How do I model 20 yr Treasury portfolio impact?
How do I model 20 yr Treasury portfolio impact?
I’ve downloaded 20year Treasury daily yield data back to 1993. How do I use this to model the expected impact of holding 20 year treasuries in a portfolio?
Let’s start simple, with a portfolio that is 100% 20 year treasuries via an ETF like TLT.
I think there’s a “rule of thumb” that a 1% increase in yield is multiplied by the duration to get the expected loss. The average effective duration of TLT is currently 16.7 years, so a 1% yield increase would in theory cause the 100% TLT portfolio to drop by 16.7%.
First question: am I applying the rule of thumb right?
Second question: how accurate has that rule of thumb actually been and is it truly scalable in a linear way? Or is there a more accurate way to do this?
By “linearly scalable” I mean, let’s say yields increase by 0.25% instead of 1%. If the rule is linearly scalable that would mean the impact is .25*16.7=4.18% decline in TLT. Is the rule of thumb only intended to apply to a 1% yield change or can you scale it linearly?
Now if TLT was only 20% of the portfolio assets, a .25% increase in 20 year yield would in theory cause a .20*4.18=0.84% decline of the full portfolio due to TLT alone. Is that right, or is there a more accurate way to accomplish this?
Thanks.
Let’s start simple, with a portfolio that is 100% 20 year treasuries via an ETF like TLT.
I think there’s a “rule of thumb” that a 1% increase in yield is multiplied by the duration to get the expected loss. The average effective duration of TLT is currently 16.7 years, so a 1% yield increase would in theory cause the 100% TLT portfolio to drop by 16.7%.
First question: am I applying the rule of thumb right?
Second question: how accurate has that rule of thumb actually been and is it truly scalable in a linear way? Or is there a more accurate way to do this?
By “linearly scalable” I mean, let’s say yields increase by 0.25% instead of 1%. If the rule is linearly scalable that would mean the impact is .25*16.7=4.18% decline in TLT. Is the rule of thumb only intended to apply to a 1% yield change or can you scale it linearly?
Now if TLT was only 20% of the portfolio assets, a .25% increase in 20 year yield would in theory cause a .20*4.18=0.84% decline of the full portfolio due to TLT alone. Is that right, or is there a more accurate way to accomplish this?
Thanks.

 Posts: 7079
 Joined: Mon Sep 07, 2009 2:57 pm
 Location: Milky Way
Re: How do I model 20 yr Treasury portfolio impact?
Yes, in the case of small interest rate changes.Tamales wrote:I’ve downloaded 20year Treasury daily yield data back to 1993. How do I use this to model the expected impact of holding 20 year treasuries in a portfolio?
Let’s start simple, with a portfolio that is 100% 20 year treasuries via an ETF like TLT.
I think there’s a “rule of thumb” that a 1% increase in yield is multiplied by the duration to get the expected loss. The average effective duration of TLT is currently 16.7 years, so a 1% yield increase would in theory cause the 100% TLT portfolio to drop by 16.7%.
First question: am I applying the rule of thumb right?
It is perfectly correct by definition only for infinitesimally small changes in rate. It does not account for the curvature of what is essentially a nonlinear function. Duration is the firstderivative of the price versus yield curve.Tamales wrote: Second question: how accurate has that rule of thumb actually been and is it truly scalable in a linear way? Or is there a more accurate way to do this?
By “linearly scalable” I mean, let’s say yields increase by 0.25% instead of 1%. If the rule is linearly scalable that would mean the impact is .25*16.7=4.18% decline in TLT. Is the rule of thumb only intended to apply to a 1% yield change or can you scale it linearly?
That's correct, but this is of limited value unless you can predict the future.Tamales wrote: Now if TLT was only 20% of the portfolio assets, a .25% increase in 20 year yield would in theory cause a .20*4.18=0.84% decline of the full portfolio due to TLT alone. Is that right, or is there a more accurate way to accomplish this?
Thanks.
Best regards, Op 

"In the middle of difficulty lies opportunity." Einstein
Re: How do I model 20 yr Treasury portfolio impact?
Not sure what you plan to do with the data; your questions don't seem to relate to it.
For small changes in interest rates, the duration rule of thumb is fine. For a fund, it assumes that rates shift the same at every relevant point on the yield curve, which is rarely the case. Each bond in the fund will change somewhat differently depending on the change in the yield curve. Even for a fund that owns all treasuries that mature in 20 years, bonds with different coupon rates will change in price somewhat differently. TLT does not own all 20year treasuries, but treasuries with maturities of 20 years or more, so the movement of the yield curve from 20 years to 30 years will determine its price change.
For a single bond, you can do an accurate calculation using the spreadsheet PRICE function. You also can determine the duration using the MDURATION function. Say you buy a 20 year treasury now at a YTM of 3% (close to what it is) with a coupon of 3%. The MDURATION function tells us the modified duration is 14.95. If YTM on 20year treasury increases to 4% right after you buy it, the price would be 86.33, for a decline of 13.67%, so somewhat less then predicted by modified duration. If yield only increased by 0.1 percentage point to 3.1%, price would drop by 1.48%, which we can see is closer to the mduration prediction. This is because the duration curve is not a straight line; it has positive convexity, so larger increases in rates result in proportionally smaller price changes.
Say in one year the rate on a 19year treasury has risen to 4% (after one year you will own a 19year treasury, so that's the relevant comparison). The PRICE function tells us that the new price is 86.78 (starting price is 100), so a loss of 13.22%. The new modified duration is 13.94.
You can use any spreadsheet to do the calculation. If you don't have Excel or something like it, you can easily and freely use a Google spreadsheet to do it.
Kevin
For small changes in interest rates, the duration rule of thumb is fine. For a fund, it assumes that rates shift the same at every relevant point on the yield curve, which is rarely the case. Each bond in the fund will change somewhat differently depending on the change in the yield curve. Even for a fund that owns all treasuries that mature in 20 years, bonds with different coupon rates will change in price somewhat differently. TLT does not own all 20year treasuries, but treasuries with maturities of 20 years or more, so the movement of the yield curve from 20 years to 30 years will determine its price change.
For a single bond, you can do an accurate calculation using the spreadsheet PRICE function. You also can determine the duration using the MDURATION function. Say you buy a 20 year treasury now at a YTM of 3% (close to what it is) with a coupon of 3%. The MDURATION function tells us the modified duration is 14.95. If YTM on 20year treasury increases to 4% right after you buy it, the price would be 86.33, for a decline of 13.67%, so somewhat less then predicted by modified duration. If yield only increased by 0.1 percentage point to 3.1%, price would drop by 1.48%, which we can see is closer to the mduration prediction. This is because the duration curve is not a straight line; it has positive convexity, so larger increases in rates result in proportionally smaller price changes.
Say in one year the rate on a 19year treasury has risen to 4% (after one year you will own a 19year treasury, so that's the relevant comparison). The PRICE function tells us that the new price is 86.78 (starting price is 100), so a loss of 13.22%. The new modified duration is 13.94.
You can use any spreadsheet to do the calculation. If you don't have Excel or something like it, you can easily and freely use a Google spreadsheet to do it.
Kevin
Last edited by Kevin M on Sun Jul 13, 2014 2:09 pm, edited 1 time in total.
....... Suggested format for Asking Portfolio Questions (edit original post)
Re: How do I model 20 yr Treasury portfolio impact?
Looking a little more closesely at TLT, it has an average maturity of 27 years, so it's more like a 27year treasury than a 20year treasury. But even if you plug in the average coupon, YTM, and coupon into the bond formulas you'll get a duration that's somewhat different than the average duration shown on the website (modified duration of 17.5 vs. effective duration shown of 16.8).
The fund only has 24 holdings, but I still doubt you want to plug in all the numbers and speculate on changes at the relevant points on the yield curve to see what the actual price change would be.
The duration rule of thumb gets you close enough into the ballpark to evaluate the term risk of this fund.
Kevin
The fund only has 24 holdings, but I still doubt you want to plug in all the numbers and speculate on changes at the relevant points on the yield curve to see what the actual price change would be.
The duration rule of thumb gets you close enough into the ballpark to evaluate the term risk of this fund.
Kevin
....... Suggested format for Asking Portfolio Questions (edit original post)
 Phineas J. Whoopee
 Posts: 7706
 Joined: Sun Dec 18, 2011 6:18 pm
Re: How do I model 20 yr Treasury portfolio impact?
The underlying mechanics are indeed interesting, but with respect to making investment decisions about the ETF you referred to (TLT, iShares 20+ Year Treasury Bond ETF), if you plan to hold it over the very long term rather than as a speculative vehicle, I'd suggest looking up the SEC yield (3.26%) and average duration (16.81 years) and calling it a day.
PJW
PJW
Re: How do I model 20 yr Treasury portfolio impact?
1. Yes, that's what the rule of thumb is. But 2 things. The reason why the duration is high in your example is because of historically low interest rate. For higher interest rates (to be precise yields), the duration will be smaller. Second, as someone pointed out this rule of thumb works for small yield change. 1% change when yield is 2% is a huge change.
2. The priceyield relationship is not linear. It's convex. In short, the price increase more, and decrease less than what the rule of thumb suggests. The more accurate price prediction formula given yield change involves both (mod) duration and convexity.
2. The priceyield relationship is not linear. It's convex. In short, the price increase more, and decrease less than what the rule of thumb suggests. The more accurate price prediction formula given yield change involves both (mod) duration and convexity.
Last edited by acegolfer on Sun Jul 13, 2014 7:22 pm, edited 1 time in total.
Re: How do I model 20 yr Treasury portfolio impact?
I gave an example of this above. It makes more sense when looking at the graph provided by acegolfer, so here it is again:acegolfer wrote: 2. The priceyield relationship is not linear. It's convex. In short, the price increase more, and decrease less than why the rule of thumb suggests. The more accurate price prediction formula given yield change involves both (mod) duration and convexity.
It's not exactly the right chart, since modified duration graphs percentage price change (not price change) as a function of change in yield (YTM). But conceptually the chart works, and you can think of modified duration at a given yield as the slope of the tangent line at that yield. A 0.1 percentage point change in price doesn't move too far off the tangent line, so modified duration predicts that % price change pretty closely in the example given. A one percentage point increase in rate moves enough off the tangent line that the difference is more noticeable.Kevin M wrote:Say you buy a 20 year treasury now at a YTM of 3% (close to what it is) with a coupon of 3%. The MDURATION function tells us the modified duration is 14.95. If YTM on 20year treasury increases to 4% right after you buy it, the price would be 86.33, for a decline of 13.67%, so somewhat less then predicted by modified duration. If yield only increased by 0.1 percentage point to 3.1%, price would drop by 1.48%, which we can see is closer to the mduration prediction. This is because the duration curve is not a straight line; it has positive convexity, so larger increases in rates result in proportionally smaller price changes.
Kevin
....... Suggested format for Asking Portfolio Questions (edit original post)
Re: How do I model 20 yr Treasury portfolio impact?
Not to put too fine a point on it, but (modified) duration is not the first derivative of the price with respect to the yield; dollar duration is the first derivative of price with respect to yield.Call_Me_Op wrote:Duration is the firstderivative of the price versus yield curve.
Simplify the complicated side; don't complify the simplicated side.
Re: How do I model 20 yr Treasury portfolio impact?
What I hate about this graph is that at left end of it suggests that as the yield approaches zero (percent), the price goes to infinity, which is silly. But finance people always draw the graph like this.acegolfer wrote:
Simplify the complicated side; don't complify the simplicated side.
Re: How do I model 20 yr Treasury portfolio impact?
Thanks for the replies and info. Below are charts of 20 yr Treasury yield and TLT (percent change from a zero point of Jan 2008). Roughly, TLT is in the range of .8 to 1.2 times the (negative of the) yield at the active/spikey points, but has generally lower volatility over time. I guess this makes the point you guys are making about convexity (isn't is concavity?) and at these large swings it's closer to 1: 1 rather than scaled by a factor of duration of 16.7:1.
Re: How do I model 20 yr Treasury portfolio impact?
Or, modified duration = (dP/dy) / P =  slope of 1st derivative of priceyield curve / Pricemagician wrote:Not to put too fine a point on it, but (modified) duration is not the first derivative of the price with respect to the yield; dollar duration is the first derivative of price with respect to yield.Call_Me_Op wrote:Duration is the firstderivative of the price versus yield curve.
This is why dP/P (% change in price) =  mod D * dy (change in yield).
Re: How do I model 20 yr Treasury portfolio impact?
Tamales, a couple of points.
As I've already pointed out, TLT is not a 20year treasury. It's a fund of treasuries with maturities of greater than 20 years, and an average maturity of about 27 years.
I'm not sure what you're trying to show with your charts, but modified duration relates percentage price change to percentage point change in YTM. You are showing charts of percentage change in yield (very different than percentage point change) of a 20year treasury next to percentage price change of a fund with an average maturity of 27 years.
Price and YTM of a single treasury are directly related. You can use the PRICE function with various yield values to observe the relationship directly. You can use the MDURATION function to see modified duration for specified price and yield (and coupon and maturity), then change the yield a little bit to see how the change in price relative to the change predicted by modified duration. You could even build a table and create the duration chart (be sure to use percentage price change on the y axis).
For the fund, you should see that there is a relationship between percentage price change and average yield to maturity change, and that it is approximated by the average duration over small yield changes. But trying to compare numbers for the fund to numbers for a 20year treasury doesn't make sense.
Kevin
As I've already pointed out, TLT is not a 20year treasury. It's a fund of treasuries with maturities of greater than 20 years, and an average maturity of about 27 years.
I'm not sure what you're trying to show with your charts, but modified duration relates percentage price change to percentage point change in YTM. You are showing charts of percentage change in yield (very different than percentage point change) of a 20year treasury next to percentage price change of a fund with an average maturity of 27 years.
Price and YTM of a single treasury are directly related. You can use the PRICE function with various yield values to observe the relationship directly. You can use the MDURATION function to see modified duration for specified price and yield (and coupon and maturity), then change the yield a little bit to see how the change in price relative to the change predicted by modified duration. You could even build a table and create the duration chart (be sure to use percentage price change on the y axis).
For the fund, you should see that there is a relationship between percentage price change and average yield to maturity change, and that it is approximated by the average duration over small yield changes. But trying to compare numbers for the fund to numbers for a 20year treasury doesn't make sense.
Kevin
....... Suggested format for Asking Portfolio Questions (edit original post)
Re: How do I model 20 yr Treasury portfolio impact?
Hi Kevin, I'm at a much more simplistic level. I read this "rule of thumb" about a 1% increase in yield leads to a factor of "duration" loss percentage, so I'm just essentially trying to judge how devastating to a portfolio the holding of something like TLT would be as yields increase. The rule of thumb sure makes it sound like it would be devastating.
From the graphs I posted, I interpret them to mean that I should ignore the rule of thumb, and a better approximation of the losses in TLT would be proportional (within the .8 to 1.2x range) to the percent increase in yield (with the understanding as you note that the ETF is not just a single 20 year treasury). Within a portfolio, that loss due to TLT is further reduced by not holding a 100% position in TLT, not to mention that equities often move in the opposite direction so when summed across the portfolio, the holding of TLT, even during fairly large trough to peak increases in yield percentage, doesn't have a devastating impact. This is what I'm seeing in the composite portfolio graphs I've looked at (i.e. periods of increase in yield of pretty large magnitudes don't cause TLT to sink the portfolio at all). Am I interpreting this right?
Edit: for example, here's one of the portfolio results over the same time period as the graph I posted. The start point was limited by the ETF inception dates, and was not intentionally cherry picked for this result relative to SP500. It will obviously not perform this way relative to SP500 across all time periods, but it does maintain the comparatively smoother ride relative to SP500 for most time periods, which is what I was after (it's more evident in a graph that is more expanded horizontally). PS ignore the VTIVEU label in the graphic.
From the graphs I posted, I interpret them to mean that I should ignore the rule of thumb, and a better approximation of the losses in TLT would be proportional (within the .8 to 1.2x range) to the percent increase in yield (with the understanding as you note that the ETF is not just a single 20 year treasury). Within a portfolio, that loss due to TLT is further reduced by not holding a 100% position in TLT, not to mention that equities often move in the opposite direction so when summed across the portfolio, the holding of TLT, even during fairly large trough to peak increases in yield percentage, doesn't have a devastating impact. This is what I'm seeing in the composite portfolio graphs I've looked at (i.e. periods of increase in yield of pretty large magnitudes don't cause TLT to sink the portfolio at all). Am I interpreting this right?
Edit: for example, here's one of the portfolio results over the same time period as the graph I posted. The start point was limited by the ETF inception dates, and was not intentionally cherry picked for this result relative to SP500. It will obviously not perform this way relative to SP500 across all time periods, but it does maintain the comparatively smoother ride relative to SP500 for most time periods, which is what I was after (it's more evident in a graph that is more expanded horizontally). PS ignore the VTIVEU label in the graphic.
Re: How do I model 20 yr Treasury portfolio impact?
This is not correct, Tamales. In a couple of posts (here and here) Kevin M has explained the right way to do this for an individual bond. For small % point changes in yieldtomaturity (YTM) you can use the modified duration [can be calculated with the Excel MDURATION() function] and the "rule of thumb". For large or small % point changes in YTM you can use the price of the bond at each YTM [can be calculated using the Excel PRICE() function] and just compute the % change in price yourself. Perhaps an example will help clarify this.Tamales wrote:... a better approximation of the losses in TLT would be proportional (within the .8 to 1.2x range) to the percent increase in yield...
The average maturity of iShares 20+ Year Treasury Bond ETF : TLT is about 27 years. One of its holdings is the 8/15/2041 maturity with a 3.75% coupon which also matures in about 27 years. So lets take this bond as representative of the fund and see what happens to its price for various changes in its YTM. According to the WSJ Treasury Quotes 7/11/2014, it had a YTM of 3.27% last Friday. This corresponds to a price of 108.58 and a modified duration of 17.15. * The table below shows the price at this yield and also at 4 higher yields, all calculated using the PRICE() function with the respective yields.
Code: Select all
 Yield Chg  Price
Yield % Point Percent Price Chg Pct
    
3.27% 108.58
3.37% 0.10% 3.1% 106.71 1.7%
4.27% 1.00% 30.6% 91.70 15.6%
5.27% 2.00% 61.2% 78.20 28.0%
6.27% 3.00% 91.7% 67.35 38.0%
Also, as several posters have said, the actual price decrease is less than the "rule of thumb" would indicate for YTM increases of 1% point or more. For example, if the YTM were to instantly jump 3% points from 3.27% to 6.27%, the rule of thumb would indicate a fall in price of 3 X 17.15 or about 51%. But the actual decrease is only 38%. This is a good estimate of the fall in price of the TLT fund were yields on all of its holdings to instantly jump 3% points.
* Here are the Excel formulas to calculate these:
Code: Select all
108.58 =PRICE(DATE(2014, 7, 11), DATE(2041, 8, 15), 3.75%, 3.27%, 100, 2, 0)
17.15 =MDURATION(DATE(2014, 7, 11), DATE(2041, 8, 15), 3.75%, 3.27%, 2, 0)
Re: How do I model 20 yr Treasury portfolio impact?
You can get exact answers to your questions by using my (free/opensource) bond simulator:
http://www.bogleheads.org/forum/viewtop ... 10&t=79874
I'd provide code but have too many deadlines right now. PM me with questions if you can't figure something out.
http://www.bogleheads.org/forum/viewtop ... 10&t=79874
I'd provide code but have too many deadlines right now. PM me with questions if you can't figure something out.

 Posts: 7079
 Joined: Mon Sep 07, 2009 2:57 pm
 Location: Milky Way
Re: How do I model 20 yr Treasury portfolio impact?
I think you may be putting too fine a point on it. I did not specify dollar price  my statement could as easily been based upon factional price.magician wrote:Not to put too fine a point on it, but (modified) duration is not the first derivative of the price with respect to the yield; dollar duration is the first derivative of price with respect to yield.Call_Me_Op wrote:Duration is the firstderivative of the price versus yield curve.
Best regards, Op 

"In the middle of difficulty lies opportunity." Einstein
Re: How do I model 20 yr Treasury portfolio impact?
Thanks #cruncher, I'll give your post a careful reading and reread Kevin's posts. I appreciate you guys sticking with me on this as I try to understand, and will probably have some followup questions later.
In the interim, I wonder if it's possible that I just did a crappy job of asking my question. It almost seems that no matter what the equations or rules of thumb predict, the actual % increase in some portion of the yield graph for the 20 yr treasury will reflect approximately in the actual price change in TLT. If you had to pick one thing to best represent price changes in TLT, wouldn't that 20 yr yield be the closest relationship I have available to judge yield change impact on TLT as a whole, even if imperfect?
Maybe if I explain a bit more on how I'm interpreting the graphs that will help either in pointing out where i'm going wrong, or better explain what I'm looking for:
Just looking at the (inverse) commonality between the rising and falling and the overall shape of those two graphs (% yield and % TLT), they have a lot in common at a superficial level. So for example the downward spike in the yield from late 2008 to early 2009 went from about +10% to 35%. Let's call that a 45% move down. The similar timeframe from the TLT curve went from about 0% to 35%. So if I'm approximating, they moved roughly the same percentage. Certainly TLT did not move up by the rule of thumb which would be 45%*duration over the same time period (that would be hundreds of percent). If I look at the other strongly trending segments of the yield graph, and do the same (i.e. noting the total percent range of movement in the yield) comparison to the same time frame for TLT, they move roughly the same percent, maybe scaled by something in the .8 to 1.2 range.
Now a .8 to 1.2 multiplier on the percent is actually a pretty big range, which probably reflects the multiple holdings in TLT, among other things, but it's tiny compared to the multiplier of duration used in the rule of thumb, which would be in the 1627 range. That observation (if correct) is the biggie for me.
If you can help me understand what is fundamentally incorrect about the simple comparison I'm making between the % yield change and % TLT change, over a given trending period, I'd appreciate it. It seems like the % change in the 20 year yield during trending periods is a close approximation to similar periods in TLT and its % change (scaled by about .81.2), and it's not the "% change in yield multiplied by duration" rule of thumb, which would be a massively larger number down at these low yields.
In the interim, I wonder if it's possible that I just did a crappy job of asking my question. It almost seems that no matter what the equations or rules of thumb predict, the actual % increase in some portion of the yield graph for the 20 yr treasury will reflect approximately in the actual price change in TLT. If you had to pick one thing to best represent price changes in TLT, wouldn't that 20 yr yield be the closest relationship I have available to judge yield change impact on TLT as a whole, even if imperfect?
Maybe if I explain a bit more on how I'm interpreting the graphs that will help either in pointing out where i'm going wrong, or better explain what I'm looking for:
Just looking at the (inverse) commonality between the rising and falling and the overall shape of those two graphs (% yield and % TLT), they have a lot in common at a superficial level. So for example the downward spike in the yield from late 2008 to early 2009 went from about +10% to 35%. Let's call that a 45% move down. The similar timeframe from the TLT curve went from about 0% to 35%. So if I'm approximating, they moved roughly the same percentage. Certainly TLT did not move up by the rule of thumb which would be 45%*duration over the same time period (that would be hundreds of percent). If I look at the other strongly trending segments of the yield graph, and do the same (i.e. noting the total percent range of movement in the yield) comparison to the same time frame for TLT, they move roughly the same percent, maybe scaled by something in the .8 to 1.2 range.
Now a .8 to 1.2 multiplier on the percent is actually a pretty big range, which probably reflects the multiple holdings in TLT, among other things, but it's tiny compared to the multiplier of duration used in the rule of thumb, which would be in the 1627 range. That observation (if correct) is the biggie for me.
If you can help me understand what is fundamentally incorrect about the simple comparison I'm making between the % yield change and % TLT change, over a given trending period, I'd appreciate it. It seems like the % change in the 20 year yield during trending periods is a close approximation to similar periods in TLT and its % change (scaled by about .81.2), and it's not the "% change in yield multiplied by duration" rule of thumb, which would be a massively larger number down at these low yields.
Re: How do I model 20 yr Treasury portfolio impact?
For those who used a representative bond and use =PRICE() to show how the bond portfolio value changes for given yield change, this approach results in inaccurate values.
The price change of a portfolio is mathematically different from the price change of one bond.
If you are not convinced, consider
A: a medium maturity bond
B: x1 number of shortterm bond and x2 number of longterm bond with the same value and same duration as A (this is possible because there are 2 unknowns, x1 and x2, and we have 2 equations to solve.)
Then use =PRICE() on A and B. For any yield change, the value of B is always higher than A. It's because B has higher convexity (also called as "benter") than A. B is known as barbell and A is known as bullet. And one bond strategy used by active managed funds is to long barbell and short bullet.
In short, you cannot use a representative bond to explain how the portfolio value changes. If you do, then you will always undervalue the true value.
The price change of a portfolio is mathematically different from the price change of one bond.
If you are not convinced, consider
A: a medium maturity bond
B: x1 number of shortterm bond and x2 number of longterm bond with the same value and same duration as A (this is possible because there are 2 unknowns, x1 and x2, and we have 2 equations to solve.)
Then use =PRICE() on A and B. For any yield change, the value of B is always higher than A. It's because B has higher convexity (also called as "benter") than A. B is known as barbell and A is known as bullet. And one bond strategy used by active managed funds is to long barbell and short bullet.
In short, you cannot use a representative bond to explain how the portfolio value changes. If you do, then you will always undervalue the true value.
Re: How do I model 20 yr Treasury portfolio impact?
Depends what you mean by a portfolio. A portfolio of identical bonds will behave exactly the same as a portfolio of a single bond. If you mean a portfolio of bonds all with different durations (or even the same duration but a different pattern of payouts), then it starts mattering. See convexity (http://www.bogleheads.org/wiki/Bonds:_A ... #Convexity) for a good way to approximate and think about these differences.acegolfer wrote:The price change of a portfolio is mathematically different from the price change of one bond.
Re: How do I model 20 yr Treasury portfolio impact?
Sorry, I didn't specify what bond portfolio meant. I was responding to those who used a single bond to represent the bond ETF, which is clearly the latter of your examples.linuxizer wrote:Depends what you mean by a portfolio. A portfolio of identical bonds will behave exactly the same as a portfolio of a single bond. If you mean a portfolio of bonds all with different durations (or even the same duration but a different pattern of payouts), then it starts mattering. See convexity (http://www.bogleheads.org/wiki/Bonds:_A ... #Convexity) for a good way to approximate and think about these differences.acegolfer wrote:The price change of a portfolio is mathematically different from the price change of one bond.
Re: How do I model 20 yr Treasury portfolio impact?
On my earlier comment: "Now a .8 to 1.2 multiplier on the percent is actually a pretty big range, which probably reflects the multiple holdings in TLT, among other things, but it's tiny compared to the multiplier of duration used in the rule of thumb, which would be in the 1627 range. That observation (if correct) is the biggie for me."
I think I see the error I'm making here (or one of the errors anyway ;o)). I'm inexplicably interchanging absolute and relative percent changes into the "rule of thumb" and in retrospect I would assume the rule of thumb is based on absolute changes. So for example the drop in yield in late 2008 went from 4.74% on 10/31/08, to 2.86% on 12/18/08. So the absolute % change in the yield is 1.88%. The change in value for TLT over that same period was a 32.5% drop. Those two data points are what they are, based on the market. No calculations or scale factors involved. If I divide those two numbers to back into the rule of thumb, 32.5/1.88=17.3, so it's in the ballpark for the avg effective yield, which is just an approximation itself. If I picked another max to min in the yield at a different time, and compared to TLT, that ratio would probably be slightly different. So I guess the rule of thumb does get you in the ballpark and what I was doing earlier was using relative percent change. The relative change (over the time frame of the graphs anyway) is closer to a 1:1 relationship +/ 20%. No idea if that holds over other time frames.
I think I see the error I'm making here (or one of the errors anyway ;o)). I'm inexplicably interchanging absolute and relative percent changes into the "rule of thumb" and in retrospect I would assume the rule of thumb is based on absolute changes. So for example the drop in yield in late 2008 went from 4.74% on 10/31/08, to 2.86% on 12/18/08. So the absolute % change in the yield is 1.88%. The change in value for TLT over that same period was a 32.5% drop. Those two data points are what they are, based on the market. No calculations or scale factors involved. If I divide those two numbers to back into the rule of thumb, 32.5/1.88=17.3, so it's in the ballpark for the avg effective yield, which is just an approximation itself. If I picked another max to min in the yield at a different time, and compared to TLT, that ratio would probably be slightly different. So I guess the rule of thumb does get you in the ballpark and what I was doing earlier was using relative percent change. The relative change (over the time frame of the graphs anyway) is closer to a 1:1 relationship +/ 20%. No idea if that holds over other time frames.
Re: How do I model 20 yr Treasury portfolio impact?
Price change is not the same as percentage price change, no matter what you specify.Call_Me_Op wrote:I think you may be putting too fine a point on it. I did not specify dollar price  my statement could as easily been based upon factional price.magician wrote:Not to put too fine a point on it, but (modified) duration is not the first derivative of the price with respect to the yield; dollar duration is the first derivative of price with respect to yield.Call_Me_Op wrote:Duration is the firstderivative of the price versus yield curve.
Simplify the complicated side; don't complify the simplicated side.
Re: How do I model 20 yr Treasury portfolio impact?
acegolfer  Agreed. FWIW, my comment was meant to be explanatory for others, not critical.
Re: How do I model 20 yr Treasury portfolio impact?
No. Why do you keep bringing up a 20year treasury when the fund has nothing to do with a 20year treasury. The fund doesn't even hold any 20year treasuries, the shortest maturity being 2/15/2036. Is it just because the number 20 appears in the fund name: iShares 20+ Year Treasury Bond ETF? The "+" in the name is just as important as the "20".Tamales wrote: If you had to pick one thing to best represent price changes in TLT, wouldn't that 20 yr yield be the closest relationship I have available to judge yield change impact on TLT as a whole, even if imperfect?
The one single thing to use to estimate price change over a small range of yield change is the average fund duration. For larger yield changes, understand that positive convexity will result in smallerthanpredicted changes. If you really want to build a more accurate model to get an idea of the convexity impact, you could enter the data for each bond in the fund into a spreadsheet, then use the PRICE function to evaluate the change in price for each bond given a change in yield for each bond. You could assume an instantaneous parallel shift in the yield curve (all yields increase or decrease by the same percentage point amount), or try nonparallel shifts for fun.
I'm sure linuxizer's program gives even better results, but the spreadsheet thing would be an instructive example that would be easier to build.
Or here's an even simpler idea. Pick 35 bonds that span the maturity range of the fund, and experiment with total price change for different yield changes. Or maybe pick a weighted sample that's somewhat representative of the different maturities and durations (durations range from about 14 to about 18).
The point is that there already is bond math to determine what you want to know. You don't have to make up your own scheme.
Have you tried using the PRICE function in a spreadsheet?
Kevin
....... Suggested format for Asking Portfolio Questions (edit original post)
Re: How do I model 20 yr Treasury portfolio impact?
Good point, Ace. Here's an illustration that confirms what you say. For Case B I used an extreme combination of a 1year bond and a 30year bond so that the effect you describe would (hopefully) be more apparent. (I assumed a settlement date of 1/1/2014 and tweaked the maturity date of the single bond to make the durations for the two cases come out the same.)acegolfer in [url=http://www.bogleheads.org/forum/viewtopic.php?p=2121091#p2121091]this post[/url] wrote:For those who used a representative bond and use =PRICE() to show how the bond portfolio value changes for given yield change, this approach results in inaccurate values. The price change of a portfolio is mathematically different from the price change of one bond. If you are not convinced, consider
A: a medium maturity bond
B: x1 number of shortterm bond and x2 number of longterm bond with the same [ total ] value and same [ average ] duration as A ...
Then use =PRICE() on A and B. For any yield change, the value of B is always higher than A. ... In short, you cannot use a representative bond to explain how the portfolio value changes. If you do, then you will always undervalue the true value.
Case A  one bond:
Code: Select all
Years to Par Market Modified Yield Market Mkt Value
Maturity Coupon Mature Value Yield Value Duration + 3% Pts Value Change
         
11/16/24 3% 10.87 100.00 3% 100.00 9.19 6% 76.28 23.7%
Code: Select all
01/01/15 1% 1.00 50.00 1% 50.00 0.99 4% 48.54 2.9%
01/01/44 4% 30.00 50.00 4% 50.00 17.38 7% 31.29 37.4%
    
Total & Wtd Avg 100.00 100.00 9.19 79.84 20.2%
The effect is not so apparent for TLT, the bond ETF the original poster is considering. For the "representative" bond I computed a 38.0% price decline for a 3% point increase in yield in my previous post. I then said, "This is a good estimate of the fall in price of the TLT fund were yields on all of its holdings to instantly jump 3% points."
I have since computed the effect of a 3% point increase in yield for each of the fund's 23 bonds. The result was a 38.14% fall in market value, pretty close to 38.0%. This seems reasonable since the fund holds only bonds with a remaining life between 21.6 and 30 years. The declines in value therefore fell in a fairly tight range between 33.6% and 40.5%. The fact that the representative bond fell slightly less than the average, doesn't discredit your point. This is because I picked one of the bonds that happened to have the same average maturity, not duration, as the portfolio average. If as a "representative" bond I'd used a hypothetical one with a modified duration exactly equal to the weighted average for the portfolio, I would expect to get a price drop slightly higher than the 38.14% for the portfolio, in conformance with your point.
Good, I hope what we've been saying has helped. By the way, if you want to use historical results to understand what might happen to the value of bonds, I suggest looking at a long period of history. It won't tell you want will happen, but it will give you a better understanding of what might happen. Your graphs (in this post and in this post) only go back to about 2008. If you look at a longer period, you will see a much bigger range.Tamales wrote:I think I see the error I'm making here (or one of the errors anyway ;o)).
For example, consider this graph. (I'm showing the 10year because the 20 and 30year have gaps. But they moved pretty much in synch with the 10year during this period.)
Re: How do I model 20 yr Treasury portfolio impact?
Thanks #cruncher for the examples, that helped. Apologies to Kevin, since you were saying the same but I didn't follow at the time.
Since I'm not sure I understand how or when you'd change some of the parameters in PRICE() such as redemption, frequency, basis, I'd like to try a different example using EDV (Vanguard extended duration treasury).
EDV has an average effective duration of 24.9 years, so in looking at its holdings I'd pick May 15, 2039 as representative of that average. Next, looking here: http://online.wsj.com/mdc/public/page/2 ... nav_2_3020 under treasury bond, stripped principle, it shows an asked yield of 3.43%:
Maturity: Bid: Ask: Chg: asked yield:
2039 May 15: 42.910: 43.014: 0.218: 3.43
Next I set up PRICE() as follows (assuming coupon should be zero, and not sure what "frequency" should be for zero coupon so I used 2 since that put the result in the bid/ask range ;o))
=PRICE(DATE(2014,7,14),DATE(2039,5,15),0%,3.43%,100,2,0)
which gives $42.96
and
=MDURATION(DATE(2014,7,14),DATE(2039,5,15),0%,3.43%,2,0)
which gives 24.42 yrs
and then created a table similar to yours from the earlier post:
Have I done this right (or maybe PRICE is not the right function to use in zero coupon case?)
PS I haven't read your most recent post directly prior to this one yet, but I will.
Since I'm not sure I understand how or when you'd change some of the parameters in PRICE() such as redemption, frequency, basis, I'd like to try a different example using EDV (Vanguard extended duration treasury).
EDV has an average effective duration of 24.9 years, so in looking at its holdings I'd pick May 15, 2039 as representative of that average. Next, looking here: http://online.wsj.com/mdc/public/page/2 ... nav_2_3020 under treasury bond, stripped principle, it shows an asked yield of 3.43%:
Maturity: Bid: Ask: Chg: asked yield:
2039 May 15: 42.910: 43.014: 0.218: 3.43
Next I set up PRICE() as follows (assuming coupon should be zero, and not sure what "frequency" should be for zero coupon so I used 2 since that put the result in the bid/ask range ;o))
=PRICE(DATE(2014,7,14),DATE(2039,5,15),0%,3.43%,100,2,0)
which gives $42.96
and
=MDURATION(DATE(2014,7,14),DATE(2039,5,15),0%,3.43%,2,0)
which gives 24.42 yrs
and then created a table similar to yours from the earlier post:
Have I done this right (or maybe PRICE is not the right function to use in zero coupon case?)
PS I haven't read your most recent post directly prior to this one yet, but I will.
Re: How do I model 20 yr Treasury portfolio impact?
@Tam,
Most of your calculations are correct but not B/A.
Look at the last row (1% yield increase). 21.62%/1% is 21.62 not 23.68.
Most of your calculations are correct but not B/A.
Look at the last row (1% yield increase). 21.62%/1% is 21.62 not 23.68.
Re: How do I model 20 yr Treasury portfolio impact?
You're making progress!Tamales wrote: Since I'm not sure I understand how or when you'd change some of the parameters in PRICE() such as redemption, frequency, basis, I'd like to try a different example using EDV (Vanguard extended duration treasury).
No reason to switch to zero coupon bonds. The only parameter you change to see impact on price of change in yield is yield; all other parameters remain the same.
Kevin
....... Suggested format for Asking Portfolio Questions (edit original post)
Re: How do I model 20 yr Treasury portfolio impact?
In his defense, he needs to use a zero coupon bond for his example because he's matching 24.9yr duration with a 24.9yr bond. He can't use coupon bonds as the duration of will not be 24.9 yrs.Kevin M wrote: No reason to switch to zero coupon bonds. The only parameter you change to see impact on price of change in yield is yield; all other parameters remain the same.
Kevin
Re: How do I model 20 yr Treasury portfolio impact?
But it may not have been a good idea to start trying to match duration before understanding that using a 20year bond is not a good choice to model a fund with an average maturity of 27 years. #cruncher showed that using a 27year bond worked out pretty well in the case of the fund OP started with. Maybe stick with that until the concepts are better understood.acegolfer wrote: In his defense, he needs to use a zero coupon bond for his example because he's matching 24.9yr duration with a 24.9yr bond. He can't use coupon bonds as the duration of will not be 24.9 yrs.
Kevin
....... Suggested format for Asking Portfolio Questions (edit original post)
Re: How do I model 20 yr Treasury portfolio impact?
I wonder whether that's a coincidence. In his example, the 27yr bond had a 3.75% coupon rate. If it had a different coupon rate such as 2% or 5%, then I don't think he would get the similar ETF's 38.14% price change, when yield increased by 3%.Kevin M wrote:But it may not have been a good idea to start trying to match duration before understanding that using a 20year bond is not a good choice to model a fund with an average maturity of 27 years. #cruncher showed that using a 27year bond worked out pretty well in the case of the fund OP started with. Maybe stick with that until the concepts are better understood.acegolfer wrote: In his defense, he needs to use a zero coupon bond for his example because he's matching 24.9yr duration with a 24.9yr bond. He can't use coupon bonds as the duration of will not be 24.9 yrs.
Kevin
Re: How do I model 20 yr Treasury portfolio impact?
When I looked up the holdings in EDV they are all zero coupon. Here's the top 20:
and the fund objective says it seeks to track the performance of an index of extended duration zero coupon US Treasuries. So I'm missing something on why that doesn't matter? Was my entry of 0% in the coupon rate field in PRICE() incorrect?
and the fund objective says it seeks to track the performance of an index of extended duration zero coupon US Treasuries. So I'm missing something on why that doesn't matter? Was my entry of 0% in the coupon rate field in PRICE() incorrect?
Re: How do I model 20 yr Treasury portfolio impact?
That's true: a 27year bond with a 2% or 5% coupon wouldn't have tracked TLT as well. But fortunately, in addition to having about the same average maturity, the Aug152041 3.75% bond has about the same coupon as the fund's 3.67% average. Thus the bond has about the same duration as the fund average. I should have said that was part of the reason I selected it.ace golfer in [url=http://www.bogleheads.org/forum/viewtopic.php?p=2121859#p2121859]this post[/url] wrote:I wonder whether that's a coincidence. In his example, the 27yr bond had a 3.75% coupon rate. If it had a different coupon rate such as 2% or 5%, then I don't think he would get the similar ETF's 38.14% price change, when yield increased by 3%.Kevin M wrote:… #cruncher showed that using a 27year bond worked out pretty well in the case of the fund OP started with. ...
You may be misinterpreting Kevin M's statement, "No reason to switch to zero coupon bonds" in this post. I think all he meant was that you didn't need to use a zerocoupon bond fund to do your calculations. I don't think he meant that there was anything wrong with doing so.Tamales in the previous post wrote:When I looked up the holdings in EDV they are all zero coupon. … So [ am I ] missing something on why that doesn't matter? Was my entry of 0% in the coupon rate field in PRICE() incorrect?
And you are correct to enter zero as the coupon rate parameter in the PRICE and MDURATION functions. Here are a couple comments about two other parameters to these functions:
Code: Select all
PRICE(settlement, maturity, coupon rate, yld, redemption, frequency, basis)
MDURATION(settlement, maturity, coupon rate, yld, frequency, basis)
42.97 =PRICE(DATE(2014, 7, 14), DATE(2039, 5, 15), 0%, 3.43%, 100, 2, 1)
24.42 =MDURATION(DATE(2014, 7, 14), DATE(2039, 5, 15), 0%, 3.43%, 2, 1)
 The frequency (next to last) parameter matters even for a zero coupon bond. Besides indicating the frequency of coupon payments, it also controls whether compounding is annual (=1) or semiannual (=2). Since Treasury yields are computed on the assumption of semiannual compounding, 2 is the correct value.
 The basis (last) parameter should be set to 1 for Treasury bonds (whether zerocoupon or not) since the Treasury computes fractional semiannual periods using the actual number of days. I incorrectly set it to zero in this post. Fortunately it doesn't make much difference.
Re: How do I model 20 yr Treasury portfolio impact?
Thanks #cruncher, and you're right, if that's what Kevin meant, I misunderstood.
Sort of a sidebar question: what's the distinction between using DURATION() and MDURATION()? I guess MDURATION is DURATION times a scale factor of 1/(1+(market yield/{coupon payments per year})) so would they give the same result in the case of a zero coupon?
Sort of a sidebar question: what's the distinction between using DURATION() and MDURATION()? I guess MDURATION is DURATION times a scale factor of 1/(1+(market yield/{coupon payments per year})) so would they give the same result in the case of a zero coupon?
Re: How do I model 20 yr Treasury portfolio impact?
They are still different. Even for zeros, there's a compounding frequency.Tamales wrote:Thanks #cruncher, and you're right, if that's what Kevin meant, I misunderstood.
Sort of a sidebar question: what's the distinction between using DURATION() and MDURATION()? I guess MDURATION is DURATION times a scale factor of 1/(1+(market yield/{coupon payments per year})) so would they give the same result in the case of a zero coupon?
Re: How do I model 20 yr Treasury portfolio impact?
Tamales,
The TLT returns that you posted earlier appear to me to be total returns (cg's + interest), not price returns  are you more interested in the impact of rising rates on your total returns? If so, then unlike price return that only depends on initial and final states, total return depends on the path that interest rates take through time. So the modeling gets a little more complicated!
MB
The TLT returns that you posted earlier appear to me to be total returns (cg's + interest), not price returns  are you more interested in the impact of rising rates on your total returns? If so, then unlike price return that only depends on initial and final states, total return depends on the path that interest rates take through time. So the modeling gets a little more complicated!
MB
Re: How do I model 20 yr Treasury portfolio impact?
Is TLT = floating rate note? If fixed coupon bond, then it's not hard to calculate the latter. It's current yield (=annual interest/current price, which is not current YTM) * # of years that one holds.mindbogle wrote:Tamales,
The TLT returns that you posted earlier appear to me to be total returns (cg's + interest), not price returns  are you more interested in the impact of rising rates on your total returns? If so, then unlike price return that only depends on initial and final states, total return depends on the path that interest rates take through time. So the modeling gets a little more complicated!
MB
However, I think all the discussions so far were about instant yield change. So they ignored the interest payment.
Re: How do I model 20 yr Treasury portfolio impact?
mindbogle: yes, TR. I was actually just looking at EDV and it's had some pretty sizable cg distributions in past years.
I think I've decided I'll abandon the modeling idea, and I'll have to figure out some other way to decide whether a portfolio with about 2025% long treasuries ETF as the only bond component and the rest in global equities is an accident waiting to happen or if the 20082014 timeframe I'm able to backtest is actually pretty representative. The desire is to minimize downside capture ratio (See the graph/table I posted earlier), and long treasuries are about the only asset class that accomplishes that on a somewhat consistent basis. EDV does a better job of it compared to TLT, when allocated at the same %. But EDV is more volatile which affects other things. Another one is ZROZ, which is a lot like EDV but has some pretty thin trading volume. It works well in backtesting though.
I think I've decided I'll abandon the modeling idea, and I'll have to figure out some other way to decide whether a portfolio with about 2025% long treasuries ETF as the only bond component and the rest in global equities is an accident waiting to happen or if the 20082014 timeframe I'm able to backtest is actually pretty representative. The desire is to minimize downside capture ratio (See the graph/table I posted earlier), and long treasuries are about the only asset class that accomplishes that on a somewhat consistent basis. EDV does a better job of it compared to TLT, when allocated at the same %. But EDV is more volatile which affects other things. Another one is ZROZ, which is a lot like EDV but has some pretty thin trading volume. It works well in backtesting though.