Filling the TIPS gap years with bracket year duration matching
Re: Filling the TIPS gap years with bracket year duration matching
I'm going through withdrawal as haven't seen any new posts on this very interesting thread in a few days lol. Longerterm TIPS yields in a range and hoping for a breakout to the upside so can accumulate additional 2040s!
Re: Filling the TIPS gap years with bracket year duration matching
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?Kevin M wrote: ↑Sat Jun 22, 2024 4:24 pm As a radical null hypothesis test, consider covering the gap years with all 2054s.Below are the tables that show this. I'll use yield changes of 2% to 4% or 0% to make them symmetrical.
 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 pre2034.
 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.
Duration matching with a decrease in yield from 2% to 0% with all gaps filled:
Gap coverage with 2054s with a decrease in yield from 2% to 0% with all gaps filled:
Duration matching with an increase in yield from 2% to 4% with all gaps filled:
Gap coverage with 2054s with an increase in yield from 2% to 4% with all gaps filled:
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.
The risk in buying 2054 TIPS to hedge future 203539 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 203539 TIPS will cost about the same.
Re: Filling the TIPS gap years with bracket year duration matching
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.MtnBiker wrote: ↑Sun Jun 30, 2024 9:36 am What should one consider to be a "worst case" nonparallel 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 25year ladder with the three coverage methods: 1: Bracket (2034s/2040s), 2: 2034s, 3: 2040s?
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.
Re: Filling the TIPS gap years with bracket year duration matching
That's too bad that a nonparallel yield shift can't be modeled. But it is easy enough to predict the conclusions using another simple thought experiment.Kevin M wrote: ↑Tue Jul 02, 2024 12:14 pmUnfortunately 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.MtnBiker wrote: ↑Sun Jun 30, 2024 9:36 am What should one consider to be a "worst case" nonparallel 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 25year ladder with the three coverage methods: 1: Bracket (2034s/2040s), 2: 2034s, 3: 2040s?
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.
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 nonparallel 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 nonparallel 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 zerocoupon approximation, the duration (years to maturity) of the 2034 is 5 and the duration of the 2040 is 11. (Use of the zerocoupon approximation is slightly inaccurate but shouldn't change the general conclusions.)
If all the excess holdings are 2034s, the bracketyear 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 bracketyear 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 bracketyear holdings decrease 5.5%. If the yield curve had instead inverted to the same degree, the bracketyear 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 bracketyear 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 nonparallel yield shifts.
In conclusion, minimizing the range of results from nonparallel 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 backoftheenvelope analysis is correct.
Re: Filling the TIPS gap years with bracket year duration matching
The tables you showed were based on an older model that has been superseded, so ignore those. No one here is promoting using very longterm TIPS for gap coverage, but let's explore some of your comments anyway.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 203539 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 203539 TIPS will cost about the same.
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 20352038, 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 shorterterm yields. Currently the yield curve is positively sloped between 2034 and 2054, although it's not particularly steep.
It wasn't too hard to add using the 2054 for gap coverage to my latest model; here are the results:
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 nonparallel 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 nongap, non2054 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:
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.
Re: Filling the TIPS gap years with bracket year duration matching
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.MtnBiker wrote: ↑Tue Jul 02, 2024 3:15 pm
That's too bad that a nonparallel 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 nonparallel 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 nonparallel 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 zerocoupon approximation, the duration (years to maturity) of the 2034 is 5 and the duration of the 2040 is 11. (Use of the zerocoupon approximation is slightly inaccurate but shouldn't change the general conclusions.)
If all the excess holdings are 2034s, the bracketyear 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 bracketyear 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 bracketyear holdings decrease 5.5%. If the yield curve had instead inverted to the same degree, the bracketyear 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 bracketyear 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 nonparallel yield shifts.
In conclusion, minimizing the range of results from nonparallel 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 backoftheenvelope analysis is correct.
Holding a 50/50 mix of 2034/2040, incurs a certain risk of loss due to nonparallel yield shifts. The risk isn't huge, but it can be quantified. With this holding ratio, the range of outcomes due to nonparallel 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 nonparallel 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 gapyears are auctioned, we do know the approximate swap dates. The holding ratios to immunize against nonparallel 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 (zerocoupon approximation) here are the calculated holding ratios (2034/2040) that immunize against nonparallel 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 zerocoupon 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 bracketyear holdings slightly heavier (715%) toward 2034s is all this needed to minimize the effects of nonparallel yield shifts. Has anyone tried to verify my calculations? Does this pass the smell test?
Re: Filling the TIPS gap years with bracket year duration matching
Shouldn't a parallel yield curve shift be within the range of nonparallel 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.
Re: Filling the TIPS gap years with bracket year duration matching
A parallel yield shift, as I am analyzing it is:Kevin M wrote: ↑Wed Jul 03, 2024 11:00 amShouldn't a parallel yield curve shift be within the range of nonparallel 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?
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 nonparallel 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 nonparallel 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 nonparallel 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?
Re: Filling the TIPS gap years with bracket year duration matching
Yeah, I figured out what you meant by nonparallel yield shift while driving to an appointmentI visualized it as a teeter totter. Another way to describe it is that the 2034 and 2040 yields move in opposite directions.MtnBiker wrote: ↑Wed Jul 03, 2024 11:49 amA parallel yield shift, as I am analyzing it is:Kevin M wrote: ↑Wed Jul 03, 2024 11:00 amShouldn't a parallel yield curve shift be within the range of nonparallel 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?
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 nonparallel 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.
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:
Here's a monthly version that makes the trends a bit easier to see:
Notes and observations:
 When the green line is 0, the yield curve is flat, and the 2032 and 2040 yields are the same.
 When the green line is above 0, we have the more typical positively sloped yield curve, with the 2040 yield higher than the 2032.
 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.
 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.
 When the slope of the green line is positive, the yield shift is nonparallel, with the gap between the yields increasing.
 When the slope of the green line is negative, the yield shift is nonparallel, with the gap between the yields decreasing.
 Perhaps most importantly, the yields generally change in the same direction!
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.
Re: Filling the TIPS gap years with bracket year duration matching
The picture I have in my mind is a teeter totter mounted on a party barge that is rising and falling with the tides.Kevin M wrote: ↑Wed Jul 03, 2024 1:44 pmYeah, I figured out what you meant by nonparallel yield shift while driving to an appointmentI visualized it as a teeter totter. Another way to describe it is that the 2034 and 2040 yields move in opposite directions.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 nonparallel 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.
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:
Here's a monthly version that makes the trends a bit easier to see:
Notes and observations:So the reality appears to be that nonparallel 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.
 When the green line is 0, the yield curve is flat, and the 2032 and 2040 yields are the same.
 When the green line is above 0, we have the more typical positively sloped yield curve, with the 2040 yield higher than the 2032.
 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.
 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.
 When the slope of the green line is positive, the yield shift is nonparallel, with the gap between the yields increasing.
 When the slope of the green line is negative, the yield shift is nonparallel, with the gap between the yields decreasing.
 Perhaps most importantly, the yields generally change in the same direction!
Since the teeter totter model doesn't seem to fit reality, how does that affect your analysis?
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 bracketyear mix. Or, if you duration matched against the teeter totter, you would feel the full effect from the rising tide for that bracketyear 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 bracketyear 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 bracketyear weighting to counter the teeter totter is well within the range of bracketyear 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 backoftheenvelope model to a combined rising tide with teeter totter case. I'm not sure that is worth the effort, since you have shown that backoftheenvelope models cannot accurately model rising tide effects.
The backoftheenvelope 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 backoftheenvelope 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.
Re: Filling the TIPS gap years with bracket year duration matching
Here's the zoomed in chart of the yield deltas between the 2040 and 2032:
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
If I make a calculation error, #Cruncher probably will let me know.
Re: Filling the TIPS gap years with bracket year duration matching
I think there is another important point that should be made regarding nonparallel 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/jointrisk ... 6freeman/) states the following:
This is a reminder that nonparallel 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 tenyear auctions), so that the bracket year bonds don't become short term bonds. It also might suggest that the 2034/2040 6year 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).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.
Re: Filling the TIPS gap years with bracket year duration matching
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.
Re: Filling the TIPS gap years with bracket year duration matching
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:MtnBiker wrote: ↑Thu Jul 04, 2024 10:36 am I think there is another important point that should be made regarding nonparallel 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/jointrisk ... 6freeman/) states the following:
This is a reminder that nonparallel 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 tenyear auctions), so that the bracket year bonds don't become short term bonds. It also might suggest that the 2034/2040 6year 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).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.
Just by inspection we can see that the 5y (red) is more volatile.
Here are some stats:
Code: Select all
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.
Re: Filling the TIPS gap years with bracket year duration matching
A random thought on this thread:
One of the complications in duration matching is the potential loss of coupon income for the pregap years ARA. That is, when you sell off the 2034 and 2040 TIPS you have to adjust the ARA for the pregap 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 preGAP 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.
One of the complications in duration matching is the potential loss of coupon income for the pregap years ARA. That is, when you sell off the 2034 and 2040 TIPS you have to adjust the ARA for the pregap 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 preGAP 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.
Re: Filling the TIPS gap years with bracket year duration matching
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.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 pregap years ARA. That is, when you sell off the 2034 and 2040 TIPS you have to adjust the ARA for the pregap 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 preGAP 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.
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.

 Posts: 449
 Joined: Mon Nov 04, 2019 3:43 am
 Location: U.K.
Re: Filling the TIPS gap years with bracket year duration matching
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)
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 postgap 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., 21061340000) 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 postgap 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
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 postgap 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
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
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
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

 Posts: 449
 Joined: Mon Nov 04, 2019 3:43 am
 Location: U.K.
Re: Filling the TIPS gap years with bracket year duration matching
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).
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=14824=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 postgap 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.
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.
Holding a fixed proportion of bonds pre and postgap, 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
Code: Select all
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
The proceeds from the sale will depend on the number of bonds sold, n (in this example n=14824=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 postgap 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
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
Cheers
StillGoing
Re: Filling the TIPS gap years with bracket year duration matching
^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 outcomesat 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 oncesay 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.
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 oncesay 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.
Re: Filling the TIPS gap years with bracket year duration matching
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: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.
Holding a fixed proportion of bonds pre and postgap, 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.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
Cheers
StillGoing
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
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

 Posts: 449
 Joined: Mon Nov 04, 2019 3:43 am
 Location: U.K.
Re: Filling the TIPS gap years with bracket year duration matching
I think it depends on exactly how you are going to fill the gap.MtnBiker wrote: ↑Tue Jul 09, 2024 4:23 pmIt 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: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.
Holding a fixed proportion of bonds pre and postgap, 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.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
Cheers
StillGoing
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 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
Does averaging give the correct result for both of these filling methods, or not?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
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

 Posts: 449
 Joined: Mon Nov 04, 2019 3:43 am
 Location: U.K.
Re: Filling the TIPS gap years with bracket year duration matching
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)
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 postgap 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+585040000=78006. This is independent of the yield prevailing in 2034.
2040 bonds: 56 of the 2040 bonds will be sold (8024, 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 postgap 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.
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 nonflat 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/policyissues ... 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
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
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 postgap 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+585040000=78006. This is independent of the yield prevailing in 2034.
2040 bonds: 56 of the 2040 bonds will be sold (8024, 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 postgap 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
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 nonflat 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/policyissues ... 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
Re: Filling the TIPS gap years with bracket year duration matching
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 nonparallel yield shift would be minimized with a 62/38 mix of 2034/2040 bracketyear holdings (multipliers of 4.1 (for 2034) and 2.9 (for 2040)).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 nonflat 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/policyissues ... 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
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 nonparallel yield shifts using more detailed analysis.
Re: Filling the TIPS gap years with bracket year duration matching
It's a different yield curve model, but I think it's probably fine to use.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 nonflat 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/policyissues ... 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?
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.
Re: Filling the TIPS gap years with bracket year duration matching
I'm trying to replicate your numbers here, and have some questions for you: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 20352039 gap year TIPS is 432,951.
 This leaves us with extra cash of 82,718.
 We can choose to buy the pre2034 TIPS that are left, 20302033, with the total being 31,396, if we want ARA to equal DARA for those rungs.
 If we do the pre2034 transactions, we are left with 51,322 in cash.
What formula do you use to determine the various TIPS prices?
Why in the bolded statement are you focusing only on the 203033 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 preGAP years appears to be understated.
Re: Filling the TIPS gap years with bracket year duration matching
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.Jaylat wrote: ↑Wed Jul 10, 2024 11:36 amI'm trying to replicate your numbers here, and have some questions for you: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 20352039 gap year TIPS is 432,951.
 This leaves us with extra cash of 82,718.
 We can choose to buy the pre2034 TIPS that are left, 20302033, with the total being 31,396, if we want ARA to equal DARA for those rungs.
 If we do the pre2034 transactions, we are left with 51,322 in cash.
What formula do you use to determine the various TIPS prices?
Why in the bolded statement are you focusing only on the 203033 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 preGAP years appears to be understated.
If I make a calculation error, #Cruncher probably will let me know.
Re: Filling the TIPS gap years with bracket year duration matching
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:
First, here is the ladder before any gaps are filled, with the rows for 20412054 hidden since they aren't of particular interest, other than providing "Interest later bonds" for the maturities that are shown.
Things to note:
Things to note:
Observations:
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:
The formulas that are different for row 2, the 2054 maturity, are:
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.
The current model has these features:
 Assumes all gap years are filled in 2029, with the 20252029 proceeds used for expenses.
 So the ladder now is a 25year ladder instead of a 30y ladder, with terms to maturity reduced by 5 years each.
First, here is the ladder before any gaps are filled, with the rows for 20412054 hidden since they aren't of particular interest, other than providing "Interest later bonds" for the maturities that are shown.
Things to note:
 The total proceeds (aka total ARA), in cell J32 with the cursor focus, now equals total DARA of 2.5M for the 25y ladder. Recall that previously this was greater than total DARA; this is the result of the change for version 2.
 The multipliers for the 2040 and 2034 of 3.43 and 3.57 are based on duration matching at 2% yields.
 The terms to maturity, in column C, are all reduced by five years, so, for example, the 2034 now is a 5y bond and the 2040 is an 11y bond.
 The multipliers for the 20252029 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 pre2034 maturities.
 The gap year coupons are irrelevant with no gaps yet filled.
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 20302040; 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 20302033 since we will buy or sell them so that ARA = DARA.
 Total proceeds still equals 2.5M, as it should.
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 20302033 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.
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).
Code: Select all
G2: 0
H2: =E2*B1/(1+F2)
If I make a calculation error, #Cruncher probably will let me know.
Re: Filling the TIPS gap years with bracket year duration matching
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.
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.
Re: Filling the TIPS gap years with bracket year duration matching
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.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.
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: ↑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 25year ladder assumes 5 years have passed since the ladder was built, so the maturities of all TIPS have decreased by 5 years, and the 20252029 have all matured with the proceeds removed from the ladder (e.g., for expenses).
<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%.
The results do change with the corrections provided in #Cruncher's Version 2 simple spreadsheet. But the changes aren't huge.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:
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.
 Assumes all gap years are filled in 2029, with the 20252029 proceeds used for expenses.
 So the ladder now is a 25year ladder instead of a 30y ladder, with terms to maturity reduced by 5 years each.
<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.
Observations:
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.
 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 20302033 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.
<snip>
Re: Filling the TIPS gap years with bracket year duration matching
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."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:
Currently there are TIPS maturing in Jan 2034 and Feb 2040, so 2034 and 2040 are the bracket years, and the gap years are 20352039 (five of them).
 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 inflationadjusted 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 DARAmultiplierweighted duration of them equals the expected duration of a gap year TIPS when it is issued.
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 20342040 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 DARAmultiplierweightedaverage 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 gapyear DARA multipliers for the 2034 and 2040, we start with this equation:
With some algebra, we solve for x to get:Code: Select all
d34 * x + d40 * (1x) = 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
I'll cover the calculation of durations in a subsequent post, and for now I'll just show the example of calculating x and (1x) for the 2035 gap year.Code: Select all
x = (d40dg) / (d40d34)
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.Code: Select all
Independent variable values: d34 = 8.75 d40 = 13.23 dg = d35 = 9.41 So, x = (d40dg) / (d40d34) x = (13.239.41) / (13.238.75) x = 0.85 and 1x = 0.15
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:
which is very close to 0.85 derived using the duration matching formula.Code: Select all
n/6 = 5/6 = 0.83
Here is the table of the DARA weights using durations of TIPS based on quotes from Schwab today, also showing the approximations using the n/6 method for the 2034 weights:
Note that the sum of the weights for each of the 2034 and 2040 are the DARA multipliers we enter into the #Cruncher spreadsheet for them respectively. Of course the sum of these multipliers equals 7, which is the total number of years covered (2034, 2040 + 5 gap years).
 EDIT 
I'm going to summarize what I've learned in doing the experiments documented in this thread. I'll add to this summary as I learn more.
Since I started the thread, #Cruncher developed a simplified ladder building spreadsheet, which I then used extensively for all analysis after that. I refer to this as "the simple tool" or just "simple".
Everything here is premised on using the simple tool, bracketyear coverage of gap years, a 30year ladder, DARA = $100K, and initial duration matching based on hypothetical gap year yield and coupon of 2%.This table summarizes the results of the experiments to date:
 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.
 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.
 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.
 Given #2, purchases or sales of the pregap rungs are required for ARA to equal DARA for the gap and pregap rungs, even after factoring in #3.
 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 30year 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.
"5 fill vs 0 fill at X%" means the numbers in that column relate to having all gap years filled (and the 20252039 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 20352039 gap year TIPS is 432,951.
 This leaves us with extra cash of 82,718.
 We can choose to buy the pre2034 TIPS that are left, 20302033, with the total being 31,396, if we want ARA to equal DARA for those rungs.
 If we do the pre2034 transactions, we are left with 51,322 in cash.
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
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 nonflat yield curve
1) Calculating a ladder payout rate for a nonflat 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.
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
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) noninverted 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
There are two components to obtaining the annual income in the gap for a nonflat yield curve
1) Calculating a ladder payout rate for a nonflat 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
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
1) The flat yield curve results from the previous post can be seen in the leading diagonal (i.e. where Y35=Y40).
2) noninverted 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.
Re: Filling the TIPS gap years with bracket year duration matching
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.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.
As MtnBiker pointed out, I had already done a 25year 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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 Location: U.K.
Re: Filling the TIPS gap years with bracket year duration matching
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.
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/policyissues ... 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
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.
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/policyissues ... 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
Re: Filling the TIPS gap years with bracket year 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.Kevin M wrote: ↑Thu Jun 27, 2024 8:58 amThe math does not bear this out. It's easy to make statements like thisit'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 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.
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.
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.
Re: Filling the TIPS gap years with bracket year duration matching
This analysis indicates it is reasonable to assume nonparallel 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.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.
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/policyissues ... 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
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 nonparallel 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 nonparallel 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?
Why duration matching fails for TIPS bought at auctions
Jaylat wrote: ↑Thu Jul 11, 2024 2:11 pmI 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.Kevin M wrote: ↑Thu Jun 27, 2024 8:58 amThe math does not bear this out. It's easy to make statements like thisit'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 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.
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.
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.
 The duration matching portfolio of bonds consists of our excess holdings of the 2034 and 2040 (bracket years), and the liabilities are the 20352039s (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 20352039s (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
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:
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):
Note the following:
 The gap year coupons are unchanged, so the values do not correspond to what they would be at auction for the indicated yields, y.
 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.
 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%.
 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.
 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.
Observations:
 The coupon now is set to the closest 0.125% less than or equal to the yield, as it would be at auction.
 Because of #1, the price, p, is pretty close to 100, as it would be at auction (for a new issue).
 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%.
 Of course the change in price of the duration match weighted price is the same as before.
 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).
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 20332035s 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: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.
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 20302033s 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.
Re: Why duration matching fails for TIPS bought at auctions
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 pregap 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.)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 20302033s 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 the pregap 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 pregap coupons. This is what I called a change in cash flow distribution at the beginning of this thread. Rebalancing the pregap years to smooth ARA is a nice solution to the cash flow redistribution issue.
Re: Why duration matching fails for TIPS bought at auctions
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?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 20302033s 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.
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 pregap 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."
Glad you finally agree with me on this.“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.”
Re: Why duration matching fails for TIPS bought at auctions
I should add that, in my opinion, there are two slightly different methods for durationmatched 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.MtnBiker wrote: ↑Fri Jul 12, 2024 4:52 pmIn 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 pregap 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.)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 20302033s 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 the pregap 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 pregap coupons. This is what I called a change in cash flow distribution at the beginning of this thread. Rebalancing the pregap years to smooth ARA is a nice solution to the cash flow redistribution issue.
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 pregap coupon payments increase and the gap principal payments decrease. This decreases the average duration of the ladder. If yields decrease before the swaps, the pregap 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 pregap years remain). That case was analyzed by StillGoing.
Re: Filling the TIPS gap years with bracket year duration matching
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.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:
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.
 Assumes all gap years are filled in 2029, with the 20252029 proceeds used for expenses.
 So the ladder now is a 25year ladder instead of a 30y ladder, with terms to maturity reduced by 5 years each.
<snip>
<snip
Here are the results:
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.
Re: Filling the TIPS gap years with bracket year duration matching
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.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.
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 bracketyear 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% gapyear coupon case (as you did for the 2.00% gapyear coupon case previously). Like the two charts shown in this earlier post:
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.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:
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.
 Assumes all gap years are filled in 2029, with the 20252029 proceeds used for expenses.
 So the ladder now is a 25year ladder instead of a 30y ladder, with terms to maturity reduced by 5 years each.
First, here is the ladder before any gaps are filled, with the rows for 20412054 hidden since they aren't of particular interest, other than providing "Interest later bonds" for the maturities that are shown.
Things to note:Here's the way it looks after filling the five gap years, assuming gap year cost and yield of 2%:
 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 20252029 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 pre2034 maturities.
 The gap year coupons are irrelevant with no gaps yet filled.
Things to note:<snip>
 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 20302040; 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 20302033 since we will buy or sell them so that ARA = DARA.
 Total proceeds still equals 2.5M, as it should.
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.
Re: Filling the TIPS gap years with bracket year duration matching
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 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% gapyear coupon case (as you did for the 2.00% gapyear coupon case previously). Like the two charts shown in this earlier post:
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.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.
And I think you're onto something here which is at the core of the differences we see between the durationmatch and all2034 coverage methods, which is that both the proceeds and principal required for the 2034 and 2040 are about 25K more for the all2034 case than for the DM case.
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.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.
If I make a calculation error, #Cruncher probably will let me know.
Re: Filling the TIPS gap years with bracket year duration matching
We could look at the 2040 and/or 2034 rows in the tables shown in the post copied 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:
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.
 Assumes all gap years are filled in 2029, with the 20252029 proceeds used for expenses.
 So the ladder now is a 25year ladder instead of a 30y ladder, with terms to maturity reduced by 5 years each.
First, here is the ladder before any gaps are filled, with the rows for 20412054 hidden since they aren't of particular interest, other than providing "Interest later bonds" for the maturities that are shown.
Things to note:Here's the way it looks after filling the five gap years, assuming gap year cost and yield of 2%:
 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 20252029 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 pre2034 maturities.
 The gap year coupons are irrelevant with no gaps yet filled.
Things to note:To determine the outcome for each cost at YTM scenario, I copy/pastevalues 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.
 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 20302040; 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 20302033 since we will buy or sell them so that ARA = DARA.
 Total proceeds still equals 2.5M, as it should.
Observations:
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.
 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 20302033 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.
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:
The formulas that are different for row 2, the 2054 maturity, are: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).
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.Code: Select all
G2: 0 H2: =E2*B1/(1+F2)
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 postswap 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 gapyear maturities (when ARA is forced to equal DARA in 2034).