First, terminology:

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

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

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

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

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

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

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

Code: Select all

```
d34 * x + d40 * (1-x) = dg,
where
d34 = modified duration (MD) of the 2034
d40 = MD of the 2040
dg = estimated MD of the gap year TIPS
x = gap year DARA multiplier for the 2034
```

Code: Select all

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

Code: Select all

```
Independent variable values:
d34 = 8.75
d40 = 13.23
dg = d35 = 9.41
So,
x = (d40-dg) / (d40-d34)
x = (13.23-9.41) / (13.23-8.75)
x = 0.85
and
1-x = 0.15
```

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

Code: Select all

```
n/6 = 5/6 = 0.83
```

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

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

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

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

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

Everything here is premised on using the simple tool, bracket-year coverage of gap years, a 30-year ladder, DARA = $100K, and initial duration matching based on hypothetical gap year yield and coupon of 2%.

- 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 pre-gap rungs are required for ARA to equal DARA for the gap and pre-gap 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 30-year ladder, all other multipliers set to 1). This is a technical detail that is not particularly important, and I assume is due to my imperfect implementation of the multiplier feature, which was not included in #Cruncher's original simplified spreadsheet.

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

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

Looking at the 0% yield case:

- The proceeds from selling all excess 2034 and 2040 bracket year TIPS are 515,669.
- The cost of buying the 2035-2039 gap year TIPS is 432,951.
- This leaves us with extra cash of 82,718.
- We can choose to buy the pre-2034 TIPS that are left, 2030-2033, with the total being 31,396, if we want ARA to equal DARA for those rungs.
- If we do the pre-2034 transactions, we are left with 51,322 in cash.