## what is an ibonds duration?

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Jaylat
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### Re: what is an ibonds duration?

LTCM wrote: Fri May 13, 2022 6:06 pm
Oregano wrote: Fri May 13, 2022 5:03 pm it can never change how much you can redeem it for.
Well it can. A savvy investor maxed out on I-bonds would offer me a lot more than \$10,000 for the economic rights to my 2021 I-bonds. To me that means the value is higher than \$10,000. The economic benefit to me if I don't sell is more than \$10,000 in present value. There's a \$10,000 floor on the value, but not a ceiling. That's the point. That affects duration. It's definitely not duration=0.
Exactly. In fact, every investor makes the same calculation in choosing to invest in an I Bond for his own portfolio. You look at other alternatives and choose the best bang for the buck, given the interest rate environment.
LTCM
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### Re: what is an ibonds duration?

vineviz wrote: Fri May 13, 2022 9:05 pm
LTCM wrote: Fri May 13, 2022 6:06 pm A duration of 0 means the holder is indifferent to interest rates rising or falling (I think). So I'm (reasonably) confident that's not the case here.
That's not what a duration of 0 means in any technical sense of the word.

If you have \$1,000 worth of savings bonds today, and tomorrow you wake up to find that yields have dropped by 3%, what is the price at which you can sell your savings bonds? \$1,000. Duration = 0.

If you have \$1,000 worth of 20-year zero coupon Treasury bond with a yield of 3% today and you wake up tomorrow to find that yields have dropped by 3%, what is the price at which you can sell your bond? \$1,806. Duration = 20.
You mean yields have risen by 3% right? You are indifferent to rates rising because of the put option. You are not indifferent to rates rising or falling. You want them to fall.

Negative duration is when the value of a bond rises with interest rates. Positive duration is when the value falls with rising interest rates. So 0 is when it has no impact.

(I think!)

Last edited by LTCM on Fri May 13, 2022 10:48 pm, edited 1 time in total.
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Jaylat
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### Re: what is an ibonds duration?

This could be a business opportunity for an enterprising investment banker:
• Sign up the rights to I Bonds from US citizens not in a position to invest (the majority of Americans, I think) for a nominal fee;
• Issue a zero coupon bond backed by the I Bond portfolio. Financing costs should be minimal as it's US Gov't risk;
Edit: For those for whom this isn't obvious - this is a joke.
Last edited by Jaylat on Sat May 14, 2022 1:59 pm, edited 1 time in total.
LTCM
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### Re: what is an ibonds duration?

vineviz wrote: Fri May 13, 2022 9:05 pm
LTCM wrote: Fri May 13, 2022 6:06 pm A duration of 0 means the holder is indifferent to interest rates rising or falling (I think). So I'm (reasonably) confident that's not the case here.
That's not what a duration of 0 means in any technical sense of the word.

If you have \$1,000 worth of savings bonds today, and tomorrow you wake up to find that yields have dropped by 3%, what is the price at which you can sell your savings bonds? \$1,000. Duration = 0.

If you have \$1,000 worth of 20-year zero coupon Treasury bond with a yield of 3% today and you wake up tomorrow to find that yields have dropped by 3%, what is the price at which you can sell your bond? \$1,806. Duration = 20.
I could sell the savings bonds to my non citizen friends and relatives for more than \$1806. If rates rose by 3% I could sell the savings bond for \$1000.

Excluding transaction costs and assuming rational investors.
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vineviz
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### Re: what is an ibonds duration?

Jaylat wrote: Fri May 13, 2022 9:40 pm This could be a business opportunity for an enterprising investment banker:
• Sign up the rights to I Bonds from US citizens not in a position to invest (the majority of Americans, I think) for a nominal fee;
• Issue a zero coupon bond backed by the I Bond portfolio. Financing costs should be minimal as it's US Gov't risk;
This is illegal.
"Far more money has been lost by investors preparing for corrections than has been lost in corrections themselves." ~~ Peter Lynch
vineviz
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### Re: what is an ibonds duration?

LTCM wrote: Fri May 13, 2022 10:54 pm I could sell the savings bonds to my non citizen friends and relatives for more than \$1806. If rates rose by 3% I could sell the savings bond for \$1000.

Excluding transaction costs and assuming rational investors.
No, because a rational investor would not pay for an asset that they cannot sell or redeem.
"Far more money has been lost by investors preparing for corrections than has been lost in corrections themselves." ~~ Peter Lynch
Oregano
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### Re: what is an ibonds duration?

LTCM wrote: Fri May 13, 2022 6:06 pm
Oregano wrote: Fri May 13, 2022 5:03 pm it can never change how much you can redeem it for.
Well it can. A savvy investor maxed out on I-bonds would offer me a lot more than \$10,000 for the economic rights to my 2021 I-bonds. To me that means the value is higher than \$10,000. The economic benefit to me if I don't sell is more than \$10,000 in present value. There's a \$10,000 floor on the value, but not a ceiling. That's the point. That affects duration. It's definitely not duration=0.

A duration of 0 means the holder is indifferent to interest rates rising or falling (I think). So I'm (reasonably) confident that's not the case here.
Nonsense. An I-bond is not marketable, so there is no value to sell it to someone else. You can imagine some world in which there is a thriving I-bond swap market, but it doesn't exist.

I already explained that your preference to hold an I-bond varies with interest rates, but it does not change the actual "market value", which for I-bonds is, and only an be, the redemption value.
Jaylat
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### Re: what is an ibonds duration?

vineviz wrote: Sat May 14, 2022 4:06 am
Jaylat wrote: Fri May 13, 2022 9:40 pm This could be a business opportunity for an enterprising investment banker:
• Sign up the rights to I Bonds from US citizens not in a position to invest (the majority of Americans, I think) for a nominal fee;
• Issue a zero coupon bond backed by the I Bond portfolio. Financing costs should be minimal as it's US Gov't risk;
This is illegal.
This was a joke.
LTCM
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### Re: what is an ibonds duration?

vineviz wrote: Sat May 14, 2022 4:10 am
LTCM wrote: Fri May 13, 2022 10:54 pm I could sell the savings bonds to my non citizen friends and relatives for more than \$1806. If rates rose by 3% I could sell the savings bond for \$1000.

Excluding transaction costs and assuming rational investors.
No, because a rational investor would not pay for an asset that they cannot sell or redeem.
I find your bullheadedness on this issue a little frustrating but I'll try one last time. If an "irrational" investor offered to pay \$1500 for your \$1000 2022 ibond why wouldn't you accept if the market value is only \$1000? (if necessary imagine you own some older paper bonds and adjust the price accordingly).

I'm honestly not sure if you know I'm right and want to avoid backing down or you just don't see it.

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LTCM
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### Re: what is an ibonds duration?

Oh. And one last thing. I don't think you even need to sell it. You can get a present value of over \$1000 from the future cash flows.
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vineviz
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### Re: what is an ibonds duration?

LTCM wrote: Sat May 14, 2022 12:11 pm I find your bullheadedness on this issue a little frustrating but I'll try one last time.
Maybe stop trying to prove you're right and absorb the following points:

1) Savings bonds have restrictions on who can own them, and transferring them even between eligible parties can be cumbersome at best. This is why I'm not enthusiastic to grant your "let's sell them to non-citizen friends and family" premise.

2) The very concept of duration depends on the instrument having cash flows, which a savings bond does not have. A savings bond is a depository account, not a capital asset. It's true that a savings bonds has some features that resemble the features of marketable bonds, but it also differs in some important ways. The fact that a table and a horse each have four legs doesn't make them interchangeable.

I don't think it ultimately matters much whether someone concludes that savings bonds have an indeterminate duration or simply have no duration. Unlike a true bond, whose modified duration and Macaulay duration are objectively measurable and nearly identical almost always, finding that a savings bond has a duration of anywhere between 0 years and 30 years doesn't strike me as helpful analytically.
"Far more money has been lost by investors preparing for corrections than has been lost in corrections themselves." ~~ Peter Lynch
LTCM
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### Re: what is an ibonds duration?

Apologies for the rudeness. I shouldn't post before breakfast. I get frustrated when either I can't explain my position to someone else or have them explain it to me in a way that convinces me.

1) you can pretend this isn't an ibond and just have a imaginary 30 year puttable bond with a floating coupon tied to the inflation rate if that helps explain it to me.

2) I struggle to see how a payment in 30 years isn't a cash flow. It doesn't have to be plural flows to be calculable. A bond paying 6% over market rates calculated bi-anually but only paid out after 30 years has a specific present value. Our problem is harder but not impossible.
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Oregano
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### Re: what is an ibonds duration?

LTCM wrote: Fri May 13, 2022 6:06 pm
Oregano wrote: Fri May 13, 2022 5:03 pm it can never change how much you can redeem it for.
Well it can. A savvy investor maxed out on I-bonds would offer me a lot more than \$10,000 for the economic rights to my 2021 I-bonds. To me that means the value is higher than \$10,000. The economic benefit to me if I don't sell is more than \$10,000 in present value. There's a \$10,000 floor on the value, but not a ceiling. That's the point. That affects duration. It's definitely not duration=0.

A duration of 0 means the holder is indifferent to interest rates rising or falling (I think). So I'm (reasonably) confident that's not the case here.
That is absolutely not what duration of 0 means. Duration of 0 means that the security's price has no sensitivity to changes in interest rates. Has nothing to do with whether the owner of the security likes the price moves.
LTCM
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### Re: what is an ibonds duration?

Oregano wrote: Sat May 14, 2022 6:03 pm That is absolutely not what duration of 0 means. Duration of 0 means that the security's price has no sensitivity to changes in interest rates. Has nothing to do with whether the owner of the security likes the price moves.
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market timer
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### Re: what is an ibonds duration?

LTCM wrote: Sat May 14, 2022 1:23 pm Apologies for the rudeness. I shouldn't post before breakfast. I get frustrated when either I can't explain my position to someone else or have them explain it to me in a way that convinces me.

1) you can pretend this isn't an ibond and just have a imaginary 30 year puttable bond with a floating coupon tied to the inflation rate if that helps explain it to me.

2) I struggle to see how a payment in 30 years isn't a cash flow. It doesn't have to be plural flows to be calculable. A bond paying 6% over market rates calculated bi-anually but only paid out after 30 years has a specific present value. Our problem is harder but not impossible.
Since this issue remains unresolved after 9 years of discussion, it is optimistic to think it will get resolved this time around. From the discussion, I think many people get bogged down in trivial details.

What I would suggest to bridge the gap is to first identify why you care about the concept of duration. For example, you might be saving to fund expenses that you'll have in 20 years. Then you could ask whether the return from investing in I bonds would depend on interest rate movements between now and then (it doesn't). A more sophisticated approach might be to ask whether investing in I bonds with an optimal redemption strategy--one where you redeem I bonds when real interest rates rise--would depend on interest rate movements between now and 20 years (it would). Ultimately, this is why people care about duration, and it avoids discussions getting mired in semantics.

Personally, I own I bonds as part of an investment strategy to fund expenses 30-50 years from now. When real interest rates were deeply negative across the yield curve, I anticipated holding these I bonds until maturity. However, due to the recent rise in interest rates, it is likely I'll redeem my I bonds to fund more appropriate long term investments with higher returns. Effectively, the "duration" of the I bond has sharply decreased. Anyone investing in EE bonds and I bonds should pay attention to prevailing Treasury yields and exercise the redemption option when the savings bonds are no longer competitive with Treasuries.
LTCM
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### Re: what is an ibonds duration?

market timer wrote: Sun May 15, 2022 2:08 am What I would suggest to bridge the gap is to first identify why you care about the concept of duration.
A couple of reasons. Firstly to educate myself about what is a good chunk of my investments. I find bond math interesting. I'd like to understand it better. Duration seems like one of the first things to try and understand. Convexity after that! Secondly to understand what it does to my specific portfolio. Does it make my bond holdings more sensitive to interest rates? Less sensitive? Does it extend my investment horizon or shorten it?

From what I can tell reading around the subject the last day or so: duration is often not fixed. It's relative to sizes of changes in interest rates and in the case of bonds with options non-symmetrical with respect to direction of change in interest rates. Floating coupon rates no doubt messes with things even further.

I feel like working through difficult examples helps to explain the workings behind what's going on with bond math and the "why do you care? Don't worry about it" answers frustrate me.

I don't expect to get a definitive answer, but I do feel for sure the duration is not ZERO and having that confirmed would go some way towards me thinking I'm understanding the concept somewhat.
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vineviz
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### Re: what is an ibonds duration?

LTCM wrote: Sun May 15, 2022 3:16 am
market timer wrote: Sun May 15, 2022 2:08 am What I would suggest to bridge the gap is to first identify why you care about the concept of duration.
A couple of reasons. Firstly to educate myself about what is a good chunk of my investments. I find bond math interesting. I'd like to understand it better. Duration seems like one of the first things to try and understand. Convexity after that! Secondly to understand what it does to my specific portfolio. Does it make my bond holdings more sensitive to interest rates? Less sensitive? Does it extend my investment horizon or shorten it?
These are commendable goals, but ones that probably can't be reached using savings bonds as an example.

Because savings bonds aren't true bonds.

They aren't just a "difficult example", but rather are separate asset type with some characteristics that resemble true bonds and other characteristics that do not resemble true bonds.

You can imagine the forced redemption of the savings bond is LIKE a maturing true bond. This effectively treats the savings bond like a zero coupon Treasury bond with a duration of 30.

But it's just as valid to say that a savings bond is LIKE a bank account in that it not marketable and has a fixed price. As such, it has no price sensitivity to changes in yield and therefore a duration of 0.

In short, you can pick your duration by choosing your analogy. This is not the case with a true bond, whose duration does not depend on looking at it from a certain point of view.
"Far more money has been lost by investors preparing for corrections than has been lost in corrections themselves." ~~ Peter Lynch
MrJedi
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### Re: what is an ibonds duration?

I treat it MUCH closer to a bank account (zero duration, cash-like) than a bond with a maturity. The reason is because after the year hold, you can cash it at any time with zero principal fluctuations other than the monthly interest payment, just like a bank account.

Just because there's a limited contribution doesn't make it have a duration in my eyes. That'd be like saying one of those bank accounts that have an extra high interest rate for the first \$5,000 or something has some non zero duration because it has a higher yield on a limited amount. I don't think many people would suggest that either.
Oregano
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### Re: what is an ibonds duration?

LTCM wrote: Sat May 14, 2022 9:53 pm
Oregano wrote: Sat May 14, 2022 6:03 pm That is absolutely not what duration of 0 means. Duration of 0 means that the security's price has no sensitivity to changes in interest rates. Has nothing to do with whether the owner of the security likes the price moves.
Nope. That is not a calculation for duration.

Reminder for everyone else next time you get into a dumb argument: make sure everyone defines what they are talking about.
LTCM
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### Re: what is an ibonds duration?

vineviz wrote: Sun May 15, 2022 5:30 am
But it's just as valid to say that a savings bond is LIKE a bank account in that it not marketable and has a fixed price. As such, it has no price sensitivity to changes in yield and therefore a duration of 0.

In short, you can pick your duration by choosing your analogy. This is not the case with a true bond, whose duration does not depend on looking at it from a certain point of view.
Most of the effective duration calculators I see don't use price as the input. They use present value. The present value can be formulated for a savings bond.

With 1 year remaining...
PV=FV/(1+i)n or \$1000
PV=(1000*29 year compounded coupon rate+final year coupon rate/(1+i)n or \$1000

With 2 years remaining
PV=FV/(1+i)n or \$1000
PV=(1000*28 year compounded coupon rate+final 2 years coupon rate/(1+i)n or \$1000

Etc. Etc.

If the US government announces a 100 year bond that pays a coupon rate of the cube root of the current presidents age will be sold from 2025 onwards I'm confident bond experts could calculate or estimate its value without needing a market to do it for them. The presidents age is not much harder to guess than the CPI or the market rate. Then add a put option and redo the calculation.

All my googling suggests that savings bonds are zero coupon bonds. Nothing weird about them. Just not vanilla because of the put option.

Whether the input (purchase) price is fixed (savings bond) or the output (redemption) price is fixed (zero coupon bond) doesn't seem to change the math involved.
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MrJedi
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### Re: what is an ibonds duration?

The key is when you try to liquidate your zero coupon bond. A "real" bond will have a fluctuating value upon liquidation depending on interest rate issued vs. the current interest rate in the market conditions. If interest rate suddenly spikes up, you will immediately see the value of your bond drop when you try to liquidate it. This is precisely the sensitivity that duration is measuring.

For a savings bond, the value you get upon liquidation does not fluctuate like that no matter what happens to interest rates. You will always be able to liquidate it at its current value and a interest rate spike does not change that.

How would you describe a savings account that pays x% interest in terms of duration?
sandan
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### Re: what is an ibonds duration?

MrJedi wrote: Sun May 15, 2022 6:25 am I treat it MUCH closer to a bank account (zero duration, cash-like) than a bond with a maturity. The reason is because after the year hold, you can cash it at any time with zero principal fluctuations other than the monthly interest payment, just like a bank account.

Just because there's a limited contribution doesn't make it have a duration in my eyes. That'd be like saying one of those bank accounts that have an extra high interest rate for the first \$5,000 or something has some non zero duration because it has a higher yield on a limited amount. I don't think many people would suggest that either.
Agreed. Its a very safe way to approximate portfolio duration for practical purposes. The alternative of completely ignoring ibond holdings seems a bit naïve, especially after Lotus 123.
LTCM
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### Re: what is an ibonds duration?

MrJedi wrote: Sun May 15, 2022 1:08 pm

For a savings bond, the value you get upon liquidation does not fluctuate like that no matter what happens to interest rates. You will always be able to liquidate it at its current value and a interest rate spike does not change that.
On the downside as rates rise you're right. That's the put option and the reason for the asymmetry.

As rates fall you can borrow from another investor an equivalent amount to your gains from the present value incease at no risk to yourself using the savings bond as collateral. The "price" is not what's relevant. It's the value.

Anything with a cash flow can be securitized. I think that includes a savings account. It's of indeterminate length but so are stocks. Stocks have duration.. They're just hard to calculate.

If I guaranteed you a savings account or CD of 10% for 100 years and then rates fell to 1% but you needed the money for something urgently would you close the account or would you borrow money at 1%? You can then easily determine the price/value/cost of this loan and its sensitivity to interest rates and work back from there.

I can't do it but I'm certain it's doable! These things are all fungible. It's not some kind of magic money that's different to everything else. It's a cash flow. It can be sliced and diced and calculated.
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vineviz
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### Re: what is an ibonds duration?

LTCM wrote: Sun May 15, 2022 11:21 am Most of the effective duration calculators I see don't use price as the input. They use present value.
That's because for a true bond, "price" and "present value of future cashflows" are the same thing. They are interchangeably terms because they describe the same concept.

That's not the case for a savings bond.

With a true bond you don't have to "imagine" the price, because the price is the price. The fact that we have to generate a hypothetical construct to change the price of a savings bond should be the first clue that it's not the same kind of asset.
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LTCM
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### Re: what is an ibonds duration?

MrJedi wrote: Sun May 15, 2022 6:25 am

That'd be like saying one of those bank accounts that have an extra high interest rate for the first \$5,000 or something has some non zero duration because it has a higher yield on a limited amount. I don't think many people would suggest that either.
I think that's the conclusion I've come to. A bank account also has duration. Its value changes as interest rates change.
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vineviz
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### Re: what is an ibonds duration?

LTCM wrote: Sun May 15, 2022 1:48 pm I can't do it but I'm certain it's doable! These things are all fungible. It's not some kind of magic money that's different to everything else. It's a cash flow. It can be sliced and diced and calculated.
Banks create structured notes all the time. If those notes are structured like a bond then they will behave like a bond and can be measured like one. But it would be the resultant note that benefits from that, not whatever the bank is using to collateralize the note.
LTCM wrote: Sun May 15, 2022 1:54 pm
MrJedi wrote: Sun May 15, 2022 6:25 am

That'd be like saying one of those bank accounts that have an extra high interest rate for the first \$5,000 or something has some non zero duration because it has a higher yield on a limited amount. I don't think many people would suggest that either.
I think that's the conclusion I've come to. A bank account also has duration. Its value changes as interest rates change.
You've reached the wrong conclusion if you think that cash has a duration of anything other than zero.
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LTCM
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### Re: what is an ibonds duration?

MrJedi wrote: Sun May 15, 2022 1:08 pm How would you describe a savings account that pays x% interest in terms of duration?
Effective Duration = [(PV–) – (PV+) / (2 * (∆Curve) * (PV0)]
Effective Duration = [(Present Value with falling rates) – (PV with rising rates) / (2 * (Change in rates) * (Present Value now)]

1 year savings account guaranteeing 10% interest in a 5% market rate environment has an effective duration of 0.46
(1046-1000)/(2*0.05*1000)

30 year savings account guaranteeing 10% interest in a 5% market rate environment has a effective duration of 28.93
(3893-1000)/(2*0.05*1000)

30 year i-bond with a 9.62 coupon in a 2.03 rate environment (current 1 year treasury yield) has an effective duration of 53.53
30 year i-bond with a 9.62 coupon in a 3.09 rate environment (current 30 year treasury yield) has an effective duration of 44.54

The i-bond duration is longer than time to maturity because effective duration is not a time unit but a sensitivity to interest rate unit and the current massive discrepancy between the coupon rate and market rate makes it unfortunate to lose (so very sensitive to interest rates) I think.

How comparable to Macauley/modified effective duration is I haven't done yet, but I could easily plug in examples and find out.
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vineviz
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### Re: what is an ibonds duration?

LTCM wrote: Sun May 15, 2022 3:33 pm The i-bond duration is longer than time to maturity because effective duration is not a time unit but a sensitivity to interest rate unit....
The duration of a bond cannot, by definition, be longer than its time to maturity.

Whenever a student gets that result, it's sure sign that the calculations have gone wrong.

Or they're trying to calculate duration for something without cashflows.
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LTCM
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### Re: what is an ibonds duration?

You can also easily see that when the coupon rate is below the market rate the duration is zero for any bond with a put option. When the coupon rate is above the market rate then it is above zero.
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LTCM
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### Re: what is an ibonds duration?

vineviz wrote: Sun May 15, 2022 3:38 pm
LTCM wrote: Sun May 15, 2022 3:33 pm The i-bond duration is longer than time to maturity because effective duration is not a time unit but a sensitivity to interest rate unit....
The duration of a bond cannot, by definition, be longer than its time to maturity.
That's not the case at all. Effective duration CAN be longer than time to maturity.
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vineviz
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### Re: what is an ibonds duration?

LTCM wrote: Sun May 15, 2022 3:39 pm
vineviz wrote: Sun May 15, 2022 3:38 pm
LTCM wrote: Sun May 15, 2022 3:33 pm The i-bond duration is longer than time to maturity because effective duration is not a time unit but a sensitivity to interest rate unit....
The duration of a bond cannot, by definition, be longer than its time to maturity.
That's not the case at all. Effective duration CAN be longer than time to maturity.
Not if you're doing it right.
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vineviz
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### Re: what is an ibonds duration?

vineviz wrote: Sun May 15, 2022 3:40 pm
LTCM wrote: Sun May 15, 2022 3:39 pm
vineviz wrote: Sun May 15, 2022 3:38 pm
LTCM wrote: Sun May 15, 2022 3:33 pm The i-bond duration is longer than time to maturity because effective duration is not a time unit but a sensitivity to interest rate unit....
The duration of a bond cannot, by definition, be longer than its time to maturity.
That's not the case at all. Effective duration CAN be longer than time to maturity.
Not if you're doing it right.
https://www.blackrock.com/fp/documents/ ... ration.pdf
Rules of Duration
When thinking about duration, a few general rules apply. With everything else being equal:
The duration of any bond that pays a coupon will be less than its maturity, because some amount of coupon payments will be received before the maturity date.
• The lower a bond’s coupon, the longer
its duration, because proportionately
less payment is received before final maturity. The higher a bond’s coupon, the shorter its duration, because proportionately more payment is received before final maturity.
• Because zero coupon bonds make no coupon payments, a zero coupon bond’s duration will be equal to its maturity.
• The longer a bond’s maturity, the longer its duration, because it takes more time to receive full payment. The shorter a bond’s maturity, the shorter its duration, because it takes less time to receive
full payment.
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LTCM
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### Re: what is an ibonds duration?

vineviz wrote: Sun May 15, 2022 3:40 pm
LTCM wrote: Sun May 15, 2022 3:39 pm
vineviz wrote: Sun May 15, 2022 3:38 pm
LTCM wrote: Sun May 15, 2022 3:33 pm The i-bond duration is longer than time to maturity because effective duration is not a time unit but a sensitivity to interest rate unit....
The duration of a bond cannot, by definition, be longer than its time to maturity.
That's not the case at all. Effective duration CAN be longer than time to maturity.
Not if you're doing it right.
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### Re: what is an ibonds duration?

vineviz wrote: Sun May 15, 2022 3:40 pm
LTCM wrote: Sun May 15, 2022 3:39 pm
vineviz wrote: Sun May 15, 2022 3:38 pm
LTCM wrote: Sun May 15, 2022 3:33 pm The i-bond duration is longer than time to maturity because effective duration is not a time unit but a sensitivity to interest rate unit....
The duration of a bond cannot, by definition, be longer than its time to maturity.
That's not the case at all. Effective duration CAN be longer than time to maturity.
Not if you're doing it right.

https://finance.zacks.com/longerduratio ... 10310.html
Effective duration is related to the maturity of the bond. The longer the bond's maturity, the greater its duration. However, the duration is always a smaller number than the maturity. If a bond has 10 years to maturity, its duration may be six or seven, for example, but it will not be 10.
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### Re: what is an ibonds duration?

I'm not sure what you're disagreeing with. Look up the effective duration formula and run the numbers. Its easily possible to get negative numbers or numbers larger than the time to maturity. Make up your own examples of extremely sensitive instruments.

I may have made some small mistakes calculating present values (regarding bi-annual interest calculations etc) but I think the big picture is right. Do you disagree with the effective durations of the saving account examples?
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### Re: what is an ibonds duration?

LTCM wrote: Sun May 15, 2022 3:43 pm
[url=https://quant.stackexchange.com/questio ... o-maturity]
some quant dude wrote:Theoretically bonds with embedded options (e.g. callable bonds) could also produce a duration greater than their maturity.

I think you should challenge "some quant dude" to produce an actual example of a bond that has an embedded option that creates an effective duration > time to maturity.

Constructing such a thing would require two things: 1) an embedded put option which allows the holder to put the bond back to the issuer at an above-market value if yields decrease; and 2) a bond market that could not accurately price that embedded option.

A callable bond would not create such result unless we're assuming that issuers are incompetent: who calls a bond yielding 9% when it would cost them 10% to issue a new bond?
Last edited by vineviz on Sun May 15, 2022 4:13 pm, edited 1 time in total.
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### Re: what is an ibonds duration?

LTCM wrote: Sun May 15, 2022 4:05 pm Do you disagree with the effective durations of the saving account examples?
Unless your duration calculation produces a result of "0" as the duration of a cash investment, then yes: it is incorrect.
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### Re: what is an ibonds duration?

vineviz wrote: Sun May 15, 2022 4:12 pm I think you should challenge "some quant dude" to produce an actual example of a bond that has an embedded option that creates an effective duration > time to maturity.
OK. I'm a generous government with AAA perfect credit trying to encourage retirement savings amongst its citizens. I'm offering a 30 year bond with a floating coupon which is the square of the market rate (whatever that is). What's the effective duration of such a bond when the market rate is 3% and rates change by 1%?

I get effective duration = 311
(85850-3243)/(2*0.01*13267)

This bond is incredibly sensitive to interest rates.
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### Re: what is an ibonds duration?

I will say the most important time frame is one year since I will have to wait for 12 months to redeem. Worthless if I want to use them before one year. Understanding in the classical sense of the term "duration" does not apply here, I would say 1 year is an estimate of duration after purchase and not 0. After 1 year the estimate of duration is 0 and can be used immediately to match my liabilities.
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### Re: what is an ibonds duration?

LTCM wrote: Sun May 15, 2022 4:46 pm
vineviz wrote: Sun May 15, 2022 4:12 pm I think you should challenge "some quant dude" to produce an actual example of a bond that has an embedded option that creates an effective duration > time to maturity.
I get effective duration = 311
(85850-3243)/(2*0.01*13267)
I've given this bond a present value of \$13,267 based on the 9% coupon over 3% market interest rate. This is obviously wrong because any wise investor would bid up the price of a bond with a significantly larger upside than downside. If the present value is bid up to \$26,534 the effective duration is halved. Do this enough and the duration falls back under maturity.

This may be the error with my i-bond calculation above. I'm using the wrong present value. I shouldn't be using the price that the government sell them at, but what they're actually worth. If I change the present value to the value based on the current coupon/market rate difference it comes back under the time to maturity.
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### Re: what is an ibonds duration?

LTCM wrote: Sun May 15, 2022 4:54 pm This may be the error with my i-bond calculation above. I'm using the wrong present value. I shouldn't be using the price that the government sell them at, but what they're actually worth. If I change the present value to the value based on the current coupon/market rate difference it comes back under the time to maturity.
Savings bonds don't have coupons.

Did you purchase any Series I savings bonds before March of this year?

If so, log into your Treasury Direct account and compare the price at which you can redeem them to the price at which you purchased them. I guaranteed the price has moved a lot less than any estimate you're getting from your PV calculations despite the sharp increase in bond yields.
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### Re: what is an ibonds duration?

vineviz wrote: Sun May 15, 2022 4:12 pm I think you should challenge "some quant dude" to produce an actual example of a bond that has an embedded option that creates an effective duration > time to maturity.
Ex: A bond with coupon rate of 5%, callable at par after 1 year, with a YTM of 1% and current price of below \$96 has an effective duration of 3.61 regardless of when it matures. So let's say it matures in 3 years.

Voila, you have a callable bond, with an effective duration above time-to-maturity, that did not use an embedded put (it's a call), and that trades at a perfectly reasonable price (it didn't even require mispricing by the market). Such a bond would almost certainly trade at less than \$96, so its effective duration would, if anything, be even greater.

IMO, "effective duration" is a bit of a misleading term because it isn't measuring the timeline for cash flows (that one is always between 0 and time-to-maturity). It just measures the sensitivity to a rate change. Given arbitrary values, you could even make it negative. It's a silly-metric IMO, I much prefer option-adjusted duration when talking about callable bonds.
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### Re: what is an ibonds duration?

No. I just got my account unlocked last week.

Sorry if my terminology is off!
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### Re: what is an ibonds duration?

It's baffling how there could be a three page thread debating the duration of a non-marketable investment that's puttable/redeemable at par plus accrued interest.
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### Re: what is an ibonds duration?

Further discussion of whether cash has duration and whether it's a type of bond here for those interested :
viewtopic.php?t=75024
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### Re: what is an ibonds duration?

Dry-Drink wrote: Sun May 15, 2022 6:23 pm
Ex: A bond with coupon rate of 5%, callable at par after 1 year, with a YTM of 1% and current price of below \$96 has an effective duration of 3.61 regardless of when it matures. So let's say it matures in 3 years.

A 5% coupon bond callable in 1 year with a YTC of 1% has a price of \$103.97 and a duration of duration of 1 year.
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### Re: what is an ibonds duration?

Thanks to everyone in this thread for one of the more interesting discussions here. I find value to both major interpretations: the theoretical IBond value can be useless or useful depending on the context.

I have been building up an IBond emergency fund. In the space of safe, liquid investments with guaranteed principle, I can choose between IBonds and a savings account. In this context, the redemption value is all that matters. I can look at short term inflation expectations and my tax situation and easily compare rates. I can switch back and forth subject to a few limitations. I don't need a theoretical net present value, as argued by nisiprius, Mel Lindauer, and vineviz among others.

I've almost filled up my emergency fund. Now I consider the different problem of deploying \$10k safely to fund an expense in five years.

Since I don't need liquidity, I have more options, namely I can purchase a 5 year treasury note or a 5 year TIPS. Conveniently, these instruments have auction mechanisms and secondary markets ensuring I will get a fair price based on how the market evaluates the various risks. However, I can also buy \$10k of IBonds, which provides inflation indexed returns similar to TIPS. Do I buy \$10k of 5 year TIPS or \$10k of IBonds?

There is no secondary market reporting an IBond price. Quite the opposite, the Treasury sets IBond terms using a mysterious secret formula. If the fixed rate is too low, the IBond might be worth (in theory) less than its \$10k purchase price, and I should buy the 5 year TIPS. If the Treasury sets it too high (as has happened recently with the 0% floor on the fixed rate), the theoretical value exceeds the \$10k purchase price and I should buy IBonds. The lack of the secondary market reporting the fair price means I have to do the value calculation myself to make the best decision.

In this context, I have found the analysis posted by LTCM, Epsilon Delta, and market timer, among others, to be very useful. Thanks again to everyone in the thread.
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### Re: what is an ibonds duration?

LTCM wrote: Sun May 15, 2022 4:54 pm
LTCM wrote: Sun May 15, 2022 4:46 pm
vineviz wrote: Sun May 15, 2022 4:12 pm I think you should challenge "some quant dude" to produce an actual example of a bond that has an embedded option that creates an effective duration > time to maturity.
I get effective duration = 311
(85850-3243)/(2*0.01*13267)
1. I've given this bond a present value of \$13,267 based on the 9% coupon over 3% market interest rate. This is obviously wrong because any wise investor would bid up the price of a bond with a significantly larger upside than downside. If the present value is bid up to \$26,534 the effective duration is halved. Do this enough and the duration falls back under maturity.

2. This may be the error with my i-bond calculation above. I'm using the wrong present value. I shouldn't be using the price that the government sell them at, but what they're actually worth. If I change the present value to the value based on the current coupon/market rate difference it comes back under the time to maturity.
1. I agree with you in concept. 'Embedded option' might make people think only of a conventional issuer call or investor put, both of which decrease duration to further beneath under the bond's stated maturity as compared to a bond with no embedded options. But once you consider *any* derivative you might jam into a bond's terms, the duration can likewise be anything. The discussion on the other thread with unknown 'quant guy' mentioned inverse floaters. Those have been a somewhat real thing, the bond pays L*market coupon-(L-1)*floating rate, at some leverage multiple L. It's a bond with interest rate swap(s) attached to it. At L=2 it's got twice the duration as at L=1, easily more than the bond's maturity in a moderate rate environment.

Interest rate swaps where dealer paid high fixed rate and end user paid Libor^2 were done at one time, gaining brief infamy in the Gibson Greeting Cards loss of \$23mil on such a swap with Bankers Trust in 1992 (quaint now how only \$23mil was a big deal in a trading debacle ). It didn't work so much for a bond coupon because the idea was the derivative dealer fully understood the value of the extra convexity and the end used didn't ('any wise investor would bid up the price', but how much? and relatively early in the history of quant finance not now) so the dealer needed to receive that flow. A swaplet between two future dates that pays floating^2 is worth, under simple assumptions in common with the Black-Scholes option valuation model, the forward rate squared times exp((vol^2)*t) where it's the volatility of that short forward rate from now to settlement. If say a short forward rate starting 5 yrs from now is 3% and short interest rate 'cap' market showed a 20% implied volatility at the money for that forward period, the fixed rate for that forward period would be ~9%*exp(.20%^2*5)~11%. For a whole swap or bond you'd need to value the correction according to the relevant vol (from the term structure of short forward rate vols) and maturity of each short forward period. Yeah, the duration of such a stream is huge, with massive positive convexity to the receiver.

2. On I-bond I'd just say the duration is zero considering 'what they are actually worth' that's important to me is what the govt will pay me to redeem them, always par (with full accrued after 5 yrs) regardless of TIPS real rates or fixed rates on newer I-bonds. And with the I-bond in recent times stuck on the 0% fixed rate floor, as long as that feature holds, the zero duration can only benefit me.
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### Re: what is an ibonds duration?

JackoC wrote: Tue May 17, 2022 10:32 am 1. I agree with you in concept. 'Embedded option' might make people think only of a conventional issuer call or investor put, both of which decrease duration to further beneath under the bond's stated maturity as compared to a bond with no embedded options. But once you consider *any* derivative you might jam into a bond's terms, the duration can likewise be anything. The discussion on the other thread with unknown 'quant guy' mentioned inverse floaters. Those have been a somewhat real thing, the bond pays L*market coupon-(L-1)*floating rate, at some leverage multiple L. It's a bond with interest rate swap(s) attached to it. At L=2 it's got twice the duration as at L=1, easily more than the bond's maturity in a moderate rate environment.
This is an interesting exception, though I think it's worth noting that applying enough leverage to ANY bond can produce an effective duration > maturity.
"Far more money has been lost by investors preparing for corrections than has been lost in corrections themselves." ~~ Peter Lynch