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

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

Kevin M wrote: Wed Jul 10, 2024 5:28 pm So we now have two tracks going to evaluate the results of different gap coverage methods. I'll continue a bit on the track I've been on, but now using #Cruncher's version 2 of the simplified ladder spreadsheet, which ensures that total ARA equals total DARA, regardless of whether or not any gap years are filled.

The current model has these features:
• Assumes all gap years are filled in 2029, with the 2025-2029 proceeds used for expenses.
• So the ladder now is a 25-year ladder instead of a 30y ladder, with terms to maturity reduced by 5 years each.
In this post I'll only cover the bracket year method, using initial multipliers for 2040 and 2034 based on duration matching as discussed in the first few posts of this thread.

First, here is the ladder before any gaps are filled, with the rows for 2041-2054 hidden since they aren't of particular interest, other than providing "Interest later bonds" for the maturities that are shown.

Things to note:
• The total proceeds (aka total ARA), in cell J32 with the cursor focus, now equals total DARA of 2.5M for the 25y ladder. Recall that previously this was greater than total DARA; this is the result of the change for version 2.
• The multipliers for the 2040 and 2034 of 3.43 and 3.57 are based on duration matching at 2% yields.
• The terms to maturity, in column C, are all reduced by five years, so, for example, the 2034 now is a 5y bond and the 2040 is an 11y bond.
• The multipliers for the 2025-2029 are all 0, since these are assumed to have been consumed. The only impact on the analysis is that these are not involved in any sales or purchases necessary to match ARA to DARA for the pre-2034 maturities.
• The gap year coupons are irrelevant with no gaps yet filled.
Here's the way it looks after filling the five gap years, assuming gap year cost and yield of 2%:

Things to note:
• The only relevant cost column for this scenario, one of the three covered here, is Cost at 2%, since I'm assuming all yields are 2% from 2030-2040; I've highlighted the relevant cost cells, which include the gap years since we're buying them, the bracket years since we're selling them, and the 2030-2033 since we will buy or sell them so that ARA = DARA.
• Total proceeds still equals 2.5M, as it should.
The two charts in this post show the Version 2 spreadsheet for 2% yield (yield unchanged) and multipliers M equal to 3.43 and 3.57 for 2040 and 2034, respectively. One can compare the principal and cost of the 2034 and 2040 rungs before and after the swaps. The Delta-principal and Delta-cost for these rungs shows the amount of excess holdings sold to make the swaps.

Delta-principal for 2040 is 238,351
Delta-principal for 2034 is 159,099
2040/2034 excess holdings ratio is 1.498:1 (based on principal)

Delta-cost for 2040 is 241,267
Delta-cost for 2034 is 157,224
2040/2034 excess holdings ratio is 1.535:1 (based on cost)

The excess holdings at 2% yield for 2040 are about 50% higher than for 2034 even though the multipliers are about equal (2040/2034 = 0.961:1). This shows why I am thinking that using multipliers fails to achieve the desired ratio of excess holdings.

EDIT: Another observation from the first chart above is that for a fixed pair of multipliers, the ratio of excess holdings costs seems to vary depending on the prevailing yield at the time. This implies that if the original ladder with excess bracket-year holdings is created at low yield (say 0%) using a certain set of bracket-year multipliers (say 3.5/3.5), the mix of 2040/2034 excess bracket holdings based on cost would be different than if the ladder was created at higher yield (say 4%). If the duration-matching goal is to create the ladder with a set of bracket year holdings with costs in a certain ratio (say 50/50) at any given starting yield, using a fixed pair of multipliers doesn't seem to be the way to do that. (The multipliers that provide best duration matching would differ somewhat depending on the initial level of interest rates.)

While multipliers of 3.88 and 3.12 for 2034 and 2040 are predicted to give an excellent duration match using #Cruncher's TIPS ladder building spreadsheet, my intuition says that these values may not be universal. These multiplier values apply for a ladder constructed at a yield level of 2% and using January 2034 and February 2040 for the excess bracket year holdings, and with the five swaps completed simultaneously in 2029 after all gap-year issues become available. Somewhat different multipliers may apply when using different TIPS for the bracket year holdings and/or when yields deviate away from 2% when the ladder is constructed. Any difference in ideal multipliers if the swaps are made annually between now and 2029 remains to be determined.

If the goal of duration matching is for the cost of the excess bracket-year holdings to track the cost of the hypothetical gap-year holdings as yields change, shouldn't the relative costs of the excess holdings of the two bracket-years be what is initially set to the desired ratio (say 50/50)?
bpg1234
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### Re: Filling the TIPS gap years with bracket year duration matching

So based on duration matching I have more than my required number of 2034s but lacking on 2040s. So based on this thread should I remain grounded on the tarmac for now until this is more clear or just continue to accumulate more 2040s based on duration matching while rates are still over 2% at this point?
MtnBiker
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### Re: Filling the TIPS gap years with bracket year duration matching

bpg1234 wrote: Thu Jul 25, 2024 1:31 pm So based on duration matching I have more than my required number of 2034s but lacking on 2040s. So based on this thread should I remain grounded on the tarmac for now until this is more clear or just continue to accumulate more 2040s based on duration matching while rates are still over 2% at this point?
The best result at 2% yield obtained so far from Kevin's model is shown here:
Kevin M wrote: Mon Jul 22, 2024 5:30 pm
At 2% yield total delta cost is 465:

In this chart, it shows that the bracket year holdings sold to make the swaps (see the "Delta cost" column) were 210,178 of 2040s and 191,490 of 2034s.

This is about 52% excess holdings of February 2040s and 48% excess holdings of January 2034s, based on cost in 2029.

These values are so close to the prediction in the original post of this thread (49% excess holdings of February 2040s and 51% excess holdings of January 2034s in 2024, based on the simplest of duration-matching principles), that the results seem believable to me. Lacking further information, I would suggest aiming for an excess holdings ratio of about 52/48 (2040/2034), based on cost.
bpg1234
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Joined: Fri Jun 24, 2011 6:53 pm

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

MtnBiker wrote: Thu Jul 25, 2024 2:32 pm
bpg1234 wrote: Thu Jul 25, 2024 1:31 pm So based on duration matching I have more than my required number of 2034s but lacking on 2040s. So based on this thread should I remain grounded on the tarmac for now until this is more clear or just continue to accumulate more 2040s based on duration matching while rates are still over 2% at this point?
The best result at 2% yield obtained so far from Kevin's model is shown here:
Kevin M wrote: Mon Jul 22, 2024 5:30 pm
At 2% yield total delta cost is 465:

In this chart, it shows that the bracket year holdings sold to make the swaps (see the "Delta cost" column) were 210,178 of 2040s and 191,490 of 2034s.

This is about 52% excess holdings of February 2040s and 48% excess holdings of January 2034s, based on cost.

These values are so close to the prediction in the original post of this thread (49% excess holdings of February 2040s and 51% excess holdings of January 2034s, based on the simplest of duration-matching principles), that the results seem believable to me. Lacking further information, I would suggest aiming for the 52/48 holdings ratio (2040/2034), based on cost.
Thanks MtnBiker. Admitedly I'm still a bit confused on all of this when you say based on cost. I presently have 200 2034 TIPS and 56 2040 TIPS. To date I just bought them regardless of the cost. The 2040s with all of the accrued inflation obviously cost a lot more.

My ladder is targeted at \$50K per year at this point but is not intended to have this specific dollar amount available to meet specfic expenses but rather just wealth preservation and potentially use for say RMDs at some point, roll back over, etc. As such, if going with duration matching of 48% 34s and 52% 40s I was just thinking I would just buy another 156 2040 TIPS costing roughly \$226K plus accrued interest and not be concerned with annual TIPS coupon payments which would just be used to roll into additional TIPS as they accummulate in settlement fund.

Is the wrong way to look at this? Thanks in advance.
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Kevin M
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### Re: Filling the TIPS gap years with bracket year duration matching

Here's another set of simple experiments to look more closely at shifting from overweighting 2040s to overweighting 2034s. The ladder has these characteristics:
• Multipliers for 2041-2054 are all 0.
• Multipliers for 2025-2029 are all 0, as they have been for most experiments lately, since I'm assuming these have matured and been used for expenses.
• All bonds are par bonds, so coupon = yield.
First, here is the ladder before and after filling the gaps at 2% yield (and coupon), with pre-fill multiplier of 3 for the 2034 and 4 for the 2040; i.e., overweighting the 2040s:

Notes and observations:
• With all par bonds, cost is equal to needed principal.
• I've added a "Deltas" row that shows the total for a column of interest in the left table (post-fill), e.g. Interest Later Bonds, minus the corresponding total in the right table (pre-fill).
• The ladder costs 6,253 more after filling the gaps, which is the amount by which the total needed principal has increased.
• The reason the total needed principal has increased is because the total interest from later bonds has decreased; total annual interest has increased, but this effect is swamped out by the decrease in total interest from later bonds.
• The change in interest from later bonds is not uniform at the individual maturity level; it's negative for the 2036-2039 and positive for the 2030-2035.
I think the impact of interest from later bonds is the most interesting thing about this exercise, so I'll add a column that shows the deltas for the individual bonds:

Now I'll change the pre-swap multipliers to 4 for 2034 and 3 for 2040; i.e., overweighting the 2034s:

The signs for the delta totals have flipped, so now the cost and needed principal are 4,133 less after the swaps, relative to the cost before the swaps. But note that what actually changed is that the cost and needed principal before the swaps increased, relative to the previous set of tables; everything in the left table is unchanged.

Now I'll increase the pre swap multipliers to 5 for 2034 and 2 for 2040:

And now just using all 2034s to cover the gap years; i.e., pre swap multiplier of 6 for 2034 and 1 for 2040:

So what we see is that the more 2034s are used, the more expensive the ladder is before the swaps. This is because with more 2034s, there is less total interest from later bonds pre-swap, so more needed principal is required.

Note that these experiments don't have anything to do with changes in yields, but just show the impact of swapping bracket years for gap years (and doing the relatively small additional transactions so that ARA = DARA for the 2030-2033) at a given yield.
If I make a calculation error, #Cruncher probably will let me know.
MtnBiker
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Joined: Sun Nov 16, 2014 3:43 pm

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

bpg1234 wrote: Thu Jul 25, 2024 3:02 pm
Thanks MtnBiker. Admitedly I'm still a bit confused on all of this when you say based on cost. I presently have 200 2034 TIPS and 56 2040 TIPS. To date I just bought them regardless of the cost. The 2040s with all of the accrued inflation obviously cost a lot more.

My ladder is targeted at \$50K per year at this point but is not intended to have this specific dollar amount available to meet specfic expenses but rather just wealth preservation and potentially use for say RMDs at some point, roll back over, etc. As such, if going with duration matching of 48% 34s and 52% 40s I was just thinking I would just buy another 156 2040 TIPS costing roughly \$226K plus accrued interest and not be concerned with annual TIPS coupon payments which would just be used to roll into additional TIPS as they accummulate in settlement fund.

Is the wrong way to look at this? Thanks in advance.
I can understand buying \$50K per year and just ignoring the coupon payments which generally would only add an additional buffer. But buying 156 more 2040s seems like overkill. Here is how I would look at the gap.

Per the Wall Street Journal table, the adjusted price (Ask price times inflation factor) for January 2034s is about 0.993. The adjusted price for 2040 is about 1.442.

To cover 2034 itself, you need 50K of the 2034s, which would be 50K/(993 per bond) = 50 of the 2034 bonds
To cover 2040 itself, you need 50K of the 2040s, which would be 50K/(1,442 per bond) = 35 of the 2040 bonds

To cover the 5-year gap, you need an additional 250K. If split 48/52, that would be 120K of excess 2034s and 130K of excess 2040s.
or, 120K/(993 per bond) = 121 of the 2034 bonds
and 130K/(1.442 per bond) = 90 of the 2040 bonds

For a grand total needed of 171 of the 2034s and 125 of the 2040s.

You already have 200 2034s, so you could sell 29 of them and use the proceeds to buy 20 more 2040s. That would leave you with 171 2034s and 76 2040s.

If you agreed with this general train of thought, you would need to use new money to buy an additional 49 2040s to get up to the desired 125 total. Does this make sense?
MtnBiker
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### Re: Filling the TIPS gap years with bracket year duration matching

Kevin M wrote: Thu Jul 25, 2024 4:02 pm
So what we see is that the more 2034s are used, the more expensive the ladder is before the swaps. This is because with more 2034s, there is less total interest from later bonds pre-swap, so more needed principal is required.

Note that these experiments don't have anything to do with changes in yields, but just show the impact of swapping bracket years for gap years (and doing the relatively small additional transactions so that ARA = DARA for the 2030-2033) at a given yield.
The more 2034s are used, the more expensive the remaining ladder is in 2029 just before the swaps are made. I think it may also be true that as more 2034s are used, the pre-swap 2025-2029 rungs become less expensive. I doubt that any decrease in the cost of the pre-swap rungs is enough to fully offset the increase in the cost of the post-swap rungs, but without showing those details it makes one wonder. Is there any easy way to include that factor in the comparison?
bpg1234
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### Re: Filling the TIPS gap years with bracket year duration matching

MtnBiker wrote: Thu Jul 25, 2024 8:56 pm
bpg1234 wrote: Thu Jul 25, 2024 3:02 pm
Thanks MtnBiker. Admitedly I'm still a bit confused on all of this when you say based on cost. I presently have 200 2034 TIPS and 56 2040 TIPS. To date I just bought them regardless of the cost. The 2040s with all of the accrued inflation obviously cost a lot more.

My ladder is targeted at \$50K per year at this point but is not intended to have this specific dollar amount available to meet specfic expenses but rather just wealth preservation and potentially use for say RMDs at some point, roll back over, etc. As such, if going with duration matching of 48% 34s and 52% 40s I was just thinking I would just buy another 156 2040 TIPS costing roughly \$226K plus accrued interest and not be concerned with annual TIPS coupon payments which would just be used to roll into additional TIPS as they accummulate in settlement fund.

Is the wrong way to look at this? Thanks in advance.
I can understand buying \$50K per year and just ignoring the coupon payments which generally would only add an additional buffer. But buying 156 more 2040s seems like overkill. Here is how I would look at the gap.

Per the Wall Street Journal table, the adjusted price (Ask price times inflation factor) for January 2034s is about 0.993. The adjusted price for 2040 is about 1.442.

To cover 2034 itself, you need 50K of the 2034s, which would be 50K/(993 per bond) = 50 of the 2034 bonds
To cover 2040 itself, you need 50K of the 2040s, which would be 50K/(1,442 per bond) = 35 of the 2040 bonds

To cover the 5-year gap, you need an additional 250K. If split 48/52, that would be 120K of excess 2034s and 130K of excess 2040s.
or, 120K/(993 per bond) = 121 of the 2034 bonds
and 130K/(1.442 per bond) = 90 of the 2040 bonds

For a grand total needed of 171 of the 2034s and 125 of the 2040s.

You already have 200 2034s, so you could sell 29 of them and use the proceeds to buy 20 more 2040s. That would leave you with 171 2034s and 76 2040s.

If you agreed with this general train of thought, you would need to use new money to buy an additional 49 2040s to get up to the desired 125 total. Does this make sense?
Thanks MtnBiker. Yes this makes sense but if I were to stop my ladder at year 2038 (so 4 Gap years and no need for 2040 in the end) I assume these .48 2034 and .52 2040 percentages for bracket years would remain and just the number of required TIPS for each would change? Lastly do you think Kevin's and your ongoing analysis will change this very much or are we dealing on the fringes at this point?

-bpg1234
MtnBiker
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### Re: Filling the TIPS gap years with bracket year duration matching

bpg1234 wrote: Fri Jul 26, 2024 10:07 am
MtnBiker wrote: Thu Jul 25, 2024 8:56 pm
bpg1234 wrote: Thu Jul 25, 2024 3:02 pm
Thanks MtnBiker. Admitedly I'm still a bit confused on all of this when you say based on cost. I presently have 200 2034 TIPS and 56 2040 TIPS. To date I just bought them regardless of the cost. The 2040s with all of the accrued inflation obviously cost a lot more.

My ladder is targeted at \$50K per year at this point but is not intended to have this specific dollar amount available to meet specfic expenses but rather just wealth preservation and potentially use for say RMDs at some point, roll back over, etc. As such, if going with duration matching of 48% 34s and 52% 40s I was just thinking I would just buy another 156 2040 TIPS costing roughly \$226K plus accrued interest and not be concerned with annual TIPS coupon payments which would just be used to roll into additional TIPS as they accummulate in settlement fund.

Is the wrong way to look at this? Thanks in advance.
I can understand buying \$50K per year and just ignoring the coupon payments which generally would only add an additional buffer. But buying 156 more 2040s seems like overkill. Here is how I would look at the gap.

Per the Wall Street Journal table, the adjusted price (Ask price times inflation factor) for January 2034s is about 0.993. The adjusted price for 2040 is about 1.442.

To cover 2034 itself, you need 50K of the 2034s, which would be 50K/(993 per bond) = 50 of the 2034 bonds
To cover 2040 itself, you need 50K of the 2040s, which would be 50K/(1,442 per bond) = 35 of the 2040 bonds

To cover the 5-year gap, you need an additional 250K. If split 48/52, that would be 120K of excess 2034s and 130K of excess 2040s.
or, 120K/(993 per bond) = 121 of the 2034 bonds
and 130K/(1.442 per bond) = 90 of the 2040 bonds

For a grand total needed of 171 of the 2034s and 125 of the 2040s.

You already have 200 2034s, so you could sell 29 of them and use the proceeds to buy 20 more 2040s. That would leave you with 171 2034s and 76 2040s.

If you agreed with this general train of thought, you would need to use new money to buy an additional 49 2040s to get up to the desired 125 total. Does this make sense?
Thanks MtnBiker. Yes this makes sense but if I were to stop my ladder at year 2038 (so 4 Gap years and no need for 2040 in the end) I assume these .48 2034 and .52 2040 percentages for bracket years would remain and just the number of required TIPS for each would change? Lastly do you think Kevin's and your ongoing analysis will change this very much or are we dealing on the fringes at this point?

-bpg1234
Personally, I think the detailed analysis is converging toward a general confirmation that what people have been doing all along is good enough. That is, if filling the gap uniformly, buy excess holdings of the bracket years in roughly equal amounts. I'm not sure if Kevin would agree at this point.

If ending the ladder in 2038, the gap isn't filled uniformly and the percentages for the bracket years would need to be adjusted accordingly. The simplest way to address that is to just use the n/6 approximation (as mentioned in the original post of this thread) for the relative weighting of the excess bracket year holdings targeted for each gap year. So, instead of about a 50/50 mix of excess bracket holdings, one would need about a 58/42 mix of 2034 and 2040, respectively:

Code: Select all

``````gap year      2034 wt       2040 wt

2035            0.83         0.17
2036            0.67         0.33
2037            0.50         0.50
2038            0.33         0.67

TOTAL           2.33         1.67

RATIO            58%          42%

``````
Note that this table also shows you how much of each bracket year excess holdings to sell each year when swapping for the gap year TIPS as it becomes available. For the 2035 gap year, you use a high percentage of 2034s since the duration of the 2034s is so close to the duration of the 2035s. For the middle of the gap (2037), you use an equal mix of 2034s and 2040s, since the duration of the 2037s is about the average of the durations of the two bracket years.
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Kevin M
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### Re: Filling the TIPS gap years with bracket year duration matching

MtnBiker wrote: Thu Jul 25, 2024 9:32 pm
Kevin M wrote: Thu Jul 25, 2024 4:02 pm So what we see is that the more 2034s are used, the more expensive the ladder is before the swaps. This is because with more 2034s, there is less total interest from later bonds pre-swap, so more needed principal is required.

Note that these experiments don't have anything to do with changes in yields, but just show the impact of swapping bracket years for gap years (and doing the relatively small additional transactions so that ARA = DARA for the 2030-2033) at a given yield.
The more 2034s are used, the more expensive the remaining ladder is in 2029 just before the swaps are made. I think it may also be true that as more 2034s are used, the pre-swap 2025-2029 rungs become less expensive. I doubt that any decrease in the cost of the pre-swap rungs is enough to fully offset the increase in the cost of the post-swap rungs, but without showing those details it makes one wonder. Is there any easy way to include that factor in the comparison?
Yesterday I downloaded #Cruncher's latest full ladder building spreadsheet, and did my modifications to use Schwab quotes. His default now is to use multiplier 3 for the Jul 2034 and 4 for the 2040. With a DARA of 100K and Schwab quotes I just pulled, the 30y ladder cost is 2,204,816. If I swap the multipliers to use 4 and 3 for 2034 and 2040, the cost increases to 2,217,402. With multipliers of 6 and 1 for 2034 and 2040, the cost is 2,237,055.

For me, the main benefit of the simplified ladder tool is that it allows me to more easily gain insight into patterns and what's causing them. Verifying the results against the full ladder tool lends credence to the results.

With a quick check of principal, final year interest and other interest in the full ladder tool, I see the same pattern, which is that as the 2034 multiplier is increased and the 2040 decreased, the 2034+2040 other interest decreases which increases the principal, and thus the cost.
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: Fri Jul 26, 2024 11:33 am
MtnBiker wrote: Thu Jul 25, 2024 9:32 pm
Kevin M wrote: Thu Jul 25, 2024 4:02 pm So what we see is that the more 2034s are used, the more expensive the ladder is before the swaps. This is because with more 2034s, there is less total interest from later bonds pre-swap, so more needed principal is required.

Note that these experiments don't have anything to do with changes in yields, but just show the impact of swapping bracket years for gap years (and doing the relatively small additional transactions so that ARA = DARA for the 2030-2033) at a given yield.
The more 2034s are used, the more expensive the remaining ladder is in 2029 just before the swaps are made. I think it may also be true that as more 2034s are used, the pre-swap 2025-2029 rungs become less expensive. I doubt that any decrease in the cost of the pre-swap rungs is enough to fully offset the increase in the cost of the post-swap rungs, but without showing those details it makes one wonder. Is there any easy way to include that factor in the comparison?
Yesterday I downloaded #Cruncher's latest full ladder building spreadsheet, and did my modifications to use Schwab quotes. His default now is to use multiplier 3 for the Jul 2034 and 4 for the 2040. With a DARA of 100K and Schwab quotes I just pulled, the 30y ladder cost is 2,204,816. If I swap the multipliers to use 4 and 3 for 2034 and 2040, the cost increases to 2,217,402. With multipliers of 6 and 1 for 2034 and 2040, the cost is 2,237,055.

For me, the main benefit of the simplified ladder tool is that it allows me to more easily gain insight into patterns and what's causing them. Verifying the results against the full ladder tool lends credence to the results.

With a quick check of principal, final year interest and other interest in the full ladder tool, I see the same pattern, which is that as the 2034 multiplier is increased and the 2040 decreased, the 2034+2040 other interest decreases which increases the principal, and thus the cost.
Yes, I agree that the decrease in the cost of the pre-swap rungs when more 2034s are used should be a really small effect. After thinking about it some more, it is a second-order effect that has to be very small.

As more 2034s are used, the interest in later years is decreased, which increases the principal in later years, which increases the interest in earlier years, which decreases the principal in earlier years. The earlier year effects are tiny compared to the later year effects, resulting in higher costs overall.
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Kevin M
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### Re: Filling the TIPS gap years with bracket year duration matching

To close the loop on this, here are some results from the simplified spreadsheet using multipliers of 1 for the 2025-2029.

Starting with multipliers of 3 and 4 for 2034 and 2040:

Changing multipliers to 4 and 3 for 2034 and 2040:

Now 6 for 2034 and 1 for 2040:

Focusing on just the right table, we see the same pattern, with later bond interest decreasing, and needed principal and cost increasing as we increase the 2034 multiplier.

We could think of the left table as representing a scenario where Treasury decided to issue all the gap years now.
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

MtnBiker wrote: Mon Jul 22, 2024 10:44 pm
Kevin M wrote: Mon Jul 22, 2024 5:30 pm
MtnBiker wrote: Sat Jul 20, 2024 10:52 am StillGoing showed that the range of results when making the swaps in 2034 was minimized using multipliers M = 3.95 for 2034 and M = 3.05 for 2040.

Assuming that Version 2 of the spreadsheet has no errors, it would be interesting to see what bracket-year multipliers should be used to minimize the range of results if making the swaps in 2029.

Here is an observation: Using Version 2, Kevin showed that the net cost of doing the swaps (with ARA = DARA) at 4% real yield changed from -5,699 to +38,534 when the 2034 multiplier increased from 3.57 to 6. Using interpolation, one can estimate that the net cost of the swaps at 4% yield should be close to zero for M = 3.85 for 2034 and M = 3.15 for 2040. Since the net cost is always zero for swaps at 0% yield, M values near this should give near-zero net cost across the full 0% to 4% yield range.

Note that the predicted optimal M values for swaps made in 2029 (optimized to minimize the range of results in ladder net cost) is not much different than for swaps made in 2034. So, if this prediction (by interpolation) is verified by results from the spreadsheet, the optimal M values should be rather insensitive to when the swaps are made.
Not a bad guess. A 2034 pre-swap multiplier of 3.8795 makes the total delta cost 0 at 4%:

At 2% yield total delta cost is 465:

The result that M = 3.88 for 2034 and M = 3.12 for 2040 provides the best duration matching was certainly not anticipated at the beginning of this thread. I think it is time to take a step back and do some critical thinking about the basic assumptions made originally.

First of all, duration matching formulas were presented in the OP which identified that the mix of the excess bracket year holdings (2034 and 2040) would need to be about 50/50 for duration matching. What does this mean, exactly?

Does this mean that the cost of buying the excess holdings should be about a 50/50 mix? Does this mean that the principal at maturity of the excess holding should be approximately a 50/50 mix? Does this mean that the DARA for the 2034 and 2040 holdings should be approximately equal? Does this mean that the DARA multipliers for the 2034 and 2040 holdings should be approximately equal?

Setting the DARAs (multipliers) equal does not appear to give equal excess holding costs or equal excess principal amounts. Setting the excess holding costs or principal amounts equal does not appear to give equal multipliers. For a given mix of excess holdings, the multipliers appear to be a function of the coupon rates of the holdings.

An assumption implicit in the OP is that a 50/50 excess holding mix corresponds to DARA multipliers of 3.5 for each. But examination of the spreadsheets presented in this thread shows that using DARA multipliers near 3.5/3.5 does not appear to require equal amounts of 2034 and 2040 excess holdings. In fact, the spreadsheet seems to calculate that a substantial portion of the 2034 DARA is assumed to be funded by the 2035 - 2039 coupons from the 2040s. Thus, the spreadsheet suggests buying considerably lesser amounts of 2034s compared to the amounts of 2040s when the multipliers are both equal at 3.5. (It seems that setting the multipliers equal would only give equal excess holdings if TIPS had zero coupons.)

Is this the reason multipliers of 3.5/3.5 don't work as well as 2034/2040 multipliers of 3.88/3.12? Are the excess holdings of 2034s and 2040s closer to 50/50 with multipliers of 3.88/3.12? Is the original duration-matching concept intact, but maybe the use of multipliers proportional to the desired excess holding amounts is a flawed concept?
Based on the detailed analysis results presented to date, I am coming to the conclusion that using roughly equal amounts of excess bracket-year holdings does minimize the range of results when filling the gap uniformly. In the second chart reproduced above, it can be seen that the excess bracket year holdings sold to buy the gap at 2% yield (see the 2040 and 2034 entries in the "Delta cost" column) are about 52.3% for 2040 and 47.7% for 2034. This approximately 50/50 mix is consistent with the simplified (but easy to understand) duration matching arguments made in the original post.

It can also be seen that using DARA multipliers to achieve a 50/50 mix of excess bracket-year holdings is imperfect, at best, and wildly misleading in general. The ARA in the bracket years consists of interest from 2034, 2040 and later-year TIPS received in all years 2034 - 2040 (gap and bracket) in addition to the principal from the bracket-year TIPS. So, equal multipliers for 2040 and 2034 does not give equal excess bracket-year holdings. In the duration-matched example shown in the charts above, the multipliers were adjusted to be 3.88 for 2034 and 3.12 for 2040 before achieving roughly equal excess bracket-year holdings. The multipliers needed to achieve duration matching will vary depending on various factors such as coupon rates, the ending date of the ladder, which TIPS are used as the bracket years, the prevailing yield at the time, and so on. (Various factors affect the coupon payments assigned to each of the bracket years in the spreadsheet.)

Furthermore, the use of multipliers is subject to the definitions selected by the author of the spreadsheet tool. In #Cruncher's spreadsheet, the interest generated in the gap years 2035-2039 is assigned to the DARA for 2034. If that assignment was ever changed, or if a different ladder tool assigned some or all of the gap-year interest to 2040, the multipliers for duration matching would change accordingly. Under the same set of conditions, different sets of multipliers might be needed for different tools.

Thus, it seems to me that using a standard set of multipliers to achieve duration matching makes no sense. If I needed to build a ladder with the tools available today, I would use #Cruncher's spreadsheet to build the ladder using multipliers of 1 for 2034 and 2040, and then add the excess bracket-year holdings manually after that, so that the excess holdings were distributed between the two bracket years in the desired proportions based on cost. And I would keep a record of the number of bonds purchased to cover each of the gap years, so that I would know how many of each to sell when the time comes to make each of the swaps for a gap year. Does anyone have a better suggestion?
bpg1234
Posts: 444
Joined: Fri Jun 24, 2011 6:53 pm

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

MtnBiker wrote: Fri Jul 26, 2024 10:57 am
bpg1234 wrote: Fri Jul 26, 2024 10:07 am
MtnBiker wrote: Thu Jul 25, 2024 8:56 pm
bpg1234 wrote: Thu Jul 25, 2024 3:02 pm
Thanks MtnBiker. Admitedly I'm still a bit confused on all of this when you say based on cost. I presently have 200 2034 TIPS and 56 2040 TIPS. To date I just bought them regardless of the cost. The 2040s with all of the accrued inflation obviously cost a lot more.

My ladder is targeted at \$50K per year at this point but is not intended to have this specific dollar amount available to meet specfic expenses but rather just wealth preservation and potentially use for say RMDs at some point, roll back over, etc. As such, if going with duration matching of 48% 34s and 52% 40s I was just thinking I would just buy another 156 2040 TIPS costing roughly \$226K plus accrued interest and not be concerned with annual TIPS coupon payments which would just be used to roll into additional TIPS as they accummulate in settlement fund.

Is the wrong way to look at this? Thanks in advance.
I can understand buying \$50K per year and just ignoring the coupon payments which generally would only add an additional buffer. But buying 156 more 2040s seems like overkill. Here is how I would look at the gap.

Per the Wall Street Journal table, the adjusted price (Ask price times inflation factor) for January 2034s is about 0.993. The adjusted price for 2040 is about 1.442.

To cover 2034 itself, you need 50K of the 2034s, which would be 50K/(993 per bond) = 50 of the 2034 bonds
To cover 2040 itself, you need 50K of the 2040s, which would be 50K/(1,442 per bond) = 35 of the 2040 bonds

To cover the 5-year gap, you need an additional 250K. If split 48/52, that would be 120K of excess 2034s and 130K of excess 2040s.
or, 120K/(993 per bond) = 121 of the 2034 bonds
and 130K/(1.442 per bond) = 90 of the 2040 bonds

For a grand total needed of 171 of the 2034s and 125 of the 2040s.

You already have 200 2034s, so you could sell 29 of them and use the proceeds to buy 20 more 2040s. That would leave you with 171 2034s and 76 2040s.

If you agreed with this general train of thought, you would need to use new money to buy an additional 49 2040s to get up to the desired 125 total. Does this make sense?
Thanks MtnBiker. Yes this makes sense but if I were to stop my ladder at year 2038 (so 4 Gap years and no need for 2040 in the end) I assume these .48 2034 and .52 2040 percentages for bracket years would remain and just the number of required TIPS for each would change? Lastly do you think Kevin's and your ongoing analysis will change this very much or are we dealing on the fringes at this point?

-bpg1234
Personally, I think the detailed analysis is converging toward a general confirmation that what people have been doing all along is good enough. That is, if filling the gap uniformly, buy excess holdings of the bracket years in roughly equal amounts. I'm not sure if Kevin would agree at this point.

If ending the ladder in 2038, the gap isn't filled uniformly and the percentages for the bracket years would need to be adjusted accordingly. The simplest way to address that is to just use the n/6 approximation (as mentioned in the original post of this thread) for the relative weighting of the excess bracket year holdings targeted for each gap year. So, instead of about a 50/50 mix of excess bracket holdings, one would need about a 58/42 mix of 2034 and 2040, respectively:

Code: Select all

``````gap year      2034 wt       2040 wt

2035            0.83         0.17
2036            0.67         0.33
2037            0.50         0.50
2038            0.33         0.67

TOTAL           2.33         1.67

RATIO            58%          42%

``````
Note that this table also shows you how much of each bracket year excess holdings to sell each year when swapping for the gap year TIPS as it becomes available. For the 2035 gap year, you use a high percentage of 2034s since the duration of the 2034s is so close to the duration of the 2035s. For the middle of the gap (2037), you use an equal mix of 2034s and 2040s, since the duration of the 2037s is about the average of the durations of the two bracket years.
MtnBiker thanks yet again. In using the n/6 approximation method above is this done by just multiplying the desired \$50,000 per year for 2035-2038 by the relative percentages to arrive at how much of each bracket year one sells to buy the corresponding gap year?

So for instance for 2035 one would sell 50000*.83 = \$41,500 from 2034 (rounded to 42 TIPS) and then 8 2040 TIPS? For 2036 one would sell 50000*.67=\$33,500 from 2034 (rounded to 34 TIPS) and then 16 2040 TIPS, etc. or does the .993 for 2034 and 1.442 for 2040 cost factors come into the equation?

Also would this be buying over time as each new 10 year TIPS is issued say in January or waiting until 2029 after all of the gap years TIPS are available?

-bpg1234
MtnBiker
Posts: 628
Joined: Sun Nov 16, 2014 3:43 pm

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

bpg1234 wrote: Sat Jul 27, 2024 10:09 am
MtnBiker wrote: Fri Jul 26, 2024 10:57 am
bpg1234 wrote: Fri Jul 26, 2024 10:07 am
MtnBiker wrote: Thu Jul 25, 2024 8:56 pm
bpg1234 wrote: Thu Jul 25, 2024 3:02 pm
Thanks MtnBiker. Admitedly I'm still a bit confused on all of this when you say based on cost. I presently have 200 2034 TIPS and 56 2040 TIPS. To date I just bought them regardless of the cost. The 2040s with all of the accrued inflation obviously cost a lot more.

My ladder is targeted at \$50K per year at this point but is not intended to have this specific dollar amount available to meet specfic expenses but rather just wealth preservation and potentially use for say RMDs at some point, roll back over, etc. As such, if going with duration matching of 48% 34s and 52% 40s I was just thinking I would just buy another 156 2040 TIPS costing roughly \$226K plus accrued interest and not be concerned with annual TIPS coupon payments which would just be used to roll into additional TIPS as they accummulate in settlement fund.

Is the wrong way to look at this? Thanks in advance.
I can understand buying \$50K per year and just ignoring the coupon payments which generally would only add an additional buffer. But buying 156 more 2040s seems like overkill. Here is how I would look at the gap.

Per the Wall Street Journal table, the adjusted price (Ask price times inflation factor) for January 2034s is about 0.993. The adjusted price for 2040 is about 1.442.

To cover 2034 itself, you need 50K of the 2034s, which would be 50K/(993 per bond) = 50 of the 2034 bonds
To cover 2040 itself, you need 50K of the 2040s, which would be 50K/(1,442 per bond) = 35 of the 2040 bonds

To cover the 5-year gap, you need an additional 250K. If split 48/52, that would be 120K of excess 2034s and 130K of excess 2040s.
or, 120K/(993 per bond) = 121 of the 2034 bonds
and 130K/(1.442 per bond) = 90 of the 2040 bonds

For a grand total needed of 171 of the 2034s and 125 of the 2040s.

You already have 200 2034s, so you could sell 29 of them and use the proceeds to buy 20 more 2040s. That would leave you with 171 2034s and 76 2040s.

If you agreed with this general train of thought, you would need to use new money to buy an additional 49 2040s to get up to the desired 125 total. Does this make sense?
Thanks MtnBiker. Yes this makes sense but if I were to stop my ladder at year 2038 (so 4 Gap years and no need for 2040 in the end) I assume these .48 2034 and .52 2040 percentages for bracket years would remain and just the number of required TIPS for each would change? Lastly do you think Kevin's and your ongoing analysis will change this very much or are we dealing on the fringes at this point?

-bpg1234
Personally, I think the detailed analysis is converging toward a general confirmation that what people have been doing all along is good enough. That is, if filling the gap uniformly, buy excess holdings of the bracket years in roughly equal amounts. I'm not sure if Kevin would agree at this point.

If ending the ladder in 2038, the gap isn't filled uniformly and the percentages for the bracket years would need to be adjusted accordingly. The simplest way to address that is to just use the n/6 approximation (as mentioned in the original post of this thread) for the relative weighting of the excess bracket year holdings targeted for each gap year. So, instead of about a 50/50 mix of excess bracket holdings, one would need about a 58/42 mix of 2034 and 2040, respectively:

Code: Select all

``````gap year      2034 wt       2040 wt

2035            0.83         0.17
2036            0.67         0.33
2037            0.50         0.50
2038            0.33         0.67

TOTAL           2.33         1.67

RATIO            58%          42%

``````
Note that this table also shows you how much of each bracket year excess holdings to sell each year when swapping for the gap year TIPS as it becomes available. For the 2035 gap year, you use a high percentage of 2034s since the duration of the 2034s is so close to the duration of the 2035s. For the middle of the gap (2037), you use an equal mix of 2034s and 2040s, since the duration of the 2037s is about the average of the durations of the two bracket years.
MtnBiker thanks yet again. In using the n/6 approximation method above is this done by just multiplying the desired \$50,000 per year for 2035-2038 by the relative percentages to arrive at how much of each bracket year one sells to buy the corresponding gap year?

So for instance for 2035 one would sell 50000*.83 = \$41,500 from 2034 (rounded to 42 TIPS) and then 8 2040 TIPS? For 2036 one would sell 50000*.67=\$33,500 from 2034 (rounded to 34 TIPS) and then 16 2040 TIPS, etc. or does the .993 for 2034 and 1.442 for 2040 cost factors come into the equation?

Also would this be buying over time as each new 10 year TIPS is issued say in January or waiting until 2029 after all of the gap years TIPS are available?

-bpg1234
I would suggest that the excess bracket year holdings should be allocated based on original cost, so the 0.993 and 1.442 adjusted prices would apply.

For example, for 2035 one would hold 50,000*0.83 = \$41,500 in 2034s and 50,000*0.17 = \$8,500 in 2040s. The number of 2034 bonds would be 41,500/(993 per bond) = 42. The number of 2040 bonds would be 8,500/(1,442 per bond) = 6 bonds. For 2036 one would hold 34 of the 2034s and 11 or 12 of the 2040s (depending on how you do the rounding), etc.

This would apply regardless of whether you swapped for each new 10-year TIPS soon after issuance or waited until 2029 (or even 2034) to do all the swaps simultaneously. For many reasons, making the swaps sooner rather than later makes the most sense to me. If you agree with swapping as soon as possible, sometime in 2025 you would sell 42 2034s and 6 2040s and use the proceeds (however much it might be at that time, more or less than 50K depending on interest rates and inflation or deflation) to buy the 2035s.
protagonist
Posts: 9527
Joined: Sun Dec 26, 2010 11:47 am

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

MtnBiker wrote: Sat Jul 27, 2024 2:29 pm
Personally, I think the detailed analysis is converging toward a general confirmation that what people have been doing all along is good enough. That is, if filling the gap uniformly, buy excess holdings of the bracket years in roughly equal amounts. I'm not sure if Kevin would agree at this point.
Upon reviewing this thread (I had been away), this is my conclusion as well.
Currently I have approx. 55% 2034s vs. 45% 2040s.
I have some TIPS maturing this October, and I will invest the proceeds into more 2040s, which will leave me roughly with 51% 2034s and 49% 2040s.
I think that is close enough.

In February I will buy my DARA of 2035s with roughly 5/6 (DARA) 2034s and 1/6 (DARA) 2040s...or, if I am still a little "2040 deficient" (vs. 2034s), maybe I will use only 2034s' which will leave me closer to the ideal balance for when I buy 2036-2039s. I see no reason to be precise.

Thank you, Kevin, MtnBiker and others, for doing the work!
MtnBiker
Posts: 628
Joined: Sun Nov 16, 2014 3:43 pm

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

protagonist wrote: Wed Jul 31, 2024 11:11 pm
MtnBiker wrote: Sat Jul 27, 2024 2:29 pm
Personally, I think the detailed analysis is converging toward a general confirmation that what people have been doing all along is good enough. That is, if filling the gap uniformly, buy excess holdings of the bracket years in roughly equal amounts. I'm not sure if Kevin would agree at this point.
Upon reviewing this thread (I had been away), this is my conclusion as well.
Currently I have approx. 55% 2034s vs. 45% 2040s.
I have some TIPS maturing this October, and I will invest the proceeds into more 2040s, which will leave me roughly with 51% 2034s and 49% 2040s.
I think that is close enough.

In February I will buy my DARA of 2035s with roughly 5/6 (DARA) 2034s and 1/6 (DARA) 2040s...or, if I am still a little "2040 deficient" (vs. 2034s), maybe I will use only 2034s' which will leave me closer to the ideal balance for when I buy 2036-2039s. I see no reason to be precise.

Thank you, Kevin, MtnBiker and others, for doing the work!
The thread about the tipsladder.com tool has recently announced (viewtopic.php?p=7976696#p7976696) that there is a new experimental UI for the TIPS ladder calculator here: https://tipsladder.com/spa. This new version offers a more manual way of creating a ladder, including assigning 2034 and 2040 TIPS directly to each gap year 2035-2039 in whatever proportions you desire.

You might find this new tool helpful for better visualization of building the excess bracket year holdings up in n/6 chunks assigned to each gap year. The tool shows the effects of coupon interest payments on each year's income, including in the gap, which is helpful for smoothing DARA when trying to get as even a balance as possible between each gap year. It also calculates the Macaulay duration of the mix of bonds assigned to each gap year, so you can confirm that the duration for the holdings for each year increases incrementally as it should from 2034 to 2040. Once completed, the ladder can be stored or printed out as a record showing how many of each bracket year bond to sell each year when swapping the proceeds for the newly issued gap-year bond.
protagonist
Posts: 9527
Joined: Sun Dec 26, 2010 11:47 am

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

MtnBiker wrote: Thu Aug 01, 2024 10:13 am
protagonist wrote: Wed Jul 31, 2024 11:11 pm
MtnBiker wrote: Sat Jul 27, 2024 2:29 pm
Personally, I think the detailed analysis is converging toward a general confirmation that what people have been doing all along is good enough. That is, if filling the gap uniformly, buy excess holdings of the bracket years in roughly equal amounts. I'm not sure if Kevin would agree at this point.
Upon reviewing this thread (I had been away), this is my conclusion as well.
Currently I have approx. 55% 2034s vs. 45% 2040s.
I have some TIPS maturing this October, and I will invest the proceeds into more 2040s, which will leave me roughly with 51% 2034s and 49% 2040s.
I think that is close enough.

In February I will buy my DARA of 2035s with roughly 5/6 (DARA) 2034s and 1/6 (DARA) 2040s...or, if I am still a little "2040 deficient" (vs. 2034s), maybe I will use only 2034s' which will leave me closer to the ideal balance for when I buy 2036-2039s. I see no reason to be precise.

Thank you, Kevin, MtnBiker and others, for doing the work!
The thread about the tipsladder.com tool has recently announced (viewtopic.php?p=7976696#p7976696) that there is a new experimental UI for the TIPS ladder calculator here: https://tipsladder.com/spa. This new version offers a more manual way of creating a ladder, including assigning 2034 and 2040 TIPS directly to each gap year 2035-2039 in whatever proportions you desire.

You might find this new tool helpful for better visualization of building the excess bracket year holdings up in n/6 chunks assigned to each gap year. The tool shows the effects of coupon interest payments on each year's income, including in the gap, which is helpful for smoothing DARA when trying to get as even a balance as possible between each gap year. It also calculates the Macaulay duration of the mix of bonds assigned to each gap year, so you can confirm that the duration for the holdings for each year increases incrementally as it should from 2034 to 2040. Once completed, the ladder can be stored or printed out as a record showing how many of each bracket year bond to sell each year when swapping the proceeds for the newly issued gap-year bond.
Thanks, MtnBiker!

I have not yet set up the TIPS ladder tool, though I downloaded it a while ago. It looks excellent.
I have been keeping track of my TIPS on an Excel spreadsheet, because I have been doing so since I started buying TIPS in mid-2022.
I mean to use the ladder tool at some point, but if and when I get around to it is unknown....the spreadsheet is not nearly as good but "good enough" for my purposes (knowing what I have, when it is maturing, and about what it is worth). I think I have excess maturing every year over what I will need to spend. I don't really need precision.