Filling the TIPS gap years with bracket year duration matching

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

Post by bpg1234 »

I'm going through withdrawal as haven't seen any new posts on this very interesting thread in a few days lol. Longer-term TIPS yields in a range and hoping for a breakout to the upside so can accumulate additional 2040s!
Jaylat
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Jaylat »

Kevin M wrote: Sat Jun 22, 2024 4:24 pm As a radical null hypothesis test, consider covering the gap years with all 2054s.
  1. From a net cash perspective, the upside is much larger if yield decreases a lot than with any other method, including duration matching. This is both for coverage + gap trades as well as for all trades including pre-2034.
  2. There is a relatively small downside for a yield increase from 2% to 4% for the 2054 coverage method, compared to a small upside for the duration matching method, if all transactions are performed to get ARA = DARA.
Below are the tables that show this. I'll use yield changes of 2% to 4% or 0% to make them symmetrical.

Duration matching with a decrease in yield from 2% to 0% with all gaps filled:

Image

Gap coverage with 2054s with a decrease in yield from 2% to 0% with all gaps filled:

Image

Duration matching with an increase in yield from 2% to 4% with all gaps filled:

Image

Gap coverage with 2054s with an increase in yield from 2% to 4% with all gaps filled:

Image

With the caveat, as always, that there are no major flaws in how well my analysis matches reality, it looks like using all 2054s for gap coverage provides good odds of doing much better than with duration matching, especially if one thinks negative yield changes are more likely than positive ones. And the downside if yields increase to levels I think most of us think we're unlikely to see is much less than the upside if yields decrease.

Duration matching works better if considering only minimizing the difference between the coverage TIPS sales and the gap year purchases, which is what I was focused on originally, but looking at the bigger picture, it appears that using longer maturities for gap coverage may be more advantageous from a net cash perspective. Very interesting. :idea:
I wanted to circle back to this with a question: Are you changing the yields equally across all maturities? So when you say a 2% yield increase you are applying that to both the 2054's and the target gap year TIPS?

The risk in buying 2054 TIPS to hedge future 2035-39 TIPS purchases is that the yield curve will change slope. Right now we have a fairly flat / slightly inverted yield curve. If that reverts to a positive or steeply positive yield curve (which is a more typical scenario) the purchases of 2054 TIPS will decline dramatically in price while the 2035-39 TIPS will cost about the same.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

MtnBiker wrote: Sun Jun 30, 2024 9:36 am What should one consider to be a "worst case" non-parallel yield shift? Maybe something like going from a flat yield curve with 2.0% uniform across the gap, to an increasing yield curve with 1% change across the gap? Maybe 1.5% in 2034, 2.0% in 2037 and 2.5% in 2040? Can that be modeled for the 25-year ladder with the three coverage methods: 1: Bracket (2034s/2040s), 2: 2034s, 3: 2040s?
Unfortunately that's not something I can easily model with the tool as it's set up. I played around a bit just trying to look at filling the 2035, and even that was too tedious.

My bigger concern is seeing the relatively large positive net cash outcomes even with yields unchanged at 2%. I think this is at least partly due to total ARA being greater than total DARA with no gaps filled, and as I think I've said, that seems to be related to my imperfect multiplier implementation. I've PM'd #Cruncher to see if he has any ideas on a better way to implement a solution for no gap years filled, and am waiting to hear back from him.

Having said that, I kind of think if we take the numbers with a big grain of salt, maybe subtracting a few thousand dollars from the net cash flows, they still provide a sense of the range of outcomes given different scenarios. So unless someone comes up with something better, we might consider using my summary in my last post, 25 year ladder for three coverage methods, to get an idea of which of the three coverage methods analyzed is most appropriate given one's outlook on future yields; i.e., which method seems to provide the best risk/reward.
If I make a calculation error, #Cruncher probably will let me know.
MtnBiker
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Re: Filling the TIPS gap years with bracket year duration matching

Post by MtnBiker »

Kevin M wrote: Tue Jul 02, 2024 12:14 pm
MtnBiker wrote: Sun Jun 30, 2024 9:36 am What should one consider to be a "worst case" non-parallel yield shift? Maybe something like going from a flat yield curve with 2.0% uniform across the gap, to an increasing yield curve with 1% change across the gap? Maybe 1.5% in 2034, 2.0% in 2037 and 2.5% in 2040? Can that be modeled for the 25-year ladder with the three coverage methods: 1: Bracket (2034s/2040s), 2: 2034s, 3: 2040s?
Unfortunately that's not something I can easily model with the tool as it's set up. I played around a bit just trying to look at filling the 2035, and even that was too tedious.

My bigger concern is seeing the relatively large positive net cash outcomes even with yields unchanged at 2%. I think this is at least partly due to total ARA being greater than total DARA with no gaps filled, and as I think I've said, that seems to be related to my imperfect multiplier implementation. I've PM'd #Cruncher to see if he has any ideas on a better way to implement a solution for no gap years filled, and am waiting to hear back from him.

Having said that, I kind of think if we take the numbers with a big grain of salt, maybe subtracting a few thousand dollars from the net cash flows, they still provide a sense of the range of outcomes given different scenarios. So unless someone comes up with something better, we might consider using my summary in my last post, 25 year ladder for three coverage methods, to get an idea of which of the three coverage methods analyzed is most appropriate given one's outlook on future yields; i.e., which method seems to provide the best risk/reward.
That's too bad that a non-parallel yield shift can't be modeled. But it is easy enough to predict the conclusions using another simple thought experiment.

Based on the detailed analysis of parallel yield shifts, the three coverage methods aren't that different. But I think they are much different when non-parallel yield shifts are considered (as Jaylat has pointed out).

Here is the thought experiment. Suppose you are filling the 2037s in year 2029. Suppose the non-parallel yield shift is -0.5% in 2034 and +0.5% in 2040 (dramatic steepening of the yield curve). The yield of the 2037s is unchanged. In the zero-coupon approximation, the duration (years to maturity) of the 2034 is 5 and the duration of the 2040 is 11. (Use of the zero-coupon approximation is slightly inaccurate but shouldn't change the general conclusions.)

If all the excess holdings are 2034s, the bracket-year holdings increase 2.5% (0.5% yield change times the 5 year duration). If the yield curve had instead inverted to the same degree, the bracket-year holdings would be decreased by 2.5%. So, the range of results is +/- 2.5% on the 2037 swap if all the excess is held in 2034s.

If all the excess holdings are 2040s, the bracket-year holdings decrease 5.5%. If the yield curve had instead inverted to the same degree, the bracket-year holdings would be increased by 5.5%. So, the range of results is -/+ 5.5% on the 2037 swap if all the excess is held in 2040s.

If the excess holdings are in a 50/50 mix of 2034s and 2040s, the change in the bracket year holdings value is the average of the two (+2.5% - 5.5%)/2 = -1.5%. If the yield curve had instead inverted to the same degree, the bracket-year holdings would be increased by 1.5%. So, the range of results is -/+ 1.5% on the 2037 swap if the excess is held in a 50/50 mix of 2034/2040. Clearly a 50/50 mix is better than all 2034s or all 2040s in this case.

Solving the algebraic equation to find what mix has zero for the range of results, gives a 69/31 mix of 2034/2040 (in this particular example, swapping in 2029). The range of results is zero for a 65/35 mix of 2034/2040 if swapping for the 2037s in 2027. The longer one waits to do the swap, having more 2034s in the mix tends to optimize the result. It seems that holding the 2034s to maturity before making any of the swaps is the only way that a 100% holding of 2034s would immunize against non-parallel yield shifts.

In conclusion, minimizing the range of results from non-parallel yield shifts would suggest using 50% or more of the holdings in 2034s. Depending on when the swap is planned to be made, one might want closer to a 65/35 mix of 2034/2040 if this back-of-the-envelope analysis is correct.
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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 »

Jaylat wrote: Tue Jul 02, 2024 12:13 pm I wanted to circle back to this with a question: Are you changing the yields equally across all maturities? So when you say a 2% yield increase you are applying that to both the 2054's and the target gap year TIPS?

The risk in buying 2054 TIPS to hedge future 2035-39 TIPS purchases is that the yield curve will change slope. Right now we have a fairly flat / slightly inverted yield curve. If that reverts to a positive or steeply positive yield curve (which is a more typical scenario) the purchases of 2054 TIPS will decline dramatically in price while the 2035-39 TIPS will cost about the same.
The tables you showed were based on an older model that has been superseded, so ignore those. No one here is promoting using very long-term TIPS for gap coverage, but let's explore some of your comments anyway.

Regardless of changes in the yield curve, the gap year TIPS costs don't change nearly as much as the TIPS already in the ladder since the gap year TIPS are essentially par bonds that always cost about 100. What changes the cost of gap year TIPS are the coupons of each TIPS and the later gap year TIPS (e.g., the 2039 coupon affects the interest received for the 2035-2038, which affects the principal required for those), and this effect is much smaller than the duration effect on the TIPS already in the ladder. For a yield increase, the gap coupons are higher, so the principal required (gap cost) is lower.

Regarding a change in yield curve slope, it depends if it steepens due to increasing longer term or decreasing shorter-term yields. Currently the yield curve is positively sloped between 2034 and 2054, although it's not particularly steep.

Image

It wasn't too hard to add using the 2054 for gap coverage to my latest model; here are the results:

Image

This model has imperfections, but I still think it provides a rough idea of what would happen under various scenarios.

To get an idea of a non-parallel yield curve shift, you could, for example, look at the gap cost at 2% and the 2055 proceeds at 3% or 4%, and yes, this makes things worse, as expected, but not as much as if the gap year TIPS cost changed in value based on duration.

We see from the table that even with an increase of 2054 yield to 3% (proceeds 414,999) and the gap years at 2% (cost 403,119), we'd still have enough to do just the gap transactions. The issue would be that we'd need to buy more of the non-gap, non-2054 TIPS to maintain ARA=DARA, because of the impact of selling 5/6 of the 2054 on the later year interest for the gap years (including all the trickle down effects); this puts us in the red.

I'll show the bracket year outcomes again for comparison:

Image

As stated when I first shared this, this has the narrowest range of outcomes, so from an uncertainty as risk perspective, it's the least risky.
If I make a calculation error, #Cruncher probably will let me know.
MtnBiker
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Re: Filling the TIPS gap years with bracket year duration matching

Post by MtnBiker »

MtnBiker wrote: Tue Jul 02, 2024 3:15 pm

That's too bad that a non-parallel yield shift can't be modeled. But it is easy enough to predict the conclusions using another simple thought experiment.

Based on the detailed analysis of parallel yield shifts, the three coverage methods aren't that different. But I think they are much different when non-parallel yield shifts are considered (as Jaylat has pointed out).

Here is the thought experiment. Suppose you are filling the 2037s in year 2029. Suppose the non-parallel yield shift is -0.5% in 2034 and +0.5% in 2040 (dramatic steepening of the yield curve). The yield of the 2037s is unchanged. In the zero-coupon approximation, the duration (years to maturity) of the 2034 is 5 and the duration of the 2040 is 11. (Use of the zero-coupon approximation is slightly inaccurate but shouldn't change the general conclusions.)

If all the excess holdings are 2034s, the bracket-year holdings increase 2.5% (0.5% yield change times the 5 year duration). If the yield curve had instead inverted to the same degree, the bracket-year holdings would be decreased by 2.5%. So, the range of results is +/- 2.5% on the 2037 swap if all the excess is held in 2034s.

If all the excess holdings are 2040s, the bracket-year holdings decrease 5.5%. If the yield curve had instead inverted to the same degree, the bracket-year holdings would be increased by 5.5%. So, the range of results is -/+ 5.5% on the 2037 swap if all the excess is held in 2040s.

If the excess holdings are in a 50/50 mix of 2034s and 2040s, the change in the bracket year holdings value is the average of the two (+2.5% - 5.5%)/2 = -1.5%. If the yield curve had instead inverted to the same degree, the bracket-year holdings would be increased by 1.5%. So, the range of results is -/+ 1.5% on the 2037 swap if the excess is held in a 50/50 mix of 2034/2040. Clearly a 50/50 mix is better than all 2034s or all 2040s in this case.

Solving the algebraic equation to find what mix has zero for the range of results, gives a 69/31 mix of 2034/2040 (in this particular example, swapping in 2029). The range of results is zero for a 65/35 mix of 2034/2040 if swapping for the 2037s in 2027. The longer one waits to do the swap, having more 2034s in the mix tends to optimize the result. It seems that holding the 2034s to maturity before making any of the swaps is the only way that a 100% holding of 2034s would immunize against non-parallel yield shifts.

In conclusion, minimizing the range of results from non-parallel yield shifts would suggest using 50% or more of the holdings in 2034s. Depending on when the swap is planned to be made, one might want closer to a 65/35 mix of 2034/2040 if this back-of-the-envelope analysis is correct.
The conventional wisdom for duration matching the gap is to hold about a 50/50 mix of 2034/2040. This is based on the concept of roughly immunizing against the interest rate risks resulting from parallel yield shifts. This ratio is valid independent of when the swaps are made.

Holding a 50/50 mix of 2034/2040, incurs a certain risk of loss due to non-parallel yield shifts. The risk isn't huge, but it can be quantified. With this holding ratio, the range of outcomes due to non-parallel yield shifts seems to be independent of the date when the swaps are made.

The concept that one could try to immunize against the risk of loss from a non-parallel yield shift is new to me. What I discovered is that this seems to be possible, if you know when you are going to make the swap. Since most of us plan to make each swap soon after the gap-years are auctioned, we do know the approximate swap dates. The holding ratios to immunize against non-parallel yield shifts can be calculated for each of the five swaps, assuming the five swaps are made sequentially in years 2025 through 2029.

Using the approximation that the duration of each TIPS is the time to maturity (zero-coupon approximation) here are the calculated holding ratios (2034/2040) that immunize against non-parallel yield shifts for each of the five swaps:

2035 90/10
2036 79/21
2037 65/35
3038 48/52
2039 27/73
TOT 62/38

These ratios can be compared to the calculated ratios for immunizing against parallel yield shifts. Again, using the zero-coupon approximation (1/n method), those ratios are:

2035 83/17
2036 67/33
2037 50/50
3038 33/67
2039 17/83
TOT 50/50

It seems that weighting the bracket-year holdings slightly heavier (7-15%) toward 2034s is all this needed to minimize the effects of non-parallel yield shifts. Has anyone tried to verify my calculations? Does this pass the smell test?
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

MtnBiker wrote: Wed Jul 03, 2024 10:50 am Does this pass the smell test?
Shouldn't a parallel yield curve shift be within the range of non-parallel yield curve shifts? Say the yield curve shift could be anywhere from -1/+1 to +1/-1 percentage points for the bracket years. If you truly are immunizing against any shift within this range, 0/0, 0.5/0.5, 1/1, etc. would be included in that range. If so, how could the parallel yield curve shift ratios be different?
If I make a calculation error, #Cruncher probably will let me know.
MtnBiker
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Re: Filling the TIPS gap years with bracket year duration matching

Post by MtnBiker »

Kevin M wrote: Wed Jul 03, 2024 11:00 am
MtnBiker wrote: Wed Jul 03, 2024 10:50 am Does this pass the smell test?
Shouldn't a parallel yield curve shift be within the range of non-parallel yield curve shifts? Say the yield curve shift could be anywhere from -1/+1 to +1/-1 percentage points for the bracket years. If you truly are immunizing against any shift within this range, 0/0, 0.5/0.5, 1/1, etc. would be included in that range. If so, how could the parallel yield curve shift ratios be different?
A parallel yield shift, as I am analyzing it is:

2034 was 2%, now is 3%, or now is 1%
2037 was 2%, now is 3%, or now is 1%
2040 was 2%, now is 3%, or now is 1%

In other words, the overall level of yields changed, but the slope didn't change.

A non-parallel yield shift, as I am analyzed it in the post above, is:

2034 was 2%, now is 3%, or now is 1%
2037 was 2%, and now is 2%
2040 was 2%, now is 1%, or now is 3%

In other words, the overall level of yields is unchanged, but the slope changed, pivoting around year 2037.

Note: Since making the previous post, I reanalyzed each swap on its own basis. That is, instead of assuming the slope changed pivoting about year 2037, for each swap the slope changes pivoting about the gap year that is being swapped. I think this is a fairer way of analyzing this, since it fully separates slope changes from changes in the overall yield level of the year being swapped. The results for the holding ratios (2034/2040) that immunize against non-parallel yield shifts for each of the five swaps changed a wee bit:

2035 89/11
2036 78/22
2037 65/35
2038 50/50
2039 30/70
TOT 62/38

The holding ratios that immunize against non-parallel yield shifts are independent of how much the slope changes to first order (which I think would mean neglecting convexity?). Does this answer your question?
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

MtnBiker wrote: Wed Jul 03, 2024 11:49 am
Kevin M wrote: Wed Jul 03, 2024 11:00 am
MtnBiker wrote: Wed Jul 03, 2024 10:50 am Does this pass the smell test?
Shouldn't a parallel yield curve shift be within the range of non-parallel yield curve shifts? Say the yield curve shift could be anywhere from -1/+1 to +1/-1 percentage points for the bracket years. If you truly are immunizing against any shift within this range, 0/0, 0.5/0.5, 1/1, etc. would be included in that range. If so, how could the parallel yield curve shift ratios be different?
A parallel yield shift, as I am analyzing it is:

2034 was 2%, now is 3%, or now is 1%
2037 was 2%, now is 3%, or now is 1%
2040 was 2%, now is 3%, or now is 1%

In other words, the overall level of yields changed, but the slope didn't change.

A non-parallel yield shift, as I am analyzed it in the post above, is:

2034 was 2%, now is 3%, or now is 1%
2037 was 2%, and now is 2%
2040 was 2%, now is 1%, or now is 3%

In other words, the overall level of yields is unchanged, but the slope changed, pivoting around year 2037.
Yeah, I figured out what you meant by non-parallel yield shift while driving to an appointment--I visualized it as a teeter totter. Another way to describe it is that the 2034 and 2040 yields move in opposite directions.

So I wondered how yield curve shifts have actually happened for TIPS in this maturity range, and to see if I could answer that I went to FRED.

FRED has history for the 2040 and the Apr 2032, so I used these two to check it out. Here's a chart that shows the two yields, 2040 red and 2032 blue, and the 2040 minus the 2032 using, green:

Image

Here's a monthly version that makes the trends a bit easier to see:
Image

Notes and observations:
  1. When the green line is 0, the yield curve is flat, and the 2032 and 2040 yields are the same.
  2. When the green line is above 0, we have the more typical positively sloped yield curve, with the 2040 yield higher than the 2032.
  3. When the green line is below 0, the yield curve is inverted, with the 2040 yield lower than the 2032; we see that this is uncommon.
  4. When the slope of the green line is 0, the yield curve shift is parallel; we see something close to this from about Apr 2012 to Mar 2013, and from Jan 2018 to Sep 2018.
  5. When the slope of the green line is positive, the yield shift is non-parallel, with the gap between the yields increasing.
  6. When the slope of the green line is negative, the yield shift is non-parallel, with the gap between the yields decreasing.
  7. Perhaps most importantly, the yields generally change in the same direction!
So the reality appears to be that non-parallel yield shifts don't happen like a teeter totter around some midpoint, but by both yields changing in the same direction but by different amounts.

Since the teeter totter model doesn't seem to fit reality, how does that affect your analysis?
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: Wed Jul 03, 2024 1:44 pm
MtnBiker wrote: Wed Jul 03, 2024 11:49 am
A parallel yield shift, as I am analyzing it is:

2034 was 2%, now is 3%, or now is 1%
2037 was 2%, now is 3%, or now is 1%
2040 was 2%, now is 3%, or now is 1%

In other words, the overall level of yields changed, but the slope didn't change.

A non-parallel yield shift, as I am analyzed it in the post above, is:

2034 was 2%, now is 3%, or now is 1%
2037 was 2%, and now is 2%
2040 was 2%, now is 1%, or now is 3%

In other words, the overall level of yields is unchanged, but the slope changed, pivoting around year 2037.
Yeah, I figured out what you meant by non-parallel yield shift while driving to an appointment--I visualized it as a teeter totter. Another way to describe it is that the 2034 and 2040 yields move in opposite directions.

So I wondered how yield curve shifts have actually happened for TIPS in this maturity range, and to see if I could answer that I went to FRED.

FRED has history for the 2040 and the Apr 2032, so I used these two to check it out. Here's a chart that shows the two yields, 2040 red and 2032 blue, and the 2040 minus the 2032 using, green:

Image

Here's a monthly version that makes the trends a bit easier to see:
Image

Notes and observations:
  1. When the green line is 0, the yield curve is flat, and the 2032 and 2040 yields are the same.
  2. When the green line is above 0, we have the more typical positively sloped yield curve, with the 2040 yield higher than the 2032.
  3. When the green line is below 0, the yield curve is inverted, with the 2040 yield lower than the 2032; we see that this is uncommon.
  4. When the slope of the green line is 0, the yield curve shift is parallel; we see something close to this from about Apr 2012 to Mar 2013, and from Jan 2018 to Sep 2018.
  5. When the slope of the green line is positive, the yield shift is non-parallel, with the gap between the yields increasing.
  6. When the slope of the green line is negative, the yield shift is non-parallel, with the gap between the yields decreasing.
  7. Perhaps most importantly, the yields generally change in the same direction!
So the reality appears to be that non-parallel yield shifts don't happen like a teeter totter around some midpoint, but by both yields changing in the same direction but by different amounts.

Since the teeter totter model doesn't seem to fit reality, how does that affect your analysis?
The picture I have in my mind is a teeter totter mounted on a party barge that is rising and falling with the tides.

Rightly or wrongly, I have been thinking that the methods for countering the two effects should be fairly independent. That is, if you duration matched against the rising tide, you would feel the full effect from the teeter totter for that bracket-year mix. Or, if you duration matched against the teeter totter, you would feel the full effect from the rising tide for that bracket-year mix.

Your analysis (with a more accurate, detailed model) seems to show that the adverse effects from the rising tide are minimal, even with extreme bracket-year weightings as much as 100/0 and 0/100. Which is why I brought up the teeter totter as another boogeyman that might be countered without adverse suffering from the rising tide. Since the bracket-year weighting to counter the teeter totter is well within the range of bracket-year weightings that seem to be acceptable for fighting the rising tide, why not do both? At least don’t make the teeter totter situation worse by using all 2040s!!!

Thanks for the charts. It is good to know that the historical yield difference between 2034 and 2040 seems to be roughly bounded between flat (0%) and modestly increasing slope (~0.5% delta between 2034 and 2040). This can be factored in when estimating the magnitude of the teeter totter effect. As I mentioned earlier, my estimate of the teeter totter effect on the 2037 swap with 50/50 2034/2040 weightings was a loss of 1.5% if the yield curve slope increased from flat to 1% delta between 2034 and 2040. So, the max realistic teeter totter effect on the 2037 swap is only half that, or a 0.75% loss.

Thus, the potential savings by using 2034/2040 weightings optimized to offset the teeter totter effect is only of order 0.75% vs using a 50/50 mix. But if that benefit comes at no expense from the rising tide effect, it may be worth doing. With 30K of DARA times 5 years, one might save a thousand dollars.

I haven't tried to apply my back-of-the-envelope model to a combined rising tide with teeter totter case. I'm not sure that is worth the effort, since you have shown that back-of-the-envelope models cannot accurately model rising tide effects.

The back-of-the-envelope model simply calculates the loss (or gain) in value of the bracket year holdings due to the movement of the teeter totter. So, you have more or less money than planned to buy the gap year. The back-of-the-envelope model makes no effort to calculate the change in the cost of the gap year. But the yield of the gap year hasn't changed, so the cost to buy the gap year shouldn't have changed.

But, as you have shown, the cost to buy the gap year does change in unexpected ways due to coupons. And, in reality, the tide rises and falls as the teeter totter tilts, so all that requires a more complex model.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

MtnBiker wrote: Wed Jul 03, 2024 3:21 pm Thanks for the charts. It is good to know that the historical yield difference between 2034 and 2040 seems to be roughly bounded between flat (0%) and modestly increasing slope (~0.5% delta between 2034 and 2040).
Here's the zoomed in chart of the yield deltas between the 2040 and 2032:

Image

And here are some stats:

Code: Select all

Delta average	 0.23
Delta min	-0.10
Delta max	 0.69
Delta Std Dev	 0.14
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Re: Filling the TIPS gap years with bracket year duration matching

Post by MtnBiker »

I think there is another important point that should be made regarding non-parallel yield shifts. The conventional duration matching analysis uses approximately at 50/50 mix of the bracket years to hedge against parallel yield shifts. This hedging becomes more imperfect when the yield curve slope changes. The FRED charts show that the slope changes between 2032 and 2040 have been relatively small to date. However, these slope changes will increase as these bonds approach maturity. This paper (https://www.soa.org/sections/joint-risk ... 6-freeman/) states the following:
Interest Rate Movements—Not Parallel
The modern definition of duration assumed a small, parallel movement of the yield curve to allow the theory to progress. However, it is unrealistic as we know that the short end of the yield curve is far more volatile than the long end.
This is a reminder that non-parallel yield shifts will become more and more important the closer the bracket years get to maturity. This suggests that the swaps should be made as soon as practical (shortly after the ten-year auctions), so that the bracket year bonds don't become short term bonds. It also might suggest that the 2034/2040 6-year spread between bracket year holdings might want to be narrowed somewhat when possible. (For example, maybe switch to 2036/2040 holdings and then 2038/2040 holdings, when available).
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

As a quick update, #Cruncher supplied me with an updated version of the simplified TIPS ladder spreadsheet using multipliers such that total proceeds always equals DARA times sum of multipliers. However, it doesn't seem to model things in what seems to me like rational way as I fill the gaps, so I'm waiting to hear back from him on that.
If I make a calculation error, #Cruncher probably will let me know.
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Kevin M
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

MtnBiker wrote: Thu Jul 04, 2024 10:36 am I think there is another important point that should be made regarding non-parallel yield shifts. The conventional duration matching analysis uses approximately at 50/50 mix of the bracket years to hedge against parallel yield shifts. This hedging becomes more imperfect when the yield curve slope changes. The FRED charts show that the slope changes between 2032 and 2040 have been relatively small to date. However, these slope changes will increase as these bonds approach maturity. This paper (https://www.soa.org/sections/joint-risk ... 6-freeman/) states the following:
Interest Rate Movements—Not Parallel
The modern definition of duration assumed a small, parallel movement of the yield curve to allow the theory to progress. However, it is unrealistic as we know that the short end of the yield curve is far more volatile than the long end.
This is a reminder that non-parallel yield shifts will become more and more important the closer the bracket years get to maturity. This suggests that the swaps should be made as soon as practical (shortly after the ten-year auctions), so that the bracket year bonds don't become short term bonds. It also might suggest that the 2034/2040 6-year spread between bracket year holdings might want to be narrowed somewhat when possible. (For example, maybe switch to 2036/2040 holdings and then 2038/2040 holdings, when available).
We can see this by looking at the 30y (FRED ID DFII30) and 5y (FRED ID DFII5) constant maturity inflation indexed yields. Here's the chart:

Image

Just by inspection we can see that the 5y (red) is more volatile.

Here are some stats:

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		DFII30	DFII5
Std Dev	 	 0.7	1.0
Max		 2.5	2.4
Min		-0.5	-1.9
Max - Min	 3.0	4.3
If I make a calculation error, #Cruncher probably will let me know.
Jaylat
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Jaylat »

A random thought on this thread:

One of the complications in duration matching is the potential loss of coupon income for the pre-gap years ARA. That is, when you sell off the 2034 and 2040 TIPS you have to adjust the ARA for the pre-gap years to reflect the loss of 2040 TIPS coupons.

One option to consider: Ignore 71.4% of the 2034 and 2040 TIPS coupons in computing pre-GAP ARA.

Why 71.4%? You know with certainty going in that:
You are buying around 3.5x DARA of 2034 and 2040 TIPS;
You are going to sell around 71.4% (2.5x DARA) of your 2034 and 2040 TIPS in the future in order to buy GAP year TIPS;
Therefore you will lose 71.4% of the coupon income from your 2034 and 2040 TIPS.

So one (perhaps overly conservative) approach would be to ignore the 71.4% of the 2034 and 2040 TIPS coupons that you know will disappear.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

Jaylat wrote: Fri Jul 05, 2024 10:26 am A random thought on this thread:

One of the complications in duration matching is the potential loss of coupon income for the pre-gap years ARA. That is, when you sell off the 2034 and 2040 TIPS you have to adjust the ARA for the pre-gap years to reflect the loss of 2040 TIPS coupons.

One option to consider: Ignore 71.4% of the 2034 and 2040 TIPS coupons in computing pre-GAP ARA.

Why 71.4%? You know with certainty going in that:
You are buying around 3.5x DARA of 2034 and 2040 TIPS;
You are going to sell around 71.4% (2.5x DARA) of your 2034 and 2040 TIPS in the future in order to buy GAP year TIPS;
Therefore you will lose 71.4% of the coupon income from your 2034 and 2040 TIPS.

So one (perhaps overly conservative) approach would be to ignore the 71.4% of the 2034 and 2040 TIPS coupons that you know will disappear.
You're right about the coupons being one of the main issues with duration matching, or any other gap year coverage scheme. The thing is that we don't know what the gap year coupons will be when they're issued, so we could be earning more or less than the bracket year coupons.

What I'm hoping is that #Cruncher will come through with another revision of the simplified model so that we can at least reliably evaluate the impact. Currently I don't think we have a model that does that, so I'm pretty much in a holding pattern for now.
If I make a calculation error, #Cruncher probably will let me know.
StillGoing
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Re: Filling the TIPS gap years with bracket year duration matching

Post by StillGoing »

Thank you for a very interesting thread – you were right (viewtopic.php?p=7931681#p7931681) that extending a ladder beyond 30 years is a similar problem to the one you have investigated here.

While you have concentrated on duration matching in this thread, in the spreadsheet #cruncher describes three ways to fill a gap, Plan 1: buy multiple amounts for the year before the gap, Plan 2: buy multiple amounts after the gap, and Plan 3: buy some of each bond before and after the gap. I was curious as to what the results would look like if either of the first two plans were followed (i.e., no attempt to even slightly duration match at all), so thought I’d have a go at calculating them.

In the analysis below, I’ve used #cruncher’s ladder construction spreadsheet with prices and yields from 1 July 2024 and a desired annual real income of $40k per year. To set up Plan 1, I’ve used a multiplier of 6 for January 2034 and 1 for February 2040 and assumed that the retiree will wait to construct a ladder to fill the gap until the 2034 TIPS matures because this will not then change the coupons and, therefore, affect the income in the years before 2034.

Using the ‘ToPaste’ sheet in the spreadsheet gives the following (for brevity I have excluded the rungs before 2034)

Code: Select all

Seq	Row		Matures 	Coupon	Price 		Yield 	CUSIP  		Mult	Bnds	Cost	 	Principal	FinalCoupon
10	43		15/01/2034	1.750%	 97.37500 	2.054%	91282CJY8	6	199	 199,297 	 202,986 	 1,776 
11	45		15/02/2040	2.125%	 99.62500 	2.154%	912810QF8	1	24	 34,964 	 34,816 	 370 
12	46		15/02/2041	2.125%	 99.78125 	2.141%	912810QP6	1	25	 36,002 	 35,795 	 380 
13	47		15/02/2042	0.750%	 78.71875 	2.216%	912810QV3	1	25	 27,406 	 34,691 	 130 
14	48		15/02/2043	0.625%	 75.68750 	2.227%	912810RA8	1	27	 27,956 	 36,822 	 115 
15	49		15/02/2044	1.375%	 86.37500 	2.237%	912810RF7	1	27	 31,563 	 36,325 	 250 
16	50		15/02/2045	0.750%	 75.53125 	2.237%	912810RL4	1	27	 27,256 	 35,951 	 135 
17	51		15/02/2046	1.000%	 78.68750 	2.249%	912810RR1	1	28	 29,295 	 37,052 	 185 
18	52		15/02/2047	0.875%	 75.78125 	2.247%	912810RW0	1	29	 28,670 	 37,668 	 165 
19	53		15/02/2048	1.000%	 77.28125 	2.245%	912810SB5	1	30	 29,623 	 38,145 	 191 
20	54		15/02/2049	1.000%	 76.68750 	2.236%	912810SG4	1	31	 29,768 	 38,627 	 193 
21	55		15/02/2050	0.250%	 61.71875 	2.216%	912810SM1	1	31	 23,370 	 37,807 	 47 
22	56		15/02/2051	0.125%	 58.53125 	2.189%	912810SV1	1	32	 22,575 	 38,539 	 24 
23	57		15/02/2052	0.125%	 57.50000 	2.183%	912810TE8	1	34	 22,038 	 38,296 	 24 
24	58		15/02/2053	1.500%	 84.87500 	2.216%	912810TP3	1	37	 33,346 	 39,028 	 293 
25	59		15/02/2054	2.125%	 98.21875 	2.208%	912810TY4	1	39	 39,453 	 39,844 	 423
In January 2034, the retiree will receive the maturing principal ($202986 – all values are rounded to the nearest dollar), the final coupon of the TIPS maturing in 2034 ($1776) and (in February and August) the combined coupons of all the TIPS in the post-gap part of the ladder (an annual total of $5850) for a total of $210613. After setting aside the income for 2034, this leaves $170613 (i.e., 210613-40000) with which to construct the ladder for the next 5 years. Assuming a flat yield curve, the annual income from the new five year ladder, calculated using pmt(ytm,5,-170613,0,0), income from the coupons provided by the post-gap ladder (i.e., from the TIPS maturing in 2040 onwards), PGC, the total annual income for the 5 gap years, and the number of rungs, N (calculated using the nper excel function) that could be constructed to provide a total income of $40k as a function of yield in 2034 (i.e. when the ladder is constructed) are as follows

Code: Select all

YTM	Ladder	PGC	Total	N
-4	30139	5850	35990	4.46
-3	31114	5850	36964	4.59
-2	32103	5850	37953	4.71
-1	33106	5850	38956	4.85
0	34123	5850	39973	5.00
1	35153	5850	41003	5.15
2	36197	5850	42047	5.32
3	37254	5850	43104	5.49
4	38324	5850	44175	5.68
These results indicate that for yields to maturity of -4%, the income in the gap years would be about 10% below target (i.e., 36k/40k) or the target income of $40k could only be provided for about 4.5 years. While I note that, in their relatively short history, 5 year TIPS yields have not fallen below about -2%, in the UK, yields for their equivalent, 5 year inflation linked gilts, have been as low as -4%. I also note that it is the income from coupons from the post-gap part of the ladder (i.e, 2040 and later), which is independent of the yields prevailing in 2034, that ensures the reduction in total income is not larger. Of course, the point of duration matching is that it should be able to improve upon these results.

Since this post is getting a bit long, the results for Plan 2 (i.e., “buy multiple amounts for the year after the gap”) will be presented later.

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

Post by StillGoing »

Following on from my previous post, using #cruncher’s spreadsheet to adopt Plan 2, i.e., buy multiple amounts of the bond maturing in the year after the gap, with multipliers of 1 and 6 for 2034 and 2040 results in no bond purchase for 2034 and an income shortfall in 2033 (in other words, the bonds maturing in 2040 would have to be sold starting in 2033 rather than in 2034). In order to provide the required income in 2033, multipliers of 1.45 (2034) and 5.55 (2040) have been used instead. Assuming a required income of $40k per year and prices as of 1 July 2024, the outcomes are given below (note that using these multipliers, no bonds maturing in 2034 are bought – I’ve started the table at 2033).

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Seq	Row		Matures 	Coupon	Price 		Yield 	CUSIP  		Mult	Bnds	Cost 		Principal	FinalCoupon
9	41		15/01/2033	1.125%	 92.71875 	2.057%	91282CGK1	1	29	 28,462 	 30,526 	 172 
10	45		15/02/2040	2.125%	 99.62500 	2.154%	912810QF8	5.55	148	 215,611 	 214,699 	 2,281 
11	46		15/02/2041	2.125%	 99.78125 	2.141%	912810QP6	1	25	 36,002 	 35,795 	 380 
12	47		15/02/2042	0.750%	 78.71875 	2.216%	912810QV3	1	25	 27,406 	 34,691 	 130 
13	48		15/02/2043	0.625%	 75.68750 	2.227%	912810RA8	1	27	 27,956 	 36,822 	 115 
14	49		15/02/2044	1.375%	 86.37500 	2.237%	912810RF7	1	27	 31,563 	 36,325 	 250 
15	50		15/02/2045	0.750%	 75.53125 	2.237%	912810RL4	1	27	 27,256 	 35,951 	 135 
16	51		15/02/2046	1.000%	 78.68750 	2.249%	912810RR1	1	28	 29,295 	 37,052 	 185 
17	52		15/02/2047	0.875%	 75.78125 	2.247%	912810RW0	1	29	 28,670 	 37,668 	 165 
18	53		15/02/2048	1.000%	 77.28125 	2.245%	912810SB5	1	30	 29,623 	 38,145 	 191 
19	54		15/02/2049	1.000%	 76.68750 	2.236%	912810SG4	1	31	 29,768 	 38,627 	 193 
20	55		15/02/2050	0.250%	 61.71875 	2.216%	912810SM1	1	31	 23,370 	 37,807 	 47 
21	56		15/02/2051	0.125%	 58.53125 	2.189%	912810SV1	1	32	 22,575 	 38,539 	 24 
22	57		15/02/2052	0.125%	 57.50000 	2.183%	912810TE8	1	34	 22,038 	 38,296 	 24 
23	58		15/02/2053	1.500%	 84.87500 	2.216%	912810TP3	1	37	 33,346 	 39,028 	 293 
24	59		15/02/2054	2.125%	 98.21875 	2.208%	912810TY4	1	39	 39,453 	 39,844 	 423 
Since no bonds maturing in 2034 were purchased, in 2034 the retiree will sell all bar 24 of the 148 bonds maturing in 2040 to provide income for 6 years, i.e., 2034, 2035, … 2039 (the 24 remaining bonds maturing in 2040 will, together with the coupons from later bonds, provide the income for 2040). I note that an alternative approach would be to sell enough of the 2040 bonds each year to provide the required income, although this comes with the risk that the income in the final year of the gap, 2039, might fall short.

The proceeds from the sale will depend on the number of bonds sold, n (in this example n=148-24=124 bonds), the yield to maturity (ytm), and therefore price, prevailing in 2034, and the CPI adjusted principal, CAP (1450.67 in July 2024 for the 2040 bond) such that

proceeds=n*CAP*price/100.

As mentioned above, the price will depend on the yield. For example, since the coupon for the bond maturing in 2040 is 2.125%, and the remaining term in 2034 is 6 years, the price will range from 142.01 (for a ytm of -4%) to 90.09 (for a ytm of +4%).

Of the proceeds from the sale, about $32.2k, i.e., required income of $40k, less the final coupon of the 2040 bonds being sold, $1911 and the coupons from the post-gap ladder, $5850, will be needed to provide income for 2034.

The following table then contains the price, the proceeds from the sale (before reducing the amount by $32.2k to provide for the 2034 income), the annual income derived from a five year ladder (i.e., for 2035, 2036, 2037, 2038, and 2039), the income from the remaining post gap coupons (PGC), the total income and the number of rungs, N that could be constructed to provide an income of $40k per year.

Code: Select all

YTM	Price	Prcds	Ladder	PGC	Total	N
-4	142.01	255452	39431	5850	45281	5.69
-3	133.97	240989	38069	5850	43919	5.53
-2	126.44	227444	36730	5850	42580	5.36
-1	119.37	214726	35410	5850	41260	5.18
0	112.75	202818	34116	5850	39966	5.00
1	106.54	191647	32845	5850	38695	4.80
2	100.7	181142	31591	5850	37441	4.61
3	95.23	171303	30365	5850	36215	4.41
4	90.09	162057	27847	5850	33697	4.21
For a ytm of 4%, the total income of $33.7k is nearly 16% below the target of $40k. Historically, the highest yield observed for 5 year TIPS has been just over 4%, although this rate was observed only briefly towards the end of November 2008. The total income for the strategy of buying bonds at the end of the gap decreases with increasing yield which is the opposite behaviour to buying bonds at the beginning of the gap (see my previous post).

Combining the two plans in a 50/50 approach (i.e. averaging the results in the above table with the equivalent one in the previous post), leads to the following outcomes for income and number of rungs N.

Code: Select all

YTM	Income	N
-4	40635	5.08
-3	40442	5.06
-2	40267	5.04
-1	40108	5.02
0	39969	5.00
1	39849	4.98
2	39744	4.96
3	39660	4.95
4	38936	4.94
Holding a fixed proportion of bonds pre- and post-gap, greatly improves the results even over this large range yields. For example, the total income only falls to about 3% below the $40k target at a yield of 4%. Of course, this is not duration matching since the proportion of before and post gap bonds is fixed at the start and does not respond to changes in yield, and hence duration, but does result in an income that may be close enough to that required.

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

Post by Kevin M »

^Nice work, StillGoing. Your results appear to further bolster confidence that something close to a 50/50 mix of overweighting the 2034 and 2040 to cover the gap years has the lowest range of outcomes--at least for parallel yield curve shifts.

I now know how many (most?) others probably feel when looking at my posts in this thread, in that it's going to take more than just a quick read to digest your method. Still, from a quick read it seems that what you're doing makes sense, and the results are intuitively appealing.

Although #Cruncher's latest tweak of his simplified spreadsheet doesn't seem to provide rational results for filling the gap years one at a time, it may do so for filling all five at once--say in 2029. I kind of got sidetracked by the one at a time scenario, and was hoping #Cruncher could come up with an even better simplified spreadsheet, but he PM'd me and told me that what I was trying to do is very complicated, and not something any of his TIPS ladder building spreadsheets are suited for. So I'll probably take another look at the filling all gaps in 2029 scenario.
If I make a calculation error, #Cruncher probably will let me know.
MtnBiker
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Re: Filling the TIPS gap years with bracket year duration matching

Post by MtnBiker »

StillGoing wrote: Tue Jul 09, 2024 3:24 am

Combining the two plans in a 50/50 approach (i.e. averaging the results in the above table with the equivalent one in the previous post), leads to the following outcomes for income and number of rungs N.

Code: Select all

YTM	Income	N
-4	40635	5.08
-3	40442	5.06
-2	40267	5.04
-1	40108	5.02
0	39969	5.00
1	39849	4.98
2	39744	4.96
3	39660	4.95
4	38936	4.94
Holding a fixed proportion of bonds pre- and post-gap, greatly improves the results even over this large range yields. For example, the total income only falls to about 3% below the $40k target at a yield of 4%. Of course, this is not duration matching since the proportion of before and post gap bonds is fixed at the start and does not respond to changes in yield, and hence duration, but does result in an income that may be close enough to that required.

Cheers
StillGoing
It seems to me that averaging the results for filling with all 2034s and filling with all 2040s would give results equivalent to filing each gap year with a 50/50 mix:

Code: Select all

year       filled with bracket years
2035       50% 2034 and 50% 2040
2036       50% 2034 and 50% 2040
2037       50% 2034 and 50% 2040
2038       50% 2034 and 50% 2040
2039       50% 2034 and 50% 2040
What isn't quite obvious to me is whether the result would be the same if one followed the actual plan which is to fill the gaps like this:

Code: Select all

year       filled with bracket years
2035       83% 2034 and 17% 2040
2036       67% 2034 and 33% 2040
2037       50% 2034 and 50% 2040
2038       33% 2034 and 67% 2040
2039       17% 2034 and 83% 2040
Does averaging give the correct result for both of these filling methods, or not?
StillGoing
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Re: Filling the TIPS gap years with bracket year duration matching

Post by StillGoing »

MtnBiker wrote: Tue Jul 09, 2024 4:23 pm
StillGoing wrote: Tue Jul 09, 2024 3:24 am

Combining the two plans in a 50/50 approach (i.e. averaging the results in the above table with the equivalent one in the previous post), leads to the following outcomes for income and number of rungs N.

Code: Select all

YTM	Income	N
-4	40635	5.08
-3	40442	5.06
-2	40267	5.04
-1	40108	5.02
0	39969	5.00
1	39849	4.98
2	39744	4.96
3	39660	4.95
4	38936	4.94
Holding a fixed proportion of bonds pre- and post-gap, greatly improves the results even over this large range yields. For example, the total income only falls to about 3% below the $40k target at a yield of 4%. Of course, this is not duration matching since the proportion of before and post gap bonds is fixed at the start and does not respond to changes in yield, and hence duration, but does result in an income that may be close enough to that required.

Cheers
StillGoing
It seems to me that averaging the results for filling with all 2034s and filling with all 2040s would give results equivalent to filing each gap year with a 50/50 mix:

Code: Select all

year       filled with bracket years
2035       50% 2034 and 50% 2040
2036       50% 2034 and 50% 2040
2037       50% 2034 and 50% 2040
2038       50% 2034 and 50% 2040
2039       50% 2034 and 50% 2040
What isn't quite obvious to me is whether the result would be the same if one followed the actual plan which is to fill the gaps like this:

Code: Select all

year       filled with bracket years
2035       83% 2034 and 17% 2040
2036       67% 2034 and 33% 2040
2037       50% 2034 and 50% 2040
2038       33% 2034 and 67% 2040
2039       17% 2034 and 83% 2040
Does averaging give the correct result for both of these filling methods, or not?
I think it depends on exactly how you are going to fill the gap.

In the analysis I've done above, I've assumed that in 2034, the retiree liquidates all of the 2034 assets (if any) and any of the 2040 assets in excess of those required to provide the income for 2040 (again, if any) in order to construct the entire 5 year ladder (and to obtain income for 2034). For that approach, I think the two filling methods you've outlined above are identical.

However, if the rungs are filled as and when assets become available to invest in (i.e., a TIPS maturing in 2035 will presumably become available next year) then the two filling methods are likely to produce different results. That the two methods will give different outcomes will also be the case if the assets are liquidated one year at a time from 2035 onwards.

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

Post by StillGoing »

While the results for Plans 1 and 2 (i.e., but multiple amounts of the bond on one side or the other of the gap) covered in previous posts are useful, there are (at least) two further questions I have
1) Is there an ‘optimum’ set of multipliers for Plan 3 (i.e., where multiple amounts of each bond on either side of the gap are bought) that produces a fairly flat income as a function of ytm?
2) What happens if the yield curve is not flat over the period 2034 and 2040?

The answer to the first of those questions is considered in this post.

Using the #cruncher spreadsheet with a required income of $40k, prices from 1 July 2024, and multipliers of 3.95 (for 2034) and 3.05 (for 2040) gives the following outcome (from the ToPaste sheet – only the 2034 and 2040 bonds are shown)

Code: Select all

Seq	Row		Matures 	Coupon	Price 		Yield 	CUSIP  		Mult	Bnds	Cost 		Principal	FinalCoupon
10	43		15/01/2034	1.750%	 97.37500 	2.054%	91282CJY8	3.95	109	 109,163 	 111,183 	 973 
11	45		15/02/2040	2.125%	 99.62500 	2.154%	912810QF8	3.05	80	 116,547 	 116,054 	 1,233 
The following assumes that the retiree waits until 2034 before constructing the ladder to cover the gap and then uses the bonds maturing in 2034 and the proceeds from selling those bonds from 2040 not required to provide income in 2040 to do so.

2034 bonds: the principal of the maturing bonds provides $111183, the final coupon of the maturing bonds $973, and the income from coupons from the post-gap ladder (PGC), taking into account the sale of the 2040 bonds (see below), is $5850. Since income of $40k is required for 2034, the net contribution to the ladder is then 111183+973+5850-40000=78006. This is independent of the yield prevailing in 2034.

2040 bonds: 56 of the 2040 bonds will be sold (80-24, where 24 bonds are required to provide the required income in 2040), while the final coupon of those sold will be $863. The amount gained from the sale will depend on the yield of the 2040 bond at the time of sale (i.e., in 2034).

The following table then shows how the proceeds from the 2034 and 2040 bonds are combined to produce an income from the ladder (a flat yield curve is assumed). The table headings are Contr(ibution) of 2034 bonds to ladder, the price and the proceeds generated from the sale of 2040 bonds (including the final coupon), Sum is the sum of 2034 and 2040 contributions, ladder is the income generated from the combined amount for the given yield, PGC is the post-gap coupons, total is the total income from ladder and PGC, and N is the number of rungs offering the full $40k target income that could be constructed.

Code: Select all

	2034	2040
YTM	Contr	Price	Prcds	Sum	Ladder	PGC	Total	N
-4	78006	142.01	116229	194235	34312	5850	40162	5.02
-3	78006	133.97	109697	187703	34231	5850	40081	5.01
-2	78006	126.44	103580	181586	34167	5850	40017	5.00
-1	78006	119.37	97836	175842	34120	5850	39970	5.00
0	78006	112.75	92458	170464	34093	5850	39943	4.99
1	78006	106.54	87414	165420	34083	5850	39933	4.99
2	78006	100.7	82669	160675	34089	5850	39939	4.99
3	78006	95.23	78226	156232	34114	5850	39964	4.99
4	78006	90.09	74050	152056	34156	5850	40006	5.00
Two things to note:
1) There is now a minimum in total income (and N) at a yield of about 1%
2) At that minimum, the income is only about $67 short of the target!

I note that this outcome is very sensitive to multiplier (and I suspect to the yields of the 2034 and 2040 bonds when the ladder is constructed at the beginning of retirement). For example, using multipliers of 3.74 and 3.26, the outcome is close to (but not quite identical with) that of the average in my previous post (viewtopic.php?p=7946700#p7946700), i.e. the income declines slightly with increasing yield. On the other hand, changing the multipliers to 4.1 and 2.9 results in the opposite behaviour since income then increases with increasing yield.

These results have been calculated assuming a flat yield curve. I think I can see a way to model the outcomes where the yield curve between 2034 and 2040 is not flat, but need to develop a couple of tools to do so. My guess is that a non-flat yield curve will make the outcomes worse - but by how much will depend on the gradient (both magnitude and sign). In terms of finding historical values of the yield curve, I note that, unlike the nominal yield curves, the real yield curves at https://home.treasury.gov/policy-issues ... statistics only start from 5 years. But has anyone used the data at https://www.federalreserve.gov/data/yie ... 805_1.html which does appear to have par and zero coupon yields for maturities from 2 to 6 years (and higher) for 1999 onwards?

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

Post by MtnBiker »

StillGoing wrote: Wed Jul 10, 2024 6:28 am While the results for Plans 1 and 2 (i.e., but multiple amounts of the bond on one side or the other of the gap) covered in previous posts are useful, there are (at least) two further questions I have
1) Is there an ‘optimum’ set of multipliers for Plan 3 (i.e., where multiple amounts of each bond on either side of the gap are bought) that produces a fairly flat income as a function of ytm?
2) What happens if the yield curve is not flat over the period 2034 and 2040?

The answer to the first of those questions is considered in this post.

Using the #cruncher spreadsheet with a required income of $40k, prices from 1 July 2024, and multipliers of 3.95 (for 2034) and 3.05 (for 2040) gives the following outcome (from the ToPaste sheet – only the 2034 and 2040 bonds are shown)

<snip>

Two things to note:
1) There is now a minimum in total income (and N) at a yield of about 1%
2) At that minimum, the income is only about $67 short of the target!

I note that this outcome is very sensitive to multiplier (and I suspect to the yields of the 2034 and 2040 bonds when the ladder is constructed at the beginning of retirement). For example, using multipliers of 3.74 and 3.26, the outcome is close to (but not quite identical with) that of the average in my previous post (viewtopic.php?p=7946700#p7946700), i.e. the income declines slightly with increasing yield. On the other hand, changing the multipliers to 4.1 and 2.9 results in the opposite behaviour since income then increases with increasing yield.

These results have been calculated assuming a flat yield curve. I think I can see a way to model the outcomes where the yield curve between 2034 and 2040 is not flat, but need to develop a couple of tools to do so. My guess is that a non-flat yield curve will make the outcomes worse - but by how much will depend on the gradient (both magnitude and sign). In terms of finding historical values of the yield curve, I note that, unlike the nominal yield curves, the real yield curves at https://home.treasury.gov/policy-issues ... statistics only start from 5 years. But has anyone used the data at https://www.federalreserve.gov/data/yie ... 805_1.html which does appear to have par and zero coupon yields for maturities from 2 to 6 years (and higher) for 1999 onwards?

cheers
StillGoing
In previous posts upthread (viewtopic.php?p=7937901#p7937901 and viewtopic.php?p=7937992#p7937992), I looked at what happens if the yield curve is initially flat when the bracket year holdings are purchased, the overall yield level doesn't change, but the yield curve is no longer flat when the swaps are made. I discovered (using a very crude calculation) that the effects from such a non-parallel yield shift would be minimized with a 62/38 mix of 2034/2040 bracket-year holdings (multipliers of 4.1 (for 2034) and 2.9 (for 2040)).

It is quite interesting to see that your detailed analysis finds that the multipliers to minimize the impact of parallel yield shifts (multipliers of 3.95 (for 2034) and 3.05 (for 2040)) are so close to these multiplier values.

Looking forward to seeing how your results turn out when analyzing the effects of non-parallel yield shifts using more detailed analysis.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

StillGoing wrote: Wed Jul 10, 2024 6:28 am I think I can see a way to model the outcomes where the yield curve between 2034 and 2040 is not flat, but need to develop a couple of tools to do so. My guess is that a non-flat yield curve will make the outcomes worse - but by how much will depend on the gradient (both magnitude and sign). In terms of finding historical values of the yield curve, I note that, unlike the nominal yield curves, the real yield curves at https://home.treasury.gov/policy-issues ... statistics only start from 5 years. But has anyone used the data at https://www.federalreserve.gov/data/yie ... 805_1.html which does appear to have par and zero coupon yields for maturities from 2 to 6 years (and higher) for 1999 onwards?
It's a different yield curve model, but I think it's probably fine to use.

Another source of historical yields for actual TIPS is from FRED. If you search on "inflation due", you'll find lots of TIPS. Even if you don't use those, you could spot check the yields from the FRB source against them.
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: Wed May 22, 2024 10:28 pm No temporal effects are considered; e.g., durations are not updated based on the passing of almost five years before all gaps are filled.

Looking at the 0% yield case:
  • The proceeds from selling all excess 2034 and 2040 bracket year TIPS are 515,669.
  • The cost of buying the 2035-2039 gap year TIPS is 432,951.
  • This leaves us with extra cash of 82,718.
  • We can choose to buy the pre-2034 TIPS that are left, 2030-2033, with the total being 31,396, if we want ARA to equal DARA for those rungs.
  • If we do the pre-2034 transactions, we are left with 51,322 in cash.
I'm trying to replicate your numbers here, and have some questions for you:

What formula do you use to determine the various TIPS prices?
Why in the bolded statement are you focusing only on the 2030-33 years? Is that because 5 years have passed? That seems inconsistent with your not considering temporal effects on the bond durations?

In reality, you would be selling off bracket year TIPS starting in 2025. So the cost of buying additional bonds in order to make ARA equal to DARA for the pre-GAP years appears to be understated.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

Jaylat wrote: Wed Jul 10, 2024 11:36 am
Kevin M wrote: Wed May 22, 2024 10:28 pm No temporal effects are considered; e.g., durations are not updated based on the passing of almost five years before all gaps are filled.

Looking at the 0% yield case:
  • The proceeds from selling all excess 2034 and 2040 bracket year TIPS are 515,669.
  • The cost of buying the 2035-2039 gap year TIPS is 432,951.
  • This leaves us with extra cash of 82,718.
  • We can choose to buy the pre-2034 TIPS that are left, 2030-2033, with the total being 31,396, if we want ARA to equal DARA for those rungs.
  • If we do the pre-2034 transactions, we are left with 51,322 in cash.
I'm trying to replicate your numbers here, and have some questions for you:

What formula do you use to determine the various TIPS prices?
Why in the bolded statement are you focusing only on the 2030-33 years? Is that because 5 years have passed? That seems inconsistent with your not considering temporal effects on the bond durations?

In reality, you would be selling off bracket year TIPS starting in 2025. So the cost of buying additional bonds in order to make ARA equal to DARA for the pre-GAP years appears to be understated.
I've moved on from this model, and I was about to post some updated numbers using the latest model, which uses a newer version of the simplified ladder spreadsheet from #Cruncher. I'll do that now, and then you can ask whatever questions are still relevant.
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 »

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

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

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

Image

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

Image

Things to note:
  • The only relevant cost column for this scenario, one of the three covered here, is Cost at 2%, since I'm assuming all yields are 2% from 2030-2040; I've highlighted the relevant cost cells, which include the gap years since we're buying them, the bracket years since we're selling them, and the 2030-2033 since we will buy or sell them so that ARA = DARA.
  • Total proceeds still equals 2.5M, as it should.
To determine the outcome for each cost at YTM scenario, I copy/paste-values the zero gaps filled version into rows below, change the gaps filled to 5 for each yield scenario, and subtract the values in the former from the later. From these deltas I create the table below, which is similar to the tables shown before, except that I've maintained the cash flow sign convention for all sales and purchases.

Image

Observations:
  • The range of Bracket/gap net is similar, but quite symmetrical, unlike before.
  • The range of the total net (after doing the not gap/bracket buys or sells) also is similar, but shifted from down from positive values to zero or negative values, and with all values much closer to zero.
  • As with version 1, the buys or sells of the 2030-2033 not only result in ARA = DARA for these maturities, but also dramatically compresses the range of net values at different yields.
  • I'm not sure what to use as the denominator, but if we look at total net as a percentage of cost to buy the gap years, for example, these values are 0%, 0.95% and 1.5% for yields of 0%, 2% and 4% respectively, which seem quite small.
It would be interesting to compare StillGoing's results to these, but I'm assuming transactions in 2029 while SG is assuming transactions in 2034, so the results are not directly comparable.

For JayLat, or anyone else who is interested in duplicating these results, here are the key formulas for row 16, which are copied to all rows except row 2, for which I'll note the differences below:

Code: Select all

G16: =SUM(I$2:I15)
H16: =MAX(0,(B$1*SUM(E$2:E16)-SUM(J$2:J15)-G16)/(1+F16))
I16: =H16*F16
J16: =SUM(G16:I16)
K16: =-PV(RIGHT(K$1,3),$C16,$I16,$H16,0) (copied to L16 and M16).
The formulas that are different for row 2, the 2054 maturity, are:

Code: Select all

G2: 0
H2: =E2*B1/(1+F2)
These are all formulas from the #Cruncher simplified ladder spreadsheet, version 2, except that I modified the formula for column K slightly so that I could put "Cost at N%" in one row, and extract the yield from the text.
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 »

What a huge change! The difference of "5 fill vs 0 fill" went from 51,322 at 0% discount to nothing! The rest of your values are pretty close to even as well.

It appears your initial calculations were so far off because they assumed the Bracket GAPS TIPS still had their initial duration. Because you ignored "temporal effects" you were assuming the 2034 and 2040 TIPS each had 5 more years' duration than they actually had, which accounts for their wild swings in values. That made the bracket year TIPS more valuable at lower discount rates and less valuable at higher discounts.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by MtnBiker »

Jaylat wrote: Wed Jul 10, 2024 6:39 pm What a huge change! The difference of "5 fill vs 0 fill" went from 51,322 at 0% discount to nothing! The rest of your values are pretty close to even as well.

It appears your initial calculations were so far off because they assumed the Bracket GAPS TIPS still had their initial duration. Because you ignored "temporal effects" you were assuming the 2034 and 2040 TIPS each had 5 more years' duration than they actually had, which accounts for their wild swings in values. That made the bracket year TIPS more valuable at lower discount rates and less valuable at higher discounts.
Here is a copy of Kevin's recent prior results with the Version 1 simple spreadsheet that had errors. These results have the correct durations for the assumed swap date of 2029. 2034 and 2040 multipliers are 3.5 and 3.5.
Kevin M wrote: Sat Jun 29, 2024 12:47 pm Below are the tables for the 25 year ladder with the three coverage methods: 1: Bracket (2034s/2040s), 2: 2034s, 3: 2040s. Again, the 25-year ladder assumes 5 years have passed since the ladder was built, so the maturities of all TIPS have decreased by 5 years, and the 2025-2029 have all matured with the proceeds removed from the ladder (e.g., for expenses).

Image

<snip>

Observations:
  • Now the bracket method, with original multipliers of 3.5 for each of the 2034 and 2040, generates the smallest range of outcomes for the totals. There's always net total cash left over, and the amounts don't change very much with different yield changes. This is the least risky if risk is defined as uncertainty of the outcomes with various yield changes.
  • The 2034 method, with original multiplier of 6 for the 2034, had the second largest range of outcomes, with the largest upside for a yield increase from 2% to 4%, but no upside or downside if yield decrease to 0%.
  • The 2040 method, with original multiplier of 6 for the 2040, has the largest range of outcomes, with significant upside if yields decrease to 0%, but still some small upside if yields increase to 4%.
For comparison, here is another copy of Kevin's latest results with the corrected Version 2 spreadsheet. The multipliers for the 2040 and 2034 are 3.43 and 3.57, based on duration matching at 2% yields. This is slightly different than the 3.5/3.5 values used above.
Kevin M wrote: Wed Jul 10, 2024 5:28 pm So we now have two tracks going to evaluate the results of different gap coverage methods. I'll continue a bit on the track I've been on, but now using #Cruncher's version 2 of the simplified ladder spreadsheet, which ensures that total ARA equals total DARA, regardless of whether or not any gap years are filled.

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

<snip>

From these deltas I create the table below, which is similar to the tables shown before, except that I've maintained the cash flow sign convention for all sales and purchases.

Image

Observations:
  • The range of Bracket/gap net is similar, but quite symmetrical, unlike before.
  • The range of the total net (after doing the not gap/bracket buys or sells) also is similar, but shifted from down from positive values to zero or negative values, and with all values much closer to zero.
  • As with version 1, the buys or sells of the 2030-2033 not only result in ARA = DARA for these maturities, but also dramatically compresses the range of net values at different yields.
  • I'm not sure what to use as the denominator, but if we look at total net as a percentage of cost to buy the gap years, for example, these values are 0%, 0.95% and 1.5% for yields of 0%, 2% and 4% respectively, which seem quite small.
It would be interesting to compare StillGoing's results to these, but I'm assuming transactions in 2029 while SG is assuming transactions in 2034, so the results are not directly comparable.

<snip>
The results do change with the corrections provided in #Cruncher's Version 2 simple spreadsheet. But the changes aren't huge.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Jaylat »

Kevin M wrote: Wed May 22, 2024 10:28 pm There have been many threads on how to fill the gap years in a TIPS ladder. If you don't know what I'm talking about, this thread is not for you, although if you want to understand it anyway, you might find it enlightening. Although we have had some discussions in other threads of the technique I'll discuss here, the questions that have come up in those threads indicate that this is a meaty enough topic to have its own thread (I hope this thread won't get merged with one of these existing, less specific threads, one or more of which I'll link to later in this thread for additional background and to answer questions asked in the other threads).

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

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

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

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

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

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

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

Code: Select all

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

where

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

Code: Select all

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

Code: Select all

Independent variable values:

d34 = 8.75
d40 = 13.23
dg = d35 = 9.41

So,

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

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

x = 0.85

and

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

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

Code: Select all

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

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

Image

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

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

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

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

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

Image

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

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

Looking at the 0% yield case:
  • The proceeds from selling all excess 2034 and 2040 bracket year TIPS are 515,669.
  • The cost of buying the 2035-2039 gap year TIPS is 432,951.
  • This leaves us with extra cash of 82,718.
  • We can choose to buy the pre-2034 TIPS that are left, 2030-2033, with the total being 31,396, if we want ARA to equal DARA for those rungs.
  • If we do the pre-2034 transactions, we are left with 51,322 in cash.
Here are the latest results still posted by Kevin on his initial post, which he refers to as follows: "I'm going to summarize what I've learned in doing the experiments documented in this thread. I'll add to this summary as I learn more."

Obviously the results on page one are way off base.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by StillGoing »

In this post, I’ve attempted to have a look at what happens if the yield curve is not flat at the time when the gap ladder is created). The same case has been modelled as in previous posts (e.g., see viewtopic.php?p=7948356#p7948356), i.e., the same multipliers (in #crunchers spreadsheet) have been used (i.e., 3.95 and 3.05 for 2034 and 2040, respectively), an income requirement of $40k per year, and prices from 1 July 2024.

There are two components to obtaining the annual income in the gap for a non-flat yield curve
1) Calculating a ladder payout rate for a non-flat yield curve
2) Using the ladder payout rate in conjunction with the amounts available from the 2034 and 2040 bonds to calculate the total income

Part 1 Calculate ladder payout rate

For a 5 year ladder, with fractional yields y1, y2, … y5, the ladder payout rate (in percent) is given by

100/[(1+y1)^1+(1+y2)^2+(1+y3)^3+(1+y4)^4+(1+y5)^5]

The following table shows the ladder payout rate, P (in percent) as a function of different yields for 2035 (Y35) and 2040 (Y40) assuming a payment at the end of each period. I’ve linearly interpolated the yields between 2035 and 2040. For example, if Y35=-1% and Y40=3%, the yields used in the five year ladder are then -1.0, -0.2, 0.6, 1.4, and 2.2% (note that the last yield is for 2039 not 2040). Practically, these would be the yields pertaining towards the beginning of 2034.

Code: Select all

	Y40								
Y35	-4	-3	-2	-1	0	1	2	3	4
-4	17.665	17.974	18.283	18.590	18.896	19.201	19.505	19.808	20.110
-3	17.924	18.237	18.548	18.859	19.169	19.478	19.785	20.092	20.397
-2	18.184	18.501	18.816	19.131	19.444	19.756	20.067	20.377	20.686
-1	18.447	18.767	19.086	19.404	19.721	20.037	20.351	20.665	20.977
0	18.711	19.035	19.358	19.679	20.000	20.319	20.638	20.954	21.270
1	18.978	19.305	19.631	19.957	20.281	20.604	20.926	21.246	21.565
2	19.246	19.577	19.907	20.236	20.564	20.891	21.216	21.540	21.862
3	19.516	19.851	20.185	20.518	20.849	21.179	21.508	21.835	22.161
4	19.789	20.127	20.465	20.801	21.136	21.470	21.802	22.133	22.463
As might be expected, the values for leading diagonal (i.e., where Y35=Y40) are the same as those given by the standard pmt function. I;ve quoted to 3 decimal places because that gives a precision of about $1 in the income.

Part 2 Calculate income

In order to calculate the total income, the contributions, C34 from the maturing bonds in 2034 (which has been calculated as $78006 in post viewtopic.php?p=7948356#p7948356), while the contributions, C40 from selling the bonds in 2040 varied from $116229 for a yield of 4% to $74050 for a yield of +4% (see table in post linked just above). The contributions of the post gap coupons, PGC ($5850) must also be included such that the total income is given by

Total income=(C35+C40)*(P/100)+PGC

The total income values then calculated are presented in the following table

Code: Select all

	Y40								
Y35	-4	-3	-2	-1	0	1	2	3	4
-4	40162	39588	39049	38539	38061	37612	36323	36796	36428
-3	40665	40081	39531	39012	38526	38070	37640	37240	36865
-2	41170	40577	40017	39490	38995	38530	38093	37685	37304
-1	41680	41076	40507	39970	39467	38995	38549	38135	37747
0	42193	41579	41001	40454	39943	39462	39010	38587	38192
1	42712	42086	41497	40943	40422	39933	39473	39043	38641
2	43232	42597	41998	41433	40904	40408	39939	39502	39092
3	43757	43111	42503	41929	41390	40884	40408	39963	39547
4	44287	43629	43012	42427	41879	41366	40880	40429	40006
There are a few points that can be made from the table:
1) The flat yield curve results from the previous post can be seen in the leading diagonal (i.e. where Y35=Y40).
2) non-inverted yield curves (i.e. where Y35<Y40) result in income below the target with larger differences in yields (i.e., greater positive gradients) leading to larger shortfalls in income.
3) Conversely, inverted yield curves (i.e., where Y35>Y40) lead to income greater than target.

However, while the worst case income of $36.4k is not great (representing a 9% shortfall in income), it is still better than the worst cases where multiple bonds were purchased on one side of the gap or the other, i.e., Plan 1 ($36.0k) or Plan 2 ($33.7k) (see posts viewtopic.php?p=7946678#p7946678 and viewtopic.php?p=7946700#p7946700) and would require what I suspect is a highly unusual yield curve (i.e., an 8 percentage point rise over the first 6 years of maturity). However, I still need to look at the historical data of the yield curve of the first few years of maturity to see what range of values have occurred (see my next post).

It must also be remembered that the method I’ve adopted here is not duration matching – a fixed ratio between the bonds on either side of the gap is set when the ladder is constructed and then maintained until 2034 when the gap is filled all in one go with the proceeds from the 2034 and 2040 bonds. There are a number of methods for filling the gap, some of which are being explored in this thread, that will lead to different outcomes that may be better (or worse) than the ones presented here.

cheers
StillGoing
Last edited by StillGoing on Thu Jul 11, 2024 12:13 pm, edited 1 time in total.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

Jaylat wrote: Wed Jul 10, 2024 9:24 pm Here are the latest results still posted by Kevin on his initial post, which he refers to as follows: "I'm going to summarize what I've learned in doing the experiments documented in this thread. I'll add to this summary as I learn more."

Obviously the results on page one are way off base.
I've been negligent about updating the OP with the latest results, so you need to read through the more recent thread posts to see the latest.

As MtnBiker pointed out, I had already done a 25-year ladder, assuming five years had elapsed since building the ladder, with V1 of #Cruncher's simplified ladder spreadsheet, and that's what I was comparing to in my most recent update using V2 of the spreadsheet.
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 StillGoing »

The yields in this post are drawn from the data set at https://www.federalreserve.gov/data/yie ... 805_1.html which has par and zero coupon (zc) yields for TIPS maturities from 2 years to 20 years starting from January 1999. For maturities of 2 and 6 years (i.e. roughly those used to construct the 5 year ladder in my previous posts) the difference between par and zc yields are minimal, so I only show par yields below.

The upper panel of the following figure shows the real yields for 2 year and 6 year TIPS as a function of time, while the lower panel shows the difference between the 6 year yield and 2 year yield.

Image

A few things to note:
1) The 2 year yield ranges from about -3% to nearly 6% (i.e., the top end is a bit more than that of 4% used in my previous posts, but only occurred during 2008).
2) The 6 year yield ranges from about -1.5% to just over 4% (i.e., not surprisingly, fairly similar to the 5 year yield published at https://home.treasury.gov/policy-issues ... statistics.
3) The difference between the two yields ranges from extremes of about -2.5 to 3.5 percentage points, but is typically confined to between -1 and 2 percentage points.

Therefore, the range of yields -4% to +4% used for maturities of both one year (2035) and 6 year (2040) in the tables in my previous post (viewtopic.php?p=7949964#p7949964) appear to be reasonably consistent with those found since 1999. However, it is worth pointing out that the 25 years since 1999 represents a relatively small slice of history and had TIPS existed earlier their yields and gradients may have exceeded those seen in the last 25 years.

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

Post by Jaylat »

Kevin M wrote: Thu Jun 27, 2024 8:58 am
Jaylat wrote: Thu Jun 27, 2024 6:28 amWhatever configuration the newly issued TIPS have at t1 (high coupon, low coupon, etc.) will also be reflected in the pricing and discount of the 2034 / 2040 TIPS you bought. So at both t1 and t2 the duration matching works just fine. If the new 2035 TIPS is issued at 4% coupon / 4% yield the 2034 / 2040 TIPS you hold will already be repriced to reflect that.
The math does not bear this out. It's easy to make statements like this--it's harder to prove it with math, which is what I started out trying to do. It turns out not to be the case.
Jaylat wrote: Thu Jun 27, 2024 6:28 amAnd I also confirmed with my trader friend there's no "special case" at auction. The TIPS bids are not made in 1/8% increments, they can go infinitesimally small.
There is no constraint on yield increments, but there is on the coupons, which are set at 0.125% increments with a minimum of 0.125%. But this is not a particularly important point.

As I've said multiple times, what's different about auctions compared to secondary is that price is relatively fixed at about 100 unless yields are negative, and for duration matching to work as intended, price must vary with yield.

I'd be glad to have you use math to show that any of the results I've generated, where I didn't later find a mistake and correct it, are wrong. The point of the thread is not to debate with words only, but to use math to show how well a particular gap coverage method works, and I think we've pretty much gotten to that point with the method we've referred to as duration matching.
I wanted to go back to this exchange, which I'd like to reexamine in the light of the significant changes in your spreadsheet results. As you refine "the math" you are discovering that the big differences you thought existed between the sales of 2034 / 2040 TIPS and the purchase of gap year TIPS under various interest rate scenarios are actually due to simple math errors.

There's nothing wrong with errors - I make plenty myself - but the problem arises when you rely on those errors to come up with incorrect conclusions. That's what you've done here in the statements highlighted in bold.

As I've said all along, duration matching works just fine with new TIPS at auction. There is absolutely nothing "special" about buying TIPS at auction as the yield can vary perfectly with market rates. The fact that the price is fixed at 100 is completely immaterial.

I'll repeat this as well: "You are concerned about a situation where a high yielding TIPS at auction doesn’t provide enough principal amount to allow the maturity year ARA to approximate DARA. If that’s the case, then the prior year coupons reinvested should be sufficient to make up the difference. You can sell off portions of the prior year TIPS accordingly, which should get you to the same result of ARA close to DARA."

Your latest results do not yet show a perfect matching of cash flows, but they are getting pretty darn close. The simplifying assumptions probably account for the small differences remaining.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by MtnBiker »

StillGoing wrote: Thu Jul 11, 2024 12:12 pm The yields in this post are drawn from the data set at https://www.federalreserve.gov/data/yie ... 805_1.html which has par and zero coupon (zc) yields for TIPS maturities from 2 years to 20 years starting from January 1999. For maturities of 2 and 6 years (i.e. roughly those used to construct the 5 year ladder in my previous posts) the difference between par and zc yields are minimal, so I only show par yields below.

The upper panel of the following figure shows the real yields for 2 year and 6 year TIPS as a function of time, while the lower panel shows the difference between the 6 year yield and 2 year yield.

Image

A few things to note:
1) The 2 year yield ranges from about -3% to nearly 6% (i.e., the top end is a bit more than that of 4% used in my previous posts, but only occurred during 2008).
2) The 6 year yield ranges from about -1.5% to just over 4% (i.e., not surprisingly, fairly similar to the 5 year yield published at https://home.treasury.gov/policy-issues ... statistics.
3) The difference between the two yields ranges from extremes of about -2.5 to 3.5 percentage points, but is typically confined to between -1 and 2 percentage points.

Therefore, the range of yields -4% to +4% used for maturities of both one year (2035) and 6 year (2040) in the tables in my previous post (viewtopic.php?p=7949964#p7949964) appear to be reasonably consistent with those found since 1999. However, it is worth pointing out that the 25 years since 1999 represents a relatively small slice of history and had TIPS existed earlier their yields and gradients may have exceeded those seen in the last 25 years.

Cheers
StillGoing
This analysis indicates it is reasonable to assume non-parallel yield shifts are generally confined within the range -1% for Y40 - Y35 (inverted yield curve) to 2% (increasing yield curve). We can compare the expected change in total income under that range of conditions.

The matrix presented in a previous post by StillGoing showed an expected income of about $40,000 for a flat yield curve, nearly independent of the yield level (see the leading diagonal (i.e. where Y35=Y40)).

For the max expectation of -1% for an inverted curve, we look at the next lower diagonal in the matrix and see that the expected income is about $40,500 for all yield levels, an increase of about 1.25%.

For the max expectation of 2% for a curve with positive slope, we look at the second higher diagonal in the matrix and see that the expected income is about $39,000 for all yield levels, a decrease of about 2.5%.

It doesn’t seem like non-parallel yield shifts are a major threat to funding the gap years. The range of results is small enough for most purposes. This result is obtained for swaps done in 2034, which is the worse case for non-parallel yield shifts due to the volatility of shorter term yields.

This result is also based on multipliers of 3.95 and 3.05 for 2034 and 2040, respectively, which optimize results for parallel yield shifts for swapping in 2034. What are the multipliers which optimize results for parallel yield shifts if the swaps are made earlier, such as in 2029? Kevin?
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Why duration matching fails for TIPS bought at auctions

Post by Kevin M »

Jaylat wrote: Thu Jul 11, 2024 2:11 pm
Kevin M wrote: Thu Jun 27, 2024 8:58 am
Jaylat wrote: Thu Jun 27, 2024 6:28 amWhatever configuration the newly issued TIPS have at t1 (high coupon, low coupon, etc.) will also be reflected in the pricing and discount of the 2034 / 2040 TIPS you bought. So at both t1 and t2 the duration matching works just fine. If the new 2035 TIPS is issued at 4% coupon / 4% yield the 2034 / 2040 TIPS you hold will already be repriced to reflect that.
The math does not bear this out. It's easy to make statements like this--it's harder to prove it with math, which is what I started out trying to do. It turns out not to be the case.
Jaylat wrote: Thu Jun 27, 2024 6:28 amAnd I also confirmed with my trader friend there's no "special case" at auction. The TIPS bids are not made in 1/8% increments, they can go infinitesimally small.
There is no constraint on yield increments, but there is on the coupons, which are set at 0.125% increments with a minimum of 0.125%. But this is not a particularly important point.

As I've said multiple times, what's different about auctions compared to secondary is that price is relatively fixed at about 100 unless yields are negative, and for duration matching to work as intended, price must vary with yield.

I'd be glad to have you use math to show that any of the results I've generated, where I didn't later find a mistake and correct it, are wrong. The point of the thread is not to debate with words only, but to use math to show how well a particular gap coverage method works, and I think we've pretty much gotten to that point with the method we've referred to as duration matching.
I wanted to go back to this exchange, which I'd like to reexamine in the light of the significant changes in your spreadsheet results. As you refine "the math" you are discovering that the big differences you thought existed between the sales of 2034 / 2040 TIPS and the purchase of gap year TIPS under various interest rate scenarios are actually due to simple math errors.

There's nothing wrong with errors - I make plenty myself - but the problem arises when you rely on those errors to come up with incorrect conclusions. That's what you've done here in the statements highlighted in bold.

As I've said all along, duration matching works just fine with new TIPS at auction. There is absolutely nothing "special" about buying TIPS at auction as the yield can vary perfectly with market rates. The fact that the price is fixed at 100 is completely immaterial.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
There's quite a bit of confusion here. The math I was referring to is not the math used in the models I've been working with the last few weeks; it's the math used in the duration match modeling discussed very early in the thread, and that has not changed. However, the results of the failure of duration matching, in the strictest sense, do show up in the more recent models; the details changed, but the failure of duration matching shows up differently between V1 and V2 of the #Cruncher simplified ladder spreadsheets.

Rather than refer to posts earlier in the thread where my understanding of this was clarified, let me summarize it all here. First, what do I mean by duration matching working on nor working?
  • We hold a portfolio of bonds with a weighted average duration that matches the duration of a liability.
  • At some later time we sell the portfolio of bonds to fund the liability.
  • Duration matching "works" to the extent the proceeds we get from selling the duration matched bond portfolio equals the cost of the liability.
How does this apply to the duration matching method of funding the gap years?
  • The duration matching portfolio of bonds consists of our excess holdings of the 2034 and 2040 (bracket years), and the liabilities are the 2035-2039s (gap years). For example, we might hold an excess of 2.5*DARA in each of the 2034 and 2040 to fund the purchases of the 2035-2039s (a total of 5*DARA) at some point in the future.
  • We could also say that the excess holdings of the bracket years intended to buy a particular gap year are the duration matched portfolio, and the specific gap year is the liability. For example, we might hold 0.85*DARA of the 2034 and 0.15*DARA of the 2040 to fund the purchase of 1*DARA of the 2035 at some point in the future
Of course the gap year TIPS do not exist, so we must make some assumptions about coupons and yields to calculate estimated durations for them. What I did in the earlier posts was to use the current yield to maturity for the 2034 and 2040 to estimate yields for the 2035-2039 using liner interpolation, and then set the coupons to what they would be at those yields; I always use the latest ask yields I've pulled from Schwab, so the actual values I'm using now are a bit different than those used earlier in the thread.

We'll also need prices to evaluate the effectiveness of duration matching, so I use a similar approach to get estimated prices for the gap year TIPS.

With the durations calculated as described above, I calculate the weights of the 2034 and 2040 required to equal the estimated duration of each gap year. Note that I used modified durations, since those are applicable to price/yield analyses, but the results would not be much different using Macaulay durations.

Here's the resulting table:

Image

All we need to set up the ladder are the sum of the 2034 and 2040 weights, since these would be the multipliers we'd use in one of #Cruncher's ladder spreadsheets. The weights for each gap year maturity indicate the expected DARA amounts of each of the bracket years to fund the purchase of that gap year; e.g., we'd expect to sell 0.82*DARA of the 2034 and 0.18*DARA of the 2040 to fund the purchase of the 2035.

When I first started the thread, I hadn't thought about the fact that coupon is not fixed for TIPS sold at auction, so I naively did the duration matching analysis assuming fixed coupon and variable price for TIPS sold at auction. Using that faulty assumption, here is the table used to evaluate the effectiveness of duration matching for a change in yield, dy, of 1 percentage point across the curve (parallel yield shift):

Image

Note the following:
  1. The gap year coupons are unchanged, so the values do not correspond to what they would be at auction for the indicated yields, y.
  2. Price, p, is calculated based on yield and coupon, so the gap year prices are closer to 90 than to 100, the latter being closer to what they would be at auction.
  3. Because of #2, the percentage change in gap year prices, dp%, are relatively large negative values, while at auction they would be close to 0%.
  4. The change in duration match portfolio weighted price, DM dp%, of the excess bracket year TIPS held for each gap year maturity is almost equal to the change in price for that gap year.
  5. Because of #4, the difference between DM dp% and dp% is very close to 0 percentage points. The closer this value is to 0 pp, the more effective the duration matching.
After MtnBiker asked the astute question about whether or not I was resetting the coupon to what it would be at auction, I realized the flaw in this model, so I updated it to reset the coupon accordingly. Here is the resulting table:

Image

Observations:
  1. The coupon now is set to the closest 0.125% less than or equal to the yield, as it would be at auction.
  2. Because of #1, the price, p, is pretty close to 100, as it would be at auction (for a new issue).
  3. Because of #2, the price change, dp%, for each gap year, relative to our estimate at the original assumed yield, is pretty close to 0%.
  4. Of course the change in price of the duration match weighted price is the same as before.
  5. Because of #3 and #4, DM dp% - dp% is not at all close to 0%, which is what I deem a failure of duration matching (in the strictest sense).
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Jaylat wrote:I'll repeat this as well: "You are concerned about a situation where a high yielding TIPS at auction doesn’t provide enough principal amount to allow the maturity year ARA to approximate DARA. If that’s the case, then the prior year coupons reinvested should be sufficient to make up the difference. You can sell off portions of the prior year TIPS accordingly, which should get you to the same result of ARA close to DARA."

Your latest results do not yet show a perfect matching of cash flows, but they are getting pretty darn close. The simplifying assumptions probably account for the small differences remaining.
I'm not sure I'd state it the same way, but it is correct that with the latest model, buying or selling some of the 2033-2035s greatly reduces the deltas between the gap year costs and the bracket year proceeds. Again, here is the table that shows all of this for various yield change scenarios using the latest model:

Image

The Bracket/gap net row shows what I'm referring to as the failure of duration matching in the strictest sense; i.e., the proceeds from the duration matching portfolio (excess 2034/2040s) do not match the amounts required to fund the liabilities (the cost of the gap year TIPS). This is consistent with the results shown earlier if we allow the coupon to float with yield, as it does in reality.

The Net row shows that buying or selling some of the 2030-2033s reduces the deltas to what we probably could consider a negligible amount.

To the extent that these results represent reality, I think it's important to understand that what we call the duration matching method of covering the gap years might require more than just selling the bracket years to buy the gap years if one wants the ARAs for each year of their ladder to match DARA; it could require some additional "rebalancing" involving the maturities prior to 2034.
If I make a calculation error, #Cruncher probably will let me know.
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Re: Why duration matching fails for TIPS bought at auctions

Post by MtnBiker »

Kevin M wrote: Fri Jul 12, 2024 2:14 pm

The Bracket/gap net row shows what I'm referring to as the failure of duration matching in the strictest sense; i.e., the proceeds from the duration matching portfolio (excess 2034/2040s) do not match the amounts required to fund the liabilities (the cost of the gap year TIPS). This is consistent with the results shown earlier if we allow the coupon to float with yield, as it does in reality.

The Net row shows that buying or selling some of the 2030-2033s reduces the deltas to what we probably could consider a negligible amount.

To the extent that these results represent reality, I think it's important to understand that what we call the duration matching method of covering the gap years might require more than just selling the bracket years to buy the gap years if one wants the ARAs for each year of their ladder to match DARA; it could require some additional "rebalancing" involving the maturities prior to 2034.
In a practical sense, the proceeds from the duration matching portfolio (excess 2034/2040s) does closely match the cost of funding the gap years if the proceeds/cost comparison is defined to include the deltas in the coupons in the pre-gap years. Effectively, this is similar to the “Net” line in the table. (Those of us who have been dealing with gaps previously (2030, 2031, 2033, 2034) are aware of this.)

If the pre-gap years get extra coupons, they can help fund the shortfall in the gap. If the gap receives extra principal, it makes up for the shortage in the pre-gap coupons. This is what I called a change in cash flow distribution at the beginning of this thread. Rebalancing the pre-gap years to smooth ARA is a nice solution to the cash flow redistribution issue.
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Re: Why duration matching fails for TIPS bought at auctions

Post by Jaylat »

Kevin M wrote: Fri Jul 12, 2024 2:14 pm The Bracket/gap net row shows what I'm referring to as the failure of duration matching in the strictest sense; i.e., the proceeds from the duration matching portfolio (excess 2034/2040s) do not match the amounts required to fund the liabilities (the cost of the gap year TIPS). This is consistent with the results shown earlier if we allow the coupon to float with yield, as it does in reality.

The Net row shows that buying or selling some of the 2030-2033s reduces the deltas to what we probably could consider a negligible amount.

To the extent that these results represent reality, I think it's important to understand that what we call the duration matching method of covering the gap years might require more than just selling the bracket years to buy the gap years if one wants the ARAs for each year of their ladder to match DARA; it could require some additional "rebalancing" involving the maturities prior to 2034.
Talk about moving the goalposts! So now you are pretending you were just talking about duration matching in "the strictest sense"? And redefining that to mean selling the 2034 / 2040 TIPS only? :oops:

To make an obvious point, under your newly contrived "strict" definition, any year TIPS (existing or not) which doesn't have a coupon that exactly matches the 2034 / 2040 TIPS sold would also create a failure of duration matching "in the strictest sense" because it would require selling other pre-gap year TIPS.

Do you really want to stick with that definition? I agree "there's quite a bit of confusion here"!

At this point Kevin I literally do not care what your opinion is. You've been a great help to many in navigating the treasury markets (including myself for which I will always be grateful). However you have a real blind spot when it comes to an intuitive understanding of how markets work.

However you have spent a lot of time on this thread lecturing everyone on how TIPS at auction are "different." That is categorically false.

You owe it to readers of this thread to dispel that notion.

On a more positive note, let's compare these two quotes:
"You are concerned about a situation where a high yielding TIPS at auction doesn’t provide enough principal amount to allow the maturity year ARA to approximate DARA. If that’s the case, then the prior year coupons reinvested should be sufficient to make up the difference. You can sell off portions of the prior year TIPS accordingly, which should get you to the same result of ARA close to DARA."
“To the extent that these results represent reality, I think it's important to understand that what we call the duration matching method of covering the gap years might require more than just selling the bracket years to buy the gap years if one wants the ARAs for each year of their ladder to match DARA; it could require some additional "rebalancing" involving the maturities prior to 2034.”
Glad you finally agree with me on this. :sharebeer
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Re: Why duration matching fails for TIPS bought at auctions

Post by MtnBiker »

MtnBiker wrote: Fri Jul 12, 2024 4:52 pm
Kevin M wrote: Fri Jul 12, 2024 2:14 pm

The Bracket/gap net row shows what I'm referring to as the failure of duration matching in the strictest sense; i.e., the proceeds from the duration matching portfolio (excess 2034/2040s) do not match the amounts required to fund the liabilities (the cost of the gap year TIPS). This is consistent with the results shown earlier if we allow the coupon to float with yield, as it does in reality.

The Net row shows that buying or selling some of the 2030-2033s reduces the deltas to what we probably could consider a negligible amount.

To the extent that these results represent reality, I think it's important to understand that what we call the duration matching method of covering the gap years might require more than just selling the bracket years to buy the gap years if one wants the ARAs for each year of their ladder to match DARA; it could require some additional "rebalancing" involving the maturities prior to 2034.
In a practical sense, the proceeds from the duration matching portfolio (excess 2034/2040s) does closely match the cost of funding the gap years if the proceeds/cost comparison is defined to include the deltas in the coupons in the pre-gap years. Effectively, this is similar to the “Net” line in the table. (Those of us who have been dealing with gaps previously (2030, 2031, 2033, 2034) are aware of this.)

If the pre-gap years get extra coupons, they can help fund the shortfall in the gap. If the gap receives extra principal, it makes up for the shortage in the pre-gap coupons. This is what I called a change in cash flow distribution at the beginning of this thread. Rebalancing the pre-gap years to smooth ARA is a nice solution to the cash flow redistribution issue.
I should add that, in my opinion, there are two slightly different methods for duration-matched filling of the gap years: (1) selling the bracket years and using the proceeds to buy the gap years (accepting the resulting redistribution of cash flows) vs. (2) what Kevin has suggested: selling the bracket years, using the proceeds to buy the gap years, and doing additional transactions to smooth ARA/DARA (possibly adding new money or taking money out of the ladder, as applicable). I would expect that when these two methods are compared mathematically, the results would be similar but certainly not equivalent.

In method (2), the payouts every year are the same as what was planned when the ladder was originally built. Thus, the average duration of the ladder changes very little, if at all, as the result of yield changes.

In method (1), the annual payouts do change. If yields increase before the swaps, the pre-gap coupon payments increase and the gap principal payments decrease. This decreases the average duration of the ladder. If yields decrease before the swaps, the pre-gap coupon payments decrease and the gap principal payments increase. This increases the average duration of the ladder.

Since method (2) preserves the average duration of the ladder, duration is better matched when the swaps are made. Thus, one would expect the range of results (in terms of any changes in the proceeds of the ladder with changes in yield) would be smaller with method (2) than with method (1).

Methods (1) and (2) converge to become identical if the swaps are delayed until 2034 (when no pre-gap years remain). That case was analyzed by StillGoing.
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Re: Filling the TIPS gap years with bracket year duration matching

Post by Kevin M »

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

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

<snip>

Image

<snip
To continue on this track, in this post I'll review the results for the gap coverage method using all 2034s; i.e., when building the ladder we use a multiplier of 6 for the 2034s and 1 for all other maturities, including the 2040, except of course for the gap years for which the multipliers are all 0.

Here are the results:

Image

From this, it appears that using all 2034s is superior from bracket/gap net cash perspective than using a duration matched combination of 2034s and 2040s, since every yield scenario generates excess cash in doing just the bracket/gap transactions. Similarly, it appears superior from a total net cash perspective, since we either end up with 0 for the 0% yield scenario, same as with duration matching, or a fairly large positive net cash position for the 2% and 4% scenarios, compared to fairly small negative values for duration matching.

Rather than try to explain this, I'll just let it marinate and see if anyone notes any obvious explanations or glaring errors.

Remember that all I'm doing is plugging numbers into #Cruncher's simplified ladder spreadsheet, version 2, and sharing the results. I'm not recommending anything, or saying how well the model reflects reality.
If I make a calculation error, #Cruncher probably will let me know.
MtnBiker
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Re: Filling the TIPS gap years with bracket year duration matching

Post by MtnBiker »

Kevin M wrote: Mon Jul 15, 2024 7:25 pm
Rather than try to explain this, I'll just let it marinate and see if anyone notes any obvious explanations or glaring errors.

Remember that all I'm doing is plugging numbers into #Cruncher's simplified ladder spreadsheet, version 2, and sharing the results. I'm not recommending anything, or saying how well the model reflects reality.
One would expect that holding all 2034s would be superior if yields climb to 4%. One would also expect that holding all 2040s would be superior if yields fell to -2%. Version 2 seems to predict that sort of behavior.

The prediction from Version 2 that I wasn't expecting is that if the yield shifts to 0%, the net is exactly zero, independent of the multipliers of the bracket-year holdings. I guess that must be a feature that is a direct consequence of maintaining ARA = DARA under all conditions.

Perhaps it might be easier to understand that result if you could display the holdings of the 2025 - 2040 ladder years for that 0% yield/0.125% gap-year coupon case (as you did for the 2.00% gap-year coupon case previously). Like the two charts shown in this earlier post:
Kevin M wrote: Wed Jul 10, 2024 5:28 pm So we now have two tracks going to evaluate the results of different gap coverage methods. I'll continue a bit on the track I've been on, but now using #Cruncher's version 2 of the simplified ladder spreadsheet, which ensures that total ARA equals total DARA, regardless of whether or not any gap years are filled.

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

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

Image

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

Image

Things to note:
  • The only relevant cost column for this scenario, one of the three covered here, is Cost at 2%, since I'm assuming all yields are 2% from 2030-2040; I've highlighted the relevant cost cells, which include the gap years since we're buying them, the bracket years since we're selling them, and the 2030-2033 since we will buy or sell them so that ARA = DARA.
  • Total proceeds still equals 2.5M, as it should.
<snip>
I would like to add that a closer examination of these tables reveals that what is labeled as the proceeds from selling the bracket years is not simply the proceeds from selling the excess bracket year holdings. Generally, it is the proceeds from selling more or less than the amount of the original excess.

Maybe this was obvious to everyone else, but I didn't realize until now that the amount of bracket year holdings sold in the first row of the summary table is the amount needed to make ARA = DARA in the bracket year (after all the other buys and sells shown in the other rows of the summary table). Basically, the summary table shows the results (changes) that would be obtained if one liquidated the entire original ladder with a gap and used those proceeds to buy a new ladder without the gap.

The rows in the summary table should not be thought of as sequential steps in the process. The results shown in each of the rows depends upon the other processes shown in all the other rows having been completed first.
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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 »

MtnBiker wrote: Mon Jul 15, 2024 11:35 pm Perhaps it might be easier to understand that result if you could display the holdings of the 2025 - 2040 ladder years for that 0% yield/0.125% gap-year coupon case (as you did for the 2.00% gap-year coupon case previously). Like the two charts shown in this earlier post:
Right. I started going down that path in working on the explanation, and got kind of bogged down. Even if you're the only one really following this, I'll work some more on sharing the details, since your insights have been invaluable.
MtnBiker wrote: Mon Jul 15, 2024 11:35 pmI would like to add that a closer examination of these tables reveals that what you have been calling the proceeds from selling the bracket years is not simply the proceeds from selling the excess bracket year holdings. Generally, it is the proceeds from selling more or less than the amount of the original excess.
Right, by "excess" I simply mean the excess in terms of the multiplier(s). If we buy use a multiplier of 6 to cover the 2034 and the five gap years, 2034 multiplier is reduced by 5 to 1 when the gaps are filled, each with a multiplier of 1.

And I think you're onto something here which is at the core of the differences we see between the duration-match and all-2034 coverage methods, which is that both the proceeds and principal required for the 2034 and 2040 are about 25K more for the all-2034 case than for the DM case.
MtnBiker wrote: Mon Jul 15, 2024 11:35 pmMaybe this was obvious to everyone else, but I didn't realize until now that the amount of bracket year holdings sold in the first row of the summary table is the amount needed to make ARA = DARA in the bracket year (after all the other buys and sells shown in the other rows of the summary table). Basically, the summary table shows the results (changes) that would be obtained if one liquidated the entire original ladder with a gap and used those proceeds to buy a new ladder without the gap.

The rows in the summary table should not be thought of as sequential steps in the process. The results shown in each of the rows depends upon the other processes shown in all the other rows having been completed first.
I'm not sure exactly what you're saying here, but hopefully when I walk through it in more detail we can determine if what you're saying matches what the spreadsheet does.
If I make a calculation error, #Cruncher probably will let me know.
MtnBiker
Posts: 614
Joined: Sun Nov 16, 2014 3:43 pm

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

Post by MtnBiker »

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

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

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

Image

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

Image

Things to note:
  • The only relevant cost column for this scenario, one of the three covered here, is Cost at 2%, since I'm assuming all yields are 2% from 2030-2040; I've highlighted the relevant cost cells, which include the gap years since we're buying them, the bracket years since we're selling them, and the 2030-2033 since we will buy or sell them so that ARA = DARA.
  • Total proceeds still equals 2.5M, as it should.
To determine the outcome for each cost at YTM scenario, I copy/paste-values the zero gaps filled version into rows below, change the gaps filled to 5 for each yield scenario, and subtract the values in the former from the later. From these deltas I create the table below, which is similar to the tables shown before, except that I've maintained the cash flow sign convention for all sales and purchases.

Image

Observations:
  • The range of Bracket/gap net is similar, but quite symmetrical, unlike before.
  • The range of the total net (after doing the not gap/bracket buys or sells) also is similar, but shifted from down from positive values to zero or negative values, and with all values much closer to zero.
  • As with version 1, the buys or sells of the 2030-2033 not only result in ARA = DARA for these maturities, but also dramatically compresses the range of net values at different yields.
  • I'm not sure what to use as the denominator, but if we look at total net as a percentage of cost to buy the gap years, for example, these values are 0%, 0.95% and 1.5% for yields of 0%, 2% and 4% respectively, which seem quite small.
It would be interesting to compare StillGoing's results to these, but I'm assuming transactions in 2029 while SG is assuming transactions in 2034, so the results are not directly comparable.

For JayLat, or anyone else who is interested in duplicating these results, here are the key formulas for row 16, which are copied to all rows except row 2, for which I'll note the differences below:

Code: Select all

G16: =SUM(I$2:I15)
H16: =MAX(0,(B$1*SUM(E$2:E16)-SUM(J$2:J15)-G16)/(1+F16))
I16: =H16*F16
J16: =SUM(G16:I16)
K16: =-PV(RIGHT(K$1,3),$C16,$I16,$H16,0) (copied to L16 and M16).
The formulas that are different for row 2, the 2054 maturity, are:

Code: Select all

G2: 0
H2: =E2*B1/(1+F2)
These are all formulas from the #Cruncher simplified ladder spreadsheet, version 2, except that I modified the formula for column K slightly so that I could put "Cost at N%" in one row, and extract the yield from the text.
We could look at the 2040 and/or 2034 rows in the tables shown in the post copied above.

For 2040, the cost for M=1.00 in the original ladder appears to be 95,580. The cost of the excess 2040 holdings (M=2.43) appears to be 232,258. (Total 327,838, M = 3.43.) This assumes the cost of the excess can be identified as a proportional amount of the total cost.

The cost for 2040 M=1 in the post-swap ladder is 86,571. The amount of the 2040 sold (241,267) exceeds what I thought was the original excess holdings.

That is what I was pointing out as the amount of bracket sales seeming to be more or less than the original excess. This is, of course, necessary to make ARA = DARA on an annual basis.

EDIT: I guess I'm probably not evaluating this correctly because the interest from later bonds helps fund the first M=1 but doesn't help fund any excess purchases. Thus, the incremental cost to add an extra M=1 is higher than the cost to fund the first M=1. Identifying the excess proportionally wasn't correct.

In any case, the cost of M=1 of the 2034s will change after the swap because of changes in the coupons from gap-year maturities (when ARA is forced to equal DARA in 2034).
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