912828VA5, Term: 7 years. Maturity date: 4/30/2020 (3 years). Coupon: 1.125%. YTM: 1.388%

Different term, different coupon, same maturity (within a few days), same YTM.

Perhaps the duration is also the same, not sure. But it makes sense for maturity to be the only determinant. If you have two investment options that mature on the same date, why would anyone ever buy the one with less yield (assuming the credit risk is the same)? The market demands that they have the same yield, yes?
Kolea (pron. ko-lay-uh). Golden plover.

alex_686
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Re: Bond fund Maturity vs Duration

kolea wrote:CUSPID: 912828ND8, Term: 10 years. Maturity date: 5/15/2020 (3 years). Coupon: 3.50%. YTM: 1.387%

CUSPID: 912828VA5, Term: 7 years. Maturity date: 4/30/2020 (3 years). Coupon: 1.125%. YTM: 1.388%

Different term, different coupon, same maturity (within a few days), same YTM.

Perhaps the duration is also the same, not sure. But it makes sense for maturity to be the only determinant. If you have two investment options that mature on the same date, why would anyone ever buy the one with less yield (assuming the credit risk is the same)? The market demands that they have the same yield, yes?
The first bond has a lower duration than the second bond. It would interesting to see the prices and quality of those prices for these 2 bonds. I think both of these bonds are off the run bonds so the pricing data is only going to be so-so, which is important part of your case. The maturity is relatively close and the yield curve is flatish so interest rate risk is not going to play a big role here.

FYI, in the industry nobody cares about term of the bond outside of new issues. Once a bond is issued the only thing people care about is the maturity date. For example, 3 year treasury futures can be settled with a newly issued 3 year note or either of the 2 bonds you quoted.

dm200
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Re: Bond fund Maturity vs Duration

Once a bond is issued the only thing people care about is the maturity date. For example, 3 year treasury futures can be settled with a newly issued 3 year note or either of the 2 bonds you quoted.
No expert here. Is this true?

alex_686
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Re: Bond fund Maturity vs Duration

dm200 wrote:
Once a bond is issued the only thing people care about is the maturity date. For example, 3 year treasury futures can be settled with a newly issued 3 year note or either of the 2 bonds you quoted.
No expert here. Is this true?
It is true. There is good money to be made if you can figure out the math behind it. See the fictional Book "Bonfire of the Vanities", the non-fiction book "Liar's Poker" or failed hedge fund "Long Term Capital".

Here is a link to the Chicago Board of Trade. They will accept 6 different Treasuries for futures settlement and they are kind of picky on what they will included. They only want liquid items. Some are 3 year notes, others are 2 year old 5 year bonds (3 years left). Most clients are not so picky. They will gladly pick up a few bps for illiquid treasuries.

kolea
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Re: Bond fund Maturity vs Duration

alex_686 wrote: FYI, in the industry nobody cares about term of the bond outside of new issues. Once a bond is issued the only thing people care about is the maturity date. For example, 3 year treasury futures can be settled with a newly issued 3 year note or either of the 2 bonds you quoted.
Yes, and that was exactly my point. The only thing that matters is maturity.
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Doc
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Re: Bond fund Maturity vs Duration

Th differences are subtle. Go to Vanguard or some other broker that has a good search criteria.

For Vanguard I used Treasury Strips/Bonds/Notes with a maturity of 3 to 7 years (the part of the yield curve currently with the most term sensitivity). Sort by maturity date and compare the yields with the coupon for issues with the same maturity. In general the higher the coupon (lower duration) the lower the yield.

Example:

jpg imagescertificity.com
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grabiner
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Re: Bond fund Maturity vs Duration

alex_686 wrote:
kolea wrote:CUSPID: 912828ND8, Term: 10 years. Maturity date: 5/15/2020 (3 years). Coupon: 3.50%. YTM: 1.387%

CUSPID: 912828VA5, Term: 7 years. Maturity date: 4/30/2020 (3 years). Coupon: 1.125%. YTM: 1.388%

Different term, different coupon, same maturity (within a few days), same YTM.

Perhaps the duration is also the same, not sure. But it makes sense for maturity to be the only determinant. If you have two investment options that mature on the same date, why would anyone ever buy the one with less yield (assuming the credit risk is the same)? The market demands that they have the same yield, yes?
The first bond has a lower duration than the second bond.
The durations are very close. The first bond, with a 3.5% coupon, has 10% of its value in coupon payments before maturity (six coupons due from May 2017 through November 2020); the other 90% of the value is from the principal. The second bond has 3% of its value in coupon payments before maturity. So 10% of the first bond and 3% of the second bond have a duration of 1.75 years (maturity minus average time to the coupon payments, weighted by present value); this is a difference of 7% of the 1.75 years, or .12 years. Meanwhile, the first bond has two more weeks to maturity, which is .04 years. Thus the difference in durations is about .08 years, which isn't enough to make much of a difference.

Different coupon rates would make more of a duration difference for bonds further from maturity (both because they have more coupons, and because the duration of the coupons is farther from the duration of the principal).
David Grabiner

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