## Filling the TIPS gap years with bracket year duration matching

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Kevin M
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### Filling the TIPS gap years with bracket year duration matching

First, terminology:
• gap year = a year in which there are no TIPS maturing with a term to maturity of 29 years or less.
• bracket year = the years immediately before and after the gap years in which there are TIPS maturing that year.
• DARA = Desired Annual Real Amount = total real principal and interest that the ladder produces each year. This is the term used in the #Cruncher TIPS Ladder Builder spreadsheet.
• DARI = Desired Annual Real Income = DARA. This is the term used in the tipsladder.com TIPS ladder building tool.
• Real amount = amount in dollar purchasing power relative to some base date, using the reference CPI as the inflation index. A typical base date is the settlement date for the day you build or evaluate the TIPS ladder. Example: if the base date ref CPI were 100, and ref CPI increased to 103 on the maturity date of the first rung, a DARA of \$10,000 would equal an inflation-adjusted value of \$10,300, and the purchasing power would be \$10,000 relative to the base date (= 10,300 / 1.03).
• DARA multiplier = a number multiplied by the DARA, and entered in the #Cruncher TIPS ladder spreadsheet row for each distinct TIPS issue (i.e., identified by a distinct CUSIP, which is a unique identifier for a bond); this is used in the calculation of how many of that distinct TIPS issue to buy. For example, if holding only one distinct TIPS issue to generate the real principal amount of the DARA for a given maturity year, the DARA multiplier for that row would be 1. If holding none of a particular distinct TIPS, the multiplier for that TIPS issue row would be 0.
• duration matching = holding some of each of the bracket year TIPS such that the DARA-multiplier-weighted duration of them equals the expected duration of a gap year TIPS when it is issued.
Currently there are TIPS maturing in Jan 2034 and Feb 2040, so 2034 and 2040 are the bracket years, and the gap years are 2035-2039 (five of them).

For purposes of this discussion I'll assume that our TIPS ladder extends from 2034 or earlier through 2040 or later. The longest TIPS ladder would hold maturities from July 2024 (or possibly Oct 2024) to Feb 2054. The current versions of the two popular TIPS ladder building tools, the #Cruncher TIPS Ladder Builder spreadsheet and tipsladder.com, support only ladders with rungs starting in 2025.

One of many techniques that have been discussed for filling the gap years is to hold some of each of the TIPS that mature before the first gap year and after the last gap year. A specific instance of this is to do it with the bracket years, so currently 2034 and 2040 (it would have been 2033 and 2040 before the Jan 2034 was issued in Jan 2024).

The default for the #Cruncher spreadsheet is use DARA multipliers of 3 for the Jan 2034s and 4 for the Feb 2040s; note that 3 + 4 = 7, which is the total number of maturity years from 2034 through 2040. The tipsladder.com tool offers several methods to fill the gap years, but if you accept the default of "Bond maturing nearest to start of rung year", you essentially end up with multipliers of 4 for the 2034 and 3 for the 2040.

You don't need to use integers as DARA multipliers with the #Cruncher spreadsheet as long as the total of the DARA multipliers for a single maturity year equals 1; e.g., you could enter multipliers of 0.5 each for the Jan and Jul 2030 TIPS for your 2030 maturity year. With this in mind, you might use 3.5 each as the multipliers for the 2034 and 2040 to cover the 7 years from 2034-2040 inclusive, for example, and one might expect this to do a better job of duration matching the gap years.

What I do is calculate estimated durations for the TIPS for each gap year, then calculate the proportions for each of the 2034 and 2040 such that the DARA-multiplier-weighted-average duration equals the estimated duration of each gap year TIPS. Currently this results in multipliers of 3.56 for the 2034s and 3.44 for the 2040s. This confirms that simply using 3.5 as the multiplier for each gets pretty close to a decent estimated duration match, at least now, with the relatively flat yield curve in this maturity range.

To derive the formulas for the gap-year DARA multipliers for the 2034 and 2040, we start with this equation:

Code: Select all

``````d34 * x + d40 * (1-x) = dg,

where

d34 = modified duration (MD) of the 2034
d40 = MD of the 2040
dg = estimated MD of the gap year TIPS
x = gap year DARA multiplier for the 2034
``````
With some algebra, we solve for x to get:

Code: Select all

``````x = (d40-dg) / (d40-d34)
``````
I'll cover the calculation of durations in a subsequent post, and for now I'll just show the example of calculating x and (1-x) for the 2035 gap year.

Code: Select all

``````Independent variable values:

d34 = 8.75
d40 = 13.23
dg = d35 = 9.41

So,

x = (d40-dg) / (d40-d34)

x = (13.23-9.41) / (13.23-8.75)

x = 0.85

and

1-x = 0.15
``````
So we'd use DARA multipliers of 0.85 for the 2034s and 0.15 for the 2035s to match the estimated modified duration of the 2035.

As we've discussed in other threads, a simple way to approximate the gap year DARA weights is to simply set x = n/6, where n = 5 for 2035, n = 4 for 2036, ... n = 1 for 2039. To compare this method to the more complicated method shown above, note that for the 2035 gap year:

Code: Select all

``````n/6 = 5/6 = 0.83
``````
which is very close to 0.85 derived using the duration matching formula.

Here is the table of the DARA weights using durations of TIPS based on quotes from Schwab today, also showing the approximations using the n/6 method for the 2034 weights:

Note that the sum of the weights for each of the 2034 and 2040 are the DARA multipliers we enter into the #Cruncher spreadsheet for them respectively. Of course the sum of these multipliers equals 7, which is the total number of years covered (2034, 2040 + 5 gap years).

------------------------------------------------------------------------ EDIT ---------------------------------------------------------------------

I'm going to summarize what I've learned in doing the experiments documented in this thread. I'll add to this summary as I learn more.

Since I started the thread, #Cruncher developed a simplified ladder building spreadsheet, which I then used extensively for all analysis after that. I refer to this as "the simple tool" or just "simple".

Everything here is premised on using the simple tool, bracket-year coverage of gap years, a 30-year ladder, DARA = \$100K, and initial duration matching based on hypothetical gap year yield and coupon of 2%.
1. Duration matching works almost perfectly if we treat the gap years as marketable bonds; i.e., fixed coupons and variable prices (or values). The figure of merit here is how close to 0 is the change in value of the duration matched bracket year holdings minus the change(s) in value of the gap year(s) being matched. This is shown early in this thread.
2. Using the same figure of merit, duration matching does not work nearly as well for the real life situation where gap year coupons are variable and price (or value) is approximately fixed unless yields drop below 0.125%. This has been shown in the earlier posts in the thread.
3. The lack of pure duration matching effectiveness is offset to some extent by the change in interest from the gap year bonds, because the coupons will be close to the yields; i.e., at higher yield the coupon interest will be higher, requiring less principal, and therefore less cost for the gap year bonds.
4. Given #2, purchases or sales of the pre-gap rungs are required for ARA to equal DARA for the gap and pre-gap rungs, even after factoring in #3.
5. With no gaps filled and the sum of bracket year multipliers = 7 (e.g., 3.5 each for 2034/40), the sum of ARAs is greater than 30 * DARA (for a 30-year ladder, all other multipliers set to 1). This is a technical detail that is not particularly important, and I assume is due to my imperfect implementation of the multiplier feature, which was not included in #Cruncher's original simplified spreadsheet.
This table summarizes the results of the experiments to date:

"5 fill vs 0 fill at X%" means the numbers in that column relate to having all gap years filled (and the 2025-2039 all matured) at a yield of X%, and excess 2034/2040 bracket year holdings sold, compared to the initial state where 0 gap years are filled, all rungs are populated, and the excess holdings to fill the gap years are held in the 2034 and 2040 bracket years.

No temporal effects are considered; e.g., durations are not updated based on the passing of almost five years before all gaps are filled.

Looking at the 0% yield case:
• The proceeds from selling all excess 2034 and 2040 bracket year TIPS are 515,669.
• The cost of buying the 2035-2039 gap year TIPS is 432,951.
• This leaves us with extra cash of 82,718.
• We can choose to buy the pre-2034 TIPS that are left, 2030-2033, with the total being 31,396, if we want ARA to equal DARA for those rungs.
• If we do the pre-2034 transactions, we are left with 51,322 in cash.
Last edited by Kevin M on Thu Jun 27, 2024 12:10 am, edited 2 times in total.
If I make a calculation error, #Cruncher probably will let me know.
bigskyguy
Posts: 408
Joined: Sat Jan 24, 2015 3:59 pm

### Re: Filling the TIPS gap years with bracket year duration matching

I’ve been anticipating your post, and I will say you have done us all a great favor. Such a well outlined explanation. So from a distant follower, thanks.
cvn74n2
Posts: 161
Joined: Sat Aug 01, 2009 3:09 am

### Re: Filling the TIPS gap years with bracket year duration matching

Kevin et al,
Before I came across your thread on the mechanics of purchasing individual TIPS, I invested in TIPS ETFs and planned on duration matching them per Vinviz's suggestions (i.e. x% of LTPZ / y% of SCHP / z% of VTIP). It just so happened my timing was extremely poor (Dec 2022) and market losses ensued thereafter. Since then, I have created a LMP ladder of individual TIPS, but held on to the ETFs with the hope of recouping some of the losses over time.

Now, I could slowly draw down those ETFs on a duration-matched basis to buy individual gap year TIPS at their 10 year auctions. This would allow me to hold on my existing LMP ladder as is while (hopefully) limiting my losses over time as the sold, duration-matched ETF monies-turned-TIPS mature.

Or I could just glide-slope the ETFs to cover each of the gap years as originally planned?

All thoughts are welcome.
dcabler
Posts: 5046
Joined: Wed Feb 19, 2014 10:30 am
Location: TX

### Re: Filling the TIPS gap years with bracket year duration matching

cvn74n2 wrote: Wed May 22, 2024 11:50 pm Kevin et al,
Before I came across your thread on the mechanics of purchasing individual TIPS, I invested in TIPS ETFs and planned on duration matching them per Vinviz's suggestions (i.e. x% of LTPZ / y% of SCHP / z% of VTIP). It just so happened my timing was extremely poor (Dec 2022) and market losses ensued thereafter. Since then, I have created a LMP ladder of individual TIPS, but held on to the ETFs with the hope of recouping some of the losses over time.

Now, I could slowly draw down those ETFs on a duration-matched basis to buy individual gap year TIPS at their 10 year auctions. This would allow me to hold on my existing LMP ladder as is while (hopefully) limiting my losses over time as the sold, duration-matched ETF monies-turned-TIPS mature.

Or I could just glide-slope the ETFs to cover each of the gap years as originally planned?

All thoughts are welcome.
Did you use the shortcut method vineviz proposed on this thread? viewtopic.php?p=6869837#p6869837
In that method you are holding all 3 funds at the same time.

Or did you use the method where you hold only 2 funds at a time and calculate their relative weights based on matching the net duration of the 2 funds to your investment horizon, updating it periodically?

With the second method above, the market value of individual TIPS in a ladder would have been affected exactly the same way as your duration matched TIPS funds were. And yet the income produced by duration matching with funds would have remained nearly the same despite the loss in market value. Just like the income from individual TIPS would have remained unaffected. Moreover, if you convert your duration matched TIPS funds to a ladder, you can expect nearly the same income from the ladder after it's constructed as you were getting from the duration matched funds, regardless of when you make the conversion. With funds, you shouldn't care about the market value except to make the calculation when it's time to rebalance between the two funds so you know how much to shift from one fund to the other. I suppose one might care if they're rebalancing their TIPS funds with other assets, but that kinda breaks the whole idea of having steady income from TIPS. Otherwise, the fact that the market value dropped because yields rose is why you can extract steady income from 2 bond funds while duration matching. It's a feature, not a bug.

The first method is much more approximate, but the market value would be affected similarly.

Cheers.
Last edited by dcabler on Thu May 23, 2024 6:43 am, edited 7 times in total.
dcabler
Posts: 5046
Joined: Wed Feb 19, 2014 10:30 am
Location: TX

### Re: Filling the TIPS gap years with bracket year duration matching

First, terminology:
• gap year = a year in which there are no TIPS maturing with a term to maturity of 29 years or less.
• bracket year = the years immediately before and after the gap years in which there are TIPS maturing that year.
• DARA = Desired Annual Real Amount = total real principal and interest that the ladder produces each year. This is the term used in the #Cruncher TIPS Ladder Builder spreadsheet.
• DARI = Desired Annual Real Income = DARA. This is the term used in the tipsladder.com TIPS ladder building tool.
• Real amount = amount in dollar purchasing power relative to some base date, using the reference CPI as the inflation index. A typical base date is the settlement date for the day you build or evaluate the TIPS ladder. Example: if the base date ref CPI were 100, and ref CPI increased to 103 on the maturity date of the first rung, a DARA of \$10,000 would equal an inflation-adjusted value of \$10,300, and the purchasing power would be \$10,000 relative to the base date (= 10,300 / 1.03).
• DARA multiplier = a number multiplied by the DARA, and entered in the #Cruncher TIPS ladder spreadsheet row for each distinct TIPS issue (i.e., identified by a distinct CUSIP, which is a unique identifier for a bond); this is used in the calculation of how many of that distinct TIPS issue to buy. For example, if holding only one distinct TIPS issue to generate the real principal amount of the DARA for a given maturity year, the DARA multiplier for that row would be 1. If holding none of a particular distinct TIPS, the multiplier for that TIPS issue row would be 0.
• duration matching = holding some of each of the bracket year TIPS such that the DARA-multiplier-weighted duration of them equals the expected duration of a gap year TIPS when it is issued.
Currently there are TIPS maturing in Jan 2034 and Feb 2040, so 2034 and 2040 are the bracket years, and the gap years are 2035-2039 (five of them).

For purposes of this discussion I'll assume that our TIPS ladder extends from 2034 or earlier through 2040 or later. The longest TIPS ladder would hold maturities from July 2024 (or possibly Oct 2024) to Feb 2054. The current versions of the two popular TIPS ladder building tools, the #Cruncher TIPS Ladder Builder spreadsheet and tipsladder.com, support only ladders with rungs starting in 2025.

One of many techniques that have been discussed for filling the gap years is to hold some of each of the TIPS that mature before the first gap year and after the last gap year. A specific instance of this is to do it with the bracket years, so currently 2034 and 2040 (it would have been 2033 and 2040 before the Jan 2034 was issued in Jan 2024).

The default for the #Cruncher spreadsheet is use DARA multipliers of 3 for the Jan 2034s and 4 for the Feb 2040s; note that 3 + 4 = 7, which is the total number of maturity years from 2034 through 2040. The tipsladder.com tool offers several methods to fill the gap years, but if you accept the default of "Bond maturing nearest to start of rung year", you essentially end up with multipliers of 4 for the 2034 and 3 for the 2040.

You don't need to use integers as DARA multipliers with the #Cruncher spreadsheet as long as the total of the DARA multipliers for a single maturity year equals 1; e.g., you could enter multipliers of 0.5 each for the Jan and Jul 2030 TIPS for your 2030 maturity year. With this in mind, you might use 3.5 each as the multipliers for the 2034 and 2040 to cover the 7 years from 2034-2040 inclusive, for example, and one might expect this to do a better job of duration matching the gap years.

What I do is calculate estimated durations for the TIPS for each gap year, then calculate the proportions for each of the 2034 and 2040 such that the DARA-multiplier-weighted-average duration equals the estimated duration of each gap year TIPS. Currently this results in multipliers of 3.56 for the 2034s and 3.44 for the 2040s. This confirms that simply using 3.5 as the multiplier for each gets pretty close to a decent estimated duration match, at least now, with the relatively flat yield curve in this maturity range.

To derive the formulas for the gap-year DARA multipliers for the 2034 and 2040, we start with this equation:

Code: Select all

``````d34 * x + d40 * (1-x) = dg,

where

d34 = modified duration (MD) of the 2034
d40 = MD of the 2040
dg = estimated MD of the gap year TIPS
x = gap year DARA multiplier for the 2034
``````
With some algebra, we solve for x to get:

Code: Select all

``````x = (d40-dg) / (d40-d34)
``````
I'll cover the calculation of durations in a subsequent post, and for now I'll just show the example of calculating x and (1-x) for the 2035 gap year.

Code: Select all

``````Independent variable values:

d34 = 8.75
d40 = 13.23
dg = d35 = 9.41

So,

x = (d40-dg) / (d40-d34)

x = (13.23-9.41) / (13.23-8.75)

x = 0.85

and

1-x = 0.15
``````
So we'd use DARA multipliers of 0.85 for the 2034s and 0.15 for the 2035s to match the estimated modified duration of the 2035.

As we've discussed in other threads, a simple way to approximate the gap year DARA weights is to simply set x = n/6, where n = 5 for 2035, n = 4 for 2036, ... n = 1 for 2039. To compare this method to the more complicated method shown above, note that for the 2035 gap year:

Code: Select all

``````n/6 = 5/6 = 0.83
``````
which is very close to 0.85 derived using the duration matching formula.

Here is the table of the DARA weights using durations of TIPS based on quotes from Schwab today, also showing the approximations using the n/6 method for the 2034 weights:

Note that the sum of the weights for each of the 2034 and 2040 are the DARA multipliers we enter into the #Cruncher spreadsheet for them respectively. Of course the sum of these multipliers equals 7, which is the total number of years covered (2034, 2040 + 5 gap years).
Why did you choose "modified duration" instead of the "macauley duration?"

Cheers.
Topic Author
Kevin M
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### Re: Filling the TIPS gap years with bracket year duration matching

dcabler wrote: Thu May 23, 2024 6:26 am Why did you choose "modified duration" instead of the "macauley duration?"
My next post will describe how I calculate the durations, so I'll address your question there. For now I'll just mention that the difference between D and MD is very small. I just tried using D instead of MD and it made no difference in the calculated numbers.
If I make a calculation error, #Cruncher probably will let me know.
dcabler
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Joined: Wed Feb 19, 2014 10:30 am
Location: TX

### Re: Filling the TIPS gap years with bracket year duration matching

Kevin M wrote: Thu May 23, 2024 9:34 am
dcabler wrote: Thu May 23, 2024 6:26 am Why did you choose "modified duration" instead of the "macauley duration?"
My next post will describe how I calculate the durations, so I'll address your question there. For now I'll just mention that the difference between D and MD is very small. I just tried using D instead of MD and it made no difference in the calculated numbers.
Correct - if you look at the last time #cruncher updated the spreadsheet, you can see that there as well.
Jaylat
Posts: 424
Joined: Sat Mar 12, 2016 10:11 am

### Re: Filling the TIPS gap years with bracket year duration matching

Kevin, this is a fantastic resource and once again all of us at BH owe you a huge debt of gratitude for sharing your seemingly boundless expertise.

That said, I’m concerned that a complex analysis like this might scare off the “average joe” who just wants to fill in a TIPS ladder and doesn’t much care about getting things exactly right to the third decimal place. Is this additional level of complexity really warranted?

For example, we have absolutely no idea what the gap TIPS will look like – they might have coupons ranging anywhere from 1/8% to 3% or more, which will make a significant difference in the gap TIPS duration, and so in the hedging technique. The duration of a (say) 2037 TIPS with a coupon of 3.5% will be very different from that of another 2037 TIPS paying 0.125%.

The change in gap TIPS duration due to coupon payments would also change the mix of 2034 / 2040 TIPS sold to hedge them, right?

I look forward to digging into the weeds in this thread, but from a practical point of view I may well just use the simple 1/7 weights for the 2034 / 2040 TIPS to estimate how many of each to sell to hedge a given gap TIPS purchase.
rockstar
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Joined: Mon Feb 03, 2020 5:51 pm

### Re: Filling the TIPS gap years with bracket year duration matching

Sounds like what you want is inflation protected income for those periods. You can could buy higher coupon TIPS that mature after this period and clip the coupons to cover the income since you don’t have any TIPS maturing. But you’d likely want the TIPS in tax deferred despite paying state tax to avoid the phantom income tax on those TIPS since you’d be over buying with longer maturity to have significant enough coupons.

Both the Feb 2040 and 2041 have 2.125% coupons.

This is not likely to be reasonable since you’d have to way over buy. So you’d need something else to supplement the income.
MtnBiker
Posts: 612
Joined: Sun Nov 16, 2014 3:43 pm

### Re: Filling the TIPS gap years with bracket year duration matching

Jaylat wrote: Thu May 23, 2024 9:44 am
I look forward to digging into the weeds in this thread, but from a practical point of view I may well just use the simple 1/7 weights for the 2034 / 2040 TIPS to estimate how many of each to sell to hedge a given gap TIPS purchase.
Perhaps this thread with dig into the weeds to the extent that estimates will be developed showing the expected range of price penalty/deviation between using rigorous duration matching vs. the simplified 1/6 weighting (maturity-matching approximation to duration matching).

I'm guessing that the maturity-matching approximation is simple enough for the "average Joe" to implement. It would be a benefit to the community if it can be shown mathematically that the maturity-matching approximation is "good enough" under just about any reasonable scenario.
Jaylat
Posts: 424
Joined: Sat Mar 12, 2016 10:11 am

### Re: Filling the TIPS gap years with bracket year duration matching

MtnBiker wrote: Thu May 23, 2024 10:23 am
Jaylat wrote: Thu May 23, 2024 9:44 am
I look forward to digging into the weeds in this thread, but from a practical point of view I may well just use the simple 1/7 weights for the 2034 / 2040 TIPS to estimate how many of each to sell to hedge a given gap TIPS purchase.
Perhaps this thread with dig into the weeds to the extent that estimates will be developed showing the expected range of price penalty/deviation between using rigorous duration matching vs. the simplified 1/6 weighting (maturity-matching approximation to duration matching).

I'm guessing that the maturity-matching approximation is simple enough for the "average Joe" to implement. It would be a benefit to the community if it can be shown mathematically that the maturity-matching approximation is "good enough" under just about any reasonable scenario.
Totally agree, that would be a significant benefit.

Also, and particularly for retirees with smaller TIPS ladders, you just don't have the ability to match the percentages perfectly. The 2040 TIPS has an inflation factor of 1.44x, so a \$10,000 real in 2040 TIPS gives you a grand total of 7 TIPS, period. You have to sell in increments of 14% of the 2040 TIPS portfolio - so they would be forced to use the 1/7 weighting no matter what the calculations show.
Raspberry-503
Posts: 1103
Joined: Sat Oct 03, 2020 6:42 am

### Re: Filling the TIPS gap years with bracket year duration matching

My ladder ends in 2039, which is a special case, and I'm doing the "n/6" calculation for now but I will definitely welcome the double-checking that I'm doing it right
MtnBiker
Posts: 612
Joined: Sun Nov 16, 2014 3:43 pm

### Re: Filling the TIPS gap years with bracket year duration matching

Jaylat wrote: Thu May 23, 2024 10:56 am

Also, and particularly for retirees with smaller TIPS ladders, you just don't have the ability to match the percentages perfectly. The 2040 TIPS has an inflation factor of 1.44x, so a \$10,000 real in 2040 TIPS gives you a grand total of 7 TIPS, period. You have to sell in increments of 14% of the 2040 TIPS portfolio - so they would be forced to use the 1/7 weighting no matter what the calculations show.
Good point about the inability of the small investor to match percentages perfectly.

My ladder is using bracket years of 2032 and 2040, and only has rungs in even years, with the final year being 2036. So far, I have only filled one gap-year rung (2034) and didn't really notice any problem matching the target percentages accurately. I guess having fewer, bigger rungs disqualifies me as a small investor, LOL.
Topic Author
Kevin M
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### Using modified duration to estimate price change relative to yield change

To calculate the modified durations mentioned in the OP I use the spreadsheet MDURATION (MD) function. I use Google Sheets, but Excel provides the same function.

I use modified duration instead of Macaulay duration (calculated with the DURATION (D) function) because modified duration relates price change to yield change directly. However, with current yields at least, using DURATION produces the same results, because there's not much difference between D and MD, and the smaller the magnitude of the yield, the smaller the difference.

You may have heard of the duration rule of thumb that lets us estimate the percentage price or NAV change of a bond or bond fund relative the percentage point (pp) change in yield. For example, we'd expect an increase in yield of 0.1 pp, say from 5.0% to 5.1%, to result in a price decline of 0.5%, say from 10 to 9.5 for a bond with a duration of 5 or fund with a weighted-average duration of 5. This rule of thumb works slightly better using modified duration than with Macaulay duration, and technically MD is what should be used.

You may also have heard that this rule of thumb works best for a bond fund or bond ladder under two constraints:
1. The change in yield is small.
2. Yields change due to a parallel shift in the yield curve; i.e., the percentage point change of all yields is the same.
The first constraint is due to the convexity of the price/yield curve, since duration is proportional to the slope of the line at a point on the curve (first derivative of price with respect to yield). Due to the convexity of the curve, the slope changes as yield changes. So we would expect the rule of thumb to work better for a 0.1 pp change than for a 1 pp change.

The second constraint is required for a fund or a ladder, because if yields don't change by the same amount, the average of the percentage price changes won't conform to the duration rule of thumb. We can demonstrate this with some calculations, but first, why is this relevant to gap year duration matching? It's relevant because the same principal applies, since if the yields of the bracket year TIPS don't change by the same amount, the estimated price of a gap year TIPS won't change as expected based on our duration matching; we'll come back to this later.

To demonstrate the parallel yield shift constraint, consider the following ladder or fund consisting of equal weights of 1y and 5y bonds:

Note that the average yield is 4.50% and the average MD is 2.81.

Now lets increase the average yield by 10 pp by increasing the yield of each bond by 10 pp:

Note that the average of the percentage price changes is -0.280%, which conforms quite nicely to the rule of thumb with an average MD of 2.81.

Now let's increase the average yield again by 10 pp, but by increasing the 5y by 20 pp and the 1y by 0 pp:

Here we see that the magnitude of the average of the percentage price changes is quite a bit larger than we'd expect based on the rule of thumb, which confirms that the rule of thumb doesn't work well for non-parallel yield curve shifts.

That's enough for this post. In the next one, I'll discuss how I estimate the expected durations of the gap year TIPS.
Last edited by Kevin M on Thu May 23, 2024 11:41 am, edited 1 time in total.
If I make a calculation error, #Cruncher probably will let me know.
Topic Author
Kevin M
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### Re: Filling the TIPS gap years with bracket year duration matching

MtnBiker wrote: Thu May 23, 2024 10:23 am
Jaylat wrote: Thu May 23, 2024 9:44 am
I look forward to digging into the weeds in this thread, but from a practical point of view I may well just use the simple 1/7 weights for the 2034 / 2040 TIPS to estimate how many of each to sell to hedge a given gap TIPS purchase.
Perhaps this thread with dig into the weeds to the extent that estimates will be developed showing the expected range of price penalty/deviation between using rigorous duration matching vs. the simplified 1/6 weighting (maturity-matching approximation to duration matching).

I'm guessing that the maturity-matching approximation is simple enough for the "average Joe" to implement. It would be a benefit to the community if it can be shown mathematically that the maturity-matching approximation is "good enough" under just about any reasonable scenario.

Or even just use a 50/50 split between the bracket years; or just use the #Cruncher default multipliers of 3 and 4 respectively, or the tipsladder.com defaults of 4 and 3.
If I make a calculation error, #Cruncher probably will let me know.
Topic Author
Kevin M
Posts: 16207
Joined: Mon Jun 29, 2009 3:24 pm
Contact:

### Estimating expected duration of gap year TIPS

To match the duration of a gap year TIPS with the bracket year TIPS, we need to estimate the expected duration of the gap year TIPS. The DURATION and MDURATION function parameters include maturity, yield and coupon, so we need to assign or estimate those.
• I assume that the gap year TIPS maturity is Jan 15 of the gap year; i.e., 1/15/2035, ..., 1/15/2039, which is what the first gap year TIPS issued for each gap year will be if Treasury continues their current auction schedule.
• To estimate yield I simply use linear interpolation of the bracket year TIPS (Jan 2034 and Feb 2040).
• I set the estimated coupon to what it would be if the TIPS were issued at the estimated yield.
I actually use the seasonally adjusted (SA) yields of the bracket year TIPS to do the linear interpolation, since I use SA yields for my purchase decisions, but the differences are small enough at these maturities as to not be significant.

Here is the table of estimated expected yields and coupons using today's TIPS yields at Schwab:

These values are used in the MDURATION function to calculate the ModDur values shown in the table in the OP.

Of course yields will change between now and when a gap year TIPS is issued, which means coupons and durations also are likely to be different than what I'm estimating now. This could require some periodic rebalancing of the bracket year TIPS holdings, depending on how accurate we want the duration matching to be. It could be that the required rebalancing could be done simply by adjusting the number of each bracket year TIPS that we sell to buy the next gap year TIPS that's issued. Also, as has been pointed out, the resolution of the duration matching is limited by the value of a single TIPS for each bracket year, and this might be larger than any shifts in the duration matching.

This raises the topic of my next post, which will discuss how well the duration matching works given different yield curve change scenarios.
If I make a calculation error, #Cruncher probably will let me know.
protagonist
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### Re: Filling the TIPS gap years with bracket year duration matching

Hi, Kevin.
This post is indirectly related to the topic.
I get how to fund the gap years , using 2034s and 2040s.
The question I have is, for those of us with very limited dry powder (it is all tied up in our TIPS ladder), and more than DARA in each of our existing rungs, what is the specifically best way to fully fund the excess 2034s and 2040s necessary to ultimately fund the gap years using funds from other maturities, or does it not matter much?.

Following is a post of mine in another thread raising that question, with specifics I would like answered:

protagonist wrote: ↑Tue May 21, 2024 2:04 am

"How important is duration and/or maturity matching with an essentially flat yield curve?

For example, consider if one was to sell TIPS maturing relatively soon and close in maturity date with widely divergent coupons ...say, for example, sell the same dollar amount of 4/15/27s with a 0.125% coupon, or 4/15/28s with a 3.625 coupon. These are two issues close in maturity date but with very different coupons.

If one was then going to use the proceeds from each sale to buy 2040s with a coupon lying between the two (2.125%), would the end result be significantly different? Would one be a "better deal" than the other? Or would they both have pretty much the same result?

And (another question): Would selling the above TIPS (at today's yields) to buy 2034's, for example, leave one with a much different result than selling the same dollar amount of 2033s (closer duration and maturity) to buy 2034s? How would it be different?

Forgive my ignorance about this....I'm asking as a person relatively unschooled in the world of bond investing. Bonds never really interested me before the recent TIPS revolution."

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### Re: Filling the TIPS gap years with bracket year duration matching

protagonist wrote: Fri May 24, 2024 11:33 am Hi, Kevin.
This post is indirectly related to the topic.
I get how to fund the gap years , using 2034s and 2040s.
The question I have is, for those of us with very limited dry powder (it is all tied up in our TIPS ladder), and more than DARA in each of our existing rungs, what is the specifically best way to fully fund the excess 2034s and 2040s necessary to ultimately fund the gap years using funds from other maturities, or does it not matter much?.

Following is a post of mine in another thread raising that question, with specifics I would like answered:

protagonist wrote: ↑Tue May 21, 2024 2:04 am

"How important is duration and/or maturity matching with an essentially flat yield curve?

For example, consider if one was to sell TIPS maturing relatively soon and close in maturity date with widely divergent coupons ...say, for example, sell the same dollar amount of 4/15/27s with a 0.125% coupon, or 4/15/28s with a 3.625 coupon. These are two issues close in maturity date but with very different coupons.

If one was then going to use the proceeds from each sale to buy 2040s with a coupon lying between the two (2.125%), would the end result be significantly different? Would one be a "better deal" than the other? Or would they both have pretty much the same result?

And (another question): Would selling the above TIPS (at today's yields) to buy 2034's, for example, leave one with a much different result than selling the same dollar amount of 2033s (closer duration and maturity) to buy 2034s? How would it be different?

Forgive my ignorance about this....I'm asking as a person relatively unschooled in the world of bond investing. Bonds never really interested me before the recent TIPS revolution."

Even though the yield curve is relatively flat, the 2040s have significantly higher duration than the 2034s; the SA yield difference is only 5 basis points, but the durations are 8.74 and 13.21 respectively. So you're still going to want to weight them differently to match the expected duration of a particular gap year TIPS.

I can't think of why the coupons matter much. I don't think much about coupons in buying and selling, other than preferring lower coupons when there's a choice, and even that doesn't really matter in a ladder, since the coupons just reduce the number of TIPS you need to buy in earlier years.

I would think more about what I want my average duration to be; do I want to increase it, keep it about the same, or decrease it? Since I'm way overweight in shorter-term TIPS, I generally want to increase it, so I've been selling the shorter maturity/duration TIPS to buy longer maturity/duration TIPS.

In your example, the durations of the Apr 2027 and Apr 2028 are 2.84 and 3.61 respectively, using SA yields, so I probably would sell the 2027 to increase my average duration.

Ditto for the comparison with the 2033s, which have durations of 8.11 for Jan and 8.46 for Jul. I'd swap the earlier 2027s to extend my average duration. If I wanted to keep my average duration close to what it was, I might swap the 2033s for 2034s, or I might just hold the 2033s.
If I make a calculation error, #Cruncher probably will let me know.
MtnBiker
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### Re: Filling the TIPS gap years with bracket year duration matching

protagonist wrote: Fri May 24, 2024 11:33 am Hi, Kevin.
This post is indirectly related to the topic.
I get how to fund the gap years , using 2034s and 2040s.
The question I have is, for those of us with very limited dry powder (it is all tied up in our TIPS ladder), and more than DARA in each of our existing rungs, what is the specifically best way to fully fund the excess 2034s and 2040s necessary to ultimately fund the gap years using funds from other maturities, or does it not matter much?.

Following is a post of mine in another thread raising that question, with specifics I would like answered:

protagonist wrote: ↑Tue May 21, 2024 2:04 am

"How important is duration and/or maturity matching with an essentially flat yield curve?

For example, consider if one was to sell TIPS maturing relatively soon and close in maturity date with widely divergent coupons ...say, for example, sell the same dollar amount of 4/15/27s with a 0.125% coupon, or 4/15/28s with a 3.625 coupon. These are two issues close in maturity date but with very different coupons.

If one was then going to use the proceeds from each sale to buy 2040s with a coupon lying between the two (2.125%), would the end result be significantly different? Would one be a "better deal" than the other? Or would they both have pretty much the same result?

And (another question): Would selling the above TIPS (at today's yields) to buy 2034's, for example, leave one with a much different result than selling the same dollar amount of 2033s (closer duration and maturity) to buy 2034s? How would it be different?

Forgive my ignorance about this....I'm asking as a person relatively unschooled in the world of bond investing. Bonds never really interested me before the recent TIPS revolution."

What was confusing about your original question was the underlined part. The rest of your post/question has nothing to do with duration matching and/or maturity matching. It is asking about selling some shorter maturities to buy some longer maturities, thereby extending duration with the associated reinvestment risk.

In this case, the shorter maturities you are considering selling were likely purchased before interest rates rose and longer-term prices fell more than shorter-term prices have fallen. Thus, the risks of reinvestment, in this case, appear to have moved in the direction that is in your favor when considering reinvestment moving toward longer average duration.

protagonist
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### Re: Filling the TIPS gap years with bracket year duration matching

MtnBiker wrote: Fri May 24, 2024 2:10 pm

Yes. Thanks, Kevin.

The main takes that I got from Kevin's response to my post are (correct me if I am wrong):
1. Long run: not much difference if I sell those with high coupons vs. low coupons. The inverse effect of coupon rate on duration is not great enough to make a difference. The only thing that should matter is whether you want increased or decreased cash flow prior to maturity.
2. Selling 4/2027s to buy 2040s will extend the mean duration of a ladder a bit more than selling 4/2028s.
3. Selling either 4/2027s or 4/2028s to buy 2040s at today's high yields is preferable to selling 2033s to buy 2040s- it extends duration more.
4. But selling correct proportions of 2034s and 2040s is better when buying 2035s, since it avoids reinvestment risk in case yields fall in 2025.

Correct?
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### Re: Filling the TIPS gap years with bracket year duration matching

protagonist wrote: Fri May 24, 2024 4:55 pm
MtnBiker wrote: Fri May 24, 2024 2:10 pm I hope Kevin's response answered your question.
Yes. Thanks, Kevin.

The main takes that I got from Kevin's response to my post are (correct me if I am wrong):
1. Long run: not much difference if I sell those with high coupons vs. low coupons. The inverse effect of coupon rate on duration is not great enough to make a difference. The only thing that should matter is whether you want increased or decreased cash flow prior to maturity.
2. Selling 4/2027s to buy 2040s will extend the mean duration of a ladder a bit more than selling 4/2028s.
3. Selling either 4/2027s or 4/2028s to buy 2040s at today's high yields is preferable to selling 2033s to buy 2040s- it extends duration more.
4. But selling correct proportions of 2034s and 2040s is better when buying 2035s, since it avoids reinvestment risk in case yields fall in 2025.

Correct?
1. Yes. To check, I changed the coupon of the Apr 2028 from 3.625% to 0.125%, and it changed the m duration from 3.61 to 3.83, so an increase of 0.22. Compare to the difference one year in maturity makes, with the Apr 2027 at an mdur of 2.85, so 98 basis points difference with both coupons at 0.125%.

2. Yes.

3. Yes, as long as increasing duration is a goal.

4. My emphasis now is more on what to hold rather than what to sell. It would translate into what to sell if yields were to change by the same amount (parallel yield curve shift); this is the topic of my next post. Also, although I'm focusing on the bracket years, 2034 and 2040, the same would apply if you were using other maturity years, as MtnBiker is doing with the 2032 and 2040.

I guess we could characterize it as minimizing reinvestment risk, but usually people are referring to reinvesting coupons or reinvesting principal as an issue matures or is sold before maturity, typically buying the same or longer maturity. I think of it more as minimizing price risk; i.e., that the average price of our bracket year TIPS, held in the correct proportions, will change by about the same amount as the hypothetical price of the gap year TIPS we'll be selling them to buy. It doesn't matter what we call as much as that we understand how it's supposed to work.

Note that any price changes happen while we're holding the bracket year TIPS, not when we sell them. I suspect that we might end up selling in slightly different proportions than what we originally intended if there were enough non-parallel yield curve shift, but I haven't thought this through yet.
If I make a calculation error, #Cruncher probably will let me know.
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### Testing duration match model with small, parallel yield curve shift

I've said that this duration matching scheme should work as long as yields at different maturities change by the same amount, which we call a "parallel yield curve shift". I've read and posted about this principle with respect to applying the duration rule of thumb that relates price or NAV change to yield change for a bond fund, and I think I must have done some calculations to verify this, but I can't remember distinctly doing so. So, I thought I'd better do that for this duration matching scheme.

Those who understand the duration rule of thumb know that it works better for smaller yield changes, so let's start with a small, parallel shift up in the yield curve, say 0.1 percentage point, or 10 basis points.

The ask yields of the 2030 and 2040 on Friday when I pulled quotes from Schwab were 2.15% and 2.23% respectively; to keep things simple, I'll use ask yields and prices here, and ignore any small seasonal adjustments.

If we increase the yields by 10 bps, to 2.25% and 2.33% for the 2034 and 2040 respectively, the prices change by -0.88% and -1.32% respectively, as calculated with the spreadsheet PRICE function. These changes are quite close to what is predicted by the duration rule of thumb; with modified durations of 8.74 and 13.20, we'd get price changes of -0.87% and -1.32%.

To calculate the estimated expected price change of the 2035 gap year TIPS, for example, and compare it to the weighted average price change of our duration matched TIPS, we do the following:
1. Increase the estimated yield of the 2035 by 10 bps from 2.16% to 2.26%.
2. Calculate the estimated price using the increased yield and the PRICE function, which comes to 98.70.
3. Calculate the percentage change in price, dp%, which is:

Code: Select all

``````dp% = 98.70/99.64 - 1
dp% = -0.94%.
``````
4. Calculate the weighted average percent price change of the duration matched TIPS, DM dp%. For the 2035, the weighs are 0.85 of the 2034 and 0.15 of the 2040, so the calculation is:

Code: Select all

``````DM dp% = 0.85 * -0.88% + 0.15 * -1.32%
DM dp% = -0.94%

(it's 0.95% with the rounded values above, but 0.94% with the unrounded values used in the spreadsheet calculations).
``````
5. Subtract dp% from DM dp%:

Code: Select all

``````DM dp% - dp% = 0.94% - 0.94%
DM dp% - dp%  = 0% (percentage points)
``````
And we have a winner!

Here is the chart showing the yields increased by 10 bps, price based on higher yields, dp%, DM dp%, and the delta between the latter two:

Looks good!

For completeness, here's the chart of estimated yields and prices for the gap year TIPS, so anyone who's interested can check my work:

In the next post I'll evaluate the effects of a larger change in yields, and of a non-parallel yield curve shift.
Last edited by Kevin M on Sun Jun 02, 2024 7:48 pm, edited 1 time in total.
If I make a calculation error, #Cruncher probably will let me know.
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Kevin M
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### Testing duration match model with non-parallel yield curve shift

How does gap duration matching work with larger yield changes? Let's try an increase of 1 percentage point, or 100 basis points. Having already walked through the calculations in my last post, I'll just show the results.

Anticipating evaluating non-parallel yield curve shifts, I've added a delta yield (dy) column.

Not bad. The delta of the duration weighted TIPS percent price change and the estimated gap year percent price change is a maximum of 4 bps. Note that this is despite the duration rule of thumb (rot) not working as well; i.e., for the 2034 the dp% was -8.38% compared to a rot estimate of -8.74%, and for the 2040 we see dp% of -12.32% compared to a rot estimate of -13.20%. So even though the duration rule of thumb doesn't work as well for larger yield changes, the duration matching seems to still work quite well for a relatively large parallel yield curve shift.

Now let's try a non-parallel yield curve shift, where the 2034 increases by 10 bps and the 2040 increases by 100 bps. Here are the results:

Here we see much larger deltas between DM dp% and dp%, with the duration weighted TIPS price falling 65 basis points more than the estimated price of the 2037.

On the other hand, the delta for the 2035 is only 23 bps, and for the 2039 it's only 27 bps. Note that the largest delta is the middle gap year, and that it falls off fairly symmetrically on either side.

One might think that a 23 bps delta for the 2035 is not such a big deal for a relatively sharp steepening of the yield curve. What do you think? I wonder how this would compare to the trading costs of rebalancing between the bracket years before the yield curve steepened too much. We might also evaluate if we can sell a different ratio than 85/15 of the 2034/2040 to reduce the delta and rebalance a bit at the same time. Haven't thought this through yet.
Last edited by Kevin M on Sun Jun 02, 2024 7:48 pm, edited 1 time in total.
If I make a calculation error, #Cruncher probably will let me know.
MtnBiker
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### Re: Filling the TIPS gap years with bracket year duration matching

Kevin M wrote: Sat May 25, 2024 3:12 pm
One might think that a 23 bps delta for the 2035 is not such a big deal for a relatively sharp steepening of the yield curve. What do you think? I wonder how this would compare to the trading costs of rebalancing between the bracket years before the yield curve steepened too much. We might also evaluate if we can sell a different ratio than 85/15 of the 2034/2040 to reduce the delta and rebalance a bit at the same time. Haven't thought this through yet.
Seems like you are on the right track. Assuming the yield curve changed significantly before the time you want to make the first swap (selling bracket years to buy 2035s), the 2034/2040 ratio to be sold may be something different than the 85/15 ratio originally calculated. And the ratios you want to hold until the next swap may be different than the original ratios when you started.

Rebalancing should help. Maybe assume annual rebalancing at the time of each swap and see how small the delta is at each of the 5 swaps. Maybe assume the yield curve alternately steepens/flattens a similar amount each year as a worst-case scenario?

EDIT: Thinking about this some more, rebalancing may not help much, if at all, since duration is fairly insensitive to yield. The duration of the bracket year TIPS will change very little as the yield changes. Changes in the duration of the gap year TIPS will primarily result from changes in coupon rates (when yields change enough to affect coupons). (I assume you included the changes in coupon rate of the gap years when you repriced them after the yield changes, correct?)

If the large deltas in the mid-gap persist even with rebalancing, minimizing the effects of non-parallel yield shifts may require rolling the bracket years (swap all excess 2034s for excess 2035s at the first opportunity, for example).
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### Re: Filling the TIPS gap years with bracket year duration matching

MtnBiker wrote: Sat May 25, 2024 4:23 pm Changes in the duration of the gap year TIPS will primarily result from changes in coupon rates (when yields change enough to affect coupons). (I assume you included the changes in coupon rate of the gap years when you repriced them after the yield changes, correct?)
I did not. I'll look at redoing the analysis with updated gap year TIPS estimated coupons.
If I make a calculation error, #Cruncher probably will let me know.
Jaylat
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### Re: Filling the TIPS gap years with bracket year duration matching

Going back to your original post, I do have a few questions / clarifications. Apologies if I’m being a little obtuse:

(1) If I understand correctly, the DARA weights you list are intended to be a multiplier for the current (2024) TIPS inflation adjusted amounts, not the TIPS face amounts or the TIPS current value. You might want to highlight that for TIPS newbies.

(2) I’m a little cross-eyed at the logic as to how you can just simply add the TIPS weights up across the gap years (which all have different inflation factors) and come up with a number exactly equal to the total number of years, i.e. seven. I get that each TIPS is inflation adjusted, but as some TIPS expire in 2034 how can you include them in a methodology that estimates an inflated amount that extends beyond that year? Certainly, if you used nominal numbers these figures would be all over the place.

(3) The tipladder.com methodology generates a different estimated amount of TIPS needed to bridge the 2034-40 gap. Part of this is the different 4:3 weighting, but much appears to be that tipladder.com uses the later year TIPS’ coupons as a part of the DARA for each year. If I understand correctly, your methodology does not include TIPS coupons, just the principal.
[I actually prefer not including the TIPS coupons, as I like to think of the TIPS coupons as being available to pay taxes on OID.]

For example, for a \$10,000 per year DARA, the tipsladder.com estimates:
\$36,000 face amount of 2034
\$21,000 face amount of 2024

3.56 x \$10,000 = \$35,600 inflation adjusted / 1.01544 = \$35,058 face amount of 2034
3.44 x \$10,000 = \$34,400 inflation adjusted / 1.44205 = \$23,850 face amount of 2040

Not a huge difference, I guess?
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Kevin M
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### Re: Filling the TIPS gap years with bracket year duration matching

Kevin M wrote: Sun May 26, 2024 11:11 am
MtnBiker wrote: Sat May 25, 2024 4:23 pm Changes in the duration of the gap year TIPS will primarily result from changes in coupon rates (when yields change enough to affect coupons). (I assume you included the changes in coupon rate of the gap years when you repriced them after the yield changes, correct?)
I did not. I'll look at redoing the analysis with updated gap year TIPS estimated coupons.
I'm very happy that you're participating in this journey, MtnBiker. Turns out that ignoring the coupon is a big hole in the analysis, not so much because of how it affects duration, but because of how it affects price.

It only took a few minutes of work to see the model blow up if I set coupon to what it would be for the different yields at auction. Then it only took a few seconds of thought to see how this should be obvious.

At auction, coupon will be very close to yield, so price will be very close to 100. So a model that hinges on theoretical price change of yet-to-be-issued TIPS based on a fixed coupon just doesn't make sense.

The next thought is that the focus on price is misplaced, since what we're buying is some multiple of DARA, where that multiple is the number of years covered by our ladder; e.g., if we have a 30 year ladder with one TIPS issue for each year, the multiplier for the row for each year will be 1, and the sum of the multipliers will be 30.

The price of the ladder is inversely related to TIPS yields, and to a much lesser extent to TIPS coupons. To illustrate and investigate this and some other points, I'll use a variation of the #Cruncher spreadsheet to model a TIPS ladder with a flat yield curve, and I'll set the yield and coupon to various values.

A 30y ladder with a DARA of \$100K, and yield = coupon = 2% costs \$2,263,425. Increase the yield to 3% but keep the coupon at 2%, and the cost decreases to \$1,996,062. If we also increase the coupon to 3%, the cost decreases further to \$1,990,953. Note that changing the yield by 100 basis points has a much larger impact on the cost than changing the coupon by 100 basis points.

However, a higher coupon means a larger contribution to earlier year cash flows, which could reduce the number of TIPS required for the earlier years to get our DARA. With a 2% par yield curve, we need 63 Jan 2033s to get close to our \$100K DARA for that year, and a total of 1,823 TIPS in our ladder, but if the 2034 had a coupon of 3%, we'd only need 60 Jan 2033s and 1,811 total (changing the 2034 yield does not impact how many we'd need).

So, I think that the tack to pursue is to investigate how change in expected yield and coupon of a gap year impacts the ladder overall.

My intuition is that there's still something to be said for at least approximately matching the durations of the gap years with the bracket years, but it seems that the analysis I've used to demonstrate this is flawed.
If I make a calculation error, #Cruncher probably will let me know.
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Kevin M
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### Re: Filling the TIPS gap years with bracket year duration matching

Jaylat wrote: Sun May 26, 2024 5:24 pm (2) I’m a little cross-eyed at the logic as to how you can just simply add the TIPS weights up across the gap years (which all have different inflation factors) and come up with a number exactly equal to the total number of years, i.e. seven.
This isn't my invention. The way the #Cruncher spreadsheet works, the total of the DARA multipliers must equal the number of years covered by the ladder. This applies to a single year or a span of years as well, if we want coverage for every year. Since there are 2 bracket years and 5 gap years, the sum of the bracket year multipliers must be 7.
Jaylat wrote: Sun May 26, 2024 5:24 pm (3) The tipladder.com methodology generates a different estimated amount of TIPS needed to bridge the 2034-40 gap. Part of this is the different 4:3 weighting, but much appears to be that tipladder.com uses the later year TIPS’ coupons as a part of the DARA for each year. If I understand correctly, your methodology does not include TIPS coupons, just the principal.
Not correct. Again, not my invention. Total annual real cash flows in the #Cruncher spreadsheet include real principal and real coupons, and the total of these is designed to be as close to DARA as possible with rounding and the resolution of individual TIPS costs.

The easiest way to see all of this is to open up the #Cruncher spreadsheet and investigate the various values.
If I make a calculation error, #Cruncher probably will let me know.
protagonist
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### Re: Filling the TIPS gap years with bracket year duration matching

Kevin, is the main reason that you want to maximize the duration of your ladder because you believe that interest rates will fall in the future?
Since increasing duration makes your ladder more interest rate sensitive, I would assume that is your logic- if rates fall , the value of your TIPS will rise proportionate to duration (if you sell prior to maturity).

If you are completely unsure about the direction of future interest rates, the result of increasing duration seems like a crapshoot. Or am I missing something?
MtnBiker
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### Re: Filling the TIPS gap years with bracket year duration matching

Kevin M wrote: Sun May 26, 2024 7:35 pm
I'm very happy that you're participating in this journey, MtnBiker. Turns out that ignoring the coupon is a big hole in the analysis, not so much because of how it affects duration, but because of how it affects price.

It only took a few minutes of work to see the model blow up if I set coupon to what it would be for the different yields at auction. Then it only took a few seconds of thought to see how this should be obvious.

At auction, coupon will be very close to yield, so price will be very close to 100. So a model that hinges on theoretical price change of yet-to-be-issued TIPS based on a fixed coupon just doesn't make sense.

The next thought is that the focus on price is misplaced, since what we're buying is some multiple of DARA, where that multiple is the number of years covered by our ladder; e.g., if we have a 30 year ladder with one TIPS issue for each year, the multiplier for the row for each year will be 1, and the sum of the multipliers will be 30.

The price of the ladder is inversely related to TIPS yields, and to a much lesser extent to TIPS coupons. To illustrate and investigate this and some other points, I'll use a variation of the #Cruncher spreadsheet to model a TIPS ladder with a flat yield curve, and I'll set the yield and coupon to various values.

A 30y ladder with a DARA of \$100K, and yield = coupon = 2% costs \$2,263,425. Increase the yield to 3% but keep the coupon at 2%, and the cost decreases to \$1,996,062. If we also increase the coupon to 3%, the cost decreases further to \$1,990,953. Note that changing the yield by 100 basis points has a much larger impact on the cost than changing the coupon by 100 basis points.

However, a higher coupon means a larger contribution to earlier year cash flows, which could reduce the number of TIPS required for the earlier years to get our DARA. With a 2% par yield curve, we need 63 Jan 2033s to get close to our \$100K DARA for that year, and a total of 1,823 TIPS in our ladder, but if the 2034 had a coupon of 3%, we'd only need 60 Jan 2033s and 1,811 total (changing the 2034 yield does not impact how many we'd need).

So, I think that the tack to pursue is to investigate how change in expected yield and coupon of a gap year impacts the ladder overall.

My intuition is that there's still something to be said for at least approximately matching the durations of the gap years with the bracket years, but it seems that the analysis I've used to demonstrate this is flawed.
I still think duration matching the gap years with the bracket years is the right approach. Let me throw out some ideas for you to consider for how to analyze this. First, as you said, the cost of a 30y ladder goes down if the yield increases from 2% to 3%.

Similarly, the cost of buying any gap year rung at auction (or soon after on the secondary market) should fall if the yield increases from 2% to 3%. (Just as the market values of the excess bracket holdings fall as yield increases.) For the purpose of evaluating the effectiveness of duration matching, shouldn't you be focusing on the changes in cost, not the changes in price? Yes, the price of the gap year TIPS purchased at auction is about 100, but the cost will vary with yield since you should need fewer or more bonds depending on which direction the yield changed.

Or maybe the figure of merit would be something like the delta in the present value of the future cash flows that occurs when you swap the bracket-year TIPS for the gap-year TIPS. If duration matching works, your buying power should be preserved after swapping. When duration matching is imperfect, buying power will either increase or decrease. (Not sure how to quantify "buying power;" just an idea.) When making the swap you can't keep the DARAs uniform because the changing coupons push the payouts around to different years. Even the expected principal payout at the gap-year maturity will change, I suppose.

Keeping the DARA from becoming non-uniform or lumpy should be a secondary consideration. When we swap bracket-year TIPS for gap-year TIPS, the ongoing coupon payments will change and that effects the payouts in the gap and in preceding years. But that is just an inevitable consequence of swapping TIPS from one coupon to another (and from changing maturity dates). (For example, when I made the swap from a mix of April 2032s/2040s to January 2034s, the coupon went from mostly 3.875%, plus some at 2.125%, to the new value of 1.75%, which defers a portion of the interest payments I would have received from my ladder from 2025-2032 until later years.) That affects the annual cash flows in various ways, but what we are trying to preserve by duration matching the gap years is the original buying power of the funds allocated toward each gap year. If the original "yield to maturity" (buying power?) is reasonably well maintained at the expense of perturbed cash flows, so be it.

Does any of this discussion help with the analysis?
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### Re: Filling the TIPS gap years with bracket year duration matching

Perhaps using I Bonds to cover the gap years might help simplify things.
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### Re: Filling the TIPS gap years with bracket year duration matching

Let me present my perspective as an "average Joe" who has made one of these swaps. Earlier this year I swapped April 2032s and 2040s for 2034s. I used the maturity-matching approximation, rather than strict duration matching, so the 2032/2040 excess holding ratio was 75/25.

It went something like this. (Not the actual numbers but generally illustrative of my experience.)
The 2032s were sold at a yield of 2.0%.
The 2040s were sold at a yield of 2.2%.
The 2034s were purchased using the proceeds of the sale at a yield of 1.95%.

From my perspective, the average yield of the TIPS that I sold to make the swap was 2.05%. The obvious "cost" of making the swap was the delta-yield of 10 basis points that I gave up (1.95 - 2.05 = -0.10%). That delta-yield has a certain delta-price which is the dollar cost of making the swap.

Totally unknown to me is any loss (or gain) in yield to maturity that I may have incurred during the initial holding period from when I made the original purchases (in 2018 and 2023) until the time of the swap (2024). At the time of the original purchases the average quoted yield to maturity was probably something like 1.2%, or thereabouts. That is the average yield to maturity I would receive if I kept the excess bracket year holdings to their respective maturities in 2032 and 2040. One might think that the average yield to maturity (from the times of original purchases) will now be close to 1.1%, since I lost 0.1% yield when making the swap. But that assumes perfect duration matching which is unlikely to have been the case. The effects of imperfect duration matching may have affected the yield to maturity in positive or negative ways that I can't quantify.

I can think of four factors that contribute to the costs of making a swap.
1) Vanguard's bid/ask spread. The bid/ask spread of whatever broker you use is unavoidable. Schwab is better than Vanguard, but I'm not going through the hassle of moving my IRA for that reason alone.
2) Nonlinearities in the yield curve. The April 2032 TIPS is an outlier with anomalously high yield. Its yield always lies above any fit to the yield curve. So, using that particular issue for holding excess bracket-year TIPS will always incur a small yield loss when it comes time to swap. This factor should be negligible for most other bracket-year holdings.
3) Changes in the slope of the yield curve (non-parallel shifts in yield). Duration-matching doesn't immunize against effects from non-parallel yield shifts.
4) Imperfect duration-matching with parallel yield shift. In my case, I used ratios for excess holding that are only an approximation to the ideal duration-matched ratio. Also, since the coupon of the gap yield TIPS is unknown ahead of time, so it is impossible to know the ideal duration-matched ratio when originally choosing the holdings ratio.

The 0.1% yield loss that I observed when making my swap was, I think, a result of factors (1) and (2). Quantifying the effect of factors (3) and (4) is the goal of this thread.
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### Re: Filling the TIPS gap years with bracket year duration matching

MtnBiker wrote: Sun May 26, 2024 11:07 pm Similarly, the cost of buying any gap year rung at auction (or soon after on the secondary market) should fall if the yield increases from 2% to 3%. (Just as the market values of the excess bracket holdings fall as yield increases.) For the purpose of evaluating the effectiveness of duration matching, shouldn't you be focusing on the changes in cost, not the changes in price? Yes, the price of the gap year TIPS purchased at auction is about 100, but the cost will vary with yield since you should need fewer or more bonds depending on which direction the yield changed.
The number of bonds needed differs only because of the change in the last year's interest due to the difference in coupon, so this change is maybe 1 bond or none for a 1 pp change in yield/coupon. This can be verified by zeroing out the Last Yr Interest per Bond value in the #Cruncher spreadsheet.

I'll just take the Jan 2025 as an example, assuming it's a par bond. Increasing yield/coupon from 2% to 3% changes the number needed from 66 to 65, but changing it to 2% to 1% doesn't change it at all.

So the cost of buying the gap year TIPS isn't going to change much due to yield changes, as it would if it had a fixed coupon, in which case price would change per my analysis. What's going to change are the values of the TIPS already in our ladder, and of course those values will change proportional to price, scaled by index ratio. Another impact of different gap year coupons is a possible change in the number of earlier year TIPS required, due to more or less interest from the gap year TIPS contributing to the annual real amounts.
If I make a calculation error, #Cruncher probably will let me know.
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### Re: Filling the TIPS gap years with bracket year duration matching

protagonist wrote: Sun May 26, 2024 8:02 pm Kevin, is the main reason that you want to maximize the duration of your ladder because you believe that interest rates will fall in the future?
Since increasing duration makes your ladder more interest rate sensitive, I would assume that is your logic- if rates fall , the value of your TIPS will rise proportionate to duration (if you sell prior to maturity).

If you are completely unsure about the direction of future interest rates, the result of increasing duration seems like a crapshoot. Or am I missing something?
It's not that I want to maximize duration, it's that I want to move toward a duration value that's appropriate for my investment horizon. If I assume a 20-year lifetime, I probably want a duration closer to 10 than to 6.

Of course just building a 20-year ladder with each rung equal to the same DARA gets me there, but I'm trying to nudge myself in that direction without just selling all of my extra 2025s-2027s in one shot to build out a full ladder. I'm doing this because of the point you raise--I'm OK with the reinvestment risk in the hopes of getting even higher yields for the longer TIPS.

Given that we haven't seen a nice TIPS yield bump in awhile, I'm leaning toward swapping some of my 2025s for more TIPS in the 2028-2030 range, since those prices are less sensitive to yield changes.

But we digress from the topic at hand.
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### Re: Filling the TIPS gap years with bracket year duration matching

Mel Lindauer wrote: Mon May 27, 2024 12:19 am Perhaps using I Bonds to cover the gap years might help simplify things.
This discussion here isn't about different ways to cover the gap years, but about one specific method to do so. There are infinite ways to cover the gap years, and there are plenty of threads discussing many of them.
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### Re: Filling the TIPS gap years with bracket year duration matching

MtnBiker wrote: Mon May 27, 2024 10:15 am 2) Nonlinearities in the yield curve. The April 2032 TIPS is an outlier with anomalously high yield. Its yield always lies above any fit to the yield curve. So, using that particular issue for holding excess bracket-year TIPS will always incur a small yield loss when it comes time to swap. This factor should be negligible for most other bracket-year holdings.
The apparently high yield of the April 2032 for a May 28 settlement is explained by seasonality, which is why I usually use seasonally-adjusted yields, perhaps modified with an outlier factor, in my analyses and purchase decisions.

We see that seasonal adjustment eliminates the anomalously high yield for the April 2032. Here are yields for it and adjacent maturities (these are from Schwab on Friday):

Although the quoted yield will often lie above a smoothed quoted yield curve, it won't always. It depends on the ratio of seasonal adjustment (SA) factors for settlement mm/dd and maturity mm/dd. Without getting too deep into it, the seasonal yield adjustment for an April TIPS will be upward for settlement dates between about Feb 3 and Apr 14, and downward the rest of the year.

However, it could be that an additional outlier factor is at play for the April 2032. Here's an SA chart I posted on Feb 28 of this year:
Kevin M wrote: Wed Feb 28, 2024 10:34 am
Here we see that although the SA yield adjustment is up, as expected, there's still a bit of a bump at Apr 2032, and I didn't apply quite enough of an outlier factor to eliminate it.

Here's another chart I posted on Mar 20, with some relevant commentary:
Kevin M wrote: Wed Mar 20, 2024 10:02 am To emphasize what jeffyscott said about seasonality, I decided to buy Jan and Jul maturities for my ladder where available. If I looked only at yields, I would favor Apr maturities over Jul maturities, but understanding that this is a result of seasonality, I'm comfortable buying the Jul maturities.

Note how much higher the Apr ask yields are than the Jul ask yields. Adjusting for seasonality mostly removes the sawtooth pattern in the ask yields.
If I make a calculation error, #Cruncher probably will let me know.
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### Re: Filling the TIPS gap years with bracket year duration matching

Kevin M wrote: Mon May 27, 2024 11:35 am
Mel Lindauer wrote: Mon May 27, 2024 12:19 am Perhaps using I Bonds to cover the gap years might help simplify things.
This discussion here isn't about different ways to cover the gap years, but about one specific method to do so. There are infinite ways to cover the gap years, and there are plenty of threads discussing many of them.
Wasn't trying to derail the discussion, Kevin. Rather, I was trying to offer a simple guaranteed inflation-protected option to cover the needed guaranteed inflation protection in the TIPS gap years when the options being discussed just seemed overly complicated.
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### Re: Filling the TIPS gap years with bracket year duration matching

Mel Lindauer wrote: Mon May 27, 2024 2:09 pm
Kevin M wrote: Mon May 27, 2024 11:35 am
Mel Lindauer wrote: Mon May 27, 2024 12:19 am Perhaps using I Bonds to cover the gap years might help simplify things.
This discussion here isn't about different ways to cover the gap years, but about one specific method to do so. There are infinite ways to cover the gap years, and there are plenty of threads discussing many of them.
Wasn't trying to derail the discussion, Kevin. Rather, I was trying to offer a simple guaranteed inflation-protected option to cover the needed guaranteed inflation protection in the TIPS gap years when the options being discussed just seemed overly complicated.
Right. There are many threads discussing I bonds vs. TIPS, and I don't want this to degenerate into one of those. Having said that, after thinking about it a bit more, I realized that someone who doesn't understand both I bonds and TIPS might not understand why I bonds wouldn't work for anything like the method we've been discussing. More importantly, after finding the fatal flaw in my analysis attempting to justify the bracket year duration approach, I've decided to expand the scope of the thread to cover mathematical analyses of gap year coverage methods, including implementation considerations. So we can discuss I bonds in that context; it may even help stimulate some useful thinking with respect to using TIPS to cover the gap years.

First, I'll just point out that the simple versions of the options that have been discussed are not complicated. You just buy extra 2034s and 2040s with the intention of later selling them to buy the gap year TIPS as they become available, which will be in 2025 through 2029 (so in five years, this discussion will be moot). The two tips ladder tools I've mentioned offer simple defaults to accomplish this, as discussed earlier. What's complicated is trying to come up with the math to determine an optimal solution, if there is one.

I bonds have a duration of 0, since value doesn't vary with changing market yields. Although this doesn't work for the duration matching scheme that has been discussed, we may find that that scheme doesn't have sufficient analytical underpinnings to support it. So we can put this aside for now.

The current I bond real rate is 1.30%, while the 2034 and 2040 TIPS yields are greater than 2%; actually all TIPS currently have yields north of 2%. The effective I bond yield is even lower if we want to buy the gap years as soon as available, since we'd be paying the 3-month interest penalty in redeeming the I bonds in less than 5 years from purchase.To favor I bonds over any TIPS at current yields, we'd have to determine a justification for sacrificing the extra yield of the TIPS.

How about getting "a simple guaranteed inflation-protected option to cover the needed guaranteed inflation protection in the TIPS gap years"? I assume the concern here is that with longer-maturity TIPS, like the 2034 or 2040, the values will fluctuate as market yields fluctuate. Yields decreasing is not an issue, since the value of our TIPS will increase, and we'd then have more than we planned on to buy the gap year TIPS when issued. However higher yields would cause a loss of value in the longer TIPS we're holding to cover the gap years. This wouldn't be an issue if the initial assumption that the gap year TIPS coupon is fixed at what it would be if issued today held, since duration matching would solve that, but obviously that assumption is incorrect.

My next thought is to buy extra Jan 2025 TIPS to buy the 2035 to be issued in Jan 2025, buy extra Jan 2026 TIPS to cover the 2036 to be issued in Jan 2026, etc. With this solution, there's no unadjusted price uncertainty, so we get the desired guaranteed inflation protection. Not only that, but we get yields between 2.20% for the Jan 2029 and 3.4% for the Jan 2025 (inverted yield curve), so even better than the yields on the 2034 and 2040, and of course much better than the I bond real rate.

Of course there's some uncertainty in the nominal values at maturity, but that's also a problem for I bonds.

The fatal flaw in using I bonds to cover the gap years, for other than the DARAs of say \$10K or less, is that pesky annual purchase limit. For a TIPS ladder that generates \$50K/year in real annual cash flows, we need roughly \$50K to cover one gap year, so to cover the five gap years we'd need roughly \$250K. A couple of years ago when the I bond composite rate was north of 9%, DW and I were able to buy \$70K between individual and entity accounts and gifts. So we could have covered one gap year with that for a DARA of up to \$70K.

The rules of the game here are that we can't go back in time when the I bond annual purchase limits were higher (and you could buy I bonds with a credit card), and we are assuming we're building our ladder today when TIPS yields are historically attractive. We need to cover the gap years with purchases today anyway if we want the inflation protection starting today.

Now for those who've bought a bunch of I bonds in the past, and who have no other particular use for them, then sure, you could use those I bonds to cover the gap years if you have enough of them. But for any I bonds with a fixed rate of less than 2%, which is any I bonds bought since Nov 2002, why not sell them and buy the 2025-2029 TIPS at significantly higher yields?

This brings to mind the topic of asset location. I bonds can't be held in IRAs, so anyone building their TIPS ladder in an IRA, which includes me, can't use I bonds. Anyone building some or all of their ladder in taxable might be able to use I bonds, in which case you'd need to compare the benefits of deferring taxes for 1-5 years on the lower yields with switching to TIPS with the higher yields. Hard to imagine deferring taxes a year or two makes much difference, but everyone's tax situation is different.

'Nuff said?
If I make a calculation error, #Cruncher probably will let me know.
Jaylat
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### Re: Filling the TIPS gap years with bracket year duration matching

Just spit-balling some ideas here:

One way to hedge the gap years would be to buy 100% 2040 TIPS and 0% 2034 TIPS. You could then hedge the early redemption of the 2040’s by shorting an equal amount of 2040 treasuries and buying the gap year treasuries. [Not sure how this would work in practice, or if it’s doable for a retail investor, but it’s possible for a financial institution. Also, you’d be hedging real TIPS with nominal T Bonds so maybe not such a great hedge after all?]

Following with the strategy of buying 100% 2040 TIPS, a retail investor could concoct a “hedge” of sorts by using a smaller portfolio of I Bonds as a hedge against early redemption of the 2040 TIPS in a high real yield / high coupon environment. If rates are down, you sell the TIPS at a profit; if rates are up, you sell the I Bonds at par and get a higher yield. Either way you’re covered.

You could also do a split of 50% 2040 TIPS and 50% I Bonds. Worst case scenario is you have to sell all the I Bonds in the first 2-3 years due to higher rates. In that case the remaining 2040 TIPS are only 2-3 years from maturity, so the hit to sell early should be significantly reduced.

The problem here is it requires a portfolio of I Bonds which not all of us have. Also, you’d be giving up a lot of yield to buy the I Bonds, making for an expensive hedge.
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### Re: Filling the TIPS gap years with bracket year duration matching

Kevin M wrote: Mon May 27, 2024 10:52 am
The number of bonds needed differs only because of the change in the last year's interest due to the difference in coupon, so this change is maybe 1 bond or none for a 1 pp change in yield/coupon. This can be verified by zeroing out the Last Yr Interest per Bond value in the #Cruncher spreadsheet.

I'll just take the Jan 2025 as an example, assuming it's a par bond. Increasing yield/coupon from 2% to 3% changes the number needed from 66 to 65, but changing it to 2% to 1% doesn't change it at all.

So the cost of buying the gap year TIPS isn't going to change much due to yield changes, as it would if it had a fixed coupon, in which case price would change per my analysis. What's going to change are the values of the TIPS already in our ladder, and of course those values will change proportional to price, scaled by index ratio. Another impact of different gap year coupons is a possible change in the number of earlier year TIPS required, due to more or less interest from the gap year TIPS contributing to the annual real amounts.
I am having trouble following what you are saying. In the TIPS Ladder Spreadsheet thread you wrote:
Kevin M wrote: Tue May 28, 2024 11:03 am
I've always assumed that this just made sense based on some sort of duration matching scheme, but in my thread, Filling the TIPS gap years with bracket year duration matching - Bogleheads.org, we've discovered that duration matching doesn't really work because the gap year TIPS price will always be close to 100. Duration matching requires that the price of the the TIPS between the bracket years varies with yields, as do the prices of the bracket year TIPS, and this assumption doesn't apply for new issues at auction.

I disagree with the statement that duration matching doesn't really work. Here is a simple thought experiment to show that it does still work.

Suppose I want to fund 2037 at the 20K level. I do this by buying 10K of 2034 at 2% yield and 10K of 2040 at 2% yield. (50/50 maturity matching approximation to duration matching.)

Suppose I want to make the swap in 2027. The "duration" (maturity) of the 2034 is 7 years. The duration of the 2040 is 13 years and the duration of the new-issue 2037 is 10 years.

Also suppose that just before I make the swap, the yield jumps 1% across the entire yield curve. The new yield is 3% for the 2034 and the 2040. The price of the 2034 falls 7%. The price of the 2040 falls 13%. The average price of the excess holdings falls 10% just before I sell to make the swap.

So, using the proceeds of the sale, I am short 10% and have to buy 10% less of the 2037 compared to what I would have bought if interest rates hadn't jumped. But the 2037 is now yielding 3% instead of 2%. Thus, I am gaining back 1% every year that I hold the 2037. By the end of the 10 years held, I will have made up 1%/year x 10 years = 10%. I will be made whole by the time of maturity in 2037.

The only thing that has changed is the cash flows. After the swap, more of the payout comes in the form of coupons, but the overall yield is the same. (If I wanted to spend all the cash flow in 2037, I would need to reinvest the excess portion of the coupons (1%/yr) to make up for the 10% shortfall in the principal at maturity.)
Jaylat
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### Re: Filling the TIPS gap years with bracket year duration matching

Agree with MtnBiker's logic, the higher coupons should offset the more expensive TIPS.

Also not following this statement:
Kevin M wrote: Tue May 28, 2024 11:03 am After thinking about it, it seems that currently we could more effectively cover the 2035 gap year with a Jan 2025 TIPS, for example, since there is more certainty of the nominal return when the 2035 is issued in Jan 2025, and the yield is much higher than the 2034 or 2040 TIPS.

The whole point of hedging the 2035 now is that we have no idea what the 10 year TIPS yield will be in Jan 2025. Buying a Jan 2025 TIPS, even with a much higher yield, does nothing for hedging longer term real yields, such as the to be issued 2035 TIPS.

Who cares if your real yields are higher for the next 7 months? It's the next 10 years you should be concerned about.
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### Re: Filling the TIPS gap years with bracket year duration matching

MtnBiker wrote: Tue May 28, 2024 2:39 pm
Kevin M wrote: Mon May 27, 2024 10:52 am The number of bonds needed differs only because of the change in the last year's interest due to the difference in coupon, so this change is maybe 1 bond or none for a 1 pp change in yield/coupon. This can be verified by zeroing out the Last Yr Interest per Bond value in the #Cruncher spreadsheet.

I'll just take the Jan 2025 as an example, assuming it's a par bond. Increasing yield/coupon from 2% to 3% changes the number needed from 66 to 65, but changing it to 2% to 1% doesn't change it at all.

So the cost of buying the gap year TIPS isn't going to change much due to yield changes, as it would if it had a fixed coupon, in which case price would change per my analysis. What's going to change are the values of the TIPS already in our ladder, and of course those values will change proportional to price, scaled by index ratio. Another impact of different gap year coupons is a possible change in the number of earlier year TIPS required, due to more or less interest from the gap year TIPS contributing to the annual real amounts.
I am having trouble following what you are saying. In the TIPS Ladder Spreadsheet thread you wrote:
Kevin M wrote: Tue May 28, 2024 11:03 am
I've always assumed that this just made sense based on some sort of duration matching scheme, but in my thread, Filling the TIPS gap years with bracket year duration matching - Bogleheads.org, we've discovered that duration matching doesn't really work because the gap year TIPS price will always be close to 100. Duration matching requires that the price of the the TIPS between the bracket years varies with yields, as do the prices of the bracket year TIPS, and this assumption doesn't apply for new issues at auction.

I disagree with the statement that duration matching doesn't really work. Here is a simple thought experiment to show that it does still work.

Suppose I want to fund 2037 at the 20K level. I do this by buying 10K of 2034 at 2% yield and 10K of 2040 at 2% yield. (50/50 maturity matching approximation to duration matching.)

Suppose I want to make the swap in 2027. The "duration" (maturity) of the 2034 is 7 years. The duration of the 2040 is 13 years and the duration of the new-issue 2037 is 10 years.

Also suppose that just before I make the swap, the yield jumps 1% across the entire yield curve. The new yield is 3% for the 2034 and the 2040. The price of the 2034 falls 7%. The price of the 2040 falls 13%. The average price of the excess holdings falls 10% just before I sell to make the swap.

So, using the proceeds of the sale, I am short 10% and have to buy 10% less of the 2037 compared to what I would have bought if interest rates hadn't jumped. But the 2037 is now yielding 3% instead of 2%. Thus, I am gaining back 1% every year that I hold the 2037. By the end of the 10 years held, I will have made up 1%/year x 10 years = 10%. I will be made whole by the time of maturity in 2037.

The only thing that has changed is the cash flows. After the swap, more of the payout comes in the form of coupons, but the overall yield is the same. (If I wanted to spend all the cash flow in 2037, I would need to reinvest the excess portion of the coupons (1%/yr) to make up for the 10% shortfall in the principal at maturity.)
Another excellent contribution to our investigation!

When I say that duration matching doesn't work, I mean that in the pure sense. By that I mean the way it would work if all securities involved were trading on the secondary market, in which case the math showed that it works almost perfectly for a parallel yield curve shift. By working perfectly, I mean that the proceeds from selling the expected amounts of the bracket year TIPS would enable us to buy the DARA amount of the gap year TIPS.

Your thought experiment results are consistent with this, in that we could only buy 90% of the DARA amount for 2037 in 2027. The rest of the thought experiment relies on the fact that new-issue TIPS bought at auction, which are close to par both unadjusted and adjusted, deliver most of their yield in the form of coupon payments.

My initial thinking about this, before reading your latest post, was along the lines of the annual real amounts (ARA) being larger than DARA, because the coupons of the 2037 (or whatever gap year we're interested in) would be larger than those of the 2034 or 2040. I hadn't thought through what we'd do with the extra ARAs, but of course your thought experiment posits that we'd basically bank them to contribute to the 2037 ARA.

Of course the ideal way to bank the coupons would be to reinvest them at the auction yield of the 2037 (approximately the coupon rate), in which case the end result would be the same as if we had used classical duration matching with all marketable securities. Of course we can't do this, so there is some uncertainty as to what the ARA contribution from the reinvested coupons will be, and I think this is the only possible quibble with your analysis.

The realistic alternatives are that we invest the coupons in a money market fund until we have enough to buy an additional 2037 (and repeat), or put them into a TIPS fund or combination of funds with an appropriate duration, and again, perhaps buy another 2037 when we have enough.

Perhaps the reinvestment rates aren't a big enough deal to worry much about, but it might be worth doing the math to investigate. What if real yields return to negative territory and remain there for an extended period of time?
If I make a calculation error, #Cruncher probably will let me know.
MtnBiker
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### Re: Filling the TIPS gap years with bracket year duration matching

Kevin M wrote: Tue May 28, 2024 5:36 pm

When I say that duration matching doesn't work, I mean that in the pure sense. By that I mean the way it would work if all securities involved were trading on the secondary market, in which case the math showed that it works almost perfectly for a parallel yield curve shift. By working perfectly, I mean that the proceeds from selling the expected amounts of the bracket year TIPS would enable us to buy the DARA amount of the gap year TIPS.

Your thought experiment results are consistent with this, in that we could only buy 90% of the DARA amount for 2037 in 2027. The rest of the thought experiment relies on the fact that new-issue TIPS bought at auction, which are close to par both unadjusted and adjusted, deliver most of their yield in the form of coupon payments.

My initial thinking about this, before reading your latest post, was along the lines of the annual real amounts (ARA) being larger than DARA, because the coupons of the 2037 (or whatever gap year we're interested in) would be larger than those of the 2034 or 2040. I hadn't thought through what we'd do with the extra ARAs, but of course your thought experiment posits that we'd basically bank them to contribute to the 2037 ARA.

Of course the ideal way to bank the coupons would be to reinvest them at the auction yield of the 2037 (approximately the coupon rate), in which case the end result would be the same as if we had used classical duration matching with all marketable securities. Of course we can't do this, so there is some uncertainty as to what the ARA contribution from the reinvested coupons will be, and I think this is the only possible quibble with your analysis.

The realistic alternatives are that we invest the coupons in a money market fund until we have enough to buy an additional 2037 (and repeat), or put them into a TIPS fund or combination of funds with an appropriate duration, and again, perhaps buy another 2037 when we have enough.

Perhaps the reinvestment rates aren't a big enough deal to worry much about, but it might be worth doing the math to investigate. What if real yields return to negative territory and remain there for an extended period of time?
If interest rates decrease, post-swap coupon payments will fall, and principal payouts will increase in the gap years. Reinvestment won't be a thing.

My suggestion for continuing this investigation is to focus on trying to find a method for evaluating how well duration matching maintains the yield to gap-year maturity and ignore how much the cash flow changes due to the swaps. The changes in coupons and principal payouts may be both positive and negative (depending on which direction interest rates move) and may partially cancel out over time. Some people will want perfectly uniform DARAs, so may reinvest coupons (if they rise instead of shrink), and others won't care and will accept the slightly lumpy payouts in their income floor. Evaluating the effects of reinvestment may be worth investigating later after the basic questions posed in this thread have been addressed.
phoroner
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### Re: Filling the TIPS gap years with bracket year duration matching

Seems to me the issue is convexity. If yields rise, the duration of new issue TIPS will decrease (due to higher coupon rates). This will introduce a duration mismatch with currently matched durations for 2035-2039 years. The market value of 2034 & 2040 bonds will be depressed, so when sold to buy new-issue bonds, the final-year maturity values won’t be able to be matched perfectly. But the higher coupon payments could be saved and re-invested in TIPS over time, restoring duration. So it would still work out. The same risk would not apply so much to decreased rates.

Do I have this right?
MtnBiker
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### Re: Filling the TIPS gap years with bracket year duration matching

MtnBiker wrote: Tue May 28, 2024 2:39 pm
I disagree with the statement that duration matching doesn't really work. Here is a simple thought experiment to show that it does still work.

Suppose I want to fund 2037 at the 20K level. I do this by buying 10K of 2034 at 2% yield and 10K of 2040 at 2% yield. (50/50 maturity matching approximation to duration matching.)

Suppose I want to make the swap in 2027. The "duration" (maturity) of the 2034 is 7 years. The duration of the 2040 is 13 years and the duration of the new-issue 2037 is 10 years.

Also suppose that just before I make the swap, the yield jumps 1% across the entire yield curve. The new yield is 3% for the 2034 and the 2040. The price of the 2034 falls 7%. The price of the 2040 falls 13%. The average price of the excess holdings falls 10% just before I sell to make the swap.

So, using the proceeds of the sale, I am short 10% and have to buy 10% less of the 2037 compared to what I would have bought if interest rates hadn't jumped. But the 2037 is now yielding 3% instead of 2%. Thus, I am gaining back 1% every year that I hold the 2037. By the end of the 10 years held, I will have made up 1%/year x 10 years = 10%. I will be made whole by the time of maturity in 2037.

The only thing that has changed is the cash flows. After the swap, more of the payout comes in the form of coupons, but the overall yield is the same. (If I wanted to spend all the cash flow in 2037, I would need to reinvest the excess portion of the coupons (1%/yr) to make up for the 10% shortfall in the principal at maturity.)
Circling back to carry this thought experiment one step further. I'm not up to speed on the spreadsheet functions needed to evaluate this scenario with full accuracy, but I can get Modified Duration (MD) from an online calculator. Using the actual duration to estimate the loss in market value when interest rates change will improve the accuracy of the calculations.

In January 2027 when the interest rates jump 1% hypothetically, the MD of the Jan 2034 TIPS (1.75% coupon) is about 6.5 yr and the MD of the Feb 2040 TIPS (2.125% coupon) is about 11.25 yr. (For the 2040, I get 11.33 yr using 2% yield and 11.17 yr using 3% yield; not knowing which to use, I took the midpoint.)

The principal values in 2027 will no longer be exactly 50/50, but ignoring that detail, and again using the rough approximation that the market value falls 1% for each year of duration when interest rates increase 1%, the net loss in market value is estimated to be about -(6.5+11.25)/2 = -8.875%

Accounting for the subsequent 1%/yr increase in yield when holding the newly-issued 2037 for 10 years, the net change in value at maturity in 2037 is estimated to be (1 - 0.08875)(1 + (10)(0.01)) = 1.002

Again, close enough.
FoolStreet
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### Re: Filling the TIPS gap years with bracket year duration matching

I feel like there is some really good stuff! In general, what is the benefit of a tips ladder vs just a plain old asset allocation and safe withdrawal rate?
protagonist
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### Re: Filling the TIPS gap years with bracket year duration matching

FoolStreet wrote: Wed May 29, 2024 1:06 pm I feel like there is some really good stuff! In general, what is the benefit of a tips ladder vs just a plain old asset allocation and safe withdrawal rate?
Inflation protection and a guaranteed real return backed by the treasury.
It does not preclude having assets allocated elsewhere as well if desired, so it is not one or the other.
Topic Author
Kevin M
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### Re: Filling the TIPS gap years with bracket year duration matching

I think there are multiple goals in attempting to duration match a new issue TIPS with bracket year TIPS, which isn't the case when duration matching using all secondary market issues; i.e., using longer-term and shorter-term TIPS or TIPS funds to match the duration of an intermediate-term TIPS or fund. This complicates the mathematical analysis.

The goals that come to mind are:
1. Generate a known nominal amount for purchase of the gap year TIPS when issued. It's a nominal amount because one new-issue 10-year TIPS will cost close to \$1,000, since unadjusted price will be close to 100, and index ratio will be close to 1. This isn't exact, because new issues aren't actual par bonds; the larger the difference between coupon and yield, the more deviation from a price of 100. The low and high prices for new-issue 10y TIPS auctioned to date are 98.881 and 100.447 respectively, so cost has ranged from \$988.81 to 1,004.47, which is a delta of +0.45% to -1.12%. The average cost has been \$994.25.
2. "Lock in" historically attractive longer-term real yields. We aren't exactly locking them in, since we'll be selling well before maturity, but this is where the duration matching theory comes into play, and needs to be mathematically analyzed for different yield-curve change scenarios, different coupon reinvestment rates, and different coupon reinvestment strategies. This may be of less interest when 10y yields are historically unattractive, as they were in recent years.
3. Hedge unexpected inflation risk from now until issuance of a gap-year TIPS. This point was emphasized by Mel wrt I bonds, but of course there there are different ways to do it with TIPS as well.
There may be more, but these come to mind immediately, and provide fodder for some discussion.

The safest way to achieve goal #1 would be with a STRIPS (zero-coupon nominal Treasury) maturing close to the gap year TIPS issue date, since it provides a known amount of nominal dollars at maturity. A low-coupon Treasury would be a reasonable alternative. Neither TIPS nor I bonds do this as reliably.

The way to achieve #2 is with longer-term TIPS. I bonds don't do this, and longer-term nominal Treasuries don't do this.

The most reliable way to achieve #3 with TIPS would be to buy TIPS maturing close to the gap-year issue date. This provides as much reliable inflation protection as I bonds, keeping in mind the different ways the inflation adjustments are done, and that both methods lag actual inflation. And of course using TIPS is a generalizable solution, unlike I bonds due to annual purchase limits.

So it seems the goal is to find a solution that optimizes achievement of the three subgoals, and to be able to use math to prove it.

Although I'd want to get to the math, consider this thought experiment.
• Shortly after purchasing the bracket year TIPS at more than 2% yield, real yields drop back into negative territory, and inflation drops close to 0%, or perhaps below. The latter hasn't happened for an extended period in a long time, but it has happened, and we're looking for certainty in ARA regardless of economic conditions.
• We reinvest all of our coupons at negative real yields, and possibly negative nominal yields.
• Shortly before the issuance of the gap year TIPS of interest, yields shoot up to historical highs or higher. The 10y TIPS hit 4.40% on Jan 18, 2000. Of course this drives long term prices way down.
• The gap year TIPS is issued at a very attractive yield, but still at a cost of close to \$1,000 per bond, and we now sell our bracket-year TIPS at extremely depressed prices, with no or negative earnings from the coupons already paid out.
How does duration matching work in this scenario?

Trust me, I want duration matching to work, since I've already bought all the 2034s and 2040s I need to buy the gap years at my current DARA.
If I make a calculation error, #Cruncher probably will let me know.
FoolStreet
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### Re: Filling the TIPS gap years with bracket year duration matching

protagonist wrote: Wed May 29, 2024 1:42 pm
FoolStreet wrote: Wed May 29, 2024 1:06 pm I feel like there is some really good stuff! In general, what is the benefit of a tips ladder vs just a plain old asset allocation and safe withdrawal rate?
Inflation protection and a guaranteed real return backed by the treasury.
It does not preclude having assets allocated elsewhere as well if desired, so it is not one or the other.
Can you elaborate?

If I want to retire with a 65/35 stock/bond mix, is the TIPS-crowd suggesting that the 35% goes 100% into the TIPS ladder? Or is the TIPS-crowd saying, put 100% of everything into a TIPS ladder? Or further, is something like, 20 years at 4%/year is 80%, so put 80% in the TIPS ladder and *then* split the remaining 20% of the portfolio into 65/35...? or??
GAAP
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### Re: Filling the TIPS gap years with bracket year duration matching

FoolStreet wrote: Wed May 29, 2024 3:32 pm
protagonist wrote: Wed May 29, 2024 1:42 pm
FoolStreet wrote: Wed May 29, 2024 1:06 pm I feel like there is some really good stuff! In general, what is the benefit of a tips ladder vs just a plain old asset allocation and safe withdrawal rate?
Inflation protection and a guaranteed real return backed by the treasury.
It does not preclude having assets allocated elsewhere as well if desired, so it is not one or the other.
Can you elaborate?

If I want to retire with a 65/35 stock/bond mix, is the TIPS-crowd suggesting that the 35% goes 100% into the TIPS ladder? Or is the TIPS-crowd saying, put 100% of everything into a TIPS ladder? Or further, is something like, 20 years at 4%/year is 80%, so put 80% in the TIPS ladder and *then* split the remaining 20% of the portfolio into 65/35...? or??
A TIPS ladder is deterministic -- it provides a known stream of real income for a known period of time.

An SWR strategy is probabilistic -- it provides a known stream of real income for an unknown period of time.

The "TIPS crowd" uses TIPS in a large variety of ways.

Some use TIPS instead of nominal bonds in a otherwise "typical" AA. Some of those folks are in accumulation, while others are using SWR or other withdrawal methods.

Some crowd members are using a mix of TIPS and nominal bonds in that otherwise "typical" AA.

Some members are building ladders, others are not. Some are duration-matching, others are not. Some are purchasing individual TIPS, others are using funds.

To determine how to use or not use TIPS, you should first determine what you wish the portfolio to achieve, then what role fixed income should play, and finally what role TIPS could play.
“Adapt what is useful, reject what is useless, and add what is specifically your own.” ― Bruce Lee