## Calculating future yield after growth of a long term treasury ETF?

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Topic Author
Mickelous
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Joined: Mon Apr 20, 2020 11:24 pm

### Calculating future yield after growth of a long term treasury ETF?

In an attempt to find this explained online, I have failed.

I am looking to find out how long term treasury yields work in relation to the growth of an ETF for the value. I understand as yields fall the value will rise, but how does it work?

Say an LTT ETF is valued at \$100, and the yield is 1.2%. If the yield were to drop over time to .6%, would the value have doubled to \$200? and then again down to .3% it doubles again? Or does it work in some other way?

Appreciate the help I can get on this one. Trying to figure out how much growth long treasuries have til zero rates and the math behind it.

muffins14
Posts: 241
Joined: Wed Oct 26, 2016 4:14 am

### Re: Calculating future yield after growth of a long term treasury ETF?

It depends on the duration of the fund. You can think of duration like the derivative of price with respect to interest rates. In other words, how much will the prince change due to a change in interest rates.

Assume your LTT ETF has a duration of 10

If interest rates drop by 1%, then your price will go up by 10%. If they drop by 1% again, the price goes up another 10%. If the yield goes up by 1%, the price will go down by 10%

There are some nuances about convexity that I can't explain well, but suffice it to say that the linear relationship isn't exactly right for some levels of yield, and this has an impact at that very low yields that you're interested in.

You can think of like this -- If I can go buy a higher-yielding bond elsewhere (because rates went up), then my bond will be worth less because no one is going to pay me the same price for fewer future dollars in cashflow than they can get elsewhere. Likewise if rates drop, your bond is worth more because people want to buy those future cashflows that are unavailable in new investment offerings. People will buy/sell the bonds until the price they pay now for dollars returned to them on future date are the same.

Topic Author
Mickelous
Posts: 75
Joined: Mon Apr 20, 2020 11:24 pm

### Re: Calculating future yield after growth of a long term treasury ETF?

Right I'm trying to figure out how to calculate value as rates drop. Like if there is some general mathematical formula I can use to say rates can go up x amount until interest hits x mark.

Seasonal
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Joined: Sun May 21, 2017 1:49 pm

### Re: Calculating future yield after growth of a long term treasury ETF?

Mickelous wrote:
Fri Jul 31, 2020 5:01 pm
Right I'm trying to figure out how to calculate value as rates drop. Like if there is some general mathematical formula I can use to say rates can go up x amount until interest hits x mark.
https://en.wikipedia.org/wiki/Bond_duration

alex_686
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Joined: Mon Feb 09, 2015 2:39 pm

### Re: Calculating future yield after growth of a long term treasury ETF?

Duration

Duration * yield change = change in price. Funds must publish their duration.

Duration measures the instantaneous impact to the change of value due to the change in interest rates. There is a fair amount of nuance here but it is a excellent place to start.

I am not sure if you know this, but a bond's yield is inverse to its price. If the yield goes up, the price must go down. This is hard math. If you know the yield you know the price. Bond quote systems are rigged to give quotes using either price or yield.

That being said, you question is a bit off. There is no magic limit. Yields can go up or down as far as they want. There is a current discussion if yields can be negative.
Former brokerage operations & mutual fund accountant. I hate risk, which is why I study and embrace it.

Topic Author
Mickelous
Posts: 75
Joined: Mon Apr 20, 2020 11:24 pm

### Re: Calculating future yield after growth of a long term treasury ETF?

alex_686 wrote:
Fri Jul 31, 2020 5:14 pm
Duration

Duration * yield change = change in price. Funds must publish their duration.

Duration measures the instantaneous impact to the change of value due to the change in interest rates. There is a fair amount of nuance here but it is a excellent place to start.

I am not sure if you know this, but a bond's yield is inverse to its price. If the yield goes up, the price must go down. This is hard math. If you know the yield you know the price. Bond quote systems are rigged to give quotes using either price or yield.

That being said, you question is a bit off. There is no magic limit. Yields can go up or down as far as they want. There is a current discussion if yields can be negative.
Just as a baseline trying to figure out the room to grow on price before yield hits zero.

Kevin M
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### Re: Calculating future yield after growth of a long term treasury ETF?

Mickelous wrote:
Fri Jul 31, 2020 4:29 pm
Say an LTT ETF is valued at \$100, and the yield is 1.2%. If the yield were to drop over time to .6%, would the value have doubled to \$200? and then again down to .3% it doubles again? Or does it work in some other way?

Appreciate the help I can get on this one. Trying to figure out how much growth long treasuries have til zero rates and the math behind it.
I think a good way to start to understand the relationship between price and yield is to play around with the spreadsheet PRICE function as it applies to an individual bond. We might even roughly approximate a long-term Treasury fund as a constant maturity 20-year Treasury. The yield today on the 20y Treasury is 0.98% according to treasury.gov. Let's just round to 1%.

This yield approximately assumes that you could buy a Treasury with exactly 20 years to maturity today at a price of 100 (100% of face value). This is a "par" bond, for which the coupon rate equals the yield (to maturity). The price of this bond can be calculated with this formula:

Code: Select all

``=PRICE("1/1/2000", "1/1/2020", 1%, 1%, 100, 2)``
As expected, this formula returns 100.00 as the price.

If we change the yield to 0.5%, we use this formula:

Code: Select all

``=PRICE("1/1/2000", "1/1/2020", 1%, 0.5%, 100, 2)``
which returns a price of 109.50--an increase of 9.5% in price for an decrease of 0.5 percentage points in yield.

Let's look at this result in the context of the duration concept that has been mentioned. The duration of the par bond can be calculated with this formula:

Code: Select all

``=DURATION("1/1/2000", "1/1/2020", 1%, 1%, 2)``
which returns a result of 18.2. To nitpick, it's actually modified duration that is used for the rule of thumb that folks have referenced, but it doesn't make much difference. We can calculate modified duration with this formula:

Code: Select all

``=MDURATION("1/1/2000", "1/1/2020", 1%, 1%, 2)``
which returns a result of 18.1, so not much difference.

So the duration rule of thumb indicates that for a decrease of 0.5 percentage points, the price should increase by about 9%. Our increase of 9.5% is due to the convexity that has been mentioned, which works in our favor.

If we change the yield to 0% in the PRICE formula, holding all other parameters constant, the price increases to 120.00, which is an increase of 9.58% relative to the 109.50 price at a yield of 0.5%. The modified duration at 0.5% is slightly higher, at 18.4, so the rule of thumb increase would be 9.2%; again, convexity works in our favor.

Kevin

Topic Author
Mickelous
Posts: 75
Joined: Mon Apr 20, 2020 11:24 pm

### Re: Calculating future yield after growth of a long term treasury ETF?

Kevin M wrote:
Fri Jul 31, 2020 6:12 pm
Mickelous wrote:
Fri Jul 31, 2020 4:29 pm
Say an LTT ETF is valued at \$100, and the yield is 1.2%. If the yield were to drop over time to .6%, would the value have doubled to \$200? and then again down to .3% it doubles again? Or does it work in some other way?

Appreciate the help I can get on this one. Trying to figure out how much growth long treasuries have til zero rates and the math behind it.
I think a good way to start to understand the relationship between price and yield is to play around with the spreadsheet PRICE function as it applies to an individual bond. We might even roughly approximate a long-term Treasury fund as a constant maturity 20-year Treasury. The yield today on the 20y Treasury is 0.98% according to treasury.gov. Let's just round to 1%.

This yield approximately assumes that you could buy a Treasury with exactly 20 years to maturity today at a price of 100 (100% of face value). This is a "par" bond, for which the coupon rate equals the yield (to maturity). The price of this bond can be calculated with this formula:

Code: Select all

``=PRICE("1/1/2000", "1/1/2020", 1%, 1%, 100, 2)``
As expected, this formula returns 100.00 as the price.

If we change the yield to 0.5%, we use this formula:

Code: Select all

``=PRICE("1/1/2000", "1/1/2020", 1%, 0.5%, 100, 2)``
which returns a price of 109.50--an increase of 9.5% in price for an decrease of 0.5 percentage points in yield.

Let's look at this result in the context of the duration concept that has been mentioned. The duration of the par bond can be calculated with this formula:

Code: Select all

``=DURATION("1/1/2000", "1/1/2020", 1%, 1%, 2)``
which returns a result of 18.2. To nitpick, it's actually modified duration that is used for the rule of thumb that folks have referenced, but it doesn't make much difference. We can calculate modified duration with this formula:

Code: Select all

``=MDURATION("1/1/2000", "1/1/2020", 1%, 1%, 2)``
which returns a result of 18.1, so not much difference.

So the duration rule of thumb indicates that for a decrease of 0.5 percentage points, the price should increase by about 9%. Our increase of 9.5% is due to the convexity that has been mentioned, which works in our favor.

If we change the yield to 0% in the PRICE formula, holding all other parameters constant, the price increases to 120.00, which is an increase of 9.58% relative to the 109.50 price at a yield of 0.5%. The modified duration at 0.5% is slightly higher, at 18.4, so the rule of thumb increase would be 9.2%; again, convexity works in our favor.

Kevin
Might have to read this a few times to understand but I appreciate it. Thanks a bunch.

Kevin M
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### Re: Calculating future yield after growth of a long term treasury ETF?

Mickelous wrote:
Fri Jul 31, 2020 5:47 pm
Just as a baseline trying to figure out the room to grow on price before yield hits zero.
Looking at the results of the PRICE calculation for 0% yield with 1% coupon rate (20-year) posted above, the upside for a 20-year bond is 20% (=120/100 - 1).

The yield on the 10-year Treasury is 0.55%, and the duration is about one half of the 20 year. Turns out the bond math works out so that we can see the upside simply by observing the price at 0% yield and coupon rate equal to the current yield to maturity; i.e., the price of the 10y at 0% yield and 0.55% coupon is 105.50. So the upside of the 10y is 5.5%.

So the upside of a long-term bond fund with average maturity somewhere between 10 and 20 years is somewhere between 5.5% and 20%.

Kevin

ChiGuy
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Joined: Thu Mar 12, 2020 5:37 pm

### Re: Calculating future yield after growth of a long term treasury ETF?

Hi Kevin M,

So, this means that long-term Treasuries still have room to have impressive returns, even though rates for LTT are so low? Currently, the bond portion of my portfolio is in VG's Intermediate Corp bond fund. However, I've been considering replacing this with 50% LLT & 50% short-term TIPs going forward as a modified "permanent portfolio."

Topic Author
Mickelous
Posts: 75
Joined: Mon Apr 20, 2020 11:24 pm

### Re: Calculating future yield after growth of a long term treasury ETF?

ChiGuy wrote:
Fri Jul 31, 2020 8:35 pm
Hi Kevin M,

So, this means that long-term Treasuries still have room to have impressive returns, even though rates for LTT are so low? Currently, the bond portion of my portfolio is in VG's Intermediate Corp bond fund. However, I've been considering replacing this with 50% LLT & 50% short-term TIPs going forward as a modified "permanent portfolio."
Looks like 1 1/2 - 2 years til it hits zero at average growth rates. Then who knows will it go negative? We'll see. Maybe corporate bonds will be the next flight to safety for 10 years while LTT suffer.

patrick013
Posts: 2950
Joined: Mon Jul 13, 2015 7:49 pm

### Re: Calculating future yield after growth of a long term treasury ETF?

.

Bond Price Calculator

Pick a bond and calc it's price. Then raise or lower it's Yield-to-Maturity
to have the program calc the new price. The difference between the old
price and new price as a percentage difference is also it's basic duration.

For an ETF the weighted duration is supplied for a 1 per cent yield change.
Duration is the expected price change in per cent.

Kevin M
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Contact:

### Re: Calculating future yield after growth of a long term treasury ETF?

ChiGuy wrote:
Fri Jul 31, 2020 8:35 pm
Hi Kevin M,

So, this means that long-term Treasuries still have room to have impressive returns, even though rates for LTT are so low?
Yes, that's what the bond math tells us.

Of course the flip side is that there is even more room for impressive losses. An increase from 1% to 2% for our 20-year par bond results in a return of -16.35%. Again, convexity is in our favor, so a 1 percentage point increase in yield results in a smaller magnitude percentage change (16.35%) than a 1 percentage point decrease in yield (20.00%). But of course there is less of a limit on the upside in yield change, at least unless you assume yields can go deeply negative. So an increase from 1% to 3% results in a price return of -29.75%. So don't forget the downside price risk just because the upside potential for longer-term bonds still is significant at current low yields.

Another fine point: the potential price return in going from 1% to 0% depends on the coupon (interest) rate, since that affects duration (higher coupon rate -> lower duration).

We saw that at a coupon rate of 1%, the upside in going from 1% yield to 0% yield is 20%. Looking at actual 20-year Treasuries, I see one with a coupon rate of 3.875%. The price return for that bond going from 1% yield to 0% yield is 16.9% (the initial modified duration is 15.3, compared to 18.1 for a 1% coupon).

Duration is higher for lower coupon rates and lower yields, so at current low yields, durations are higher than at higher yields, and more so for bonds that also have low coupon rates (which will be the newer issues).

Kevin