Filling the TIPS gap years with bracket year duration matching

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

Post by protagonist »

exodusing wrote: Mon Jun 03, 2024 9:38 am
protagonist wrote: Mon Jun 03, 2024 9:16 am DARA is a very vague estimate of needs, I doubt if anybody will know in 2027 with much accuracy how much money they will need in 2037 for expenses.
This is a common objection to TIPS ladders. I almost never see it as an objection to any other planning or withdrawal method, even though it seems equally applicable to just about every method. For example, the standard 4% "SWR" analyses assume a uniform inflation adjusted annual withdrawal.

The best solution would seem to be to keep some extra funds for the unanticipated and perhaps hope things will even out over time. Given that current TIPS rates provide for a 4.6% inflation adjusted withdrawal rate, you could use TIPS for the oft-cited 4% and invest the rest in something else to deal with the occasional additional need. Note that 4/4.6 is 87%, so you'd have an additional 13% for an emergency fund or stocks or whatever.
Exactly.

I don't see it as a valid argument at all against a TIPS ladder.

Kevin's explanation above for what he is doing is a good one. It is uncharted territory. If I understand correctly (he will correct me if not), it is somewhat an academic exercise, to understand the underlying mathematics. Quoting Kevin: "Right now I'm not so concerned with real-life issues, but simply showing that whatever gap year coverage scheme we use is supported by bond math. "

For those of us more removed from academic finance, just figuring out a sensible way to plan our retirement, all this may not be very relevant. We are all pretty much on the same page, with small differences. Unless, of course, he finds a particular glaring error in something that was previously assumed. But, given the high level of brain power, and time and energy, that has gone into this so far (Kevin, #Cruncher and others), I think the likelihood of that is low.

I don't think I am exaggerating in saying that we are all intelligent and thoughtful investors. Realistically, barring a financial apocalypse, I am pretty sure all of us will outlive our savings and have a secure retirement, despite the differences in our portfolios, lifestyles and net worth.

That said, I continue to follow the thread out of curiosity.
Last edited by protagonist on Mon Jun 03, 2024 11:20 am, edited 1 time in total.
protagonist
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Re: Filling the TIPS gap years with bracket year duration matching

Post by protagonist »

MtnBiker wrote: Mon Jun 03, 2024 10:24 am
protagonist wrote: Mon Jun 03, 2024 9:16 am
Is tweaking our DARA for 2037 within a relatively small range ten years in advance based on these tweaks serving any purpose , with all the unknowns? DARA ten or more years in advance is, in my mind, just a shot in the dark. Especially at our age. We choose one because we have to, not because we know much.
I would say that Kevin is attempting an academic exercise to find the optimum way to duration match the gap years. I think it is useful to know what the best way is in terms of locking in today's yield in the gap.

This is not to say that the best way academically is the best way for everyone in practice. As I stated above, the minor tweaking involved in the secondary transactions to smooth ARA/DARA mismatch is optional. I can imagine that the average investor would want to do something simpler (forego the tweaking, for example), and that is fine if the cost of the simpler approach is close enough to the cost of the optimal approach. Hopefully we will soon have an idea of the relative costs of the various approaches so that individuals can decide for themselves.
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Kevin M
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

Here's the insight I had related to why the gap year fixed coupon duration matching model seems to work even though the gap year coupon is not fixed. It's based on time value of money (TVM) analysis of cash flows, which is relevant to bond math.

Start with the present value formula for a series of cash flows:

PV = SUMt=1n{Ct / (1+r)t}

where n is the number of periods, Ct is the cash flow at the end of period t, and r is the discount rate.

Note: I assume annual coupon payments in this and all of my other posts in this thread, unless stated otherwise. This simplifies the math slightly , but doesn't impact the results significantly compared to semi-annual coupon payments.

For bonds, present value, PV, is bond value, V, the discount rate, r, is yield (to maturity), y, and Ct is a constant equal to coupon, c, times total face value, f, except for the final cash flow, Cn, which also includes face value, f, discounted by 1/(1+r)^n . So we can rewrite the equation for bonds as:

V = SUMt=1n{c*f / (1+y)t} + f/(1+y)n

We can make the following observations about this equation:
  • Since y is in the denominator of the present value of each of the cash flows, an increase in y results in a decrease in the cash flows and in V, and vice versa.
    • This is the foundation of the inverse relationship between bond price or value and yield for a bond trading on the secondary market.
  • If we hold V constant, i.e., make it an independent variable, and consider c as a dependent variable rather than a constant, an increase in y requires an increase in c for the equation to hold.
    • This is why the coupon of a gap year TIPS will be higher than it would be if issued now if market yield increases enough, since price is held approximately constant.
    • If we allow c to be continuous, rather than discrete (due to the 0.125 percentage point increment in Treasury coupons), then we could hold price exactly constant, and any increase in y would result in an increase in c.
  • Intuitively, it seems that an increase in c when viewed as a dependent variable should deliver the same benefit for an increase in y as does the increase in V when it is the dependent variable.
That's the insight, and that's as far as I've gotten. This isn't a mathematical proof, but is uses TVM analysis to bolster the intuition that the gap year fixed coupon duration matching model might apply even though the gap year coupon is not fixed.
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

Post by Jaylat »

Kevin M wrote: Mon Jun 03, 2024 10:08 am I think you're onto something with your coupon analysis. I'll think more about this.

My way of thinking about your situation is that taxes on the OID (inflation adjustments) are part of your residual expenses, so part of your DARA. After all, income taxes are an expense, just like housing, food and energy.
I hold TIPS in taxable, so taxes have to be paid every year on the OID inflation accrual, which is completely unpredictable. Other non-TIPS income is much more predictable, and so treating it like an expense makes sense. And I like having the coupons available to act as a buffer against a big tax shock.

I believe your TIPS are all in tax deferred accounts?
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Re: Filling the TIPS gap years with bracket year duration matching

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Jaylat wrote: Mon Jun 03, 2024 7:14 pm
Kevin M wrote: Mon Jun 03, 2024 10:08 am I think you're onto something with your coupon analysis. I'll think more about this.

My way of thinking about your situation is that taxes on the OID (inflation adjustments) are part of your residual expenses, so part of your DARA. After all, income taxes are an expense, just like housing, food and energy.
I hold TIPS in taxable, so taxes have to be paid every year on the OID inflation accrual, which is completely unpredictable. Other non-TIPS income is much more predictable, and so treating it like an expense makes sense. And I like having the coupons available to act as a buffer against a big tax shock.
Ignoring the coupons for your DARA is just mental accounting for increasing your DARA. Either way, you need to have enough ARA to cover expected and unexpected expenses, with unexpectedly large income taxes being in the latter category.

I don't know how you determined the construction of your TIPS ladder, but #Cruncher's spreadsheet will tell you how many TIPS are needed to meet the DARA for each year. To increase the ARA, you can either increase DARA or make the sum of the multipliers for each year > 1.

Example:
  • With a DARA of $100K and a multiplier of 1 for the Oct 2025 TIPS (and only that for 2025), I need 79, and the ARA is 100,715, consisting of 95,132 in principal and 5,583 in interest.
  • If leave the DARA at $100K but increase the multiplier to 1.3, I need 104, which generates 130,857 ARA, consisting of 125,237 principal and 5,620 interest.
    • The principal/coupon split isn't correct if I use a multiplier of 1.3 for every year, but the ARA still is about $130K
  • If I increase DARA to $130K, and use a multiplier of 1.0, I need 102, which generates 130,091 ARA, consisting of 122,828 principal and 7,262 interest.
  • Either way I get an ARA of about $130K.
I'm not suggesting that there's anything wrong about what you're doing, but just that it's different way of doing the mental accounting to get the same result.
Jaylat wrote: Mon Jun 03, 2024 7:14 pmI believe your TIPS are all in tax deferred accounts?
Sure, but with high unexpected inflation, my RMDs would be bigger, resulting in more income tax. The impact might not be as large as it is for you, but the principal is the same. If I plan for a DARA that's too low, I would come up short for the RMD, and I'd have to withdraw from later-year rungs to cover the RMD and the additional taxes.
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

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Kevin M wrote: Mon Jun 03, 2024 4:08 pm <snip>

So we can rewrite the equation for bonds as:

V = SUMt=1n{c*f / (1+y)t} + f/(1+y)n

We can make the following observations about this equation:
  • <snip>
  • If we hold V constant, i.e., make it an independent variable, and consider c as a dependent variable rather than a constant, an increase in y requires an increase in c for the equation to hold.
  • <snip>
I had a mental block earlier today in doing the algebra to make c the dependent variable in the equation above. Unless I've made a mistake, here it is:

c = ( V - f/(1+y)n ) / SUMt=1n{ f / (1+y)t }

Here we see that increasing y increases the numerator and decreases the denominator, with the latter having a larger impact, both making c larger.

Later I'll plug some values into both equations to check their accuracy.
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

Post by Jaylat »

Kevin M wrote: Mon Jun 03, 2024 8:40 pm
Jaylat wrote: Mon Jun 03, 2024 7:14 pm
Kevin M wrote: Mon Jun 03, 2024 10:08 am I think you're onto something with your coupon analysis. I'll think more about this.

My way of thinking about your situation is that taxes on the OID (inflation adjustments) are part of your residual expenses, so part of your DARA. After all, income taxes are an expense, just like housing, food and energy.
I hold TIPS in taxable, so taxes have to be paid every year on the OID inflation accrual, which is completely unpredictable. Other non-TIPS income is much more predictable, and so treating it like an expense makes sense. And I like having the coupons available to act as a buffer against a big tax shock.
Ignoring the coupons for your DARA is just mental accounting for increasing your DARA. Either way, you need to have enough ARA to cover expected and unexpected expenses, with unexpectedly large income taxes being in the latter category.

I don't know how you determined the construction of your TIPS ladder, but #Cruncher's spreadsheet will tell you how many TIPS are needed to meet the DARA for each year. To increase the ARA, you can either increase DARA or make the sum of the multipliers for each year > 1.

Example:
  • With a DARA of $100K and a multiplier of 1 for the Oct 2025 TIPS (and only that for 2025), I need 79, and the ARA is 100,715, consisting of 95,132 in principal and 5,583 in interest.
  • If leave the DARA at $100K but increase the multiplier to 1.3, I need 104, which generates 130,857 ARA, consisting of 125,237 principal and 5,620 interest.
    • The principal/coupon split isn't correct if I use a multiplier of 1.3 for every year, but the ARA still is about $130K
  • If I increase DARA to $130K, and use a multiplier of 1.0, I need 102, which generates 130,091 ARA, consisting of 122,828 principal and 7,262 interest.
  • Either way I get an ARA of about $130K.
I'm not suggesting that there's anything wrong about what you're doing, but just that it's different way of doing the mental accounting to get the same result.
Jaylat wrote: Mon Jun 03, 2024 7:14 pmI believe your TIPS are all in tax deferred accounts?
Sure, but with high unexpected inflation, my RMDs would be bigger, resulting in more income tax. The impact might not be as large as it is for you, but the principal is the same. If I plan for a DARA that's too low, I would come up short for the RMD, and I'd have to withdraw from later-year rungs to cover the RMD and the additional taxes.
You're right , it is just mental accounting, but the actual accounting is much more complex than what you've posted here (and thanks so much for taking the time to post it!).

Keep in mind that we're talking about taxes not on just one TIPS at maturity (your example) but on the annual OID interest on every single TIPS in your ladder. For a 20 year ladder that's 20 TIPS you need to pay taxes on every year - not just one. So for TIPS in taxable you actually have relatively few taxes due at maturity - it's all paid on an ongoing basis.

Do the coupons cover this? Maybe, depending on the inflation rate. But at least the coupons are somewhat correlated with the annual tax liability as they both grow with inflation.

Your 1.3x example would work fine for a distribution from a tax deferred account where all taxes are due at once.
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Funding income taxes on OID with a TIPS ladder

Post by Kevin M »

Jaylat wrote: Tue Jun 04, 2024 7:23 am
Kevin M wrote: Mon Jun 03, 2024 8:40 pm
Jaylat wrote: Mon Jun 03, 2024 7:14 pm
Kevin M wrote: Mon Jun 03, 2024 10:08 am I think you're onto something with your coupon analysis. I'll think more about this.

My way of thinking about your situation is that taxes on the OID (inflation adjustments) are part of your residual expenses, so part of your DARA. After all, income taxes are an expense, just like housing, food and energy.
I hold TIPS in taxable, so taxes have to be paid every year on the OID inflation accrual, which is completely unpredictable. Other non-TIPS income is much more predictable, and so treating it like an expense makes sense. And I like having the coupons available to act as a buffer against a big tax shock.
Ignoring the coupons for your DARA is just mental accounting for increasing your DARA. Either way, you need to have enough ARA to cover expected and unexpected expenses, with unexpectedly large income taxes being in the latter category.

I don't know how you determined the construction of your TIPS ladder, but #Cruncher's spreadsheet will tell you how many TIPS are needed to meet the DARA for each year. To increase the ARA, you can either increase DARA or make the sum of the multipliers for each year > 1.

Example:
  • With a DARA of $100K and a multiplier of 1 for the Oct 2025 TIPS (and only that for 2025), I need 79, and the ARA is 100,715, consisting of 95,132 in principal and 5,583 in interest.
  • If leave the DARA at $100K but increase the multiplier to 1.3, I need 104, which generates 130,857 ARA, consisting of 125,237 principal and 5,620 interest.
    • The principal/coupon split isn't correct if I use a multiplier of 1.3 for every year, but the ARA still is about $130K
  • If I increase DARA to $130K, and use a multiplier of 1.0, I need 102, which generates 130,091 ARA, consisting of 122,828 principal and 7,262 interest.
  • Either way I get an ARA of about $130K.
I'm not suggesting that there's anything wrong about what you're doing, but just that it's different way of doing the mental accounting to get the same result.
Jaylat wrote: Mon Jun 03, 2024 7:14 pmI believe your TIPS are all in tax deferred accounts?
Sure, but with high unexpected inflation, my RMDs would be bigger, resulting in more income tax. The impact might not be as large as it is for you, but the principal is the same. If I plan for a DARA that's too low, I would come up short for the RMD, and I'd have to withdraw from later-year rungs to cover the RMD and the additional taxes.
You're right , it is just mental accounting, but the actual accounting is much more complex than what you've posted here (and thanks so much for taking the time to post it!).

Keep in mind that we're talking about taxes not on just one TIPS at maturity (your example) but on the annual OID interest on every single TIPS in your ladder. For a 20 year ladder that's 20 TIPS you need to pay taxes on every year - not just one. So for TIPS in taxable you actually have relatively few taxes due at maturity - it's all paid on an ongoing basis.

Do the coupons cover this? Maybe, depending on the inflation rate. But at least the coupons are somewhat correlated with the annual tax liability as they both grow with inflation.

Your 1.3x example would work fine for a distribution from a tax deferred account where all taxes are due at once.
I don't think the point I'm making is getting across.
  1. I realize that you're taxed on the inflation adjustments (reported on 1099-OID) for all TIPS in the ladder.
  2. If you want to pay your taxes on all TIPS OID with the ARA from the ladder, you'll need more ARA than you would for your expenses excluding taxes.
  3. One way or the other, you'll need to estimate the worst income tax case you want to cover.
  4. Then you'll need to estimate ARA to cover OID taxes and all other expenses.
  5. There are at least three ways to ensure that you have the required ARA you estimated in #4.
    1. Increase your DARA.
    2. Increase your multiplier for each year.
    3. Ignore coupons and set DARA to non-OID expenses with multipliers = 1 for each year
  6. It seems to me that 1 and 2 are more reliable than 3.
I just looked at what thread we're posting in, and realized that we're way off topic. If you want to continue this discussion, I recommend we continue in either:
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

Post by MtnBiker »

Kevin M wrote: Mon Jun 03, 2024 9:59 am
My goal is maintaining ARA/DARA match, so I don't care what the durations of anything are when I do the swaps. I only care about duration matching in that it optimizes my ability to maintain the ARA/DARA match when I do any swaps related to buying the gap years. To state it another way, I care about duration matching the bracket year TIPS to the gap year TIPS so that any yield changes between ladder construction and gap year TIPS purchase do not negatively impact my ability to maintain ARA/DARA match when I do the swaps related to buying the gap years.
Speaking of duration matching, I have a question about this chart:
Kevin M wrote: Sat Jun 01, 2024 4:24 pm
Here is the updated table with the correct values, and the IRR values.

Image

Note that geo return equals initial yield (and coupon, for gap year TIPS) when reinvestment rate = coupon (light green background), and note that IRR equals initial yield with 0% reinvestment rate, which is the only scenario that applies for an IRR calculation, since it's calculated based on cash flows with no reinvestment assumption.
I've forgotten which gap year this chart is modeling. Anyway, my question is, what ratio(s) of 2034/2040 bracket year was assumed in this calculation? Was the same ratio used independent of the interest rate at the time of the swap? We initially pick a ratio for each gap year based on the interest rate when we buy the excess bracket year holdings. But then interest rates change.

Obviously, the duration of the gap year to be purchased varies with its coupon rate, so strict duration matching at the time of the swap would suggest using a different ratio of 2034/2040 depending on the coupon of the gap year.

If you used the same 2034/2040 ratio for the 3% and 1% coupon cases, I wonder how much different the terminal value would be if you used a duration-matched ratio instead?
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

MtnBiker wrote: Tue Jun 04, 2024 10:05 am
Kevin M wrote: Mon Jun 03, 2024 9:59 am
My goal is maintaining ARA/DARA match, so I don't care what the durations of anything are when I do the swaps. I only care about duration matching in that it optimizes my ability to maintain the ARA/DARA match when I do any swaps related to buying the gap years. To state it another way, I care about duration matching the bracket year TIPS to the gap year TIPS so that any yield changes between ladder construction and gap year TIPS purchase do not negatively impact my ability to maintain ARA/DARA match when I do the swaps related to buying the gap years.
Speaking of duration matching, I have a question about this chart:
Kevin M wrote: Sat Jun 01, 2024 4:24 pm
Here is the updated table with the correct values, and the IRR values.

Image

Note that geo return equals initial yield (and coupon, for gap year TIPS) when reinvestment rate = coupon (light green background), and note that IRR equals initial yield with 0% reinvestment rate, which is the only scenario that applies for an IRR calculation, since it's calculated based on cash flows with no reinvestment assumption.
I've forgotten which gap year this chart is modeling. Anyway, my question is, what ratio(s) of 2034/2040 bracket year was assumed in this calculation? Was the same ratio used independent of the interest rate at the time of the swap? We initially pick a ratio for each gap year based on the interest rate when we buy the excess bracket year holdings. But then interest rates change.

Obviously, the duration of the gap year to be purchased varies with its coupon rate, so strict duration matching at the time of the swap would suggest using a different ratio of 2034/2040 depending on the coupon of the gap year.

If you used the same 2034/2040 ratio for the 3% and 1% coupon cases, I wonder how much different the terminal value would be if you used a duration-matched ratio instead?
I'm not sure this question even makes sense, given the assumptions for this model. You might want to reread the post in which I first explained it: viewtopic.php?p=7890951#p7890951.

At any rate, I've moved on from this, and want to focus on the new track. Working on that now ...
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

Post by Jaylat »

Kevin and I have moved our discussion on TIPS taxation to the thread "Taxation of Treasury bills, notes and bonds." You can follow it here:

viewtopic.php?p=7898027#p7898027
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

Kevin M wrote: Mon Jun 03, 2024 9:03 pm
Kevin M wrote: Mon Jun 03, 2024 4:08 pm <snip>
So we can rewrite the equation for bonds as:

V = SUMt=1n{c*f / (1+y)t} + f/(1+y)n

We can make the following observations about this equation:
  • <snip>
  • If we hold V constant, i.e., make it an independent variable, and consider c as a dependent variable rather than a constant, an increase in y requires an increase in c for the equation to hold.
  • <snip>
I had a mental block earlier today in doing the algebra to make c the dependent variable in the equation above. Unless I've made a mistake, here it is:

c = ( V - f/(1+y)n ) / SUMt=1n{ f / (1+y)t }

Here we see that increasing y increases the numerator and decreases the denominator, with the latter having a larger impact, both making c larger.

Later I'll plug some values into both equations to check their accuracy.
Finally got around to doing this, and formatting it so hopefully it's somewhat understandable.

First, I just developed a spreadsheet approach to calculate the standard bond version of the present value formula to calculate bond value or price, which is the first formula shown above. The purpose here was just to make sure I implemented the summation part correctly.

Image

The formula is used to calculate the values in row 16, and the spreadsheet PV formula is used to calculate the values in row 17. Row 18 just subtracts the row 17 values from the row 16 values as a quick check.

So I got the summation bit correct.

Next I used a similar approach to do the spreadsheet version of the second equation, with coupon as the independent variable. Again, the idea here is that:
  • Price or value will be fairly constant: price close to 100, and standard bond value close to 1,000. In the model I simply use 1.000 as the value for simplicity, for example in the PV formula.
  • Coupon will closely match yield, the match accuracy limited only by the 0.125 pp coupon resolution and the lower coupon limit of 0.125%.
In the model I drop these constraints, so assume that coupon can be continuous and negative, like yield. Here are the results:

Image

The values in row 16 are calculated with the formula. Since the verification is that coupon = yield, or coupon - yield = 0, the values in row 17 calculate the latter as the check.

This is as far as I've gotten, and I'm not sure where to go next, but the fact that the standard bond value as PV equation can be rearranged such that coupon is the independent variable, and the formula gives the expected result, increases my warm fuzzy feeling that the original duration model works for fixed price and variable coupon, even though it calculates the duration match bracket year ratios using the standard fixed coupon and variable price.
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

Post by Kevin M »

I had requested inputs from the builder's of the two TIPS ladder tools that are commonly used by Bogleheads in the thread dedicated to those tools. No replies yet, but I summarized our progress in this thread in a reply in that thread just now. I think it's worth replicating that reply in this thread, so here it is.
Kevin M wrote: Thu Jun 06, 2024 10:36 am
Kevin M wrote: Tue May 28, 2024 11:03 am I have a question about both tools. Why does each one use a combination of 2034s and 2040s to cover the gap years? The #Cruncher spreadsheet uses multipliers of 3 and 4 respectively for these bracket years, and tipsladder.com uses different combinations of them depending on the options chosen for "Which TIPS to use to fund a year without a maturing TIPS:".

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 guess I could see some sense in this if the yield curve had the normal, positively-sloped shape, especially if it was fairly steep, since we'd be taking advantage of higher yields at longer maturities. However, with the currently inverted yield curve, we get higher yields at shorter maturities.

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.

:?
All quiet on the TIPS tool builders' front.

I feel like we've made progress in the thread linked above. To summarize, I'm coming around to the view that duration matching based on the assumption that the hypothetical gap year TIPS trade on the secondary market is valid, even though the assumption that coupon is fixed, and so is an independent variable and price is the dependent variable in the PV equation for bond price/value, doesn't match reality. Here's my quick summary of my most recent work:
  1. The PV equation for bond value can be rearranged so that coupon is the dependent variable, and price is an independent variable.
  2. Solving this equation results in coupon = yield, which is closer to the reality of a Treasury auction.
  3. This gives me a warm, fuzzy feeling that the same financial benefit is derived from a change in coupon from what it would be now to what it actually is at auction is the same as that derived from a change in yield from what it is now to what it is at auction.
  4. However, I haven't taken the next step, which I think is to show that by duration matching the bracket year TIPS (e.g., 2034 and 2040) to a gap year TIPS, e.g., 2035, one or more swaps (selling ladder TIPS to buy auction TIPS) can be done such that annual real income (ARA) remains close to DARA for each issue held in the ladder.
  5. As implied in #4, I've zeroed in on ARA = DARA after the gap year swap(s) as the primary goal of optimizing a duration matching strategy.
I've had difficulty trying to simulate this with the #Cruncher spreadsheet, due to the special handling of the 2034 to cover the gap years--at least it seems that way to me. The last bit of work I did on this was to work with a 10-year ladder, and treat one of the earlier years as a gap year, say gap - 2032, with bracket years 2031 and 2033. I got sidetracked on 1 and 2 above.

Still interested in your thoughts, #Cruncher, since you're the granddaddy of TIPS ladder building tools as far as I'm concerned. Usually when you don't reply to challenges like this, you're busily working on a spreadsheet solution in the background. Fingers crossed, but if not, I'm thinking I'll get there eventually.
If I make a calculation error, #Cruncher probably will let me know.
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Re: Intutition on why duration matching works with unknown gap year coupon

Post by Kevin M »

Kevin M wrote: Sun Jun 02, 2024 8:28 pm
  • <snip>
  • The model doesn't apply exactly to gap year TIPS bought at auction, because the price will always be close to par. This is because the coupon is not predetermined, but depends on the auction yield.
  • Although the model doesn't apply directly to the auction case, the principles seem to work their magic in other ways.
  • Example 1:
    • If yields increase, the average price of the duration matched bracket year TIPS decreases, so we can buy fewer gap year TIPS than we need for the gap year. This is because the cost of the gap year TIPS is almost constant (close to 100% of inflation-adjusted par).
    • <snip>
  • Example 2.
    • <snip>
  • <snip>
I've been working a bit more on this, and I've realize that while the first underlined statement is true, the second underlined statement is not true, since we might need to buy a larger quantity of the gap year TIPS; i.e., the price is relatively constant but the total cost of the gap years TIPS is not. I just wanted to bring this toward the end of the thread so I can find it easily and provide more detail when I've made more progress on evaluating it.
If I make a calculation error, #Cruncher probably will let me know.
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Re: Intutition on why duration matching works with unknown gap year coupon

Post by kaesler »

Kevin M wrote: Fri Jun 07, 2024 9:40 am I've been working a bit more on this, and I've realize that while the first underlined statement is true, the second underlined statement is not true, since we might need to buy a larger quantity of the gap year TIPS; i.e., the price is relatively constant but the total cost of the gap years TIPS is not. I just wanted to bring this toward the end of the thread so I can find it easily and provide more detail when I've made more progress on evaluating it.
Thanks for periodic summaries like this. I'm watching this thread when I can for ideas to apply to tipsladder.com 2.0.
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Re: Intutition on why duration matching works with unknown gap year coupon

Post by kaesler »

kaesler wrote: Fri Jun 07, 2024 5:07 pm Thanks for periodic summaries like this. I'm watching this thread when I can for ideas to apply to tipsladder.com 2.0.

As it stands now, I've assumed that duration can be good to know so it offers a manual mode of ladder construction in which, for each rung:
  • A list of candidate bonds is offered
  • As you select the desired number of each candidate bond, the Macauley duration is updated immediately in a field on the screen
  • Macauley duration is everywhere displayed as a pair (numberOfYears, endDate), where endDate is settlement date + numberOfYears
  • For non-gap years the candidate bonds are just the bonds maturing in the rung year
  • For gap years, the candidate bonds are the bonds maturing in the years before and after the gap year period
So for a given rung, you can select a duration that suits your needs.
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Scenario 1: No yield changes, yields 2%

Post by Kevin M »

More progress.

I set up a version of the #Cruncher tips ladder spreadsheet such that 2032/32 are the gap years, and 2030/33 are the bracket years, which avoids messing with the 2034 and gap years with no rows in the spreadsheet, although I do include the 2034 as the longest maturity in the ladder. So for modeling purposes
  • We have a 10-year ladder from 2025-2034.
  • 2031/32 are yet to be issued in the pre-swap scenarios; i.e., they are modeled as the gap years.
  • All TIPS except the gap year TIPS have coupons that are their actual coupons.
  • Temporal considerations are suspended, so all yield changes are instantaneous, and both gap year TIPS are issued at the same time.
  • We'll hold 2 * DARA each in 2030/2033 to cover the gap years, 1 each for the bracket years and 1 each for the gap years, before the swaps when the gap years are issued. In #Cruncher spreadsheet lingo, the multipliers for 2030/33 are each 2 before the swap.
  • Gap year TIPS have coupon = yield when issued, 0% coupon yield is allowed, and coupon rate is continuous (i.e., no 0.125 pp resolution limit).
  • Initial yield of all TIPS is 2.00%.
So I'll run three scenarios:
  1. No yield changes before the swaps when gap year TIPS are issued, gap year TIPS yields = coupons = 2%.
  2. Yields increase to 4% before the swaps, so gap year TIPS yields = coupons = 4% 2%. (Edited to correct typo)
  3. Yields decrease to 0% before the swaps, so gap year TIPS yields = coupons = 0%.
I've recorded the before and after values of multipliers, number of TIPS needed, and costs of TIPS for each scenario. So I'll share those, with my observations.

In this post I'll only cover Scenario 1, no change in yields, so that the post isn't too long. I'll cover scenarios 2 and 3 in subsequent posts.

Scenario 1: no change in yields.

Imagehow to screenshot on windows

Observations:
  • I've highlighted the bracket and gap years to facilitate visual interpretation.
  • The middle-left table on is pre-swap, the middle-right is post-swap, and the far-right table shows the differences in quantities and $ amounts needed.
  • For the 2030 and 2033 Diff needed and Diff total calcs I divided the pre-swap values by 2, since the multipliers changed from 2 to 1.
  • The quantities and costs (or values) of TIPS needed for 2033 and 2034 don't change; for the 2033 I divide the pre-swap values by 2 for the check.
  • The quantities and costs (or values) of TIPS needed for 2025-2030 decrease; for the 2030 I divide the pre-swap values by 2 for the check.
    • I think this is because the 2% coupon for the gap year TIPS is greater than coupon of most of the 2025-2030 TIPS.
    • I'm thinking I should run another trial for which all coupons are assumed to be 2% pre-swap.
  • The diff total of 12,368, in the merged 2031/32 rows is the sum of the post-swap 2030-2033 totals minus the sum of the pre-swap 2030-2033 totals.
    • This means that the 2030-2033 TIPS cost (or are valued at) this much more post-swap than pre-swap.
    • All of this extra cost is because the gap year TIPS cost more than the proceeds from selling half each of the bracket year TIPS.
  • The last value in the Diff total column indicates that the ladder costs 1,543 more than it did before the swap.
  • So, if we sold the indicated diff total values for the 2025-2029, we'd only be short 1,543 after buying the gap year TIPS, and this is about the rounding error for the cost of one TIPS.
Last edited by Kevin M on Tue Jun 11, 2024 8:14 am, edited 2 times in total.
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

Post by MtnBiker »

Kevin M wrote: Sat Jun 08, 2024 4:25 pm

*The last value in the Diff total column indicates that the ladder costs 1,543 more than it did before the swap.
Very interesting.

Do you feel it would be worth the effort to calculate the average duration of the 10-yr ladder holdings before and after the indicated swaps?
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Jaylat »

Kevin M wrote: Sat Jun 08, 2024 4:25 pm More progress.

I set up a version of the #Cruncher tips ladder spreadsheet such that 2032/32 are the gap years, and 2030/33 are the bracket years, which avoids messing with the 2034 and gap years with no rows in the spreadsheet, although I do include the 2034 as the longest maturity in the ladder. So for modeling purposes
  • We have a 10-year ladder from 2025-2034.
  • 2031/32 are yet to be issued in the pre-swap scenarios; i.e., they are modeled as the gap years.
  • All TIPS except the gap year TIPS have coupons that are their actual coupons.
  • Temporal considerations are suspended, so all yield changes are instantaneous, and both gap year TIPS are issued at the same time.
  • We'll hold 2 * DARA each in 2030/2033 to cover the gap years, 1 each for the bracket years and 1 each for the gap years, before the swaps when the gap years are issued. In #Cruncher spreadsheet lingo, the multipliers for 2030/33 are each 2 before the swap.
  • Gap year TIPS have coupon = yield when issued, 0% coupon yield is allowed, and coupon rate is continuous (i.e., no 0.125 pp resolution limit).
  • Initial yield of all TIPS is 2.00%.
So I'll run three scenarios:
  1. No yield changes before the swaps when gap year TIPS are issued, gap year TIPS yields = coupons = 2%.
  2. Yields increase to 4% before the swaps, so gap year TIPS yields = coupons = 4% 2%. (Edited to correct typo)
  3. Yields decrease to 0% before the swaps, so gap year TIPS yields = coupons = 0%.
I've recorded the before and after values of multipliers, number of TIPS needed, and costs of TIPS for each scenario. So I'll share those, with my observations.

In this post I'll only cover Scenario 1, no change in yields, so that the post isn't too long. I'll cover scenarios 2 and 3 in subsequent posts.

Scenario 1: no change in yields.

Imagehow to screenshot on windows

Observations:
  • I've highlighted the bracket and gap years to facilitate visual interpretation.
  • The middle-left table on is pre-swap, the middle-right is post-swap, and the far-right table shows the differences in quantities and $ amounts needed.
  • For the 2030 and 2033 Diff needed and Diff total calcs I divided the pre-swap values by 2, since the multipliers changed from 2 to 1.
  • The quantities and costs (or values) of TIPS needed for 2033 and 2034 don't change; for the 2033 I divide the pre-swap values by 2 for the check.
  • The quantities and costs (or values) of TIPS needed for 2025-2030 decrease; for the 2030 I divide the pre-swap values by 2 for the check.
    • I think this is because the 2% coupon for the gap year TIPS is greater than coupon of most of the 2025-2030 TIPS.
    • I'm thinking I should run another trial for which all coupons are assumed to be 2% pre-swap.
  • The diff total of 12,368, in the merged 2031/32 rows is the sum of the post-swap 2030-2033 totals minus the sum of the pre-swap 2030-2033 totals.
    • This means that the 2030-2033 TIPS cost (or are valued at) this much more post-swap than pre-swap.
    • All of this extra cost is because the gap year TIPS cost more than the proceeds from selling half each of the bracket year TIPS.
  • The last value in the Diff total column indicates that the ladder costs 1,543 more than it did before the swap.
  • So, if we sold the indicated diff total values for the 2025-2029, we'd only be short 1,543 after buying the gap year TIPS, and this is about the rounding error for the cost of one TIPS.
Not sure I understand any of this. Why would the TIPS needed for 2025-30 change at all? Unless your DARA changes, the face amount of the TIPS should be exactly the same. The yield changes would change their price but would have zero effect on the real DARA amount needed for each year.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Jaylat »

Jaylat wrote: Sun Jun 09, 2024 7:55 am Not sure I understand any of this. Why would the TIPS needed for 2025-30 change at all? Unless your DARA changes, the face amount of the TIPS should be exactly the same. The yield changes would change their price but would have zero effect on the real DARA amount needed for each year.
Okay, I get it - it's the higher coupons from the 2031-32 years. So you need to go out and sell off small portions of the 2025-30 TIPS to offset the coupons.

That's another good reason from my perspective not to count the coupons for DARA, but to instead use them for tax payments. It saves making all these extra trades. If you hold TIPS in taxable you'd also have to recalculate all of the withholding gross up for 2025-30 TIPS taxes for each year.

Are you assuming the passage of time in this calculation? In real life we would wait 2 years for the issuance of the 2031 and 2032 TIPS. So the duration of all the bonds would be reduced by 2 years.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

Jaylat wrote: Sun Jun 09, 2024 8:06 am
Jaylat wrote: Sun Jun 09, 2024 7:55 am Not sure I understand any of this. Why would the TIPS needed for 2025-30 change at all? Unless your DARA changes, the face amount of the TIPS should be exactly the same. The yield changes would change their price but would have zero effect on the real DARA amount needed for each year.
Okay, I get it - it's the higher coupons from the 2031-32 years. So you need to go out and sell off small portions of the 2025-30 TIPS to offset the coupons.
Correct.
Jaylat wrote: Sun Jun 09, 2024 8:06 am Are you assuming the passage of time in this calculation? In real life we would wait 2 years for the issuance of the 2031 and 2032 TIPS. So the duration of all the bonds would be reduced by 2 years.
No. As stated, for modeling purposes "Temporal considerations are suspended, so all yield changes are instantaneous, and both gap year TIPS are issued at the same time."

This is a first approximation model, to just get a handle on how well duration matching works in a purely theoretical sense.
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

Post by Jaylat »

Kevin M wrote: Sun Jun 09, 2024 9:39 am
Jaylat wrote: Sun Jun 09, 2024 8:06 am
Jaylat wrote: Sun Jun 09, 2024 7:55 am Not sure I understand any of this. Why would the TIPS needed for 2025-30 change at all? Unless your DARA changes, the face amount of the TIPS should be exactly the same. The yield changes would change their price but would have zero effect on the real DARA amount needed for each year.
Okay, I get it - it's the higher coupons from the 2031-32 years. So you need to go out and sell off small portions of the 2025-30 TIPS to offset the coupons.
Correct.
Jaylat wrote: Sun Jun 09, 2024 8:06 am Are you assuming the passage of time in this calculation? In real life we would wait 2 years for the issuance of the 2031 and 2032 TIPS. So the duration of all the bonds would be reduced by 2 years.
No. As stated, for modeling purposes "Temporal considerations are suspended, so all yield changes are instantaneous, and both gap year TIPS are issued at the same time."

This is a first approximation model, to just get a handle on how well duration matching works in a purely theoretical sense.
How would you characterize the $1,543 difference as a percentage? Would it be a percent of the combined 2031-32 TIPS?

Basically this spreadsheet is saying given the current TIPS available and assuming no changes in TIPS values whatsoever the closest you can get to 100% accuracy is around 99.2%.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

Jaylat wrote: Sun Jun 09, 2024 9:52 am
Kevin M wrote: Sun Jun 09, 2024 9:39 am
Jaylat wrote: Sun Jun 09, 2024 8:06 am
Jaylat wrote: Sun Jun 09, 2024 7:55 am Not sure I understand any of this. Why would the TIPS needed for 2025-30 change at all? Unless your DARA changes, the face amount of the TIPS should be exactly the same. The yield changes would change their price but would have zero effect on the real DARA amount needed for each year.
Okay, I get it - it's the higher coupons from the 2031-32 years. So you need to go out and sell off small portions of the 2025-30 TIPS to offset the coupons.
Correct.
Jaylat wrote: Sun Jun 09, 2024 8:06 am Are you assuming the passage of time in this calculation? In real life we would wait 2 years for the issuance of the 2031 and 2032 TIPS. So the duration of all the bonds would be reduced by 2 years.
No. As stated, for modeling purposes "Temporal considerations are suspended, so all yield changes are instantaneous, and both gap year TIPS are issued at the same time."

This is a first approximation model, to just get a handle on how well duration matching works in a purely theoretical sense.
How would you characterize the $1,543 difference as a percentage? Would it be a percent of the combined 2031-32 TIPS?

Basically this spreadsheet is saying given the current TIPS available and assuming no changes in TIPS values whatsoever the closest you can get to 100% accuracy is around 99.2%.
I'm really not concerned with that, as it's basically a rounding error. The goal is to look at the scenarios where there's a yield change, and that number's a lot bigger. I was just working on Scenario 2, yields increase to 4%, but #Cruncher found an error in of my versions of his spreadsheet that's similar to the one I'm using for these analyses, so I need to work on that issue first. He's updated his spreadsheet anyway since I last copied it (added the new Apr 2029), for example, so I need to rework my versions anyway to incorporate his latest changes.
If I make a calculation error, #Cruncher probably will let me know.
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Comparing pre-swap ladders at 2% and 4% yields

Post by Kevin M »

Now I'll begin to evaluate scenario 2, yields increase to 4% before the swaps, so gap year TIPS yields = coupons = 4%. I'll break this into two parts:
  • Compare pre-swap ladder at 2% to pre-swap ladder at 4%, which I'll do in this post.
  • Compare pre-swap ladder at 2% to post-swap ladder at 4%, which I'll do in the next post.
I've made the following changes to the tables since the last post in which I shared them (see table below):
  • Added notations for yield and pre swap or post swap.
  • Added columns for ARA (Annual Real Amount) and ARA/DARA, the ratio of ARA to DARA, which ideally is 1.0000, except for the pre-swap bracket years where it's multiplier * 1.0000. This allows us to ensure that we're meeting the ARA/DARA goal after the swaps.
  • Changed the column names for Needed and Total to Qty and Amt respectively, because I think the new names make more sense in the context of this analysis.
  • Similarly, changed Diff needed and Diff total to Diff qty n% and Diff amt n%, where n is the yield for the case we're comparing to. This is because I compare the 4% post swap, for example, to both the 2% pre swap, as seen below, and the 4% pre swap, so we can distinguish the ladder changes due to the yield change from those due to the swaps.
  • For the pre-swap tables I've added Qty and Amount totals for the bracket years to the merged gap year rows; e.g., 343 in Qty column and the merged 2031/32 rows is 153 in the 2030 row +190 in the 2033 row. The Diff amt value shown in the merged gap year rows is the difference between these amounts for the cases we're comparing, in this case the pre-swap 2% and pre-swap 4% cases.
There are a few more changes that apply only to post swap cases, but here's we're just looking at at pre-swap ladder changes, specifically, after the yield increases from 2% to 4%, but before the swaps.

Image

Observations:
  • The following values do not change after the yield increase but before the swaps: Qty, ARA and DARA, and of course also ARA/DARA.
  • All amounts decrease, due to the increase in yields.
  • Total ladder value decreases by 90,538, or -10.1%.
  • The bracket year TIPS decrease in value by -47,503, or -13.4%.
I'll continue scenario 2 in the next post, where I'll compare 2% pre-swap to 4% post-swap. Understanding the changes that occur after the increase in yields but before the swap provides some context for the post-swap analysis.
If I make a calculation error, #Cruncher probably will let me know.
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Comparing pre-swap ladder at 2% to post-swap at 4%

Post by Kevin M »

Below are the pre-swap and post-swap ladder values at 2% and 4% respectively. Notes:
  • For the spreadsheet-curious, all the hidden rows and columns are because I copy from the Ladder sheet, then paste-values into the gap year analysis sheet.
  • In the post-swap 4% table, the value of -18,451 in merged cell AG37/40, the Diff amt 2% column and merged 2031/32 rows, is sum of the values in the Amount column for the 2030-34 rows minus the 353,295 value in the Amount column and merged 2031/32 rows in the 2% pre-swap table. Here is the formula for merged cells AG37:AG40:

    Code: Select all

    =SUM(AC35:AC42)-$T37
  • Similarly, In the post-swap 4% table, the value of -12.0 in the Diff amt 2% column and merged 2031/32 rows is sum of the values in the Qty column for the 2030-34 rows minus the 343 value in the Qty column and merged 2031/32 rows in the 2% pre-swap table. Here's the formula for merged cells AF37:40

    Code: Select all

    =SUM(Z35:Z42)-$Q37
  • Note that the bracket year Diff qty and Diff amt values also are included in the table, but these are subsumed by the corresponding values in the merged 2031/31 rows, since these values are based on all four 2030-33 rows. This is a bit weird, but it's the best way I've come up with to show the numbers. Some ancillary calculations will be required to make sense all of this.
Image

Observations:
  • ETA: The ARA/DARA match is fine post-swap, as long as we do all trades indicated by the quantity differences.
  • All quantities and amounts for the 2025-2030 are lower post-swap. This is because the 4% coupon is larger than the the existing coupons or the assumed 2% coupons for the 2031/32 pre-swap.
  • Due to the higher coupon, we need fewer of the earlier maturities, and we also need fewer of the gap year TIPS than anticipated.
    • I still need to work through the math to see how this all works out if we were to sell and buy the required quantities of each maturity to maintain ARA/DARA at close to 1, but so far it looks like we're in good shape.
    • I think the next step is to compare the pre-swap 4% with the post-swap 4% cases.
  • The 2034 qty is unchanged, since the 4% 2031/32 coupons don't affect the 2034 cash flows.
  • The post-swap qty for the 2033 is exactly half of the pre-swap qty, since the multiplier is halved, and 190 is divisible by 2. I divide the bracket year pre-swap values by 2 for the post-swap comparisons, which is why cell AF42 is 0
  • Similarly, the Diff qty of -3.5 for the 2030 is calculated as 73 - 153/2. Of course this is just a theoretical value, since we can't buy half of a TIPS.
  • We can derive the hypothetical gap year diff qty and diff amt by subtracting the sum of the bracket year totals from the values in cells AF37 and AG38 respectively. The results of these calculations are shown in cells AF62:AG63; here are the formulas for those cells:

    Code: Select all

    			   Qty	   	   Amount
    			--------------	-------------------
    Bracket years total	=SUM(AF35,AF42)	=SUM(AG35,AG42)
    Gap years total		=AF37-AF62	=AG37-AG62
    
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

Post by MtnBiker »

Kevin M wrote: Sat Jun 08, 2024 4:25 pm

So I'll run three scenarios:
  1. No yield changes before the swaps when gap year TIPS are issued, gap year TIPS yields = coupons = 2%.
  2. Yields increase to 4% before the swaps, so gap year TIPS yields = coupons = 4% 2%. (Edited to correct typo)
  3. Yields decrease to 0% before the swaps, so gap year TIPS yields = coupons = 0%.
I've recorded the before and after values of multipliers, number of TIPS needed, and costs of TIPS for each scenario. So I'll share those, with my observations.
Since "duration matching" is in the title of the thread, I took a look at the durations of the bracket-year and gap-year TIPS used in these initial scenarios. In calculating modified duration, I used June 15, 2024 as "today's date" simply so that each TIPS would be an even number of months to maturity (TIPS mature on the 15th of the month).

The July 2030 bracket year TIPS has a coupon of 0.125% and a duration of about 5.94 yr. The July 2033 bracket year TIPS has a coupon of 1.375% and a duration of about 8.31 yrs.

Assuming the coupon rate of the gap-year TIPS is equal to the interest rate, one can calculate the durations of the gap year TIPS as a function of the interest rate. Furthermore, the "ideal" multipliers to be used for the bracket-year holdings can be calculated if attempting to duration-match the excess holdings.

If the interest rate is 2%, the durations of the 2% coupon 2031 and 2032 gap year TIPS would be 6.43 and 7.26 yr, respectively. If my calculations are correct, the corresponding duration-matching multipliers for the 2030 and 2033 holdings would be 2.42 and 1.58, respectively.

If the interest rate jumps to 4%, the duration of the gap-year TIPS decreases due to the larger coupons, and the ideal multipliers are 2.71 and 1.29, respectively.

If the interest rate falls to 0%, the ideal multipliers are 2.04 and 1.96, much closer to the 2.0 values used so far in the scenarios.

I don't know how to use this information, other than to suggest that the holding multipliers might want to be varied to see if the results are in any way sensitive to those parameters. In these scenarios, using a 50/50 mix of the bracket years is a duration mismatch of about 3/4 yr if the gap-year coupon rate goes as high as 4.0%.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

MtnBiker wrote: Sun Jun 09, 2024 11:15 pm
Kevin M wrote: Sat Jun 08, 2024 4:25 pm

So I'll run three scenarios:
  1. No yield changes before the swaps when gap year TIPS are issued, gap year TIPS yields = coupons = 2%.
  2. Yields increase to 4% before the swaps, so gap year TIPS yields = coupons = 4% 2%. (Edited to correct typo)
  3. Yields decrease to 0% before the swaps, so gap year TIPS yields = coupons = 0%.
I've recorded the before and after values of multipliers, number of TIPS needed, and costs of TIPS for each scenario. So I'll share those, with my observations.
Since "duration matching" is in the title of the thread, I took a look at the durations of the bracket-year and gap-year TIPS used in these initial scenarios. In calculating modified duration, I used June 15, 2024 as "today's date" simply so that each TIPS would be an even number of months to maturity (TIPS mature on the 15th of the month).

The July 2030 bracket year TIPS has a coupon of 0.125% and a duration of about 5.94 yr. The July 2033 bracket year TIPS has a coupon of 1.375% and a duration of about 8.31 yrs.

Assuming the coupon rate of the gap-year TIPS is equal to the interest rate, one can calculate the durations of the gap year TIPS as a function of the interest rate. Furthermore, the "ideal" multipliers to be used for the bracket-year holdings can be calculated if attempting to duration-match the excess holdings.

If the interest rate is 2%, the durations of the 2% coupon 2031 and 2032 gap year TIPS would be 6.43 and 7.26 yr, respectively. If my calculations are correct, the corresponding duration-matching multipliers for the 2030 and 2033 holdings would be 2.42 and 1.58, respectively.

If the interest rate jumps to 4%, the duration of the gap-year TIPS decreases due to the larger coupons, and the ideal multipliers are 2.71 and 1.29, respectively.

If the interest rate falls to 0%, the ideal multipliers are 2.04 and 1.96, much closer to the 2.0 values used so far in the scenarios.

I don't know how to use this information, other than to suggest that the holding multipliers might want to be varied to see if the results are in any way sensitive to those parameters. In these scenarios, using a 50/50 mix of the bracket years is a duration mismatch of about 3/4 yr if the gap-year coupon rate goes as high as 4.0%.
Excellent feedback.

I'll continue with the current setup until I get to the conclusions, since this probably is representative of what would typically be done. Once I get through that, perhaps I'll redo the analyses with better duration matching.

Given the assumptions of the model, we'd establish the bracket year multipliers before yields increased, and wouldn't be able to adjust them after the yield increase and before the swaps. And the quantities of each that we sell for the swaps are dictated by optimizing ARA/DARA = 1, so we can't vary what we sell based on duration or multiplier changes, although the latter might affect the former.
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

Post by MtnBiker »

Kevin M wrote: Mon Jun 10, 2024 8:41 am
MtnBiker wrote: Sun Jun 09, 2024 11:15 pm
Kevin M wrote: Sat Jun 08, 2024 4:25 pm

So I'll run three scenarios:
  1. No yield changes before the swaps when gap year TIPS are issued, gap year TIPS yields = coupons = 2%.
  2. Yields increase to 4% before the swaps, so gap year TIPS yields = coupons = 4% 2%. (Edited to correct typo)
  3. Yields decrease to 0% before the swaps, so gap year TIPS yields = coupons = 0%.
I've recorded the before and after values of multipliers, number of TIPS needed, and costs of TIPS for each scenario. So I'll share those, with my observations.
Since "duration matching" is in the title of the thread, I took a look at the durations of the bracket-year and gap-year TIPS used in these initial scenarios. In calculating modified duration, I used June 15, 2024 as "today's date" simply so that each TIPS would be an even number of months to maturity (TIPS mature on the 15th of the month).

The July 2030 bracket year TIPS has a coupon of 0.125% and a duration of about 5.94 yr. The July 2033 bracket year TIPS has a coupon of 1.375% and a duration of about 8.31 yrs.

Assuming the coupon rate of the gap-year TIPS is equal to the interest rate, one can calculate the durations of the gap year TIPS as a function of the interest rate. Furthermore, the "ideal" multipliers to be used for the bracket-year holdings can be calculated if attempting to duration-match the excess holdings.

If the interest rate is 2%, the durations of the 2% coupon 2031 and 2032 gap year TIPS would be 6.43 and 7.26 yr, respectively. If my calculations are correct, the corresponding duration-matching multipliers for the 2030 and 2033 holdings would be 2.42 and 1.58, respectively.

If the interest rate jumps to 4%, the duration of the gap-year TIPS decreases due to the larger coupons, and the ideal multipliers are 2.71 and 1.29, respectively.

If the interest rate falls to 0%, the ideal multipliers are 2.04 and 1.96, much closer to the 2.0 values used so far in the scenarios.

I don't know how to use this information, other than to suggest that the holding multipliers might want to be varied to see if the results are in any way sensitive to those parameters. In these scenarios, using a 50/50 mix of the bracket years is a duration mismatch of about 3/4 yr if the gap-year coupon rate goes as high as 4.0%.
Excellent feedback.

I'll continue with the current setup until I get to the conclusions, since this probably is representative of what would typically be done. Once I get through that, perhaps I'll redo the analyses with better duration matching.

Given the assumptions of the model, we'd establish the bracket year multipliers before yields increased, and wouldn't be able to adjust them after the yield increase and before the swaps. And the quantities of each that we sell for the swaps are dictated by optimizing ARA/DARA = 1, so we can't vary what we sell based on duration or multiplier changes, although the latter might affect the former.
Sounds like a good plan on how to proceed from here.

Carrying my duration evaluation a bit further, I should say that what I called "ideal multipliers" in my previous post (immediately above) are what would be ideal if one was doing the swap in the conventional "one step" manner, that is selling the excess bracket years and using the proceeds to buy the gap years (without regards to smoothing the ARA/DARA mismatch).

The subsequent step of smoothing ARA/DARA, selling shorter duration TIPS to buy more of the gap-year TIPS, tends to increase the average duration of the ladder. Thus, the ideal multipliers with ARA/DARA smoothing are going to be somewhat different from what I listed.

For your scenario with an interest rate of 2%, the weighted-average duration of the pre-swap ladder (weighted by the amount in each rung) is 5.34 years. After selling the 1x excess 2030 and 1x excess 2033 and buying the gap years in the usual 1/3 and 2/3 ratios, the weighted average duration of the ladder is 5.27 yr (delta of 0.07 yr). After then smoothing to eliminate ARA/DARA mismatch, the weighted average duration of the ladder is back up to 5.32 yr (delta of only 0.02 yr). Thus, it appears that the 2x multipliers you have been using are close to ideal for the 2% interest rate case with ARA/DARA smoothing, which is consistent with your result that the cost of making the swaps was essentially zero to within rounding error.

For your scenario with an interest rate of 4%, the weighted average duration of the pre-swap ladder (after the interest rate increase) is 5.08 yr (was 5.09). After selling the excess bracket years and buying the gap years (without smoothing), the weighted average duration of the ladder is 4.94 yr (delta of 0.14 yr). After smoothing, the weighted average duration of the ladder is 5.04 yr (delta of 0.04 yr).

EDIT: Final paragraph above edited to show the minor changes in numerical values after the corrections noted in Kevin's next post.
Last edited by MtnBiker on Mon Jun 10, 2024 10:01 pm, edited 1 time in total.
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Scenario 2: Yields increase from 2% to 4%

Post by Kevin M »

EDIT: I temporarily deleted this post after finding an error in the analysis. The error was that I neglected to adjust the ref CPIs and hence the index ratios for the 2031/32 TIPS, which the model treats as the gap years. In using the actual index ratios, there was a large mismatch between quantity and adjusted cost for these TIPS. I fixed the error, and although it affected gap year quantities significantly, it had minimal impact on any of the values.

-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

I think I've got some "final" results for scenario #2, 2%->4% right before swap. What I'll show now is the values after yields have increased to 4%, but before the swap (pre-swap), and the post-swap values. Remember that there were some minimal deltas even if there were no yield increases before the swap, but those were small enough that I think we can ignore them, at least for a first pass analysis.

So to summarize the scenario:
  • Yields increase instantaneously from 2% to 4%, at which point we hold the same ladder, but it's worth less due to the yield increase.
  • Gap year TIPS will be issued at the 4% yield with a 4% coupon; all non-gap TIPS coupons remain the same.
  • We first sell the number of bracket year TIPS such that the bracket year ARAs remain close to DARA; i.e., we let the #Cruncher spreadsheet determine the number of bracket year TIPS we must continue to hold
  • We buy gap year TIPS with the proceeds.
  • If we still need more gap year TIPS, we sell other maturities such that the ARAs remain close to DARA; i.e., we let the #Cruncher spreadsheet determine the number of non-bracket year TIPS we must continue to hold, and sell the rest
  • We buy gap year TIPS with the proceeds.
I made some changes to the tables, but I'll only mention them in explaining the results.

Image

Notes and observations:
  • Some of the table changes were made to facilitate the swap transactions analysis; e.g., the values in the merged cells in rows 37:40 are for the gap years only, and the bracket year values in columns AF and AG (light green background) are the full deltas compared to the pre-swap values.
  • The swap transactions are shown below the tables.
  • Note that if only bracket year TIPS are sold to buy gap years, there is a negative cash balance; i.e., there $29,533 less cash than required.
  • If the prior year maturities also are sold to minimize ARA/DARA mismatch, the quantities being indicated in column AF, the cash shortfall is reduced to $3,891. Note that this is the same as the amount by which the portfolio cost/value increases from 4% pre-swap to 4% post-swap, shown in cell AG61.
  • So to fully update the ladder to minimize ARA/DARA mismatch requires coming up with a bit of extra cash. The alternative is to accept a relatively small ARA shortfall relative to DARA for one or more years.
I think this works out pretty well, but an important point is that non-bracket year TIPS would need to be sold to maintain our ARA/DARA matches, and this would not be the case for the original model, in which I treated the gap year TIPS as marketable;i.e., fixed coupon, variable price.
Last edited by Kevin M on Tue Jun 11, 2024 8:12 am, edited 1 time in total.
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Add transaction amounts to post swap table

Post by Kevin M »

Kevin M wrote: Mon Jun 10, 2024 5:25 pm <snip>
  • <snip>
  • Note that if only bracket year TIPS are sold to buy gap years, there is a negative cash balance; i.e., there $29,533 less cash than required.
  • If the prior year maturities also are sold to minimize ARA/DARA mismatch, the quantities being indicated in column AF, the cash shortfall is reduced to $3,891. Note that this is the same as the amount by which the portfolio cost/value increases from 4% pre-swap to 4% post-swap, shown in cell AG61.
  • <snip>
For some reason I hadn't noticed that the transaction amounts for the non-gap years equal the opposite of the dAmt 4% pre values, and that the transaction amounts can easily be added to the post-swap table:

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

Post by Jaylat »

This is a complicated issue, so I’d like to state clearly what I would like to see out of these calculations (you may disagree):

What is the optimal relative weighting of 2034 and 2040 TIPS that will (1) allow the gap year TIPS to have an ARA close to DARA while (2) minimizing the trading losses resulting from selling the 2034 and 2040 TIPS and buying the gap year TIPS, plus any associated trades. The answer to this should take into account the passage of time, the resultant change in duration, etc. across various assumptions for TIPS coupons and real TIPS yields.

I understand that you are focusing on a simplified theoretical case in order to build the model. However I hope you can eventually come to this scenario?
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Scenario 2: Yields decrease from 2% to 0%

Post by Kevin M »

With a yield decrease from 2% to 0%, the value of the ladder increases by about 12%:

Image

This implies a duration rule of thumb value of 6, which seems reasonable given MtnBiker's 5.34 2% pre-swap value, and remembering that convexity works in our favor.

Here is the table for the 0% post-swap case:

Image

The transaction amounts are shown in the table in the Xact Amt column, but I've broken them out by bracket and non bracket year transactions below as well.

So, with only selling the bracket years to buy the gap years, we end up with excess cash of $7,877. Due to the lower gap year coupon (0% instead of 2%), we'd need to buy a few of the earlier year TIPS to maintain ARA/DARA match, and after doing that, we're left with $907 in cash.

Contrast this to the 2%-4% case where we would be $3,891 short on cash. So the 2%->0% scenario works better for the ladder holder, but either scenario works out pretty well, although for the 2%-4% case we had to sell a relatively large number of the non-bracket year TIPS to avoid a fairly large cash shortfall and to maintain ARA/DARA match.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

Jaylat wrote: Tue Jun 11, 2024 8:26 am This is a complicated issue, so I’d like to state clearly what I would like to see out of these calculations (you may disagree):

What is the optimal relative weighting of 2034 and 2040 TIPS that will (1) allow the gap year TIPS to have an ARA close to DARA while (2) minimizing the trading losses resulting from selling the 2034 and 2040 TIPS and buying the gap year TIPS, plus any associated trades. The answer to this should take into account the passage of time, the resultant change in duration, etc. across various assumptions for TIPS coupons and real TIPS yields.

I understand that you are focusing on a simplified theoretical case in order to build the model. However I hope you can eventually come to this scenario?
We'd want to know ARA/DARA match for each year in the ladder, not just the gap years, but yeah, that would be nice to know. Hopefully I'll get there eventually, but seeing that the first-order approximation model results are pretty good increases the warmth of my warn, fuzzy feeling about this gap year coverage method.

Before using more realistic gap and bracket years and introducing temporality, I'm kind of interested in testing some null hypothesis cases (I think I'm using that term correctly); e.g., what if we cover the 2031/32 gap years by holding 4X of the 2025? If we're on the right track by using maturities that are closest to the gap years to cover the gap years, then this experiment should result in a much larger net cash difference after doing the swaps.

Speaking of first-approximation models, #Cruncher has magnanimously developed a simplified ladder spreadsheet that I think would be much easier to work with for these kinds of experiments. I think with that it would be easier, for example, to at least evaluate the actual gap years and bracket years, but I haven't started experimenting with it yet.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Raspberry-503 »

Jaylat wrote: Tue Jun 11, 2024 8:26 am This is a complicated issue, so I’d like to state clearly what I would like to see out of these calculations (you may disagree):

What is the optimal relative weighting of 2034 and 2040 TIPS that will (1) allow the gap year TIPS to have an ARA close to DARA while (2) minimizing the trading losses resulting from selling the 2034 and 2040 TIPS and buying the gap year TIPS, plus any associated trades. The answer to this should take into account the passage of time, the resultant change in duration, etc. across various assumptions for TIPS coupons and real TIPS yields.

I understand that you are focusing on a simplified theoretical case in order to build the model. However I hope you can eventually come to this scenario?
I've been away fora couple of weeks and this thread has gotten really complicated, need to get over jetlag before I can even try to start parsing the new posts, but I like this. For us dumb people, I'd love a tool where I can tell you what I have, what my start/end years of need are, and it tells me what to do :)
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

Raspberry-503 wrote: Tue Jun 11, 2024 11:24 am I've been away fora couple of weeks and this thread has gotten really complicated, need to get over jetlag before I can even try to start parsing the new posts, but I like this. For us dumb people, I'd love a tool where I can tell you what I have, what my start/end years of need are, and it tells me what to do :)
If we're lucky, we may get there! :wink:

For now most of us seem to think that a simple starting point is to use the one or both bracket years to cover the gap years. If you only want your ladder to go through 2035, so only one gap year, you're probably fine just doubling up on Jan 2034.

If you want to do what some of us think might be a bit better, you might use the ratios of 2034 and 2040 I showed in the OP. That may end up being all you need.

The main purpose of this thread is to try and come up with the bond math that evaluates the effectiveness of whatever method we select.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Raspberry-503 »

yeah I use the ratio method for now. What I hope to get out of this really is to understand the risk/cost of trading the bracket years to fill in the gap as they become available and make sure I somehow don't lose money stupidly.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

Wanted to share that I've been getting deep into the weeds with #Cruncher's new simplified TIPS ladder builder spreadsheet; one goal is to use it for more effective modeling of gap year coverage methods. I highly recommend it to all TIPS ladder spreadsheet crunchers.

I've found it quite educational to do various gap swaps with the simplified ladder, making it easier to see and understand the effects on various numbers. With the simple spreadsheet it's easy to model the actual gap years, 2035-2039, and bracket years, 2034 and 2040. It's almost like #Cruncher was reading my mind about the kind of tool I wanted to work on this project. :wink:

I thought I should step back and "calibrate" the simple tool against the complex tool, so I just did that. I set both up for 2% par bonds, using the same ladder I've been using in the last few posts; i.e., DARA = $1,000,000, maturities 2025-2034, with the gap and bracket years mentioned above. Here are some comparisons:

Code: Select all

		Simple		Complex		S - C		S/C - 1 %
		--------------	--------------	-------------	-------------
Cost at 2%	898,259		904,638		-6,379		-0.71%
Total ARA	1,000,000	1,000,167	-167		-0.02%
I think these numbers are great, considering the simplifications; as #C describes them in his post about it:
#Cruncher wrote: Mon Jun 10, 2024 7:28 pm I thought it might be good to step back and see the underlying method of building a hypothetical ladder without the complications of a real-world TIPS ladder where ...
  • In many years more than one TIPS matures.
  • But none at all currently mature 2035-2039.
  • They are only sold in $1,000 face value increments.
  • In the year of maturity some have one six-month interest payment and some have two.
In this hypothetical ladder ...
  • Exactly one TIPS matures each of the next 30 years.
  • They can be purchased in any fractional amount.
  • They have only one interest payment each year on the anniversary of their maturity date.
  • Accrued and partial year interest is ignored.
The following table shows how, working backward from year 30, we build this hypothetical ladder to produce 30,000 constant dollars each year. Each year it deducts from the 30K total required, the interest on bonds maturing in later years. (E.g., in year 30 there will be no such bonds; in year 29 there will be the interest from the bond maturing in year 30; and in year 1 there will be the interest from all the bonds maturing in years 2-30.) This balance must be met by the principal and final year interest of the TIPS maturing that year. These calculations are shown in columns C:F below.

<code snipped>

I've added columns G and H to calculate the cost of each bond assuming either a 0% or 2% yield-to-maturity (YTM).
The fractional amount criteria allows generations of ARAs equal to DARA, which is why total ARA simple is exactly $1,000,000.

One challenge with Simple is that it's upside down compared to Complex; i.e., Simple maturities are sorted in descending order, while Complex are sorted in ascending maturity order, as most of us are familiar with. As an example, here's the sheet I used to do the calibration:

Image

Since the simple and complex rows are sorted oppositely, to get the complex data into the simple sheet I did this:
  • Copy/Paste-Values from ladder cost column in Complex spreadsheet into column O (only partially shown) in the pictured, simple spreadsheet.
  • Used this formula in cell L23 to filter the numbers and reverse the order:

    Code: Select all

    =SORT(FILTER(O23:O55,O23:O55<>"-"),1,0)
I should note that this is my version of the Simple spreadsheet, having built it based on the code snippet that #C shared in the linked post, and then adding a few things:
  • Calendar year column.
  • Added a Multiplier column, so I can use multipliers the way their used in the Complex SS.
  • A dropdown and some formulas to enable quick entry of different multipliers for the gap years; e.g., to quickly 1 as a multiplier for 2034, or for multiple gap years, and have the bracket year quantities adjust accordingly -- used in gap year version.
  • A dropdown to select the gap year coupon -- used in gap year version.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

My next step is to use the simple ladder spreadsheet to repeat the simulated gap year run with a 10y (2025-2039), 2% par ladder, with gap years 2031/32 and bracket years 2030, 2033, then compare the results to the results obtained using the more complicated ladder spreadsheet; so it's kind of another calibration step before plunging into the 2035-39 actual gap years analysis.

In this post I'll show the pre-swap and post-swap ladders, and explain the key calculations in the simple spreadsheet. In the next post I'll show the comparison table.

Here's what the ladder looks like pre-swap:

Image

And post swap:

Image

Remember that the TIPS are ordered by descending maturity, so upside down compared to the complex spreadsheet. Other notes:
  • Neither YTM nor price are used in the simple model. The only independent variables in the calculations are DARA and coupon.
  • Total proceeds (ARA) = DARA for all years with multiplier = 1. This is because of allowing fractional quantities.
  • Just eyeballing the Totals (row 33) for Cost at X% (columns H:J), we see that the differences between pre-swap and post-swap are quite small.
The only calculations that rely directly on the independent variables, coupon and DARA, are the ones for for Needed Principal, Annual Interest, and the Cost at x% values.

One of the two non-trivial formulas is for Needed Principal. Here's the formula for cell E23, which is for the longest maturity, 2034:

Code: Select all

=($B$1-D23)/(1+C23)*K23
In words, this is:
  1. Subtract older bonds interest from DARA. Since 2034 is the oldest bond, interest from older bonds is 0, and we're left with just DARA.
  2. Divide #1 result by (1 + cpn%), so 1+2% = 1+0.02 = 1.02. The quantity 1/(1+cpn%) = 1/1.02 is the discount factor, so here we're essentially discounting DARA by 2% for one year.
  3. Multiply #2 result by the multiplier, which is 1 for 2034 both pre and post swap. Adding a multiplier was my addition to #Cruncher's spreadsheet.
So ignoring the multiplier, Needed Principal is the present value calculated from the single period present value formula:

Code: Select all

PV = FV/(1+r)
And in our case:

Code: Select all

PV = Needed principal
FV = (DARA - older bonds interest) = 100,000 - 0 for the 2034.
r = cpn = 2.00%

so

Needed principal = 100,000/1.02 = 98,039.
Because the simple model assumes one coupon payment at the end of each year, the Annual Interest is just coupon times needed principal, so for the 2034 it's:

Code: Select all

2.000% * 98,039 = 1,961
The formula for Cost at 2% in cell I23 (Cost at 2% for the 2034) is:

Code: Select all

=-PV(VALUE(RIGHT(I$2,2)),$B23,$F23,$E23,0)
  • The VALUE(RIGHT ... bit just extracts the X% as a number from the header, and this value is the rate parameter for the PV function. This was a small tweak of mine so that I could use one row for the headers, and not put YTMs in a different row.
  • The second parameter, $B23, is the number_of_periods, which is 10 in the Year num column.
  • The third parameter, $F23, is payment_amount, which is the annual interest, 1,961.
  • The fourth parameter, $E23, is future_value, which is the needed principal, 98,039.
  • The fifth parameter, 0, is end_or_beginning, with 0 indicating that payments are at the end of the period.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

Here are tables that summarize the post-swap vs. pre-swap ladders:

Image

I think the most important part is that the post minus pre total deltas are all small--less than 0.5%, which is similar to the results from the complex spreadsheet.

With this model we end up with extra cash in all three coupon cases after selling the bracket years and buying the gap years. Some of that cash is used to buy small amounts of the other maturities (2025-2029) to maintain ARA at $100K, but there's still cash left over after that.

I think at this point I'm comfortable enough with the results of the simplified ladder model to use it for analysis of the real life gap year situation.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

I'm starting to work on the real life gap situation using the simplified spreadsheet now. I decided we can start with a simpler ladder, where all that's held are the gap and bracket years, 2034-2040, the idea being that the gap year coverage should stand on its own, and not rely on doing anything with TIPS maturing in other years.

There was some puzzlement over why there were deltas when we did a swap with no yield or coupon changes; i.e., yields all 2% before and after the swap, and gap year coupons = pre-swap expectations of 2%. The simplified model greatly simplifies investigating puzzles like this, so I'll start with that.

Here are the specs:
  • Gap year and bracket year coupons = 2% both pre and post swap; I set bracket year coupons to 2% to eliminate coupon as a variable in understanding the swap impacts.
  • Multiplier for each bracket year, 2034 and 2040, is 3.5 pre-swap, which is the default we seem to think should provide decent duration matching.
  • All temporal considerations suspended, so we can move from 2024 to 2027, for example, with no changes to the model values.
  • In 2027 we'll sell 0.5 DARA of each bracket years to buy 1 DARA of the 2037, chosen because it's 3 years different than both bracket years, so selling half a DARA for each bracket year makes sense.
Here's the sheet before the swap:

Image

I've included a few maturities before and after the bracket years, just to show that the multipliers are set to 0.

The Cost at YTM 2% column is highlighted because this exercise relates to swapping with no change to yields, so yield = coupon = 2%. Note that the values in this column match those in column F, Principal required; this follows the rule that the value of a par bond, where coupon = yield, is 100% of face value. Note that the values in column F are face values (remember, no limit to face value resolution in this model).

My new favorite toy is a dropdown in cells where I want to quickly change something. The dropdown in cell D18 changes the gap year coupon in that cell, then the cells for the gap years below reference it. The dropdown in cell N14 selects a number that's used as an offset in L17:L23 to reference the cells in one of the "M pre/post N" columns to the right; e.g., with 0 selected, the values in cells N17:N23 are used as the multipliers.

Here's the post-swap sheet:

Image

Note that gap M select is 1, so the M values in M post 1, cells O17:O23 are used as the multipliers in cells L17:L23.

Rather than try to draw conclusions by staring at the two sheets, here are the gap and bracket year rows pre and post, with a deltas table below:

Image

The value of -4,947 in cell J50 indicates that the ladder costs 4,947 less after the swap, (I swap the sign on this to show the net cash after the swap in cell J52), and this is a percentage difference of -0.7465%, shown in cell J51. I've highlighted these delta cells along with those in the Principal required, column F, because this is the key to answering the puzzle.

As noted earlier, for coupon = 2%, cost at YTM 2% = principal required = face value . So now we can focus on why total principal required decreased.

EDIT: It was late and I wasn't thinking clearly when I wrote what I've stricken out below. I think I've gotten closer to the truth of what's going on here, which I'll share in my next post.
In general, less principal is required for earlier maturity years in the ladder, since more of the proceeds (ARA) are provided by coupons from later years in the ladder. Note that interest from later bonds is 0 for the 2040, but non-zero for the 2037, in any of the sheets shown above. Looking at the math, as reviewed in a previous post, the formula for principal required, using the 2037 as an example, is:

Code: Select all

=($D$1-E20)/(1+D20)*L20
which in words is:

Code: Select all

(DARA - int later bonds) / (1 + cpn) * multiplier

From this we see that as total interest from later bonds increases due to the swap from 2040s to 2037s, total principal required decreases, and this is why swapping 2040s for 2037s with no yield or coupon changes results in less total principal required after the swap than before.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

I decided to start working with a full 30y ladder using the simplified model, which avoids dealing with interest in non-gap years with no TIPS holdings (multiplier = 0). All ladders discussed here are this ladder, either with the gap years filled with TIPS with a specified coupon (full ladder), or with some or no gap years filled. Everything refers to the simple model unless otherwise noted.

Here are some observations:
  1. With the 30y ladder fully populated, total proceeds = 30 * ARA = 30 * DARA , because ARA = DARA for each year; e.g., with DARA = 100K, total proceeds = 30 * 100K = 3,000,000.
    • Similarly, using the complex model (#Cruncher standard TIPS ladder builder), 30 * ARA is very close to 30 * DARA; e.g., with all TIPS yields set to 2%, it's 2,999,892 compared to 30 * DARA = 3,000,000 with DARA = 100,000
  2. With the gap years unpopulated, and using multiplier 3.5 for each of the 2034 and 2040, total proceeds = 3,014,985 <> 30 * 100K.
It seemed to me that things would work better if the bracket year multipliers were modified so that total proceeds = 30 * DARA, which for our case is 3,000,000. Note that with unpopulated gap years, total proceeds includes the interest from later bonds for the gap years. Any thoughts on this?

So I tweaked the gap year multipliers so that the bracket year multipliers are the same, but total proceeds = 3M with the gap years empty; this value is 3.41344, so less than 3.5. With this tweak, after fully populating the gap years, the differences between the gap and fully populated costs at any YTM are tiny, which is ideal, since it means that very little cash would be involved, either added to or removed from the ladder

Next observation: If I change the gap year coupon, say from 2% to 3%, although individual rung costs change for 2025-2039, the total costs remain exactly the same at all YTMs. The table below summarizes this for coupons ranging from -1% to 3%:

Image

I only included delta % for the last case since we know that the deltas are the same for all cases.

By contrast, the deltas are larger if I set bracket year multipliers to 3.5 each:

Image

Although the deltas still are small as a % change, they are much larger than if the multipliers are tweaked. OTOH, the deltas are all negative, meaning that the ladder would cost less at any YTM after the swap; i.e., you could pull cash out of the ladder and still have ARA=DARA for every year.

This all looks promising to me. I think the next step is to look at filling in the gap years one at a time.

Thoughts?
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

Post by MtnBiker »

Kevin M wrote: Sun Jun 16, 2024 4:04 pm
This all looks promising to me. I think the next step is to look at filling in the gap years one at a time.

Thoughts?

It seems like you have developed an excellent model that should be able address the fundamental questions about filling the gap years. My thought is that what we are looking to see is the magnitude of the change in proceeds after making the gap swap(s). Whether the initial proceeds are initially 3,015K or fine-tuned to exactly 3,000K isn't critical to the analysis. What matters is how the proceeds change after making (1) the swap for the gap years and (2) after the subsequent smoothing to remove ARA/DARA mismatch. In other words, how much money needs to be added to the ladder (or can be pulled from the ladder) in order to preserve the original total proceeds (under various conditions)? Please report both (1) and (2), to help us understand the cost/benefit of the ARA/DARA smoothing step. Please carry on...
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

MtnBiker wrote: Mon Jun 17, 2024 9:45 am
Kevin M wrote: Sun Jun 16, 2024 4:04 pm
This all looks promising to me. I think the next step is to look at filling in the gap years one at a time.

Thoughts?
It seems like you have developed an excellent model that should be able address the fundamental questions about filling the gap years. My thought is that what we are looking to see is the magnitude of the change in proceeds after making the gap swap(s). Whether the initial proceeds are initially 3,015K or fine-tuned to exactly 3,000K isn't critical to the analysis. What matters is how the proceeds change after making (1) the swap for the gap years and (2) after the subsequent smoothing to remove ARA/DARA mismatch. In other words, how much money needs to be added to the ladder (or can be pulled from the ladder) in order to preserve the original total proceeds (under various conditions)? Please report both (1) and (2), to help us understand the cost/benefit of the ARA/DARA smoothing step. Please carry on...
Thanks for the feedback; nice to know that at least you are hanging in there--I know this stuff is complicated. The tack I'm taking now is to tweak the multipliers to minimize both:
  1. SUM(ARAs) minus 30*DARA, and
  2. post-swap ladder cost minus pre-swap ladder cost,
which are the two figures of merit that now make sense to me.

I think I'm zeroing in on a solution, and the tldr is that a good starting point is to use the weightings we've come up with, like sell 85% of 2034 and 15% of 2040 to buy the 2035, but by tweaking them a bit we can improve both figures of merit.
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

Post by Raspberry-503 »

Just to make sure I'm following, "swap" here refer to exchanging a number of fencepost TIPS (e.g. 2024 and 2040) for TIPS maturing during the gap year as they become available (e.g. sell some 2024 and 2040s next January to buy some 2025 which becomes the new fence post)
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

Raspberry-503 wrote: Mon Jun 17, 2024 1:43 pm Just to make sure I'm following, "swap" here refer to exchanging a number of fencepost TIPS (e.g. 2024 and 2040) for TIPS maturing during the gap year as they become available (e.g. sell some 2024 and 2040s next January to buy some 2025 which becomes the new fence post)
Yes, except the bracket year is 2034, and we'd be buying the 2035.
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

Post by Raspberry-503 »

oops typo, you're right
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Re: Filling the TIPS gap years with bracket year duration matching

Post by MtnBiker »

Raspberry-503 wrote: Mon Jun 17, 2024 1:43 pm Just to make sure I'm following, "swap" here refer to exchanging a number of fencepost TIPS (e.g. 2024 and 2040) for TIPS maturing during the gap year as they become available (e.g. sell some 2024 and 2040s next January to buy some 2025 which becomes the new fence post)
Moving the fencepost from 2034 to 2035 is one option. Or it could stay where it is and one could sell some 2034s and 2040s to buy 2036s the following year.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

Changes I've made to continue the experiments:
  1. Changed bracket year coupons to actual values instead of 2%; makes the simulations more realistic.
  2. #1 required changing the tweaked bracket multipliers to 3.4081 to get SUM(ARAs) = 3M with gap years empty.
  3. Simplified the table to show only the most important values.
In this experiment, I'll do the following:
  1. Increase gap year coupon/yield from 2% to 3%.
  2. Add multiplier 1 to the 2035; i.e., buy one DARA of it.
  3. Reduce the 2034 multiplier by 0.85, and reduce the 2040 by 0.15; these are close to the ideal duration-matched weights for the 2035.
  4. Observe the two figures of merit: 1. sum of ARAs minus 30*DARA; 2. post-swap ladder cost minus pre-swap ladder cost.
  5. Tweak the bracket year weights to subtract from the 2034 and 2040 to optimize the figures of merit.
The table below shows the results of steps 1-3 above:

Image

The figures of merit are in bold font; i.e., the ladder costs 4,031 less, and sum of ARAs is 5,290 less than 30*DARA. The figures of merit expressed as ratios (delta %) instead of as differences are shown below the deltas.

Also, note that the amounts by which the bracket year multipliers are decreased are shown in the multipliers column for the Add 2035 row.

Figuring that the 2034 sale dominates, I'll tweak that to get the figures of merit closer to 0. Results:

Image

So, by reducing the 2034 multiplier by 0.78 instead of 0.85, and leaving the 2040 reduction at 0.15, the figures of merit have been reduced to almost 0%, expressed as a ratio instead of a difference.

Remember that in addition to the bracket year and gap year transactions, the principal required, annual interest, and thus the cost, change for each of the 2025-2033 as well; i.e., they will be bought (+) or sold (-) to achieve the optimization of the figures of merit. Here are all of the cost at 3% changes:

Image
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

Post by MtnBiker »

Kevin M wrote: Mon Jun 17, 2024 7:46 pm Changes I've made to continue the experiments:
  1. Changed bracket year coupons to actual values instead of 2%; makes the simulations more realistic.
  2. #1 required changing the tweaked bracket multipliers to 3.4081 to get SUM(ARAs) = 3M with gap years empty.
  3. Simplified the table to show only the most important values.
In this experiment, I'll do the following:
  1. Increase gap year coupon/yield from 2% to 3%.
  2. Add multiplier 1 to the 2035; i.e., buy one DARA of it.
  3. Reduce the 2034 multiplier by 0.85, and reduce the 2040 by 0.15; these are close to the ideal duration-matched weights for the 2035.
  4. Observe the two figures of merit: 1. sum of ARAs minus 30*DARA; 2. post-swap ladder cost minus pre-swap ladder cost.
  5. Tweak the bracket year weights to subtract from the 2034 and 2040 to optimize the figures of merit.
The table below shows the results of steps 1-3 above:

Image

The figures of merit are in bold font; i.e., the ladder costs 4,031 less, and sum of ARAs is 5,290 less than 30*DARA. The figures of merit expressed as ratios (delta %) instead of as differences are shown below the deltas.

Also, note that the amounts by which the bracket year multipliers are decreased are shown in the multipliers column for the Add 2035 row.
Trying to understand what I am seeing here. Let me state what I think this means and you can correct any misconceptions.
You sold one DARA of the bracket years (about 100K?) in the 85/15 ratio and bought the 2035 gap year. The proceeds dropped by $5,290, or about 5.3% of the 100K amount of the swap. (Your figure of merit shows a loss of only -0.18%, but that is diluted by the rest of the 29 rungs of the 30-year ladder which are all basically unchanged?)

Was this simply a swap from the bracket years to a gap year, or did all the pre-gap years change too?
Kevin M wrote: Mon Jun 17, 2024 7:46 pm Figuring that the 2034 sale dominates, I'll tweak that to get the figures of merit closer to 0. Results:

Image

So, by reducing the 2034 multiplier by 0.78 instead of 0.85, and leaving the 2040 reduction at 0.15, the figures of merit have been reduced to almost 0%, expressed as a ratio instead of a difference.

Remember that in addition to the bracket year and gap year transactions, the principal required, annual interest, and thus the cost, change for each of the 2025-2033 as well; i.e., they will be bought (+) or sold (-) to achieve the optimization of the figures of merit. Here are all of the cost at 3% changes:

Image
Here you adjusted the bracket multipliers (selling less 2034 than before), also adjusted all the pre-gap years to smooth ARA/DARA, and thus eliminated the $5.3K loss in proceeds found in the first iteration.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

MtnBiker wrote: Tue Jun 18, 2024 9:14 am
Kevin M wrote: Mon Jun 17, 2024 7:46 pm Changes I've made to continue the experiments:
  1. Changed bracket year coupons to actual values instead of 2%; makes the simulations more realistic.
  2. #1 required changing the tweaked bracket multipliers to 3.4081 to get SUM(ARAs) = 3M with gap years empty.
  3. Simplified the table to show only the most important values.
In this experiment, I'll do the following:
  1. Increase gap year coupon/yield from 2% to 3%.
  2. Add multiplier 1 to the 2035; i.e., buy one DARA of it.
  3. Reduce the 2034 multiplier by 0.85, and reduce the 2040 by 0.15; these are close to the ideal duration-matched weights for the 2035.
  4. Observe the two figures of merit: 1. sum of ARAs minus 30*DARA; 2. post-swap ladder cost minus pre-swap ladder cost.
  5. Tweak the bracket year weights to subtract from the 2034 and 2040 to optimize the figures of merit.
The table below shows the results of steps 1-3 above:

Image

The figures of merit are in bold font; i.e., the ladder costs 4,031 less, and sum of ARAs is 5,290 less than 30*DARA. The figures of merit expressed as ratios (delta %) instead of as differences are shown below the deltas.

Also, note that the amounts by which the bracket year multipliers are decreased are shown in the multipliers column for the Add 2035 row.
Trying to understand what I am seeing here. Let me state what I think this means and you can correct any misconceptions.
You sold one DARA of the bracket years (about 100K?) in the 85/15 ratio and bought the 2035 gap year. The proceeds dropped by $5,290, or about 5.3% of the 100K amount of the swap. (Your figure of merit shows a loss of only -0.18%, but that is diluted by the rest of the 29 rungs of the 30-year ladder which are all basically unchanged?)

Was this simply a swap from the bracket years to a gap year, or did all the pre-gap years change too?
I underlined your questions.

The table below shows the same thing I showed for the case after I tweaked the post swap 2034 multiplier, but this one shows it before the tweak (first table above). This table is useful in answering your questions. I've also added a buy/sell table to clarify the results from a trades perspective.

Image

Yes, one DARA is sold because the sum of the 2034 and 2040 multipliers is reduced by 1 (=0.85+0.15), and we end with 100K of ARA for the 2035. The dollar value of the bracket year sales is shown in the table above (it's not 100K).

The figure of merit is how far away we are from 30 * DARA = 300K, which I express as both a difference and a ratio.

As shown in the table, non-bracket year costs/values also were reduced, which in transaction terms mean that we'd sell some to keep ARA = DARA = 100K. This is done automatically by the spreadsheet--the multipliers for these all are left at 1.
MtnBiker wrote: Tue Jun 18, 2024 9:14 am
Kevin M wrote: Mon Jun 17, 2024 7:46 pm Figuring that the 2034 sale dominates, I'll tweak that to get the figures of merit closer to 0. Results:

Image

So, by reducing the 2034 multiplier by 0.78 instead of 0.85, and leaving the 2040 reduction at 0.15, the figures of merit have been reduced to almost 0%, expressed as a ratio instead of a difference.

Remember that in addition to the bracket year and gap year transactions, the principal required, annual interest, and thus the cost, change for each of the 2025-2033 as well; i.e., they will be bought (+) or sold (-) to achieve the optimization of the figures of merit. Here are all of the cost at 3% changes:

Image
Here you adjusted the bracket multipliers (selling less 2034 than before), also adjusted all the pre-gap years to smooth ARA/DARA, and thus eliminated the $5.3K loss in proceeds found in the first iteration.
Yes, the 2034 multiplier was tweaked to optimize the figures of merit. As before the tweak, the 2025-2033 costs/values were adjusted automatically by the spreadsheet to maintain the ARAs at 100K each.

The tables I shared were paired down to try and make the concepts easier to digest, but it might help to see the entire table. Here it is post-swap at 3% with the tweaked 2034 multiplier:

Image

Thanks for hanging in there with me.
If I make a calculation error, #Cruncher probably will let me know.
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