This post is available as a PDF here.
Summary
- From 1981 to 2017, 10-year U.S. Treasury rates declined from north of 15% to below 2%.
- Since bond prices appreciate when rates decline, many have pointed towards this secular decline as a tailwind that created an unprecedented bull market in bonds.
- Exactly how much declining rates contributed, however, is rarely quantified. An informal poll, however, tells us that people generally believe the impact was significant (explaining >50% of bond returns).
- We find that while, in theory, investors should be indifferent to rate changes, high turnover in bond portfolios means that a structural mis-estimation of rate changes could be harvested.
- Despite the positive long-term impact of declining rates, coupon yield had a much more significant impact on long-term returns.
- The bull market in bonds was caused more by the high average rates over the past 30 years than declining rates.
On 9/30/1981, the 10-year U.S. Treasury rate peaked at an all-time high of 15.84%. Over the next 30 years, it deflated to an all-time low of 1.37% on 7/5/2016.
Source: Federal Reserve of St. Louis
It has been repeated in financial circles that this decline in rates caused a bull market in bond returns that makes past returns a particularly poor indicator of future results.
But exactly how much did those declining rates contribute?
We turned to our financial circle on Twitter[1] with a question: For a constant maturity, 10-year U.S. Treasury index, what percent of total return from 12/1981 through 12/2012 could be attributed to declining rates?
Little consensus was found.
Clearly there is a large disparity in views about exactly how much declining rates actually contributed to bond returns over the last 30 years. What we can see is that people generally think it is a lot: over 50% of people said over 50% of returns can be attributed to declining rates.
Well let’s dig in and find out.
Rates Down, Bonds Up
To begin, let’s remind ourselves why the bond / rate relationship exists in the first place.
Imagine you buy a 10-year U.S. Treasury bond for $100 at the prevailing 5% rate. Immediately after you buy, interest rates drop: all available 10-year U.S. Treasury bonds – still selling for $100 – are now offering only a 4% yield.
In every other way, except the yield being offered, the bond you now hold and the bonds being offered in the market are identical. Except yours provides a higher yield.
Therefore, it should be more valuable. After all, you are getting more return for your investment. And hence we get the inverse relationship between bonds and interest rates. As rates fall, existing bond values go up and as rates rise, existing bond values go down.
With rates falling by an average of 42 basis points a year over the last 35 years, we can imagine a pretty steady, and potentially sizable tailwind to returns.
Just How Much More Valuable?
In our example, exactly how much did our bond appreciate when rates fell? Or, to ask the question another way: how much would someone now be willing to buy our bond for?
The answer arises from the fact that markets loathe an arbitrage opportunity. Scratch that: markets love arbitrage. So much so that they are quickly wiped away as market participants jump to exploit them.
We mentioned that in the example, the bond you held and the bonds now being offered by the market were identical in every fashion except the coupon yield they offer.
Consider what would happen if the 4% bonds and your 5% bonds were both still selling for $100. Someone could come to the market, short-sell a 4% bond and use the $100 to buy your 5% bond from you. Each coupon period, they would collect $5 from the bond they bought from you, pay $4 to cover the coupon payment they owe from the short-sale, and pocket $1.
Effectively, they’ve created a free stream of $1 bills.
Knowing this to be the case, someone else might step in first and try to offer you $101 for your bond to sweeten the deal. Now they must finance by short-selling 1.01 shares of the 4% bonds, owing $4.04 each period and $101 at maturity. While less profitable, they would still pocket a free $0.86 per coupon payment.[2]
The scramble to sweeten the offer continues until it reaches the magic figure of $108.11. At this price, the arbitrage disappears: the cost of financing exactly offsets the extra yield earned by the bond.
Another way of saying this is that the yield-to-maturity of both bonds is identical. If someone pays $108.11 for the 5% coupon bond, they may receive a $5 coupon each period, but there will be a “pull-to-par” effect as the bond matures, causing the bond to decline in value. This effect occurs because the bond has a pre-defined payout stream: at maturity, you are only going to receive your $100 back.
Therefore, while your coupon yield may be 5%, your effective yield – which accounts for this loss in value over time – is 4%, perfectly matching what is available to other investors.
And so everyone becomes indifferent[3] to which bond they hold. The bond you hold may be worth more on paper, but if we try to sell it to lock in our profit, we have to reinvest at a lower yield and offsets our gain.
In a strange way, then, other than mark-to-market gains and losses, we should be largely indifferent to rate changes.
The Impact of Time
One very important aspect ignored by our previous example is time. Interest rates rarely gap up or down instantaneously: rather they move over time.
We therefore need to consider the yield curve. The yield curve tells us what rate is being offered for bonds of different maturities.
Source: Federal Reserve of St. Louis.
In the yield curve plotted above, we see an upward sloping trend. Buying a 7-year U.S. Treasury earns us a 2.25% rate, while the 10-year U.S. Treasury offers 2.45%.
Which introduces an interesting dynamic: if rates do not change whatsoever, if we buy a 10-year bond today and wait three years, our bond will appreciate in value.
Why?
The answer is because it is now a 7-year bond, and compared to other 7-year bonds it is offering 0.20% more yield.
In fact, depending on the shape of the yield curve, it can continue to appreciate until the pull-to-par effect becomes too strong. Below we plot the value of a 10-year U.S. Treasury as it matures, assuming that the yield curve stays entirely constant over time.
Source: Federal Reserve of St. Louis. Calculations by Newfound Research.
Unfortunately, like in our previous example, the amount of the bond gains in value is exactly equal to the level required to make us indifferent to holding the bond to maturity or selling it and reinvesting at the prevailing rate. For all intents and purposes, we could simply pretend we bought a 7-year bond at 2.45% and rates fell instantly to 2.25%. By the same logic as before, we’re no better off.
We simply cannot escape the fact that markets are not going to give us a free return.
The Impact of Choice
Again, reality is more textured than theory. We are ignoring an important component: choice.
In our prior examples, our choice was between continuing to hold our bond, or selling it and reinvesting in the equivalent bond. What if we chose to reinvest in something else?
For example:
- We buy a 2.45% 10-year U.S. Treasury for $100
- We wait three years
- We sell the now 7-year U.S. Treasury for $101.28 (assuming the yield curve did not change)
- We reinvest in 2.45% 10-year U.S. Treasuries, sold at $100
If the yield curve never changes, we can keep capturing this roll return by simply waiting, selling, and buying what we previously owned.
What’s the catch? The catch, of course, is that we’re assuming rates won’t change. If we stop for a moment, however, and consider what the yield curve is telling us, we realize this assumption may be quite poor.
The yield curve provides several rates at which we can invest. What if we are only interested in investing over the next year? Well, we can buy a 1-year U.S. Treasury at 0.85% and just hold it to maturity, or we could buy a 10-year U.S. Treasury for 2.45% and sell it after a year.
That is a pretty remarkable difference in 1-year return potential.
If the market is even reasonably efficient, then the expected 1-year return, no matter where we buy on the curve, should be the same. Therefore, the only way the 10-year U.S. Treasury yield should be so much higher than the 1-year is if the market is predicting that rates are going to go up such that the extra yield is exactly offset by the price loss we take when we sell the bond.
Hence a rising yield curve tells us the market is expecting rising rates. At least, that’s what the pure expectations hypothesis (“PEH”) says. Competing theories argue that investors should earn at least some premium for bearing term risk. Nevertheless, there should be some component of a rising yield curve that tells us rates should go up.
However, over the past 35 years, the average slope of the yield curve (measured as 10-year yields minus 2-year yields) has been over 100bp. The market was, in theory, was consistently predicting rising rates over a period rates fell.
Source: Federal Reserve of St. Louis. Calculations by Newfound Research.
Not only could an investor potentially harvest roll-yield, but also the added bump from declining rates.
Unfortunately, doing so would require significant turnover. We would have to constantly sell our bonds to harvest the gains.
While this may have created opportunity for active bond managers, a total bond market index typically holds bonds until maturity.
Turnover in a Bond Index
Have you ever looked at the turnover in a total bond market index fund? You might be surprised.
While the S&P 500 has turnover of approximately 5% per year, the Bloomberg Barclay’s U.S. Aggregate often averages between 40-60% per year.
Where is all that turnover coming from?
- Index additions (e.g. new issuances)
- Index deletions (e.g. maturing bonds)
- Paydowns
- Coupon reinvestment
If the general structure of the fixed income market does not change considerably over time, this level of turnover implies that a total bond market index will behave very similarly to a constant duration bond fund.
Bonds are technically held to maturity, but roll return and profit/loss from shifts in interest rates are booked along the way as positions are rebalanced.
Which means that falling rates could matter. Even better, we can test how much falling rates mattered by proxying a total bond index with a constant maturity bond index[4].
Specifically, we will look at a constant maturity 10-year U.S. Treasury index. We will assume 10-year Treasuries are bought at the beginning of each year, held for a year, and sold as 9-year Treasuries[5]. The proceeds will then be reinvested back into the new 10-year Treasuries. We will also assume that coupons are paid annually.
We ran the test from 12/1981 to 12/2012, since those dates represented both the highest and lowest end-of-year rates.
We will then decompose returns into three components:
- Coupon yield (“Coupon”)
- Roll return (“Roll”)
- Rate changes (“Shift”)
Coupon yield is, simply, the return we get from the coupon itself. Roll return is equal to the slope between 10-year and 9-year U.S. Treasuries at the point of purchase adjusted by the duration of the bond. Rate changes are measured as price return we achieve due to shifts in the 9-year rate from the point at which we purchased the bond and the point at which we are selling it.
This allows us to create a return stream for each component as well as identify each component’s contribution to the total return of the index.
Source: Federal Reserve of St. Louis. Calculations by Newfound Research
What we can see is that coupon return dominates roll and shift. On an annualized basis, coupon was 6.24%, while roll only contributed 0.24% and shift contributed 2.22%.
Which leaves us with a final decomposition: coupon yield accounted for 71% of return, roll accounted for 3%, and shift accounted for 26%.
We can perform a similar test for constant maturity indices constructed at different points on the curve as well.
| Total Return | % Contribution |
| Coupon | Roll | Shift | Coupon | Roll | Shift |
| 10-year | 6.24% | 0.24% | 2.22% | 71.60% | 2.84% | 25.55% |
| 7-year | 6.08% | 0.62% | 1.72% | 72.16% | 7.37% | 20.47% |
| 5-year | 5.81% | 0.65% | 1.29% | 75.01% | 8.38% | 16.61% |
Conclusion: Were Declining Rates Important?
A resounding yes. An extra 2.22% per year over 30+ years is nothing to sneeze at. Especially when you consider that this was the result of a very unique period unlikely to be repeated over the next 30 years.
Just as important to consider, however, is that it was not the most important contributor to total returns. While most people in our poll answered that decline in rates would account for 50%+ of total return, the shift factor only came in at 26%.
The honor of the highest contributor goes to coupon yield. Even though rates deflated over 30 years, the average yield was high enough to be, by far and away, the biggest contributor to returns.
The bond bull was not due to declining rates, in our opinion, but rather the unusually high rates we saw over the period.
A fact which is changing today. We can see this by plotting the annual sources of returns year-by-year.
Source: St. Louis Federal Reserve. Calculations by Newfound Research.
Note that while coupon is always a positive contributor, its role has significantly diminished in recent years compared to the influence of rate changes.
The consistency of coupon and the varying influence of shift on returns (i.e. both positive and negative) means that coupon yield actually makes an excellent predictor of future returns. Lozada (2015)[6] finds that the optimal horizon to use yield as a predictor of return in constant duration or constant-maturity bond funds is at twice the duration.
Which paints a potentially bleak picture for fixed income investors.
| Fund | Asset | Duration | TTM Yield | Predicted Return |
| AGG | U.S. Aggregate Bonds | 5.74 | 2.37% | 2.37% per year through 2028 |
| IEI | 3-7 Year U.S. Treasuries | 4.48 | 1.31% | 1.31% per year through 2025 |
| IEF | 7-10 Year U.S. Treasuries | 7.59 | 1.77% | 1.77% per year through 2032 |
| TLT | 20+ Year U.S. Treasuries | 17.39 | 2.56% | 2.56% per year through 2051 |
| LQD | Investment Grade Bonds | 8.24 | 3.28% | 3.28% per year through 2033 |
Source: iShares. Calculations by Newfound Research.
Note that we are using trailing 12-month distribution yield for the ETFs here. We do this because ETF issuers often amortize coupon yield to account for pull-to-par effects, making it an approximation of yield-to-worst. It is not perfect, but we don’t think the results materially differ in magnitude with any other measure: it’s still ugly.
The story remains largely the same as we’ve echoed over the past year: when it comes to fixed income, your current yield will be a much better predictor of returns than trying to guess about changing rates.
Coupon yield had 3x the influence on total return over the last 30 years than changes in rates did.
What we should be concerned about today is not rising rates: rather, we should be concerned about the returns that present low rates imply for the future.
And we should be asking ourselves: are there other ways we can look to manage risk or find return?
[1] Find us on Twitter! Newfound is @thinknewfound and Corey is @choffstein.
[2] It is $0.86 instead of $0.96 because they need to set aside $0.10 to cover the extra dollar they owe at maturity.
[3] This is a bit of a simplification as the bonds will have different risk characteristics (e.g. different durations and convexity) which could cause investors, especially those with views on future rate changes, to prefer one bond over the other.
[4] We made the leap here from total bond index to constant duration index to constant maturity index. Each step introduces some error, but we believe for our purposes the error is de minimis and a constant maturity index allows for greater ease of implementation.
[5] Since no 9-year U.S. Treasury is offered, we create a model for the yield curve using cubic splines and then estimate the 9-year rate.
[6] http://content.csbs.utah.edu/~lozada/Research/IniYld_6.pdf
Did Declining Rates Actually Matter?
By Corey Hoffstein
On April 10, 2017
In Popular, Risk & Style Premia, Term, Weekly Commentary
This post is available as a PDF here.
Summary
On 9/30/1981, the 10-year U.S. Treasury rate peaked at an all-time high of 15.84%. Over the next 30 years, it deflated to an all-time low of 1.37% on 7/5/2016.
Source: Federal Reserve of St. Louis
It has been repeated in financial circles that this decline in rates caused a bull market in bond returns that makes past returns a particularly poor indicator of future results.
But exactly how much did those declining rates contribute?
We turned to our financial circle on Twitter[1] with a question: For a constant maturity, 10-year U.S. Treasury index, what percent of total return from 12/1981 through 12/2012 could be attributed to declining rates?
Little consensus was found.
Clearly there is a large disparity in views about exactly how much declining rates actually contributed to bond returns over the last 30 years. What we can see is that people generally think it is a lot: over 50% of people said over 50% of returns can be attributed to declining rates.
Well let’s dig in and find out.
Rates Down, Bonds Up
To begin, let’s remind ourselves why the bond / rate relationship exists in the first place.
Imagine you buy a 10-year U.S. Treasury bond for $100 at the prevailing 5% rate. Immediately after you buy, interest rates drop: all available 10-year U.S. Treasury bonds – still selling for $100 – are now offering only a 4% yield.
In every other way, except the yield being offered, the bond you now hold and the bonds being offered in the market are identical. Except yours provides a higher yield.
Therefore, it should be more valuable. After all, you are getting more return for your investment. And hence we get the inverse relationship between bonds and interest rates. As rates fall, existing bond values go up and as rates rise, existing bond values go down.
With rates falling by an average of 42 basis points a year over the last 35 years, we can imagine a pretty steady, and potentially sizable tailwind to returns.
Just How Much More Valuable?
In our example, exactly how much did our bond appreciate when rates fell? Or, to ask the question another way: how much would someone now be willing to buy our bond for?
The answer arises from the fact that markets loathe an arbitrage opportunity. Scratch that: markets love arbitrage. So much so that they are quickly wiped away as market participants jump to exploit them.
We mentioned that in the example, the bond you held and the bonds now being offered by the market were identical in every fashion except the coupon yield they offer.
Consider what would happen if the 4% bonds and your 5% bonds were both still selling for $100. Someone could come to the market, short-sell a 4% bond and use the $100 to buy your 5% bond from you. Each coupon period, they would collect $5 from the bond they bought from you, pay $4 to cover the coupon payment they owe from the short-sale, and pocket $1.
Effectively, they’ve created a free stream of $1 bills.
Knowing this to be the case, someone else might step in first and try to offer you $101 for your bond to sweeten the deal. Now they must finance by short-selling 1.01 shares of the 4% bonds, owing $4.04 each period and $101 at maturity. While less profitable, they would still pocket a free $0.86 per coupon payment.[2]
The scramble to sweeten the offer continues until it reaches the magic figure of $108.11. At this price, the arbitrage disappears: the cost of financing exactly offsets the extra yield earned by the bond.
Another way of saying this is that the yield-to-maturity of both bonds is identical. If someone pays $108.11 for the 5% coupon bond, they may receive a $5 coupon each period, but there will be a “pull-to-par” effect as the bond matures, causing the bond to decline in value. This effect occurs because the bond has a pre-defined payout stream: at maturity, you are only going to receive your $100 back.
Therefore, while your coupon yield may be 5%, your effective yield – which accounts for this loss in value over time – is 4%, perfectly matching what is available to other investors.
And so everyone becomes indifferent[3] to which bond they hold. The bond you hold may be worth more on paper, but if we try to sell it to lock in our profit, we have to reinvest at a lower yield and offsets our gain.
In a strange way, then, other than mark-to-market gains and losses, we should be largely indifferent to rate changes.
The Impact of Time
One very important aspect ignored by our previous example is time. Interest rates rarely gap up or down instantaneously: rather they move over time.
We therefore need to consider the yield curve. The yield curve tells us what rate is being offered for bonds of different maturities.
Source: Federal Reserve of St. Louis.
In the yield curve plotted above, we see an upward sloping trend. Buying a 7-year U.S. Treasury earns us a 2.25% rate, while the 10-year U.S. Treasury offers 2.45%.
Which introduces an interesting dynamic: if rates do not change whatsoever, if we buy a 10-year bond today and wait three years, our bond will appreciate in value.
Why?
The answer is because it is now a 7-year bond, and compared to other 7-year bonds it is offering 0.20% more yield.
In fact, depending on the shape of the yield curve, it can continue to appreciate until the pull-to-par effect becomes too strong. Below we plot the value of a 10-year U.S. Treasury as it matures, assuming that the yield curve stays entirely constant over time.
Source: Federal Reserve of St. Louis. Calculations by Newfound Research.
Unfortunately, like in our previous example, the amount of the bond gains in value is exactly equal to the level required to make us indifferent to holding the bond to maturity or selling it and reinvesting at the prevailing rate. For all intents and purposes, we could simply pretend we bought a 7-year bond at 2.45% and rates fell instantly to 2.25%. By the same logic as before, we’re no better off.
We simply cannot escape the fact that markets are not going to give us a free return.
The Impact of Choice
Again, reality is more textured than theory. We are ignoring an important component: choice.
In our prior examples, our choice was between continuing to hold our bond, or selling it and reinvesting in the equivalent bond. What if we chose to reinvest in something else?
For example:
If the yield curve never changes, we can keep capturing this roll return by simply waiting, selling, and buying what we previously owned.
What’s the catch? The catch, of course, is that we’re assuming rates won’t change. If we stop for a moment, however, and consider what the yield curve is telling us, we realize this assumption may be quite poor.
The yield curve provides several rates at which we can invest. What if we are only interested in investing over the next year? Well, we can buy a 1-year U.S. Treasury at 0.85% and just hold it to maturity, or we could buy a 10-year U.S. Treasury for 2.45% and sell it after a year.
That is a pretty remarkable difference in 1-year return potential.
If the market is even reasonably efficient, then the expected 1-year return, no matter where we buy on the curve, should be the same. Therefore, the only way the 10-year U.S. Treasury yield should be so much higher than the 1-year is if the market is predicting that rates are going to go up such that the extra yield is exactly offset by the price loss we take when we sell the bond.
Hence a rising yield curve tells us the market is expecting rising rates. At least, that’s what the pure expectations hypothesis (“PEH”) says. Competing theories argue that investors should earn at least some premium for bearing term risk. Nevertheless, there should be some component of a rising yield curve that tells us rates should go up.
However, over the past 35 years, the average slope of the yield curve (measured as 10-year yields minus 2-year yields) has been over 100bp. The market was, in theory, was consistently predicting rising rates over a period rates fell.
Source: Federal Reserve of St. Louis. Calculations by Newfound Research.
Not only could an investor potentially harvest roll-yield, but also the added bump from declining rates.
Unfortunately, doing so would require significant turnover. We would have to constantly sell our bonds to harvest the gains.
While this may have created opportunity for active bond managers, a total bond market index typically holds bonds until maturity.
Turnover in a Bond Index
Have you ever looked at the turnover in a total bond market index fund? You might be surprised.
While the S&P 500 has turnover of approximately 5% per year, the Bloomberg Barclay’s U.S. Aggregate often averages between 40-60% per year.
Where is all that turnover coming from?
If the general structure of the fixed income market does not change considerably over time, this level of turnover implies that a total bond market index will behave very similarly to a constant duration bond fund.
Bonds are technically held to maturity, but roll return and profit/loss from shifts in interest rates are booked along the way as positions are rebalanced.
Which means that falling rates could matter. Even better, we can test how much falling rates mattered by proxying a total bond index with a constant maturity bond index[4].
Specifically, we will look at a constant maturity 10-year U.S. Treasury index. We will assume 10-year Treasuries are bought at the beginning of each year, held for a year, and sold as 9-year Treasuries[5]. The proceeds will then be reinvested back into the new 10-year Treasuries. We will also assume that coupons are paid annually.
We ran the test from 12/1981 to 12/2012, since those dates represented both the highest and lowest end-of-year rates.
We will then decompose returns into three components:
Coupon yield is, simply, the return we get from the coupon itself. Roll return is equal to the slope between 10-year and 9-year U.S. Treasuries at the point of purchase adjusted by the duration of the bond. Rate changes are measured as price return we achieve due to shifts in the 9-year rate from the point at which we purchased the bond and the point at which we are selling it.
This allows us to create a return stream for each component as well as identify each component’s contribution to the total return of the index.
Source: Federal Reserve of St. Louis. Calculations by Newfound Research
What we can see is that coupon return dominates roll and shift. On an annualized basis, coupon was 6.24%, while roll only contributed 0.24% and shift contributed 2.22%.
Which leaves us with a final decomposition: coupon yield accounted for 71% of return, roll accounted for 3%, and shift accounted for 26%.
We can perform a similar test for constant maturity indices constructed at different points on the curve as well.
Conclusion: Were Declining Rates Important?
A resounding yes. An extra 2.22% per year over 30+ years is nothing to sneeze at. Especially when you consider that this was the result of a very unique period unlikely to be repeated over the next 30 years.
Just as important to consider, however, is that it was not the most important contributor to total returns. While most people in our poll answered that decline in rates would account for 50%+ of total return, the shift factor only came in at 26%.
The honor of the highest contributor goes to coupon yield. Even though rates deflated over 30 years, the average yield was high enough to be, by far and away, the biggest contributor to returns.
The bond bull was not due to declining rates, in our opinion, but rather the unusually high rates we saw over the period.
A fact which is changing today. We can see this by plotting the annual sources of returns year-by-year.
Source: St. Louis Federal Reserve. Calculations by Newfound Research.
Note that while coupon is always a positive contributor, its role has significantly diminished in recent years compared to the influence of rate changes.
The consistency of coupon and the varying influence of shift on returns (i.e. both positive and negative) means that coupon yield actually makes an excellent predictor of future returns. Lozada (2015)[6] finds that the optimal horizon to use yield as a predictor of return in constant duration or constant-maturity bond funds is at twice the duration.
Which paints a potentially bleak picture for fixed income investors.
Source: iShares. Calculations by Newfound Research.
Note that we are using trailing 12-month distribution yield for the ETFs here. We do this because ETF issuers often amortize coupon yield to account for pull-to-par effects, making it an approximation of yield-to-worst. It is not perfect, but we don’t think the results materially differ in magnitude with any other measure: it’s still ugly.
The story remains largely the same as we’ve echoed over the past year: when it comes to fixed income, your current yield will be a much better predictor of returns than trying to guess about changing rates.
Coupon yield had 3x the influence on total return over the last 30 years than changes in rates did.
What we should be concerned about today is not rising rates: rather, we should be concerned about the returns that present low rates imply for the future.
And we should be asking ourselves: are there other ways we can look to manage risk or find return?
[1] Find us on Twitter! Newfound is @thinknewfound and Corey is @choffstein.
[2] It is $0.86 instead of $0.96 because they need to set aside $0.10 to cover the extra dollar they owe at maturity.
[3] This is a bit of a simplification as the bonds will have different risk characteristics (e.g. different durations and convexity) which could cause investors, especially those with views on future rate changes, to prefer one bond over the other.
[4] We made the leap here from total bond index to constant duration index to constant maturity index. Each step introduces some error, but we believe for our purposes the error is de minimis and a constant maturity index allows for greater ease of implementation.
[5] Since no 9-year U.S. Treasury is offered, we create a model for the yield curve using cubic splines and then estimate the 9-year rate.
[6] http://content.csbs.utah.edu/~lozada/Research/IniYld_6.pdf