Modified versions of HFEA with ITT and Futures / Lifecycle Investing with Modern Portfolio Theory

[comeinvest]

An interesting study, although it doesn't answer the question of risk/return profiles of treasuries depending on initial yield over longer timeframes.
https://verdadcap.com/archive/on-low-treasury-yields

[comeinvest]

Educational data from another Verdad article https://verdadcap.com/archive/do-treasuries-still-work
How treasuries depend on inflation vs. growth
First table is "Decomposed Treasury Returns by Decade, Annualized, Jan 1962 – March 2021"
Second table is "10-Year Treasury Returns Following Three Month Changes in BBB Spread, 1962–2021"

[unemployed_pysicist]

After a bit of a learning curve, I am now finally able to use the IBKR API. Here is a comparison of the STRIPS yield curve (inherently a zero coupon yield curve) versus SOFR yield curve from 11 March 2025:

STRIPS curve:


Note the lack of liquidity of shorter duration STRIPS. I did not keep track of which ones are principal or interest STRIPS, which I might want to show on these kinds of plots in the future.

SOFR 3M curves:


I cut off data where there is no last price as a quick fix for liquidity issues in the later dated SOFR futures. Something I have learned from using the IBKR API is that these longer term contracts are often missing a bid, an ask, or both (IBKR returns -1 value for unknown values).

My thoughts on the IBKR API and the various python packages, in case this may be helpful for others who want to go down this path for live market data and historical data from interactive brokers:

-The IBKR API is very different from any other "normal" REST API that I have worked with before. You have to have trader work station installed and turned on in order to use it. It all seems very strange and not at all user friendly to me, perhaps a throwback to an earlier age. From a quick look around the internet, I see that I am not alone in this opinion.

-Maybe the IBAPI package for Python is decent, but I could not get anything to work using it. I spent quite a bit of time trying to figure it out, but to no avail.

-The ib_insync package for Python is excellent. I found it relatively easy to get running for some simple tests. So far I find it relatively user-friendly, compared to the ibapi package. One important thing I discovered is that ib_insync had trouble running on a Jupyter notebook, and some other commonly used IDEs with interpreters, like Spyder, etc. So I run everything via some scripts in the command line, which works fine. Chat GPT provided a great starting point for making simple, working scripts.

One thing I did find disappointing about IBKR's API (in conjunction with ib_insync, because it is the only package I could get to work in python) is how slow it is to retrieve data for all the bonds. At least, in the way I'm doing things now. It takes about the same amount of time that it used to take me to scrape marketwatch. I thought that using a dedicated API through a broker would be a little faster, but as I understand it you have to allow a 1-2 second delay (at least) for data to be populated. Perhaps there is a faster way. There is also the 100 ticker limit through IBKR that you have to deal with. Kind of annoying.

I am still new to the IBKR API, but despite my initial difficulties, I see a lot of potential for accessing/tracking live market or historical data. I can get a lot more data fields than I could with marketwatch (Wallstreet Journal, data ultimately from tullet prebon) and I trust it more.

The next step is to get IRRs again, but this time using the futures bid price and treasury ask price, instead of just using the midpoint or last price.

[comeinvest]

Thank you physicist!

Is it possible with your tools to adjust the treasury curve for bond convexity based on a volatility model, for a more unbiased estimate of future short-term rates? I'm particularly thinking about the inverted 2048+ section, whether the inversion can be purely explained with bond convexity.

I think the treasury department uses not only STRIPS for their interpolated CMT treasury yield curve. Is the curve constructed with STRIPS (almost) the same as the one from that the treasury department produces, except for the interpolation?

What volatility model are you using for the SOFR futures convexity adjustment, and does this model reflect the possibility of an occurrence of infrequent spikes in volatility (typically market crashes, every few years) during the period until futures expiration based on historical statistics, or does it just extrapolate the most recent volatility?

Sorry I'm not too familiar with financial mathematics conventions, but it strikes me that in your chart there is a relatively large (ca. 6 bps) gap between "SOFR forward rates after convexity adjustment" and "SOFR continuously compounded forward rates". The 3-month SOFR futures rates are not settled to 3-month Term SOFR rates, but they are settled to the average of the daily rates over a 3-month period. So you need to convert from "quarterly daily compounded" daily rates to "continuously compounded" daily rates, right? If true, per the screenshot below "quarterly" is almost the same as "continuous" (only ca. 2 bps difference).
I think nowhere in the process do we have to convert from discount rates to par rates.
So if you can please clarify how you did this conversion, thanks!

After a bit of a learning curve, I am now finally able to use the IBKR API. Here is a comparison of the STRIPS yield curve (inherently a zero coupon yield curve) versus SOFR yield curve from 11 March 2025:

[comeinvest]

The next step is to get IRRs again, but this time using the futures bid price and treasury ask price, instead of just using the midpoint or last price.

Why the futures bid and treasury ask?

Also, I don't know if you also follow the related thread in the Rational Reminder forum https://community.rationalreminder.ca/t ... -investing , but there was a discussion recently with some more insight on treasury futures implied financing rate, or futures replication cost. My understanding is that the replication cost of the continuous futures strategy really needs to be calculated from the futures roll spread between front and deferred month contracts. The "implied repo rates" of the individual legs, which I think by definition refer to the period until physical futures delivery, are almost meaningless for the purpose of estimating the replication cost of the quarterly rolled continuous futures strategy, as can be inferred from the quarterly sawtooth patterns in the implied repo rate charts from Bloomberg. They would only be meaningful for folks who hold the contract until expiration / delivery.
In fact the IRR (implied repo rates) of the individual futures contracts seem to be in the 3.x% range lately, below the risk-free rate. I have yet to rationalize the logic behind this; I think it has to do with the aversion of market participants to holding physical treasuries on their balance sheet; so the IRR decrease (the futures contracts cheapen) between two roll periods, and whoever is willing to hold a long position beyond the first notice date until delivery gets "rewarded" with an IRR below comparable risk-free rates as a reward for holding the bag.
I found an example here https://www.cmegroup.com/markets/intere ... ytics.html (screenshot below) of how to calculate replication cost expressed in IRR based on the cost of the calendar roll. I think the method should work similarly for treasury futures, except for treasury futures front and back month generally have different underlyings, so we would need to calculate the fair price for both front and back month, take the difference, compare to the actually paid roll cost, and calculate the resulting replication cost as a spread to a "replication IRR" from it. The difficult part for me is calculating the fair value of the front and back month contracts; if you could help, it would be terrific!
The implied IRR of the front and back month contracts are also on the treasury analytics page (albeit not reflecting the quality option, and probably with stale cash treasury data), but I forgot to take a screenshot during the roll period.

[unemployed_pysicist]

The next step is to get IRRs again, but this time using the futures bid price and treasury ask price, instead of just using the midpoint or last price.

Why the futures bid and treasury ask?

Also, I don't know if you also follow the related thread in the Rational Reminder forum https://community.rationalreminder.ca/t ... -investing , but there was a discussion recently with some more insight on treasury futures implied financing rate, or futures replication cost. My understanding is that the replication cost of the continuous futures strategy really needs to be calculated from the futures roll spread between front and deferred month contracts. The "implied repo rates" of the individual legs, which I think by definition refer to the period until physical futures delivery, are almost meaningless for the purpose of estimating the replication cost of the quarterly rolled continuous futures strategy, as can be inferred from the quarterly sawtooth patterns in the implied repo rate charts from Bloomberg. They would only be meaningful for folks who hold the contract until expiration / delivery.

For the IRR formula, I think I need to use the bid for futures because the basis trader simultaneously sells futures (enters the trade at the bid price) and buys cash treasuries (pays the ask price). It may be a small difference from just using the midpoint or last price that I used in the past, but it could be meaningful for the IRR calculations.

And yes, I agree, the implied repo rates of the individual legs is likely meaningless for the quarterly rolled futures strategy. I think we have both hinted at, or discussed this point before. But it is a step along the way to calculating the option adjusted basis, finding the option adjusted DV01, futures fair value, extracting the implied volatility from bond futures, and all the other things I find interesting.

Additionally, Burghardt outlines a method for assessing whether the calendar spread is rich or cheap relative to the futures "fair" value. I wonder if this might be an interesting quantity to track over time.

I can't remember if I saw that thread in the Rational Reminder forum, but I will try to check it. Provided that I can remember my log in information.

[unemployed_pysicist]

Is it possible with your tools to adjust the treasury curve for bond convexity based on a volatility model, for a more unbiased estimate of future rates? I'm particularly thinking about the inverted 2048+ section, whether the inversion can be purely explained with bond convexity.

At present, I do not have anything in place to do this. I suspect that it should be possible to calculate, and then subtract (remove) the convexity bias as per the formula given by Antti Ilmanen in Understanding the Yield Curve. This requires a volatility estimate, as you mentioned. This could be as simple as using recent volatility (Ilmanen makes a case for this, and that is the approach I currently use in my calculation of the SOFR convexity adjustment). Or, by using implied volatility, which makes more conceptual sense to me. Implied volatility was out of reach from publicly available data before, but data from IBKR opens up a lot of potential possibilities. I will have to see what is available on IBKR. Do you know off hand if we have access to OTC products like Interest rate Caps and Floors on IBKR??

I think the treasury department uses not only STRIPS for their interpolated CMT treasury yield curve. Is the curve constructed with STRIPS (almost) the same as the one from that the treasury department produces, except for the interpolation?

I don't even know if the treasury department uses STRIPS at all in their CMT treasury yield curve. I suspect that they don't. The STRIPS market may be a little bit "segmented" from the more liquid notes and bonds. But how dislocated can it really get though, before arbitrage forces bring it back in-line with the prices on coupon-bearing notes and bonds? Note that the STRIPS curve is a zero coupon curve, whereas the CMT yield curve is more like par equivalent yield curve, so they will look a little different.

What volatility model are you using for the SOFR futures convexity adjustment, and does this model reflect the possibility of an occurrence of infrequent spikes in volatility (typically market crashes, every few years) during the period until futures expiration based on historical statistics, or does it just extrapolate the most recent volatility?

For the SOFR convexity adjustment, I use the formula given in Fabio Mercurio's 2018 seminar, where convexity correction goes like volatility**2/2*T**2, where T is the time to expiration. And, to develop what I mentioned earlier in this post, I do a one month look back on the volatility for each SOFR contract to estimate the volatility - an Ilmanen approved method That is the extent of my volatility "modelling" at present. So yes, I am currently just extrapolating from the most recent volatility. I would like to eventually incorporate implied volatility from SOFR options, which I think should be a better gauge of what the market is actually saying.

Sorry I'm not too familiar with financial mathematics conventions, but it strikes me that in your chart there is a relatively large (ca. 6 bps) gap between "SOFR forward rates after convexity adjustment" and "SOFR continuously compounded forward rates". The 3-month SOFR futures rates are not settled to 3-month Term SOFR rates, but they are settled to the average of the daily rates over a 3-month period. So you need to convert from "quarterly compounded" daily rates to "continuously compounded" daily rates, right? If true, per the screenshot below "quarterly" is almost the same as "continuous" (only ca. 2 bps difference).
I think nowhere in the process do we have to convert from discount rates to par rates.
So if you can please clarify how you did this conversion, thanks!

It's been a while since I thought about this, but I think that the larger than 2 bps difference comes from the fact that SOFR uses a 360 day count, when there are actually more days in the year. Even though SOFR rates are given with the 360 day count convention, if you actually rolled it over every day for a year you will have gained a tiny extra amount because there are more than 360 days in a year. Here are the calculations of the SOFR forward rate after convexity adjustment and the SOFR continuously compounded forward rate, directly from my code:

SOFR Forward Rate after Convexity Adjustment = (100-sofr_price)-100*(sofr_vol)**2/2*t_exp**2

SOFR Continuously Compounded Forward Rate = 36500*log(1+sofr_rate/36000)

It's not necessary to make this conversion if you are okay sticking with a 360 day count convention. But when making yield curves, I like to have everything in the same format (compounding, day count, etc.) so that I can make apples to apples comparisons.

[unemployed_pysicist]

The next step is to get IRRs again, but this time using the futures bid price and treasury ask price, instead of just using the midpoint or last price.

Also, I don't know if you also follow the related thread in the Rational Reminder forum https://community.rationalreminder.ca/t ... -investing , but there was a discussion recently with some more insight on treasury futures implied financing rate, or futures replication cost.

I have always thought that the most straightforward way to calculate the futures replication cost is to track the returns of the CTD of a Treasury Futures contract, and compare them to the returns of that Treasury Futures contract fully collateralized by Tbills or reverse repo or whatever. If the CTD changes, then track the returns of the new CTD from that point onward. The only problem is getting the data. But this kind of thing should be possible in principle with the IB API.

EDIT: By "possible in principle with the IB API", I mean getting the real time data and monitoring it over time, moving forward. IB allows some ability to get historical data, but I don't think there is access to the data for the 5 year note futures contract expiring in march of 2013, for example.

[comeinvest]

Also, I don't know if you also follow the related thread in the Rational Reminder forum https://community.rationalreminder.ca/t ... -investing , but there was a discussion recently with some more insight on treasury futures implied financing rate, or futures replication cost.

I have always thought that the most straightforward way to calculate the futures replication cost is to track the returns of the CTD of a Treasury Futures contract, and compare them to the returns of that Treasury Futures contract fully collateralized by Tbills or reverse repo or whatever. If the CTD changes, then track the returns of the new CTD from that point onward. The only problem is getting the data. But this kind of thing should be possible in principle with the IB API.

That is what Corey in the RR forum did using Bloomberg data; but the result was not consistent with other data points regarding futures replication cost.
A caveat with this methodology are that CTD bonds might represent systematic biases in comparison to an unbiased set of constant or similar maturity treasury bonds, for example that they are typically more expensive as they are in higher demand; but that effect is not big according to a chart from a TBAC study that I posted earlier in this thread. Another caveat is that CTD switches when they occur might introduce some systematic bias. If the current CTD at any given time is the "cheapest" (highest IRR by definition), then a CTD switch means the former CTD "overperformed" the new CTD, implying that there is a systematic bias to "overperforming" bonds? I'm not sure if this is true, or if "expensive" and "cheap" is mostly an artifact of the conversion factor convention.

[comeinvest]

Why the futures bid and treasury ask?

Also, I don't know if you also follow the related thread in the Rational Reminder forum https://community.rationalreminder.ca/t ... -investing , but there was a discussion recently with some more insight on treasury futures implied financing rate, or futures replication cost. My understanding is that the replication cost of the continuous futures strategy really needs to be calculated from the futures roll spread between front and deferred month contracts. The "implied repo rates" of the individual legs, which I think by definition refer to the period until physical futures delivery, are almost meaningless for the purpose of estimating the replication cost of the quarterly rolled continuous futures strategy, as can be inferred from the quarterly sawtooth patterns in the implied repo rate charts from Bloomberg. They would only be meaningful for folks who hold the contract until expiration / delivery.

For the IRR formula, I think I need to use the bid for futures because the basis trader simultaneously sells futures (enters the trade at the bid price) and buys cash treasuries (pays the ask price). It may be a small difference from just using the midpoint or last price that I used in the past, but it could be meaningful for the IRR calculations.

And yes, I agree, the implied repo rates of the individual legs is likely meaningless for the quarterly rolled futures strategy. I think we have both hinted at, or discussed this point before. But it is a step along the way to calculating the option adjusted basis, finding the option adjusted DV01, futures fair value, extracting the implied volatility from bond futures, and all the other things I find interesting.

Additionally, Burghardt outlines a method for assessing whether the calendar spread is rich or cheap relative to the futures "fair" value. I wonder if this might be an interesting quantity to track over time.

I can't remember if I saw that thread in the Rational Reminder forum, but I will try to check it. Provided that I can remember my log in information.

When analyzing the calendar roll cost, the option value of the various embedded options for the short futures position holder would also need to be accounted for.

[comeinvest]

At present, I do not have anything in place to do this. I suspect that it should be possible to calculate, and then subtract (remove) the convexity bias as per the formula given by Antti Ilmanen in Understanding the Yield Curve. This requires a volatility estimate, as you mentioned. This could be as simple as using recent volatility (Ilmanen makes a case for this, and that is the approach I currently use in my calculation of the SOFR convexity adjustment). Or, by using implied volatility, which makes more conceptual sense to me. Implied volatility was out of reach from publicly available data before, but data from IBKR opens up a lot of potential possibilities. I will have to see what is available on IBKR. Do you know off hand if we have access to OTC products like Interest rate Caps and Floors on IBKR??

I don't think OTC products are available at IBKR, but I think options on interest rate futures are available.

Using recent volatility for the convexity bias of a 30-year bond? Shouldn't the volatility events that typically occur within 30 years be used, along with some estimation of black swan tail risk that didn't materialize in the past (similar to how risk premia in other markets, like equity markets or options, are estimated)? If long-term treasuries are the ultimate insurance for undefined equity market and economic growth tail risk.

[unemployed_pysicist]

That is what Corey in the RR forum did using Bloomberg data; but the result was not consistent with other data points regarding futures replication cost.
A caveat with this methodology are that CTD bonds might represent systematic biases in comparison to an unbiased set of constant or similar maturity treasury bonds, for example that they are typically more expensive as they are in higher demand; but that effect is not big according to a chart from a TBAC study that I posted earlier in this thread. Another caveat is that CTD switches when they occur might introduce some systematic bias. If the current CTD at any given time is the "most expensive" (highest IRR by definition), then a CTD switch means the former CTD "underperformed" the new CTD, implying that there is a systematic bias to "underperforming" bonds? I'm not sure if this is true, or if "expensive" and "cheap" is mostly an artifact of the conversion factor convention.

I just finished skimming through that rational reminder thread and reading the most recent posts. Indeed, it looks like Corey already did exactly what I would have done (if I had Bloomberg data). The only improvement I could think of is to do a weighting based on the probability of each deliverable in the basket to be the CTD. But this seems like a lot of effort that would probably not change the result by very much. As he suggests in that thread, I would not use VFITX or any other bond fund for this exercise. Comparison of CTD and futures is the most direct option.

Having a portfolio performance that is tied to the behavior of the CTD (and other deliverables, weighted by their probability to be CTD) is just an inherent feature of this strategy. I don't think that there is any escape from this.

I think "expensive" and "cheap" of each deliverable is mostly an artifact of the conversion factor convention. But indeed, the yield of the CTD probably "suffers" a little bit from the attached liquidity premium. I can't exactly remember offhand what the spread between the CTD and other off-the-runs are, but I thought it was in the neighborhood of 2-6 bps for notes.

How is his result inconsistent with the other data points regarding futures replication cost? What exactly are the other data points?

[unemployed_pysicist]

Using recent volatility for the convexity bias of a 30-year bond? Shouldn't the volatility events that typically occur within 30 years be used, along with some estimation of black swan tail risk that didn't materialize in the past (similar to how risk premia in other markets, like equity markets or options, are estimated)?

Yeah, maybe

Or, use the implied volatility from Interest rate caps and floors and assume that bond market will factor in all of this stuff for you To be fair though, I have no idea what the OTC interest rate cap and floor option market looks like.

[comeinvest]

I just finished skimming through that rational reminder thread and reading the most recent posts. Indeed, it looks like Corey already did exactly what I would have done (if I had Bloomberg data). The only improvement I could think of is to do a weighting based on the probability of each deliverable in the basket to be the CTD. But this seems like a lot of effort that would probably not change the result by very much. As he suggests in that thread, I would not use VFITX or any other bond fund for this exercise. Comparison of CTD and futures is the most direct option.

Having a portfolio performance that is tied to the behavior of the CTD (and other deliverables, weighted by their probability to be CTD) is just an inherent feature of this strategy. I don't think that there is any escape from this.

I think "expensive" and "cheap" of each deliverable is mostly an artifact of the conversion factor convention. But indeed, the yield of the CTD probably "suffers" a little bit from the attached liquidity premium. I can't exactly remember offhand what the spread between the CTD and other off-the-runs are, but I thought it was in the neighborhood of 2-6 bps for notes.

How is his result inconsistent with the other data points regarding futures replication cost? What exactly are the other data points?

"How is his result inconsistent with the other data points regarding futures replication cost? What exactly are the other data points?
"

- The TBAC treasury futures replication cost studies, chart in this post viewtopic.php?p=8273285#p8273285 and numerous similar papers and articles

- Various comparison backtest futures vs. ETF (I still stand by my standpoint that a comparison of futures to ETFs [properly adjusted for duration, yield curve curvature / point approximation, etc.] should by definition be the ultimate observable / indication of "futures replication cost", as those are the 2 alternatives readily available to an investor for implementing the same duration risk exposure. Any and all artifacts arising from the CTD or any other artifacts of the futures delivery mechanism should be attributed to the replication cost.)
Additionally, it cannot be that ETFs consistently and "mysteriously" outperform futures or CTDs by more than what can be rationally explained via any systematic biases.

- Calendar roll analysis - effective roll based replication IRR inferred from the differences in "futures yield" and/or from the differences in IRR of front and back month (Unfortunately I missed to take screenshots of the last roll data on the treasury analytics page. So this is a "to do" item.)

I am not taking sides or anything, and I don't mean to panic; but the long-run expected performance of leveraged treasury futures is heavily dependent on the replication cost, along with and in relation to the term premium. I would be more confident if I can reconcile or confirm the data points, for example by a roll yield analysis, or a replication cost study per futures tenor across the spectrum (ZT, ZF, ZN, TN, ZB, UB) to understand the pattern. (The average across the tenors would also hopefully reconcile with the 0.45% [2018-2025] to ca. 1% [2022-2025] drag from the TBAC study and from other simulations.) It would also help with evaluating the implementation alternatives (SOFR futures, ZT futures, ZF/ZN futures). In any case, thanks for weighing in and for your contributions in this thread, I really appreciate it!

[Paca]

Comeinvest,
You said “I'm thinking of switching to an implementation of duration exposure using a combination of a short (ca. 12 months) SOFR futures strip where the swap spread is near zero (and the Sharpe ratio the highest per studies referenced earlier in this thread), and ZB futures which seem to have much higher "futures carry" (and in particular positive and not negative carry) and even higher carry per duration (which was the original rationale of mHFEA vs. HFEA) than STT and ITT futures under the current circumstances. Or possibly, to hedge the bets, a combination of a SOFR futures strip, ZN, and ZB.”

Thanks, as always. Two questions about your proposed alternative implementation:
1. Is TN a viable alternative to ZB and/or ZN?
2. The first of the recent posts by physicist suggested high SOFR volume out to five years. Could the SOFR strip be extended beyond 1-2 years to replace the ITTs?

[comeinvest]

Comeinvest,
You said “I'm thinking of switching to an implementation of duration exposure using a combination of a short (ca. 12 months) SOFR futures strip where the swap spread is near zero (and the Sharpe ratio the highest per studies referenced earlier in this thread), and ZB futures which seem to have much higher "futures carry" (and in particular positive and not negative carry) and even higher carry per duration (which was the original rationale of mHFEA vs. HFEA) than STT and ITT futures under the current circumstances. Or possibly, to hedge the bets, a combination of a SOFR futures strip, ZN, and ZB.”

Thanks, as always. Two questions about your proposed alternative implementation:
1. Is TN a viable alternative to ZB and/or ZN?
2. The first of the recent posts by physicist suggested high SOFR volume out to five years. Could the SOFR strip be extended beyond 1-2 years to replace the ITTs?

- TN would be another alternative. At this point, the idea as well as the entire concern are still speculation and brainstorming, which I clearly said. Everything depends on the replication cost per each of the 6 futures contracts, which I would like to see but which I'm having a hard time getting hold of. This would enable us to see the pattern, and act accordingly based on estimated risk and returns (which are mostly carry returns in the long run) after cost.
The most important thing that I learned recently, also by looking at other products like high yield credit swaps and futures, is that futures replication cost matters as it might be more liquidity driven than I previously thought, and should be assessed and monitored for a continuous strategy. Blindly following the strategy for 30 years is not a good idea in my opinion.

- SOFR futures are in my opinion only viable up to ca. 2 years, and perhaps only up to 1 year out, based on the current swap spread curve. However the swap spread curve as changed significantly in history, and established plateaus at various levels for a long time. So in my opinion a SOFR based strategy would also need periodic monitoring. I have yet to fully understand the determinants of the swap spread. There are many papers available for free to read.
Here is a 2022 article on SOFR swap spread valuation and historical statistics: https://www.kamakuraco.com/the-reduced- ... valuation/

I am currently still evaluating the options, but you can do the same. It's a bit harder without Bloomberg, but possible. You could for example calculate the "effective futures roll yield" based on the last calendar roll data from the treasury futures analytics site, along the lines that we outlined in recent posts. Try to capture the cash treasury data and the futures data when markets are relatively stable, calculate the fair roll cost (calendar spread), and compare to the actual roll cost. If you trust the IRR data on the analytics site, it makes the calc a bit easier. (Although I'm not sure if they take the most likely of first or last delivery day.)

[comeinvest]

I think the futures replication cost can be relatively easily calculated from the IRR charts. What we need is the IRR of the front month contract and its remaining time to expiration, and the IRR of the back month contract and its remaining time to expiration. The vertical drops in the IRR charts seem to be at the roll dates (e.g. end of November). I'm pretty sure that the vertical drops are the switch of the active contract from front to back month, and the exponential climbs correspond to a linear CAGR of the basis trade return, exponential in the IRR chart due to the shrinking time to expiration. A rough calculation would also assume that the invoice price of the different underlying cash treasuries is roughly the same, which I think is usually the case (assumption not needed if we start from the IRR instead of the absolute roll spread). One unknown is whether delivery optionalities are already reflected in the Bloomberg IRR numbers. Another unknown is what roll date Bloomberg assumes. In the RR forum Corey says in regards to another (performance) chart "ratio adjusted using using the Bloomberg default roll schedule; e.g. the March 2025 contract (TYH5) expires on March 21, 2025, so Bloomberg’s convention would roll be to the June 2025 contract on March 1, 2025." But https://www.cmegroup.com/education/cour ... ocess.html says long futures position holders who want to avoid delivery roll the week before First Intention Day, which we all know is where most of the calendar roll volume happens, and First Intention Day is 2 business days prior to delivery month. Therefore I'm confused why Bloomberg's default roll date is after the First Intention Date.
Another unknown is the delivery date assumption during delivery month. https://www.icmagroup.org/assets/ICMA-E ... y-hill.pdf says the condition for last day delivery by a rational short futures position holder is "ARR > IRR", but I'm not sure if this makes sense, because IRR is a function of the delivery date. I think the delivery date depends on the coupon vs. ARR, as physicist mentioned earlier.
I hope it's ok to re-post the IRR charts from the RR forum for discussion.
ARR: actual repo rate
IRR: futures implied repo rate
The second chart below is "ARR minus IRR" (actual minus implied repo rate).
The chart form the TBAC publication is option-adjusted, "net of carry" which I take to mean the carry from the "net basis" i.e. relative to ARR (actual repo rates), and seems to indicate the net basis in percentage (of the notional?), not annualized like the IRR charts.
The fourth chart below is "actual repo rate vs T-bills", which would have to be added to the carry from the "net basis" to arrive at an implied repo rate relative to T-bill rates.
The RR forum has a chart of the CAGR of the basis trade, but I'm not sure about all the assumptions that went into it.

Per https://www.quantitativebrokers.com/blo ... ostructure , the Treasury futures roll period generally occurs during the 2–4 days before the First Intention Day, which would be ca. 7 calendar dates before the Bloomberg "default" roll date. I will do a second calc with this alternate assumption, as I don't know which assumption went into the Bloomberg IRR chart.

Futures replication cost of ZN (TY) during 2023:
Assuming last date delivery, and roll date on the first day of delivery month (Bloomberg's "default"?), and ignoring the other unknowns above like the value of delivery options (in case they are not already reflected in Bloomberg's IRR), I get the following estimated futures replication cost for the 2023 period:
IRR-ARR of new active contract right after calendar roll per "ARR minus IRR" chart (second chart below): ca. -25 bps
29 + 91 = 120 remaining days to delivery at time of roll -> active contract is 25 bps / 365 * 120 -> 8 bps "cheap" relative to the cash treasury forward price [6 bps in a first day delivery scenario]
IRR-ARR of active contract right before calendar roll per "ARR minus IRR" chart (second chart below): ca. -175 bps
29 remaining days to delivery at time of roll -> active contract is 175 bps / 365 * 29 -> 14 bps "cheap" relative to the cash treasury forward price [0 bps in a first day delivery scenario] [17 bps if the roll date in the chart is 7 days prior to Bloomberg default]
-> I'm selling each futures contract on average ca. 14 bps - 8 bps = 6 bps (0.06% of the cash treasury) "cheaper" than when I bought it [9 bps if the roll date in the chart is 7 days prior to Bloomberg default]
This happens 4 times per year -> replication cost during 2023 was 6 bps * 4 = 24 bps p.a. relative to ARR [36 bps if the roll date in the chart is 7 days prior to Bloomberg default]
Add to this the ca. 25 bps average from the "actual repo rate vs T-bills" chart (fourth chart below) during 2023
-> total replication cost relative to T-bill rates of 24 bps + 25 bps = 49 bps during 2023 [61 bps if the roll date in the chart is 7 days prior to Bloomberg default]

Futures replication cost of ZN (TY) during 2024:
For 2024 IRR vs ARR per the chart was ca. -11 bps on average at time of purchase, and ca. -120 bps on average at time of sale.
11 bps / 365 * 120 -> 4 bps of principal (cash treasury forward price) [3 bps in a first day delivery scenario]
120 bps / 365 * 29 -> 10 bps of principal (cash treasury forward price) [0 bps in a first day delivery scenario] [12 bps if the roll date in the chart is 7 days prior to Bloomberg default]
-> selling the futures contracts 10 bps - 4 bps = 6 bps "cheaper" than bought [3 bps in a first day delivery scenario] [8 bps if the roll date in the chart is 7 days prior to Bloomberg default]
-> replication cost relative to ARR: 6 bps * 4 = 24 bps (same as for 2023) [12 bps in a first day delivery scenario] [32 bps if the roll date in the chart is 7 days prior to Bloomberg default]
Ca. 10 bps average per the "actual repo rate vs T-bills" chart during 2024
-> total replication cost relative to T-bill rates of 24 bps + 10 bps = 34 bps during 2024 [22 bps in a first day delivery scenario] [42 bps if the roll date in the chart is 7 days prior to Bloomberg default]

No warranty for my math; I hope somebody can verify it. I'm always confused about day counts and compounding methods, but the calc seems to be robust with respect to those. I also hope I didn't miss to apply the conversion factor somewhere in the calc.
The results are consistent with Corey's telltale and basis trade gain/loss chart in the RR forum (ca. 0.5% p.a. for TY in 2023 per his charts, and US 23 bps, TY 33 bps, FV 30 bps, TU 14 bps for 2015-2024), but he provided no charts or data for US, FV, and TU and there were some previous inconsistencies and back and forth in the thread, so I'm not 100% confident in the stated FV and TU results.
I have no data points for the longer maturity contracts (TN, ZB, UB). If a participant in this thread with access to Bloomberg can produce similar charts for the other futures tenors, we can do the same calc to see if there is a pattern (e.g. is the replication cost constant, or proportional to the duration risk).

At this point I am relatively confident that the funding spread of the ZN (TY) to T-bills was ca. 34 bps during the last 10 years and spiked to ca. 49 bps in 2023 [24 bps and 61 bps respectively depending on which roll date assumptions went into the Bloomberg chart]. EDIT: Not entirely confident as long as we don't have data for the value of delivery options, which might be 0.25%-0.75%. Comparable to the implied financing cost of SPX options box trades, and higher than what I thought when this thread started, it would consume a signifiant part of the expected term premium; but not as high as the TBAC basis trade chart "Cost of Replicating Bloomberg Treasury Index with Futures" suggests for the 2018-2025 period (-0.45% for 2018-2023 and ca. -1% for 2022+). The difference to TBAC might come from the longer maturity contracts, although Corey states 23 bps spread for US (ZB) so there are not many left. (Note that the TBAC result is relative to actual repo, so the actual repo relative to T-bills would have to be added.) If we have charts or data for all futures contract tenors, we can try to reconcile all results.

P.S.: The ZT net basis chart in the third screenshot below looks scary to me.



ARR minus IRR:





Actual repo rate vs T-bills:

[unemployed_pysicist]

I still stand by my standpoint that a comparison of futures to ETFs [properly adjusted for duration, yield curve curvature / point approximation, etc.] should by definition be the ultimate observable / indication of "futures replication cost", as those are the 2 alternatives readily available to an investor for implementing the same duration risk exposure. Any and all artifacts arising from the CTD or any other artifacts of the futures delivery mechanism should be attributed to the replication cost.

Ok, NOW I think I see what you are getting at You are not necessarily only talking about the performance drag/gain associated with just the physical CTD versus a fully collaterized futures position over time, but rather the drag/gain associated with a diversified basket of treasuries and the fully collateralized futures position. Comparing futures performance to the CTD should be straightforward (if you have a Bloomberg terminal), but comparing duration adjusted futures performance to an ETF like VGIT or VFITX is trickier, unless we know the duration and composition of these funds every day. Otherwise, we have to make assumptions that could introduce uncertainty - a small amount probably, but meaningful when we are discussing a possible difference of 45 bps.

To your point, I think there is a very real risk that rolling futures contracts will not completely replicate a bond fund, suitably adjusted for duration, precisely because of artifacts arising from the CTD and futures delivery mechanisms.

I am still trying to wrap my head around the TBAC charts and Corey's chart of the relative performance of the CTD vs futures on the RR forum. The TBAC chart seems vague at first glance; Which bloomberg treasury index? What is the methodology for this index? Has it been duration adjusted? Which futures contract is it being compared to? I did not find quick answers when I skimmed the TBAC presentation, though I did notice that their results were presented as being approximate in nature.

I have some questions about the performance of the CTD vs futures in Corey's plot also. We know that IRR spreads to Tbill rates were largely low and negative for the 2022-2024-ish period, yet the CTD has gained relative to futures during this time. The gain does look more subdued than for e.g. 2017-2019, when the IRR spread to the Tbill rate was largely positive (as I recall). I suspect that the continuing outperformance of the CTD vs futures during the 2022-2024 period comes from the value of the options held by the basis trader - perhaps a sizable chunk of the return is coming from the expansion in implied volatility. Just my guess right now.

[comeinvest]

I have some questions about the performance of the CTD vs futures in Corey's plot also. We know that IRR spreads to Tbill rates were largely low and negative for the 2022-2024-ish period, yet the CTD has gained relative to futures during this time. The gain does look more subdued than for e.g. 2017-2019, when the IRR spread to the Tbill rate was largely positive (as I recall). I suspect that the continuing outperformance of the CTD vs futures during the 2022-2024 period comes from the value of the options held by the basis trader - perhaps a sizable chunk of the return is coming from the expansion in implied volatility. Just my guess right now.

No, I think you didn't read my observation and explanations that I discovered here
https://community.rationalreminder.ca/t ... /18692/900
and then elaborated on here
https://community.rationalreminder.ca/t ... /18692/935
and then summarized in the bold face portion of my comment here
https://community.rationalreminder.ca/t ... /18692/944
when I asked myself the same question that I think you are asking now.

Please note I'm not working in the finance industry, so I explained my observation intuitively in layman's terms; but you will get the point.

Please also read me previous post viewtopic.php?p=8295325#p8295325 right before your post that I am citing, where I attempt a calculation of the futures replication cost based on the IRR chart. In short, the absolute level of the IRR is rather irrelevant to the futures replication cost. What matters is mostly the magnitude of the quarterly oscillations. While holding a specific futures contract between two roll dates, it goes from "cheap" to "cheaper", i.e. you buy cheap, but sell cheaper. You are losing money in the process, regardless if your purchase price was already slightly cheap.
Or phrased differently, you really have to examine the actual calendar roll spread relative to its fair value, not the IRR of the individual legs, to arrive at the continuous futures replication cost.

Treasury futures exhibit specific dynamics not visible in other futures products like equity index futures, due to the "delayed gratification" lag between First Intention day and actual delivery, and due to asymmetric "natural" supply and demand for treasury futures which is in turn due to the aversion of market participants to cash treasuries. Whoever holds a long position in the contract beyond First Intention day, will be force-fed and be stuck with a hated physical treasury sooner or later, hence the IRR becomes cheaper and cheaper as First Intention day approaches.

[comeinvest]

I still stand by my standpoint that a comparison of futures to ETFs [properly adjusted for duration, yield curve curvature / point approximation, etc.] should by definition be the ultimate observable / indication of "futures replication cost", as those are the 2 alternatives readily available to an investor for implementing the same duration risk exposure. Any and all artifacts arising from the CTD or any other artifacts of the futures delivery mechanism should be attributed to the replication cost.

Ok, NOW I think I see what you are getting at You are not necessarily only talking about the performance drag/gain associated with just the physical CTD versus a fully collaterized futures position over time, but rather the drag/gain associated with a diversified basket of treasuries and the fully collateralized futures position. Comparing futures performance to the CTD should be straightforward (if you have a Bloomberg terminal), but comparing duration adjusted futures performance to an ETF like VGIT or VFITX is trickier, unless we know the duration and composition of these funds every day. Otherwise, we have to make assumptions that could introduce uncertainty - a small amount probably, but meaningful when we are discussing a possible difference of 45 bps.

To your point, I think there is a very real risk that rolling futures contracts will not completely replicate a bond fund, suitably adjusted for duration, precisely because of artifacts arising from the CTD and futures delivery mechanisms.

Great minds think alike. Yes the correct problem statement is mathematically more difficult. And yet, we want to solve the actual real world problem, not a problem that is mathematically most convenient.

[comeinvest]

I have some questions about the performance of the CTD vs futures in Corey's plot also. We know that IRR spreads to Tbill rates were largely low and negative for the 2022-2024-ish period, yet the CTD has gained relative to futures during this time. The gain does look more subdued than for e.g. 2017-2019, when the IRR spread to the Tbill rate was largely positive (as I recall). I suspect that the continuing outperformance of the CTD vs futures during the 2022-2024 period comes from the value of the options held by the basis trader - perhaps a sizable chunk of the return is coming from the expansion in implied volatility. Just my guess right now.

I think the major part of the discrepancy that you are observating is explained by the sawtooth pattern of the IRR, which you will see if you read my post before your last one. viewtopic.php?p=8295325#p8295325

Having that said, the ingredient to my calculation that I miss most is the value of the delivery options, most importantly the quality option. Like you said correctly, only differences between realized and expected volatility would affect the result of the empirical performance comparison in the form of Corey's telltale chart, as we must assume that market participants assigned a discount to the futures price based on their estimates of the delivery options, that would be reflected in the futures performance along with the effect of "realized" CTD switches; in the long run the effects of unexpected realized volatility would offset and smooth out. By contrast, in my theoretical estimate based on actual calendar roll spread vs. fair value, the delivery options are currently not reflected at all, when they should be reflected.

I searched a bit, but hard a hard time finding either empirical evidence (historical long-run averages) from after the 1980ies, or theoretical estimates of the value of the delivery options. Do you have any factual source of information? If so, it would have to be added to my estimate based on roll cost. I guess there is no direct observable of the historical value of the delivery options, as you can't distinguish the value of delivery options from generally cheap or expensive rolls due to other factors like liquidity or market imbalances. (I guess only the frequency of CTD switches could be directly observed or at least easily calculated without volatility model, but not the a priori value of the optionality.) So I guess some theoretical model is needed to calculate it.

[comeinvest]

I have some questions about the performance of the CTD vs futures in Corey's plot also. We know that IRR spreads to Tbill rates were largely low and negative for the 2022-2024-ish period, yet the CTD has gained relative to futures during this time. The gain does look more subdued than for e.g. 2017-2019, when the IRR spread to the Tbill rate was largely positive (as I recall). I suspect that the continuing outperformance of the CTD vs futures during the 2022-2024 period comes from the value of the options held by the basis trader - perhaps a sizable chunk of the return is coming from the expansion in implied volatility. Just my guess right now.

I think the major part of the discrepancy that you are observating is explained by the sawtooth pattern of the IRR, which you will see if you read my post before your last one. viewtopic.php?p=8295325#p8295325

Having that said, the ingredient to my calculation that I miss most is the value of the delivery options, most importantly the quality option. Like you said correctly, only differences between realized and expected volatility would affect the result of the empirical performance comparison in the form of Corey's telltale chart, as we must assume that market participants assigned a discount to the futures price based on their estimates of the delivery options, that would be reflected in the futures performance along with the effect of "realized" CTD switches; in the long run the effects of unexpected realized volatility would offset and smooth out. By contrast, in my theoretical estimate based on actual calendar roll spread vs. fair value, the delivery options are currently not reflected at all, when they should be reflected.

I searched a bit, but hard a hard time finding either empirical evidence (historical long-run averages) from after the 1980ies, or theoretical estimates of the value of the delivery options. Do you have any factual source of information? If so, it would have to be added to my estimate based on roll cost. I guess there is no direct observable of the historical value of the delivery options, as you can't distinguish the value of delivery options from generally cheap or expensive rolls due to other factors like liquidity or market imbalances. (I guess only the frequency of CTD switches could be directly observed or at least easily calculated without volatility model, but not the a priori value of the optionality.) So I guess some theoretical model is needed to calculate it.

I edited my grand total futures replication cost estimate in my recent post viewtopic.php?p=8295325#p8295325 in light of the unknown value of delivery options as well as perhaps 0.05% from the CTD premium vs. other treasuries. Perhaps the modern era grand total treasury futures replication cost is more like 0.75%, consistent with some ETF comparison backtests and all-in replication cost studies. If true, it might eat most if not all of the expected term premia. If I had Bloomberg access, I could probably find the answer. It's a shame that pros in the RR forum are quick to show IRR without context, which at 3.x% if not 2.x% is below the risk-free rate and irrelevant to the problem at hand, as is ARR.

[comeinvest]

Comparing futures performance to the CTD should be straightforward (if you have a Bloomberg terminal), but comparing duration adjusted futures performance to an ETF like VGIT or VFITX is trickier, unless we know the duration and composition of these funds every day.

Most of the differences will be from changes of duration over time of both ETF and futures contract / CTD. I think one could estimate historical duration dynamically, and adjust for it, by creating a time series of the standard deviations over a dynamic lookback period. The historical duration could be inferred and reconstructed by interpolation from known data sets with known durations, like the CMT yield data or the constant maturity time series from S&P Global. A "theoretical" performance could be calculated for each time segment, and compared to the actual performance. Not perfect, but probably good enough for purpose of "fair" risk-adjusted performance comparisons; errors from higher derivatives beyond duration (curvature, etc.) would be smoothed over time as long as the method doesn't introduce systematic errors. Actually forget the ETF; we could compare the futures performance to the performance of a dynamically constructed, replicating portfolio of CMT securities that would dynamically approximate the maturity and duration of the CTD by interpolation at any given time, right?

[unemployed_pysicist]

I think the futures replication cost can be relatively easily calculated from the IRR charts. What we need is the IRR of the front month contract and its remaining time to expiration, and the IRR of the back month contract and its remaining time to expiration. The vertical drops in the IRR charts seem to be at the roll dates (e.g. end of November). I'm pretty sure that the vertical drops are the switch of the active contract from front to back month, and the exponential climbs correspond to a linear CAGR of the basis trade return, exponential in the IRR chart due to the shrinking time to expiration. A rough calculation would also assume that the invoice price of the different underlying cash treasuries is roughly the same, which I think is usually the case (assumption not needed if we start from the IRR instead of the absolute roll spread). One unknown is whether delivery optionalities are already reflected in the Bloomberg IRR numbers. Another unknown is what roll date Bloomberg assumes. In the RR forum Corey says in regards to another (performance) chart "ratio adjusted using using the Bloomberg default roll schedule; e.g. the March 2025 contract (TYH5) expires on March 21, 2025, so Bloomberg’s convention would roll be to the June 2025 contract on March 1, 2025." But https://www.cmegroup.com/education/cour ... ocess.html says long futures position holders who want to avoid delivery roll the week before First Intention Day, which we all know is where most of the calendar roll volume happens, and First Intention Day is 2 business days prior to delivery month. Therefore I'm confused why Bloomberg's default roll date is after the First Intention Date.
Another unknown is the delivery date assumption during delivery month. https://www.icmagroup.org/assets/ICMA-E ... y-hill.pdf says the condition for last day delivery by a rational short futures position holder is "ARR > IRR", but I'm not sure if this makes sense, because IRR is a function of the delivery date. I think the delivery date depends on the coupon vs. ARR, as physicist mentioned earlier.
I hope it's ok to re-post the IRR charts from the RR forum for discussion.
ARR: actual repo rate
IRR: futures implied repo rate
The second chart below is "ARR minus IRR" (actual minus implied repo rate).
The chart form the TBAC publication is option-adjusted, "net of carry" which I take to mean the carry from the "net basis" i.e. relative to ARR (actual repo rates), and seems to indicate the net basis in percentage (of the notional?), not annualized like the IRR charts.
The fourth chart below is "actual repo rate vs T-bills", which would have to be added to the carry from the "net basis" to arrive at an implied repo rate relative to T-bill rates.
The RR forum has a chart of the CAGR of the basis trade, but I'm not sure about all the assumptions that went into it.

Per https://www.quantitativebrokers.com/blo ... ostructure , the Treasury futures roll period generally occurs during the 2–4 days before the First Intention Day, which would be ca. 7 calendar dates before the Bloomberg "default" roll date. I will do a second calc with this alternate assumption, as I don't know which assumption went into the Bloomberg IRR chart.

Futures replication cost of ZN (TY) during 2023:
Assuming last date delivery, and roll date on the first day of delivery month (Bloomberg's "default"?), and ignoring the other unknowns above like the value of delivery options (in case they are not already reflected in Bloomberg's IRR), I get the following estimated futures replication cost for the 2023 period:
IRR-ARR of new active contract right after calendar roll per "ARR minus IRR" chart (second chart below): ca. -25 bps
29 + 91 = 120 remaining days to delivery at time of roll -> active contract is 25 bps / 365 * 120 -> 8 bps "cheap" relative to the cash treasury forward price [6 bps in a first day delivery scenario]
IRR-ARR of active contract right before calendar roll per "ARR minus IRR" chart (second chart below): ca. -175 bps
29 remaining days to delivery at time of roll -> active contract is 175 bps / 365 * 29 -> 14 bps "cheap" relative to the cash treasury forward price [0 bps in a first day delivery scenario] [17 bps if the roll date in the chart is 7 days prior to Bloomberg default]
-> I'm selling each futures contract on average ca. 14 bps - 8 bps = 6 bps (0.06% of the cash treasury) "cheaper" than when I bought it [9 bps if the roll date in the chart is 7 days prior to Bloomberg default]
This happens 4 times per year -> replication cost during 2023 was 6 bps * 4 = 24 bps p.a. relative to ARR [36 bps if the roll date in the chart is 7 days prior to Bloomberg default]
Add to this the ca. 25 bps average from the "actual repo rate vs T-bills" chart (fourth chart below) during 2023
-> total replication cost relative to T-bill rates of 24 bps + 25 bps = 49 bps during 2023 [61 bps if the roll date in the chart is 7 days prior to Bloomberg default]

Futures replication cost of ZN (TY) during 2024:
For 2024 IRR vs ARR per the chart was ca. -11 bps on average at time of purchase, and ca. -120 bps on average at time of sale.
11 bps / 365 * 120 -> 4 bps of principal (cash treasury forward price) [3 bps in a first day delivery scenario]
120 bps / 365 * 29 -> 10 bps of principal (cash treasury forward price) [0 bps in a first day delivery scenario] [12 bps if the roll date in the chart is 7 days prior to Bloomberg default]
-> selling the futures contracts 10 bps - 4 bps = 6 bps "cheaper" than bought [3 bps in a first day delivery scenario] [8 bps if the roll date in the chart is 7 days prior to Bloomberg default]
-> replication cost relative to ARR: 6 bps * 4 = 24 bps (same as for 2023) [12 bps in a first day delivery scenario] [32 bps if the roll date in the chart is 7 days prior to Bloomberg default]
Ca. 10 bps average per the "actual repo rate vs T-bills" chart during 2024
-> total replication cost relative to T-bill rates of 24 bps + 10 bps = 34 bps during 2024 [22 bps in a first day delivery scenario] [42 bps if the roll date in the chart is 7 days prior to Bloomberg default]

No warranty for my math; I hope somebody can verify it. I'm always confused about day counts and compounding methods, but the calc seems to be robust with respect to those. I also hope I didn't miss to apply the conversion factor somewhere in the calc.
The results are consistent with Corey's telltale and basis trade gain/loss chart in the RR forum (ca. 0.5% p.a. for TY in 2023 per his charts, and US 23 bps, TY 33 bps, FV 30 bps, TU 14 bps for 2015-2024), but he provided no charts or data for US, FV, and TU and there were some previous inconsistencies and back and forth in the thread, so I'm not 100% confident in the stated FV and TU results.
I have no data points for the longer maturity contracts (TN, ZB, UB). If a participant in this thread with access to Bloomberg can produce similar charts for the other futures tenors, we can do the same calc to see if there is a pattern (e.g. is the replication cost constant, or proportional to the duration risk).

I think I understand your method using charts 2 and 4. Allow me to restate, to confirm.

The sawtooth pattern indicates a roll period. We assume that we start tracking the "new" contract at the bottom of the sawtooth pattern, and this comes right after the switch from the "old" to "new" contract. We can assume that the "new" contract had similar (IRR-ARR) values before we observe it - not strictly necessary if we use the Bloomberg roll date, but we would have to extrapolate backward if we want to know the values of (IRR-ARR) in the week before first intention day. By eyeballing the chart, I estimate IRR-ARR for the beginning of the new contract to be -0.3, -0.2, -0.25, -0.1, for 2023Q1, 2023Q2, 2023Q3, 2023Q4 respectively. I assume this is how you arrive at the -0.25 average estimate for the IRR-ARR at the start of holding a new futures contract through 2023.

When we get to the top of the sawtooth pattern again, this is when we have to sell the old contract if we want to roll. I eyeball IRR-ARR to be -1.35, -1.8, -1.8, -1.8 for the 4 quarters in 2023. Presumably close enough to your -1.75 average estimate. You then evaluate the amount gained or lost by time-weighting the relative richness or cheapness of the contracts according to:

(IRR-ARR of old contract)*(days to old contract expiration at time of roll)/(day count convention)
- (IRR-ARR of new contract)*(days to new contract expiration at time of roll)/(day count convention)

Finally, you include the spread of ARR to the Tbill rate, so that the Tbill rate becomes your reference instead of the ARR.

I am not entirely sure if these calculations based off IRR completely work, because of the long only futures position's inability to guarantee convergence for the time period in question. But maybe it does not matter. I have to think more about it. Also, is the ARR used in the plot a term repo rate or an overnight rate? I want to make sure the ARR used aligns with the period that the futures contract is held. One quick note: I think we should use a 360 day count for money market instruments like IRR, ARR, Tbills for basis point "spreads", and then convert to a 365 day count for the "total return" at the end. This does not change the results of your calculations by a meaningful amount, like 1-2 basis points at most I think.

Nonetheless, I do have some relevant data from this period. Unfortunately it is incomplete for all roll periods for the years 2023 and 2024. I have data for ZT, Z3N, ZF, ZN, ZB, and UB for all the dates shown below. Here are the dates:

tym23-tyu23 "roll period": 23,24,25,26,31 May 2023,
tyu23-tyz23 "roll period": 24,25 August 2023, 1,2 September 2023
tyz23-tyh24 "roll period": 30 November 2023, 1 December 2023
tyh24-tyu24 "roll period": 23 February 2024

My initial inspection of the numbers, following your basic procedure of finding the replication cost by taking into account the "time-weighted" IRR of the "old" contract and the IRR of the "new" contract, yields similar results. In my calculations, I use the IRR spread to Tbills directly, bypassing the Iast part of your method. I can share the numbers using your method if you like, but the downside here is that I effectively only have 2 "true" roll periods: tym23->tyu23 and tyu23->tyz23. Please let me know which of those dates are most relevant for the "actual roll period", and which could be used for the "Bloomberg roll day". As an aside, I thought that tyh5 expires on 31 March? Is 21 March a typo? My calcs show a last trade date of 20 March, and last delivery date of 31 March. I take the last delivery date to mean the same as expiration. If my calcs are incorrect, I will want to adjust them, moving forward.

I conducted another, slightly different investigation: I took the tyu24 CTD, and looked at the performance from 25 May until 25 August. I compared it to the cash and futures position. The CTD I calculated for 25 May was 91282CGS4, which had a 3.625% coupon. There were no coupons paid in the 25 May - 25 August period. I precalculated the dirty price, so that I did not have to worry about accrued interest calculations. Caveat: there was a CTD switch from *CGS4 to *CHF1 during that time period. *CHF1 was first issued as a 7 year note on 31 May 2023, so I don't have any data for it (does not enter the Monthly statement of the public debt until June). Nonetheless, we can look at the 92 day period of relative performance of holding the initial CTD, versus the futures and cash position.

TYU23

Code: Select all

Date:		Treasury dirty price:	Futures Price:	Closest Tbill Delta:	Closest Tbill Yield:
2023-05-25	99.05			113.313		5 days			5.135
2023-08-25	97.1551			109.453		-				-

The Tbill available closest to the expiration of the TYU23 contract was 5 days; I think a difference of 5 days in the yield curve is not much, so we can assume a yield of 5.135 on a cash instrument for 92 days. This acts as the Tbill rate for the entire period that the futures contract is held.

CTD return: 97.1551/99.0504-1 = -0.0191 -> -1.91%
Futures return + Tbill return: 109.453/113.313-1 + (92 days/360)*0.05135 = -0.0209 -> -2.09%

"Simplified" cost of replicating the return of *CGS4 with TY futures:
CTD return - Futures return = -1.91 + 2.09 = 0.18% in Q3.
Naively annualizing this, we get 0.72% (in 360 day count "units").
It's only 1 data point for one note in the basket, and not averaged over 4 roll periods. The note used for this analysis also rotates out of being CTD, so maybe not the fairest comparison. Also, I believe this takes place during the actual roll period, correct? Not the 1 June to 1 August roll period that Bloomberg uses, that I infer from your post.

Is there anything I am missing with the CTD vs Futures and Tbill calculations above? Also, feel free to check the futures and note prices in that calculation, if you have access to them. I suspect prices like 113.313 actually correspond to 113.3125; let me know if you can confirm. I might have a look at *CGS4 and *CHF1 with the IBKR API, to see if my treasury prices that I calculated from Tullet Prebon yield data are substantially different. My data is not Bloomberg data, but maybe it is at least indicative.

[comeinvest]

I think the futures replication cost can be relatively easily calculated from the IRR charts. What we need is the IRR of the front month contract and its remaining time to expiration, and the IRR of the back month contract and its remaining time to expiration. The vertical drops in the IRR charts seem to be at the roll dates (e.g. end of November). I'm pretty sure that the vertical drops are the switch of the active contract from front to back month, and the exponential climbs correspond to a linear CAGR of the basis trade return, exponential in the IRR chart due to the shrinking time to expiration. A rough calculation would also assume that the invoice price of the different underlying cash treasuries is roughly the same, which I think is usually the case (assumption not needed if we start from the IRR instead of the absolute roll spread). One unknown is whether delivery optionalities are already reflected in the Bloomberg IRR numbers. Another unknown is what roll date Bloomberg assumes. In the RR forum Corey says in regards to another (performance) chart "ratio adjusted using using the Bloomberg default roll schedule; e.g. the March 2025 contract (TYH5) expires on March 21, 2025, so Bloomberg’s convention would roll be to the June 2025 contract on March 1, 2025." But https://www.cmegroup.com/education/cour ... ocess.html says long futures position holders who want to avoid delivery roll the week before First Intention Day, which we all know is where most of the calendar roll volume happens, and First Intention Day is 2 business days prior to delivery month. Therefore I'm confused why Bloomberg's default roll date is after the First Intention Date.
Another unknown is the delivery date assumption during delivery month. https://www.icmagroup.org/assets/ICMA-E ... y-hill.pdf says the condition for last day delivery by a rational short futures position holder is "ARR > IRR", but I'm not sure if this makes sense, because IRR is a function of the delivery date. I think the delivery date depends on the coupon vs. ARR, as physicist mentioned earlier.
I hope it's ok to re-post the IRR charts from the RR forum for discussion.
ARR: actual repo rate
IRR: futures implied repo rate
The second chart below is "ARR minus IRR" (actual minus implied repo rate).
The chart form the TBAC publication is option-adjusted, "net of carry" which I take to mean the carry from the "net basis" i.e. relative to ARR (actual repo rates), and seems to indicate the net basis in percentage (of the notional?), not annualized like the IRR charts.
The fourth chart below is "actual repo rate vs T-bills", which would have to be added to the carry from the "net basis" to arrive at an implied repo rate relative to T-bill rates.
The RR forum has a chart of the CAGR of the basis trade, but I'm not sure about all the assumptions that went into it.

Per https://www.quantitativebrokers.com/blo ... ostructure , the Treasury futures roll period generally occurs during the 2–4 days before the First Intention Day, which would be ca. 7 calendar dates before the Bloomberg "default" roll date. I will do a second calc with this alternate assumption, as I don't know which assumption went into the Bloomberg IRR chart.

Futures replication cost of ZN (TY) during 2023:
Assuming last date delivery, and roll date on the first day of delivery month (Bloomberg's "default"?), and ignoring the other unknowns above like the value of delivery options (in case they are not already reflected in Bloomberg's IRR), I get the following estimated futures replication cost for the 2023 period:
IRR-ARR of new active contract right after calendar roll per "ARR minus IRR" chart (second chart below): ca. -25 bps
29 + 91 = 120 remaining days to delivery at time of roll -> active contract is 25 bps / 365 * 120 -> 8 bps "cheap" relative to the cash treasury forward price [6 bps in a first day delivery scenario]
IRR-ARR of active contract right before calendar roll per "ARR minus IRR" chart (second chart below): ca. -175 bps
29 remaining days to delivery at time of roll -> active contract is 175 bps / 365 * 29 -> 14 bps "cheap" relative to the cash treasury forward price [0 bps in a first day delivery scenario] [17 bps if the roll date in the chart is 7 days prior to Bloomberg default]
-> I'm selling each futures contract on average ca. 14 bps - 8 bps = 6 bps (0.06% of the cash treasury) "cheaper" than when I bought it [9 bps if the roll date in the chart is 7 days prior to Bloomberg default]
This happens 4 times per year -> replication cost during 2023 was 6 bps * 4 = 24 bps p.a. relative to ARR [36 bps if the roll date in the chart is 7 days prior to Bloomberg default]
Add to this the ca. 25 bps average from the "actual repo rate vs T-bills" chart (fourth chart below) during 2023
-> total replication cost relative to T-bill rates of 24 bps + 25 bps = 49 bps during 2023 [61 bps if the roll date in the chart is 7 days prior to Bloomberg default]

Futures replication cost of ZN (TY) during 2024:
For 2024 IRR vs ARR per the chart was ca. -11 bps on average at time of purchase, and ca. -120 bps on average at time of sale.
11 bps / 365 * 120 -> 4 bps of principal (cash treasury forward price) [3 bps in a first day delivery scenario]
120 bps / 365 * 29 -> 10 bps of principal (cash treasury forward price) [0 bps in a first day delivery scenario] [12 bps if the roll date in the chart is 7 days prior to Bloomberg default]
-> selling the futures contracts 10 bps - 4 bps = 6 bps "cheaper" than bought [3 bps in a first day delivery scenario] [8 bps if the roll date in the chart is 7 days prior to Bloomberg default]
-> replication cost relative to ARR: 6 bps * 4 = 24 bps (same as for 2023) [12 bps in a first day delivery scenario] [32 bps if the roll date in the chart is 7 days prior to Bloomberg default]
Ca. 10 bps average per the "actual repo rate vs T-bills" chart during 2024
-> total replication cost relative to T-bill rates of 24 bps + 10 bps = 34 bps during 2024 [22 bps in a first day delivery scenario] [42 bps if the roll date in the chart is 7 days prior to Bloomberg default]

No warranty for my math; I hope somebody can verify it. I'm always confused about day counts and compounding methods, but the calc seems to be robust with respect to those. I also hope I didn't miss to apply the conversion factor somewhere in the calc.
The results are consistent with Corey's telltale and basis trade gain/loss chart in the RR forum (ca. 0.5% p.a. for TY in 2023 per his charts, and US 23 bps, TY 33 bps, FV 30 bps, TU 14 bps for 2015-2024), but he provided no charts or data for US, FV, and TU and there were some previous inconsistencies and back and forth in the thread, so I'm not 100% confident in the stated FV and TU results.
I have no data points for the longer maturity contracts (TN, ZB, UB). If a participant in this thread with access to Bloomberg can produce similar charts for the other futures tenors, we can do the same calc to see if there is a pattern (e.g. is the replication cost constant, or proportional to the duration risk).

I think I understand your method using charts 2 and 4. Allow me to restate, to confirm.

The sawtooth pattern indicates a roll period. We assume that we start tracking the "new" contract at the bottom of the sawtooth pattern, and this comes right after the switch from the "old" to "new" contract. We can assume that the "new" contract had similar (IRR-ARR) values before we observe it - not strictly necessary if we use the Bloomberg roll date, but we would have to extrapolate backward if we want to know the values of (IRR-ARR) in the week before first intention day. By eyeballing the chart, I estimate IRR-ARR for the beginning of the new contract to be -0.3, -0.2, -0.25, -0.1, for 2023Q1, 2023Q2, 2023Q3, 2023Q4 respectively. I assume this is how you arrive at the -0.25 average estimate for the IRR-ARR at the start of holding a new futures contract through 2023.

When we get to the top of the sawtooth pattern again, this is when we have to sell the old contract if we want to roll. I eyeball IRR-ARR to be -1.35, -1.8, -1.8, -1.8 for the 4 quarters in 2023. Presumably close enough to your -1.75 average estimate. You then evaluate the amount gained or lost by time-weighting the relative richness or cheapness of the contracts according to:

(IRR-ARR of old contract)*(days to old contract expiration at time of roll)/(day count convention)
- (IRR-ARR of new contract)*(days to new contract expiration at time of roll)/(day count convention)

Finally, you include the spread of ARR to the Tbill rate, so that the Tbill rate becomes your reference instead of the ARR.

I am not entirely sure if these calculations based off IRR completely work, because of the long only futures position's inability to guarantee convergence for the time period in question. But maybe it does not matter. I have to think more about it. Also, is the ARR used in the plot a term repo rate or an overnight rate? I want to make sure the ARR used aligns with the period that the futures contract is held. One quick note: I think we should use a 360 day count for money market instruments like IRR, ARR, Tbills for basis point "spreads", and then convert to a 365 day count for the "total return" at the end. This does not change the results of your calculations by a meaningful amount, like 1-2 basis points at most I think.

I think you understood my method. Just for clarification, you can probably think of it as time-weighted richness and cheapness, but a little more rigorously, if you read my math again, by multiplying the IRR by "remaining day count to delivery" divided by day count convention, I basically convert IRR numbers to percentages of "principal", where "principal" is the cash treasury forward price (forward price using ARR as discount rate). So basically I think of the result as richness or cheapness of the futures contract relative to its "fair" price, where its "fair" price would be the forward price of the cash treasury at time of delivery, and expressed as a percentage of the leveraged notional.

You could think of the richness or cheapness in two ways:
1. You are looking at one calendar roll in isolation (a vertical drop in the charts). Calculating the fair value of each leg at time of roll, you get the fair value of the roll. Compare the actual roll to the fair value of the roll. This is your quarterly cost. Multiple by 4 to arrive at the annual cost. Since the charts don't show the roll cost in dollars but in IRR, I express all results as a percentage of "principal" (the cash treasury forward price at delivery), and the "principal" disappears in the calc of the final cost expressed as an annual rate.
2. You are looking at one period in the chart between two roll dates. (You are climbing up the chart from a trough to a peak.) You calculate the fair value of the active contract between those two roll dates at the beginning, compare the actual contract price to its fair value, that gives you the richness or cheapness at the time when you bought the active contract. Do the same at the end of the period, that gives you the richness or cheapness at the time when you sell it. The difference between the two richness or cheapness is "consumed" while the active contract is active. This will be your replication cost during the quarter. Multiply by 4 to get the annual cost. Again, since we start with IRRs instead of dollars, we express everything as a percentage of principal, which will cancel out at the end.

So I think the method is scientifically sound and accurate unless you find a flaw, and I think it is basically how "others" also calculate roll cost or replication cost, except we start from the IRR instead of dollar amounts, so we reverse-engineer the absolute richness or cheapness from the IRR numbers. IRR is nothing else but the funding cost until delivery expressed as an annualized rate. I do the reverse: I go from the IRR back to the funding cost for one quarter.

You don't need guaranteed convergence during the time period in question, because you are considering the actual roll cost vs. the fair value of the roll (or the replication drag of one active contract while it was active, based on whether the contract became richer or cheaper during that time).

I don't know if the ARR in the plot is a term repo rate or an overnight rate. I also didn't adjust for different day count conventions.

[comeinvest]

Nonetheless, I do have some relevant data from this period. Unfortunately it is incomplete for all roll periods for the years 2023 and 2024. I have data for ZT, Z3N, ZF, ZN, ZB, and UB for all the dates shown below. Here are the dates:

tyh23-tyu23 "roll period": 23,24,25,26,31 May 2023,
tyu23-tyz23 "roll period": 24,25 August 2023, 1,2 September 2023
tyz23-tyh24 "roll period": 30 November 2023, 1 December 2023
tyh24-tyu24 "roll period": 23 February 2024

My initial inspection of the numbers, following your basic procedure of finding the replication cost by taking into account the "time-weighted" IRR of the "old" contract and the IRR of the "new" contract, yields similar results. In my calculations, I use the IRR spread to Tbills directly, bypassing the Iast part of your method. I can share the numbers using your method if you like, but the downside here is that I effectively only have 2 "true" roll periods: tyh23->tyu23 and tyu23->tyz23. Please let me know which of those dates are most relevant for the "actual roll period", and which could be used for the "Bloomberg roll day". As an aside, I thought that tyh5 expires on 31 March? Is 21 March a typo? My calcs show a last trade date of 20 March, and last delivery date of 31 March. I take the last delivery date to mean the same as expiration. If my calcs are incorrect, I will want to adjust them, moving forward.

I conducted another, slightly different investigation: I took the tyu24 CTD, and looked at the performance from 25 May until 25 August. I compared it to the cash and futures position. The CTD I calculated for 25 May was 91282CGS4, which had a 3.625% coupon. There were no coupons paid in the 25 May - 25 August period. I precalculated the dirty price, so that I did not have to worry about accrued interest calculations. Caveat: there was a CTD switch from *CGS4 to *CHF1 during that time period. *CHF1 was first issued as a 7 year note on 31 May 2023, so I don't have any data for it (does not enter the Monthly statement of the public debt until June). Nonetheless, we can look at the 92 day period of relative performance of holding the initial CTD, versus the futures and cash position.

TYU23

Code: Select all

Date:		Treasury dirty price:	Futures Price:	Closest Tbill Delta:	Closest Tbill Yield:
2023-05-25	99.05			113.313		5 days			5.135
2023-08-25	97.1551			109.453		-				-

The Tbill available closest to the expiration of the TYU23 contract was 5 days; I think a difference of 5 days in the yield curve is not much, so we can assume a yield of 5.135 on a cash instrument for 92 days. This acts as the Tbill rate for the entire period that the futures contract is held.

CTD return: 97.1551/99.0504-1 = -0.0191 -> -1.91%
Futures return + Tbill return: 109.453/113.313-1 + (92 days/360)*0.05135 = -0.0209 -> -2.09%

"Simplified" cost of replicating the return of *CGS4 with TY futures:
CTD return - Futures return = -1.91 + 2.09 = 0.18% in Q3.
Naively annualizing this, we get 0.72% (in 360 day count "units").
It's only 1 data point for one note in the basket, and not averaged over 4 roll periods. The note used for this analysis also rotates out of being CTD, so maybe not the fairest comparison. Also, I believe this takes place during the actual roll period, correct? Not the 1 June to 1 August roll period that Bloomberg uses, that I infer from your post.

Is there anything I am missing with the CTD vs Futures and Tbill calculations above? Also, feel free to check the futures and note prices in that calculation, if you have access to them. I suspect prices like 113.313 actually correspond to 113.3125; let me know if you can confirm. I might have a look at *CGS4 and *CHF1 with the IBKR API, to see if my treasury prices that I calculated from Tullet Prebon yield data are substantially different. My data is not Bloomberg data, but maybe it is at least indicative.

Interactive Brokers shows the last trading date as "expiration date", but it really doesn't matter because it's just terminology. I used the days to delivery in my calculation.
I think there is a typo in your contracts: tyh23-tyu23 does not really exist, you are missing "m".
My understanding is Bloomberg has a default roll date on the first of the delivery month, which is hard to understand as it is after the first intention day. I assumed a most active roll day 7 days before that day (5 calendar days before first intention day), but like you know it is a roll period, and the price of the roll can fluctuate, so you will get different results, or you can take volume-weighted averages for the roll cost in backtests. It would be nice if you can independently replicate the results; but because of all the unknowns and delivery options, I think comparing CTD, cash treasuries or even an ETF or CMT data set to actual futures performance might be more realistic.
I'm not sure about prices like 113.313. Interactive Brokers would show 113'3125 as 113'312 which I thought is a general convention, not 113'313. Perhaps with the decimal point it is actually a decimal.

I am still missing data regarding the typical value of delivery options, do you have any references? I only found studies from the 1980ies and early 1990ies which showed ca. 0.25% to 0.75% p.a. fair adjustment of the roll price for delivery optionality.

[comeinvest]

A 2023 paper on the quality option value: https://faculty.fordham.edu/rchen/JFI-Chen-Leisikow.pdf
It suggests that the option value is lower for shorter maturity tenors. However, the quality option value seems to rise with the level of interest rates.
The 2015 paper https://www.stern.nyu.edu/sites/default ... 0paper.pdf has estimates for the Schatz, Bobl, Bund, and Buxl futures. I think the Bobl is similar to the ZF, and the Bund is similar to the ZN.

[unemployed_pysicist]

I think there is a typo in your contracts: tyh23-tyu23 does not really exist, you are missing "m".

...

I am still missing data regarding the typical value of delivery options, do you have any references? I only found studies from the 1980ies and early 1990ies which showed ca. 0.25% to 0.75% p.a. fair adjustment of the roll price for delivery optionality.

Yes, that is indeed a typo. I will fix in my original post.

Unfortunately, I don't have any concrete data for the value of the delivery options expressed as a percentage per annum over long time frames. I don't recall if Burghardt gives any numbers. The edition I have is from 2000, so if there are any numbers in his book, it would not include the 2000-present period. I am still investigating how to calculate them from live market data using the methods provided in his book.

[comeinvest]

I think there is a typo in your contracts: tyh23-tyu23 does not really exist, you are missing "m".

...

I am still missing data regarding the typical value of delivery options, do you have any references? I only found studies from the 1980ies and early 1990ies which showed ca. 0.25% to 0.75% p.a. fair adjustment of the roll price for delivery optionality.

Yes, that is indeed a typo. I will fix in my original post.

Unfortunately, I don't have any concrete data for the value of the delivery options expressed as a percentage per annum over long time frames. I don't recall if Burghardt gives any numbers. The edition I have is from 2000, so if there are any numbers in his book, it would not include the 2000-present period. I am still investigating how to calculate them from live market data using the methods provided in his book.

To be fair, after glancing over some of the papers that examine the value of delivery options, I think this is not an easy task. CME basically created a field of computational science in and by itself by way of their treasury futures delivery process definition. I'd be happy to see your estimates, but in the interim I have a feeling that for practical purposes of mHFEA practitioners in this thread, estimating or monitoring futures replication cost from backtest comparisons to cash treasury datasets, or from comparisons to the actual performance of active CTDs (or by tracking the performance of the treasury that was the CTD when a particular futures contract became active) with the caveat that an additional adjustment may be required to adjust for the different performance of CTDs vs. an unbiased selection of cash treasuries, might be more realistic.
I would however say that the theoretical replication cost calculated from roll data before considering delivery optionalities would be a lower limit of the actual replication cost.
I don't know who came up with that stuff, but why didn't they create futures that are cash settled to an index of treasuries within a maturity range, and that's it. Life could be easy.

[unemployed_pysicist]

I calculated some more returns for the May 2023 to August 2023 roll period.

I wanted to check if there was an appreciable difference between the roll dates to avoid physical delivery (e.g., 25 May to 25 August) versus the Bloomberg roll dates for ZN (TYU23). I have some data for 31 May and 1 September 2023, probably close enough to the Bloomberg roll date. I think these days are after the first intention day, where we might expect different behavior in the CTD vs cash and futures, since those who are rolling long positions in futures contracts have "moved on" by first intention day.

TYU23

Code: Select all

Date:		Cusip:	Treasury dirty price:	Futures Price:		Closest Tbill Delta:	Closest Tbill Yield:
2023-05-31	*CGS4	100.1041		114.484			5 days			5.2
2023-09-01	*CGS4	97.761			110.125			-				-

*CGS4 was the CTD on 31 May, replaced by *CHF1 some time between 31 May and 25 August. There were no coupons for *CGS4 during this period. Here is the return of just holding the original CTD vs the futures contract + tbills:

CTD return: 97.761/100.1041 - 1 = -0.0234
Futures + Cash return: 110.125/114.484 - 1 + 0.052*93/360 = -0.0246

CTD return - (Futures + Cash return) = 0.0012%, or 12 bps. Naively Annualizing this number gives 48 bps, comparable to the 49 bps result that forums poster Comeinvest arrived at. Note that in my previous calculation, using more realistic roll dates of 25 May and 25 August, I found a 18 bps difference between the return of the original CTD vs cash and futures, which is 6 bps higher than when using the Bloomberg roll dates. Please let me know if you see anything wrong with these return calculations.

Again, this is just one note, for one roll period, so we should be careful drawing too many conclusions from this one calculation. I will try to post some additional returns for different contracts from this roll period in a follow up post. I am having difficulty believing the results that I am finding for the shorter contracts for this particular roll period, so I would appreciate another set of eyes on these calculations.

[comeinvest]

I calculated some more returns for the May 2023 to August 2023 roll period.

I wanted to check if there was an appreciable difference between the roll dates to avoid physical delivery (e.g., 25 May to 25 August) versus the Bloomberg roll dates for ZN (TYU23). I have some data for 31 May and 1 September 2023, probably close enough to the Bloomberg roll date. I think these days are after the first intention day, where we might expect different behavior in the CTD vs cash and futures, since those who are rolling long positions in futures contracts have "moved on" by first intention day.

TYU23

Code: Select all

Date:		Cusip:	Treasury dirty price:	Futures Price:		Closest Tbill Delta:	Closest Tbill Yield:
2023-05-31	*CGS4	100.1041		114.484			5 days			5.2
2023-09-01	*CGS4	97.761			110.125			-				-

*CGS4 was the CTD on 31 May, replaced by *CHF1 some time between 31 May and 25 August. There were no coupons for *CGS4 during this period. Here is the return of just holding the original CTD vs the futures contract + tbills:

CTD return: 97.761/100.1041 - 1 = -0.0234
Futures + Cash return: 110.125/114.484 - 1 + 0.052*93/360 = -0.0246

CTD return - (Futures + Cash return) = 0.0012%, or 12 bps. Naively Annualizing this number gives 48 bps, comparable to the 49 bps result that forums poster Comeinvest arrived at. Note that in my previous calculation, using more realistic roll dates of 25 May and 25 August, I found a 18 bps difference between the return of the original CTD vs cash and futures, which is 6 bps higher than when using the Bloomberg roll dates. Please let me know if you see anything wrong with these return calculations.

Again, this is just one note, for one roll period, so we should be careful drawing too many conclusions from this one calculation. I will try to post some additional returns for different contracts from this roll period in a follow up post. I am having difficulty believing the results that I am finding for the shorter contracts for this particular roll period, so I would appreciate another set of eyes on these calculations.

1. I'm not certain if Corey's sawtooth chart actually assumes a roll date on the first of the delivery month, nor if Bloomberg actually has an "official" default roll date, nor if and when the user can deviate from this default. I just inferred that as a possibility from another remark by Corey in the RR forum regarding another chart.
2. As a base scenario I would assume that after First Intention day, the remaining return from the IRR is amortized by the "bagholders" at constant CAGR until delivery, and that the IRR of the active CTD is amortized at constant CAGR, which is what the exponential upward slope of the IRR sawtooth suggests (due to shrinking time to first intention day, and time being in the denominator). Common sense would also dictate that there is never a jump in IRR, as this would lead to chaotic market behavior.
3. However special supply/demand dynamics might be going on during the roll period.
4. If your data point is the rule, and Corey's charts actually reflect a roll date after first intention day, it would mean that Corey's futures drag vs. CTD chart and CAGR estimates are much too optimistic, which might explain that his results are inconsistent with other backtests and data points. 18 bps is 50% more than 12 bps, so a naive adjustment of Corey's replication drag estimate which I think was around 35 bps, would be 53 bps, still missing the effects of delivery options and others, but closer to other backtests and data points.
5. I have yet to fully rationalize the rates that the sawtooth pattern establishes from the micro-economical viewpoint of rational actors. It would appear that the long position "bagholders" after first intention day get rewarded with a positive risk-free return from the inverse basis trade, and the post-intention-day long basis traders with negative returns, which leads me to believe that the hedge funds implementing the basis trade are out at this point. So who are the people how hold long and the short positions until delivery? The long position holders must be playing some inverse basis trade arbitrage, taking over at roll time from the hedge funds that play the traditional basis trade. And why do the hedge funds implementing the basis trade not hold the short positions until delivery and deliver their cash treasuries that they use to hedge their short futures into the contracts at time of delivery, in which case the sawtooth would never develop, and the original IRR would be amortized at linear CAGR until delivery. Is it because it would create a supply shock at time of delivery, when the bagholders would dump their CTDs that they never wanted? Perhaps the "bagholders" are different kind of actors that are allowed to hold physical treasuries.
6. Never forget that none of the futures vs. CTD simulations reflect the value of delivery options, nor the bias from tracking CTDs that have special supply/demand dynamics, and not non-CTD treasuries.

This definitely requires more investigation.

This is interesting, but for practical purposes I find the simulations of futures vs. ETF or high quality CMT based return data series more realistic, when carefully crafted and adjusted.

[comeinvest]

My 91-day 1000-11000 SPX options spreads filled today at 9888.15 with consistent fills at the same price throughout the day, which is a 4.615% annualized daily compounded rate (365 day count), or 4.537% annualized coupon equivalent yield, if my math and my understanding of compounded yields is right.
I compute the annualized daily rate as "(ending balance - purchase price) ^{(365 / #days)} - 1", and the coupon equivalent yield as "((ending balance - purchase price) / purchase price) * 365 / #days".

The treasury data show a 13-week T-bills coupon equivalent yield of 4.29%, and CME shows a 3-month Term SOFR of 4.30%.

I remember the SPX options had some "issues" at the end of last year; but overall it would appear that futures have more issues since the increased regulation post GFC. Were it not for tax reasons, at some point it might be worth considering buying VGIT (0.05% ER) financed with SPX options spreads instead of ZF and ZN futures in the taxable accounts. On top of possibly lower replication cost, you know and can monitor your financing rate on an ongoing basis with complete transparency, which based on our recent discussion can hardly be said of treasury futures. But people paying federal taxes are probably still better off using futures (as discussed early in this thread).

[klaus14]

My 91-day 1000-11000 SPX options spreads filled today at 9888.15 with consistent fills at the same price throughout the day, which is a 4.615% annualized daily compounded rate (365 day count), or 4.537% annualized coupon equivalent yield, if my math and my understanding of compounded yields is right.
I compute the annualized daily rate as "(ending balance - purchase price) ^{(365 / #days)} - 1", and the coupon equivalent yield as "((ending balance - purchase price) / purchase price) * 365 / #days".

The treasury data show a 13-week T-bills coupon equivalent yield of 4.29%, and CME shows a 3-month Term SOFR of 4.30%.

I remember the SPX options had some "issues" at the end of last year; but overall it would appear that futures have more issues since the increased regulation post GFC. Were it not for tax reasons, at some point it might be worth considering buying VGIT (0.05% ER) financed with SPX options spreads instead of ZF and ZN futures in the taxable accounts. On top of possibly lower replication cost, you know and can monitor your financing rate on an ongoing basis with complete transparency, which based on our recent discussion can hardly be said of treasury futures. But people paying federal taxes are probably still better off using futures (as discussed early in this thread).

You can buy BOXA

[comeinvest]

My 91-day 1000-11000 SPX options spreads filled today at 9888.15 with consistent fills at the same price throughout the day, which is a 4.615% annualized daily compounded rate (365 day count), or 4.537% annualized coupon equivalent yield, if my math and my understanding of compounded yields is right.
I compute the annualized daily rate as "(ending balance - purchase price) ^{(365 / #days)} - 1", and the coupon equivalent yield as "((ending balance - purchase price) / purchase price) * 365 / #days".

The treasury data show a 13-week T-bills coupon equivalent yield of 4.29%, and CME shows a 3-month Term SOFR of 4.30%.

I remember the SPX options had some "issues" at the end of last year; but overall it would appear that futures have more issues since the increased regulation post GFC. Were it not for tax reasons, at some point it might be worth considering buying VGIT (0.05% ER) financed with SPX options spreads instead of ZF and ZN futures in the taxable accounts. On top of possibly lower replication cost, you know and can monitor your financing rate on an ongoing basis with complete transparency, which based on our recent discussion can hardly be said of treasury futures. But people paying federal taxes are probably still better off using futures (as discussed early in this thread).

You can buy BOXA

BOXA provides a tax-advantaged exposure to the aggregate bond index. It has nothing to do with portfolio level leverage or with the topic of this thread, nothing that I can see.

[klaus14]

You can buy BOXA

BOXA provides a tax-advantaged exposure to the aggregate bond index. It has nothing to do with portfolio level leverage or with the topic of this thread, nothing that I can see.

i am saying instead of VGIT, you can buy BOXA then you won't have the tax issue since it doesn't distribute.

[comeinvest]

BOXA provides a tax-advantaged exposure to the aggregate bond index. It has nothing to do with portfolio level leverage or with the topic of this thread, nothing that I can see.

i am saying instead of VGIT, you can buy BOXA then you won't have the tax issue since it doesn't distribute.

If you assume that BOXA both survives 30 years or whatever your investment horizon, and that it fits your needs during that entire time, or else you will face a horrendous tax bill in some year, along with annual fees all along. I will never do anything like that.

[comeinvest]

Data from this paper https://www.anderson.ucla.edu/sites/def ... File70.pdf

Looks like even the long-run average is ca. 60 bps relative to actual repo rates (ARR). (The average size of the funding basis is 58.70 basis points for the 1991–2018 period, 58.79 basis points for the 1991–2007 pre-crisis period, and 58.56 basis points for the 2008–2018 post-crisis period.)
Roll your treasury futures on one of the 300-400 bps crisis points in the Funding Basis chart, and you are out 1% of notional value for one quarterly roll only, possibly the entire expected annual term premium at the 5y level or perhaps twice the term premium depending on your assumptions.

How about ARR vs. T-bill rates? Corey from the "Return Stacked ETFs" says in the RR forum https://community.rationalreminder.ca/t ... vestwithme "If you take the “Actual Repo Rate” and subtract the T-Bill rate over the last 10 years and take the long-term average, guess what you get? About 25bp." (He also seems to be trying to make a point in the RR forum using the ARR as a validation of his empirical IRR, implying that both should be comparable. Given the evidence in the charts below and the theoretical rationalization of IRR spreads in the literature, this makes absolutely no sense, the logic seems plain wrong. This argumentation and citing the ARR as futures replication cost is very misleading, frankly.)

I personally am not quite understanding conceptually how ARR (repo) can be different from SOFR; isn't SOFR by definition the observed rate of borrowing money secured by treasuries, and isn't repo by definition the mechanism of lending or borrowing money secured by treasuries? If somebody can help me here.

In any case, I think we have to recalibrate our expectations of mHFEA, as the replication cost seems to be significantly higher than the 0% to 0.25% originally cited in this thread. Perhaps treasury futures are only suitable for short-term speculation or hedging against interest rate movements, but not for efficient long-term tracking to capture the term premia? (At least not if term premia are muted in the future, which is widely predicted.)
I have to look up what evidence precisely was presented early in the thread for a low replication cost, and reconcile the evidence. I remember there was a publication by CME that showed treasury futures performance replicating cash treasuries very closely, but I can't find it any more at the moment.

It also puzzles me that none of the literature that I saw that examines IRR elaborates on the sawthooth pattern of the IRR around First Intention days. Currently the absolute IRR seems to be in the 3.x% range, below other risk-free rates, and would definitely not tell the full truth about futures replication cost, quite the opposite. I'm curious if a similar pattern existed in the 1991-2018 period that is examined in the paper.

The charts in the UCLA paper seems to contradict the charts in the "Hedge Funds and the Treasury Cash-Futures
Disconnect" paper https://www.financialresearch.gov/worki ... onnect.pdf .





They cite these numbers for the value of delivery options of the 5-year note contract, saying that it is higher for longer-term contracts. (I'm not sure if those are per quarter or annualized. Taking the footnote literally, I read it so that it is per quarter, or until expiration.)

[comeinvest]

In light of the evidence from the previous post let's compare treasury futures to SOFR futures (or swaps) by way of the SOFR forward rate chart. The SOFR chart is created from SOFR futures, which means the replication cost is already "baked into" the chart: "what you see is what you get". I think it is also adjusted for futures convexity (physicist, can you confirm?) We can see that the average SOFR forward rate in the 0-5y segment (compare to the 5-year swap rate of 3.721%) is less than the 5-year treasury rate (4.006%), but we can roll down the curve in the 2-5y (2027-2030) segment towards the currently implied terminal rate (ca. 3.5% where the curve bottoms out) even after replication cost, replication cost relative to treasuries being the swap spread in this case. Remember that the short-term rates are controlled by the expected short-term Fed policy path, but the slope in the intermediate part of the curve is thought to be an indication of the term premium.
The swap spread at 5y seems to be about 0.28% (more details later), and the replication cost with SOFR swaps (the swap spread) or with a strip of SOFR futures is "locked in" and 100% known in advance when the position is established, for the entire length of the swap or strip, similar to a bond yield to maturity which is 100% known in advance.
Compare to quarterly treasury futures rolls that are very intransparent, sometimes unstable even within one roll period (as per my Interactive Brokers calendar roll chart a few posts up) as trillions(?) of $ are rolled over quarterly with unstable supply/demand liquidity, almost impossible to reverse-engineer without institutional data, subject to spikes and elevated levels over years as seen in the chart in the previous post, and which seem to shift money into the pockets of highly profitable hedge funds.

With a SOFR forward rate at 5y (2030) of 3.721% and assuming a "terminal" SOFR bottom at 3.4%, the carry of a SOFR strip economically equivalent to a 5y zero coupon bond after complete disinversion of the curve would be 3.721% - 3.4% = 0.321% p.a. (The carry return of a zero coupon bond is mathematically the forward rate x years forward where x is the maturity of the bond. The implied financing rate of SOFR futures or swaps is 91-day SOFR compounded in arrears, because that is what the futures and swaps settle to. The SOFR financing rate under assumption of a static yield curve [after complete disinversion] would be the current short-term rate. [I am taking the bottom of the forward curve, minus 10 bps, as estimated short-term rate after complete disinversion of the curve.])
Compare this SOFR carry to the ca. 0.28% replication cost (swap spread) relative to treasuries (already baked into above numbers which are net of replication cost). If equivalent 5y treasuries have a carry of 0.321% + 0.28% -> ca. 0.6%, and the long-run replication cost of treasury futures is for example 0.65% (ca. 0.6% based on what the paper cited in the previous post suggests, plus 0.05% for CTD idiosyncrasies and a small amount of excess trading cost), then with treasury futures I would barely break even in the long run, actually slowly lose money over time; the treasury futures replication cost would just about eat all the carry returns from the treasuries rolling down the yield curve.
This is based on the IRR estimate relative to ARR in the UCLA paper. I does not reflect the ARR premium to T-bill rates, nor the curious phenomenon of the sawtooth pattern in the IRR chart. If both of those additional drags turn out to be real, then there would be a significantly negative long-run net return from treasury futures, even with the relatively "nice" slope of the current yield curve with a 3.4% terminal short-term SOFR rate assumption. A bit of an uphill battle.

Lately I thought I may get a bit fleeced with my SOFR futures due to the negative swap spread. Now I'm realizing they may be the best of two evils.

[ipparkos]

In any case, I think we have to recalibrate our expectations of mHFEA, as the replication cost seems to be significantly higher than the 0% to 0.25% originally cited in this thread. Perhaps treasury futures are only suitable for short-term speculation or hedging against interest rate movements, but not for efficient long-term tracking to capture the term premia? (At least not if term premia are muted in the future, which is widely predicted.)
I have to look up what evidence precisely was presented early in the thread for a low replication cost, and reconcile the evidence. I remember there was a publication by CME that showed treasury futures performance replicating cash treasuries very closely, but I can't find it any more at the moment.

I cannot follow your detailed calculations, but a clear conclusion is that the replication cost is much higher than the earlier numbers on this thread. Even Corey Hoffstein's calculation indicates that, and you find the cost to be even higher if I have some grasp of the latest discussion. I think the main problem with all these, even beyond the high costs, is the need to constantly monitor the treasury futures strategy. Replication performance is variable, and we might have even higher costs in the future. At the safest level, I should give up on treasury futures altogether. I rely on others' analysis right now, and I might not even want to bother for that in the future. But if I still want to take some risk on this front, do you think rolling TN alone is sensible?.

My logic is, 10-years do not optimize the theoretical premium, but
- they are better than the LTTs in the original HFEA, hence keep the spirit of the "m" in mHFEA
- they currently survive the replication cost, and are expected (but of course not guaranteed) to be more resilient than shorter terms in the future.

[comeinvest]

In any case, I think we have to recalibrate our expectations of mHFEA, as the replication cost seems to be significantly higher than the 0% to 0.25% originally cited in this thread. Perhaps treasury futures are only suitable for short-term speculation or hedging against interest rate movements, but not for efficient long-term tracking to capture the term premia? (At least not if term premia are muted in the future, which is widely predicted.)
I have to look up what evidence precisely was presented early in the thread for a low replication cost, and reconcile the evidence. I remember there was a publication by CME that showed treasury futures performance replicating cash treasuries very closely, but I can't find it any more at the moment.

I cannot follow your detailed calculations, but a clear conclusion is that the replication cost is much higher than the earlier numbers on this thread. Even Corey Hoffstein's calculation indicates that, and you find the cost to be even higher if I have some grasp of the latest discussion. I think the main problem with all these, even beyond the high costs, is the need to constantly monitor the treasury futures strategy. Replication performance is variable, and we might have even higher costs in the future. At the safest level, I should give up on treasury futures altogether. I rely on others' analysis right now, and I might not even want to bother for that in the future. But if I still want to take some risk on this front, do you think rolling TN alone is sensible?.

My logic is, 10-years do not optimize the theoretical premium, but
- they are better than the LTTs in the original HFEA, hence keep the spirit of the "m" in mHFEA
- they currently survive the replication cost, and are expected (but of course not guaranteed) to be more resilient than shorter terms in the future.

I personally won't make a change at the moment until some of the conflicting results are reconciled.

I have yet to reconcile the UCLA paper (recent post) with the papers "Basis Trades and Treasury Market Illiquidity" https://www.financialresearch.gov/brief ... Trades.pdf and "Hedge Funds and the Treasury Cash-Futures Disconnect" https://www.financialresearch.gov/worki ... onnect.pdf by OFR. The "Hedge Funds and the Treasury Cash-Futures Disconnect" was cited earlier in this thread for evidence of low replication cost. At first glance it would appear that only one (UCLA, OFR) can be correct. I can't see the reason for the difference at first glance. The papers are quite detailed and should answer our questions. Have to read the paper, how they treat CTD switches and the value of delivery options, etc.
Surprisingly, I can't see any evidence of the sawtooth pattern in any of the charts in the OFR papers, which per Corey's IRR charts from Bloomberg also happened during the 2018-2019 period which is part of the OFR backtest period.

Regarding your suggestion to use TN: I think your reasoning is similar to that of my earlier suggestion to use ZN, i.e. to maximize the "futures carry" defined as treasuries carry minus implementation carry. Which futures contract is optimal, or if SOFR futures are better altogether, can only be answered after further examination. Reportedly swap spreads and basis spreads have similar origins. This paper https://home.treasury.gov/system/files/ ... Q32021.pdf has a good summary of swap spreads, and quantifies if and when they would be arbitraged away. In the OFR charts I see some evidence that the futures replication cost is relatively constant across the different contract tenors, while the swap spread seems to grow about linearly with maturity (except short swap maturities, which have different market forces as explained in the TBAC paper).
I still don't have any Bloomberg data. People with Bloomberg data could generate IRR charts or treasury minus replicating portfolio performance charts similar to Corey's chart, but with more details regarding the roll date, CTD switches, and other details, and for each tenor. I think that would provide useful information.

I any case based on my recent ad-hoc comparison, I think SOFR futures, up to 1 or 2 years but perhaps all the way to 5 years, are probably a viable mHFEA implementation choice as a fallback.

First 4 charts are from "Basis Trades and Treasury Market Illiquidity". Remaining 5 charts are from "Hedge Funds and the Treasury Cash-Futures Disconnect".

P.S.: Does anybody understand why the high open interest in Figure 3 (Convergence of cash and Treasury prices) only covers ca. 2 months and not 3 months? I thought it would cover the period while a contract is the active contract between the calendar rolls, i.e. ca. 120 to 30 days before delivery.

[ipparkos]

I any case based on my recent ad-hoc comparison, I think SOFR futures, up to 1 or 2 years but perhaps all the way to 5 years, are probably a viable mHFEA implementation choice as a fallback.

My concern about this is the long run. As I mentioned before, even if SOFR futures are fine now, who knows about tomorrow? I am OK giving up some of the return for the sake of being more hands off, if that is possible at all. Even if I am willing to follow these forums, who knows whether you and others will be willing to go through all this effort for decades? The reason I am considering TN is having some amount of premium over the cash rate that is unlikely to be cancelled by the replication cost, without going all the way to UB. Even if TN does not optimize "futures carry" right now, I might be OK sticking with it as long as it is likely to provide some premium that is better than that of LTTs.

[comeinvest]

I any case based on my recent ad-hoc comparison, I think SOFR futures, up to 1 or 2 years but perhaps all the way to 5 years, are probably a viable mHFEA implementation choice as a fallback.

My concern about this is the long run. As I mentioned before, even if SOFR futures are fine now, who knows about tomorrow? I am OK giving up some of the return for the sake of being more hands off, if that is possible at all. Even if I am willing to follow these forums, who knows whether you and others will be willing to go through all this effort for decades? The reason I am considering TN is having some amount of premium over the cash rate that is unlikely to be cancelled by the replication cost, without going all the way to UB. Even if TN does not optimize "futures carry" right now, I might be OK sticking with it as long as it is likely to provide some premium that is better than that of LTTs.

I have not seen evidence that TN is consistently better than ZF, ZN, or ZB. I'm also not sure why you think TN futures would require less ongoing monitoring than SOFR futures for example.

[unemployed_pysicist]

I think it is also adjusted for futures convexity (physicist, can you confirm?)

I believe this is the case.

Keep in mind that although the forward curve is downward sloping for the 2030-2027 segment, the zero coupon curve is flat to slightly upward sloping for this segment:



Treasury yield to maturity curve, for comparison (and sanity check):



It looks about flat to my eyes for 2030-2027. I lost my connection to the API when I was getting data, which is why you see a bunch of 0% yielding notes and bonds.

[unemployed_pysicist]

I have yet to reconcile the UCLA paper (recent post) with the papers "Basis Trades and Treasury Market Illiquidity" https://www.financialresearch.gov/brief ... Trades.pdf and "Hedge Funds and the Treasury Cash-Futures Disconnect" https://www.financialresearch.gov/worki ... onnect.pdf by OFR. The "Hedge Funds and the Treasury Cash-Futures Disconnect" was cited earlier in this thread for evidence of low replication cost. At first glance it would appear that only one (UCLA, OFR) can be correct. I can't see the reason for the difference at first glance. The papers are quite detailed and should answer our questions. Have to read the paper, how they treat CTD switches and the value of delivery options, etc.
Surprisingly, I can't see any evidence of the sawtooth pattern in any of the charts in the OFR papers, which per Corey's IRR charts from Bloomberg also happened during the 2018-2019 period which is part of the OFR backtest period.

I saw something like a sawtooth pattern some years ago when I tried to replicate the OFR papers (as best as I could with publicly available data):



At the time, I figured it was a problem with how the futures contracts were stitched together from the data source (yahoo finance). Additionally, if felt like there were too many other flaws in my approach, so I ultimately gave up on trying to replicate the figure. I think this was for five year note futures, but I'm not entirely sure.

I am making no claims for the accuracy of this plot and the one below, merely curious if you also see a quarterly sawtooth pattern or if my eyes are fooling me. It sure looks like it for 2012-2013, but other periods I am not so sure.

Using a 14 day sma like in the OFR paper did not bring it much closer to their results:

[comeinvest]

I think it is also adjusted for futures convexity (physicist, can you confirm?)

I believe this is the case.

Keep in mind that although the forward curve is downward sloping for the 2030-2027 segment, the zero coupon curve is flat to slightly upward sloping for this segment:



Treasury yield to maturity curve, for comparison (and sanity check):



It looks about flat to my eyes for 2030-2027. I lost my connection to the API when I was getting data, which is why you see a bunch of 0% yielding notes and bonds.

Yes, but my point was that the slope of the forward curve in the intermediate-term is indicative of a term premium, not the slope in the short term. The short-term is controlled by expectations of immediately impending Fed policy, where the existence or the magnitude of a term premium is hard to observe. I therefore approximated the short-term forward rates ca. 2 years forward by the current bottom of the forward rate curve, minus a small additional estimated term premium (not observable) for the short term, as a "target" for SOFR futures' expected final settlement, and as a target for a static yield curve that can serve as a base scenario for the term structure in the future. I copied this general approach from the ING "Rates Spark" yield curve analysis newsletters which I read regularly.
With each one SOFR futures contract you are literally and intuitively rolling down the forward curve, not the zero coupon curve.
So for example the current 5y SOFR futures contract would ride the forward curve from the ca. 3.75% at the 5y mark (after convexity adjustment) down to 3.4% as a target assuming complete uninversion of the front end and a static yield curve thereafter. The term premium and the profit arising from holding the 5y SOFR futures contract would therefore be positive in this base scenario. (Anything can happen to interest rates between now and then, and due to interest rate fluctuations this point target will very unlikely be realized. But I think it is a reasonable base case assumption for the long-run carry returns. 3.4% would also be within the range of terminal short-term rates per the dot plot [current dot plot average indicates 3%], and near what the Fed thinks might be the "neutral" rate in the future.)
The SOFR futures strip would initially generate less carry than that because of the 0-18 month inversion (where less carry might be compensated by higher returns from implied rate changes i.e. expected rate cuts, for perhaps equal total expected returns), but with the above interest rate model it would eventually (after complete disinversion) also generate exactly that carry: 3.75% minus 3.4%, as futures would be fed into the strip at the 5y mark and eventually be "consumed" (settled) at the short-term 3.4%. This is also exactly the carry return of an equivalent zero coupon bond financed at short-term rates.
Due to the inversion (and generally due to unknown Fed policy) we currently cannot easily observe the expected return of a 2y SOFR futures contract or strip, but I would assume that it will similarly be positive after uninversion of the curve, simply as an extrapolation of the generally positive slope of the forward curve between 1.5 years and up to ca. 15 years.

[comeinvest]

I have yet to reconcile the UCLA paper (recent post) with the papers "Basis Trades and Treasury Market Illiquidity" https://www.financialresearch.gov/brief ... Trades.pdf and "Hedge Funds and the Treasury Cash-Futures Disconnect" https://www.financialresearch.gov/worki ... onnect.pdf by OFR. The "Hedge Funds and the Treasury Cash-Futures Disconnect" was cited earlier in this thread for evidence of low replication cost. At first glance it would appear that only one (UCLA, OFR) can be correct. I can't see the reason for the difference at first glance. The papers are quite detailed and should answer our questions. Have to read the paper, how they treat CTD switches and the value of delivery options, etc.
Surprisingly, I can't see any evidence of the sawtooth pattern in any of the charts in the OFR papers, which per Corey's IRR charts from Bloomberg also happened during the 2018-2019 period which is part of the OFR backtest period.

I saw something like a sawtooth pattern some years ago when I tried to replicate the OFR papers (as best as I could with publicly available data):



At the time, I figured it was a problem with how the futures contracts were stitched together from the data source (yahoo finance). Additionally, if felt like there were too many other flaws in my approach, so I ultimately gave up on trying to replicate the figure. I think this was for five year note futures, but I'm not entirely sure.

I am making no claims for the accuracy of this plot and the one below, merely curious if you also see a quarterly sawtooth pattern or if my eyes are fooling me. It sure looks like it for 2012-2013, but other periods I am not so sure.

Using a 14 day sma like in the OFR paper did not bring it much closer to their results:

I can definitely see a quarterly sawtooth pattern in both charts, also for example in 2010 in your second chart. Keep in mind that the sawtooth movement is an overlay over an already fluctuating "base level" in Corey's IRR chart from Bloomberg, so it's naturally hard to identify, but I can definitely see it.

The curiosity is that I can't see anything like that in any of the OFR charts. The futures to treasury spread seems to converge more or less smoothly and at a constant pace between 120 days and delivery. Definitely something worth examining. I would have to read the approach in the paper in detail.

Not only the sawtooth pattern is missing, but also the IRR is inconsistent with the UCLA paper although they use almost the same backtest period. Per the OFR papers the IRR is in the 0% to 0.25% range in the long-run average. OFR uses "futures implied yields" comparing to T-bill yields in many charts and tables, so we have to examine and interpret their methodology. They also often use "second-to-nearest" or "second-to-deliver" contracts, the reason for which I have yet to understand. But regardless of whether front month contract or second-to-deliever contract, I can't see sawtooth patterns in their charts that show convergence to delivery.

[ipparkos]

My concern about this is the long run. As I mentioned before, even if SOFR futures are fine now, who knows about tomorrow? I am OK giving up some of the return for the sake of being more hands off, if that is possible at all. Even if I am willing to follow these forums, who knows whether you and others will be willing to go through all this effort for decades? The reason I am considering TN is having some amount of premium over the cash rate that is unlikely to be cancelled by the replication cost, without going all the way to UB. Even if TN does not optimize "futures carry" right now, I might be OK sticking with it as long as it is likely to provide some premium that is better than that of LTTs.

I have not seen evidence that TN is consistently better than ZF, ZN, or ZB. I'm also not sure why you think TN futures would require less ongoing monitoring than SOFR futures for example.

I naively assumed that the replication costs will not vary too much after a certain duration and there will be a ceiling to their long term average. This would mean the increased premium with the increased duration more likely will overcome the replication costs. I think especially the latter ceiling assumption is natural, I would not worry about replication costs if I had an asset whose expected return above the cash rate is 4%, for example. To be clear though, I do not have concrete numbers for any of this.

The closest quantitative "evidence" is Corey Hoffstein's numbers on RR: 2-years had a replication cost of 14 bp, 5-year 30bp, 10-year 33bp and 15-25 year (ZB) 23 bp. Costs are roughly proportional to duration at first, but taper off later (even decrease in this case) as in my naive assumption. You think these calculations are incomplete, but I would expect some pattern like this, or at the least the existence of a ceiling to the replication cost as I mentioned. Not to mislead anyone, I am pretty surprised by having this much replication cost in the first place, so the market very clearly does not behave as in my assumptions and expectations.

[comeinvest]

About 1/3 into the current active contract period, the June contract cheapest-to-deliver IRR are near risk-free rates of 4.3%, at least according to treasury analytics. Let's see what happens until end of May roll. Unfortunately I didn't take a screenshot right after the roll when the current contracts became active.
On the treasury analytics site futures prices seem to be relatively current, but treasury prices are somewhat stale.

[comeinvest]

I have not seen evidence that TN is consistently better than ZF, ZN, or ZB. I'm also not sure why you think TN futures would require less ongoing monitoring than SOFR futures for example.

I naively assumed that the replication costs will not vary too much after a certain duration and there will be a ceiling to their long term average. This would mean the increased premium with the increased duration more likely will overcome the replication costs. I think especially the latter ceiling assumption is natural, I would not worry about replication costs if I had an asset whose expected return above the cash rate is 4%, for example. To be clear though, I do not have concrete numbers for any of this.

The closest quantitative "evidence" is Corey Hoffstein's numbers on RR: 2-years had a replication cost of 14 bp, 5-year 30bp, 10-year 33bp and 15-25 year (ZB) 23 bp. Costs are roughly proportional to duration at first, but taper off later (even decrease in this case) as in my naive assumption. You think these calculations are incomplete, but I would expect some pattern like this, or at the least the existence of a ceiling to the replication cost as I mentioned. Not to mislead anyone, I am pretty surprised by having this much replication cost in the first place, so the market very clearly does not behave as in my assumptions and expectations.

I get your point, and that the replication cost might be relatively constant per tenor was my hope too. However I find the evidence is too weak to make a change in an impulsive reaction. Swap spreads for example are governed by some of the same underlying forces as leveraged treasuries implied financing cost, and swaps and treasury futures are indeed interchangeable for some market participants, which would suggest that leveraged treasuries should have replication cost more proportional to duration exposure, like swaps. I read that both swaps and long futures basis holders hedge their exposure with treasuries among other things.
Corey delivered just 4 data points kind of in a side note when pressed, not for all 6 tenors which might allow to see a pattern, and he obviously chose to not further engage in the discussion, even though (or perhaps exactly because) the discussion is very relevant to the viability / utility of the public funds that I think he sponsors. His reasoning logic was definitely both incomplete and flawed on multiple levels, and he did not provide the details of the data that his performance chart was based on, e.g. the roll date assumption (or the justification for the choice of roll dates outside the normal roll period), the delivery options, CTD switches, etc., so based on the sparse data with several unknowns, I personally don't give much credibility to this data point (just as a factual and evidence-based assessment, no personal offense to anybody). I mean if his data were only halfway correct, then I would be much less concerned about using treasury futures in the first place, including ZF and ZN.

[ipparkos]

I naively assumed that the replication costs will not vary too much after a certain duration and there will be a ceiling to their long term average. This would mean the increased premium with the increased duration more likely will overcome the replication costs. I think especially the latter ceiling assumption is natural, I would not worry about replication costs if I had an asset whose expected return above the cash rate is 4%, for example. To be clear though, I do not have concrete numbers for any of this.

The closest quantitative "evidence" is Corey Hoffstein's numbers on RR: 2-years had a replication cost of 14 bp, 5-year 30bp, 10-year 33bp and 15-25 year (ZB) 23 bp. Costs are roughly proportional to duration at first, but taper off later (even decrease in this case) as in my naive assumption. You think these calculations are incomplete, but I would expect some pattern like this, or at the least the existence of a ceiling to the replication cost as I mentioned. Not to mislead anyone, I am pretty surprised by having this much replication cost in the first place, so the market very clearly does not behave as in my assumptions and expectations.

I get your point, and that the replication cost might be relatively constant per tenor was my hope too. However I find the evidence is too weak to make a change in an impulsive reaction. Swap spreads for example are governed by some of the same underlying forces as leveraged treasuries implied financing cost, and swaps and treasury futures are indeed interchangeable for some market participants, which would suggest that leveraged treasuries should have replication cost more proportional to duration exposure, like swaps. I read that both swaps and long futures basis holders hedge their exposure with treasuries among other things.
Corey delivered just 4 data points kind of in a side note when pressed, not for all 6 tenors which might allow to see a pattern, and he obviously chose to not further engage in the discussion, even though (or perhaps exactly because) the discussion is very relevant to the viability / utility of the public funds that I think he sponsors. His reasoning logic was definitely both incomplete and flawed on multiple levels, and he did not provide the details of the data that his performance chart was based on, e.g. the roll date assumption (or the justification for the choice of roll dates outside the normal roll period), the delivery options, CTD switches, etc., so based on the sparse data with several unknowns, I personally don't give much credibility to this data point (just as a factual and evidence-based assessment, no personal offense to anybody). I mean if his data were only halfway correct, then I would be much less concerned about using treasury futures in the first place, including ZF and ZN.

I haven't touched my ZT and ZF yet, will wait until the roll at the least. Thanks for all the effort again.

In Corey's partial defense, RSSB only uses the 4 contracts whose replication costs he calculated and not the others, I think. Though your additional concerns about his methods remain.