So do we care more about change in bond price or bond yield?

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Kevin M
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Re: So do we care more about change in bond price or bond yi

Post by Kevin M »

stlutz wrote:Got it--I see what you're doing now. I've never actually messed much with zero-coupon bonds before.
Cool. The principle is the same for coupon bonds. I thought of a better way to explain what settlement date to use. Using zero coupon bonds just simplifies the algebra, but the principle with respect to the maturity date and settlement date is the same.

Recall the the PRICE formula is just a form of the PV (present value) formula, customized to provide parameters specific to bonds; e.g., rate (coupon rate), yield (YTM) and frequency (frequency of coupon payments). For zero coupon bonds the PRICE formula simplifies to the PV formula for a single payment at the end of the holding period:

PV = FV/(1+y)^n

where y is yield to maturity per period and n is the number of periods. However, instead of providing n directly to the PRICE function, we provide it with maturity date, settlement date, and frequency, from which n is determined:

n = (maturity - settlement)/frequency

where (maturity - settlement) is in years (so in a spreadsheet multiply by 365, since the difference between dates is returned as number of days). Making the simplifying assumption of one coupon payment per year, n = years to maturity (and remembering that zero coupon bonds are priced as if they were paying coupons).

So if you price a 9-year bond today, you will use today as settlement date and a date 9 years from now as maturity date. If you price a 9-year bond in one year, you will use a settlement date one year from now, and a maturity date 9 years from that settlement date. Either way you are telling the PRICE function that the time to maturity, n, is 9 years. If you assume a static yield curve you will use the YTM for a 9-year bond from today's yield curve in both calculations. Both calculations will provide the same result, which you can easily verify with the PRICE function.

So I didn't try to figure out exactly what you were doing, since not all values were provided, but if you change the settlement date to one year from now you also must change the maturity date to one year from now if pricing a bond of the same term to maturity.
stlutz wrote:I think that return "hump" around year 6 may disappear if you use the zero-coupon yield curve as opposed to the "regular bond" yield curve. A regular 10 year bond yields 2.2% but a 10-year zero yields 2.47%.
That's a good point. Also, if I download the quotes from Fidelity I don't have to interpolate to get the yields for maturities not reported on treasury.gov. I've already downloaded quotes for zeros and non-zeros, and will work on using those instead of the treasury.gov rates.

Thanks,

Kevin
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Re: So do we care more about change in bond price or bond yi

Post by Doc »

Kevin M wrote:Also, if I download the quotes from Fidelity I don't have to interpolate to get the yields for maturities not reported on treasury.gov.
I think the Treasury.gov charts are from the constant maturity calculation which is a multi (5?) factor spline fit. They changed their method just about the time of the Lehman collapse to force the model to hit the on the run numbers exactly. Their change in method might have an impact if you are looking at periods on both sides of the change. I had some kind of calculation a few years ago that was thwarted by their change but I don't remember what it was. Short term memory is not as good as long term as you age.

And Kevin it is very possible to accurately do the discounted cash flow equations in your head if you had taken the time to memorize the data in the log table book when you were in high school which is very helpful if you are now of the age that you have to relearn Excel every time you load it. :beer
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Re: So do we care more about change in bond price or bond yi

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Doc wrote: And Kevin it is very possible to accurately do the discounted cash flow equations in your head if you had taken the time to memorize the data in the log table book when you were in high school which is very helpful if you are now of the age that you have to relearn Excel every time you load it. :beer
Funny. So I had to go find the statement by that poster whose member name rhymes with Spock (who I'm sure could do the calculations in his head). Turns out that it was in the stlutz post that started this discussion more than a year ago:
Doc wrote: This is true but if you reinvest all cash flows in exactly the same portfolio you essentially wind up with a zero with infinite maturity so the SEC yield remains constant forever which is usually longer than the original duration. (At least I think so but I'm breaking a cardinal rule to not do discounted cash flow calculations in my head.) :beer
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Re: So do we care more about change in bond price or bond yi

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Kevin M wrote:
Doc wrote: And Kevin it is very possible to accurately do the discounted cash flow equations in your head if you had taken the time to memorize the data in the log table book when you were in high school which is very helpful if you are now of the age that you have to relearn Excel every time you load it. :beer
Funny. So I had to go find the statement by that poster whose member name rhymes with Spock (who I'm sure could do the calculations in his head). Turns out that it was in the stlutz post that started this discussion more than a year ago:
Doc wrote: This is true but if you reinvest all cash flows in exactly the same portfolio you essentially wind up with a zero with infinite maturity so the SEC yield remains constant forever which is usually longer than the original duration. (At least I think so but I'm breaking a cardinal rule to not do discounted cash flow calculations in my head.) :beer
Kevin
That's not fair. A year or two ago is short term. I just forgot.

Once upon a time in a prior life I did a lot of cash flow studies and got pretty good at estimating the results in my head. But we were doing Net Present Values and the discount rate was always 10% so there were a lot fewer cases to worry about. And unlike bond funds where 10 basis point may be the decision point we were working on development projects where close was good enough.
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Re: So do we care more about change in bond price or bond yi

Post by Kevin M »

A few updates ...

Below is a snapshot of the zero and non-zero 3-10 year yield curves based on average Fidelity bid yields for each maturity as of a few minutes ago. Red curve is for zero-coupon bonds, blue is for non-zero coupon bonds.

Image

Note that there are many more maturities for non-zeros (hence the gaps in the red curve), and note the little wiggles in the curve for non-zeros. I don't see any obvious reason for the wiggles, e.g., due to different coupons.

Regarding the uneven (zig-zag) pattern of annual returns in the chart I posted earlier, the use of the treasury.gov yield curve vs. a yield curve for zero-coupon bonds does not seem to explain it. Rather than using a zero yield curve, I went the other way, and modified my spreadsheet to work with a coupon bond. Then I plugged in Fidelity bid yields for coupon bonds for maturities surrounding the term to maturities in question (i.e., 5-7 years), and the roller-coaster effect is still there, although it varies depending on the assumed coupon rate. No time to post graphs now, but later if any interest.

Finally, I realized that the YIELD function can be used not just for calculating YTM, but also the equivalent yield measure over any holding period (investment horizon). Based on terminology used in one of my investment textbooks, I dub this YTH = Yield To Horizon. For example, you can calculate the YTH for holding a 10-year bond for 3 years assuming a static yield curve.

For a zero coupon bond this is the exact same number as the CAGR I showed in the table and graph above (if I set frequency = 1). For a coupon bond it is the same for shorter holding periods (e.g., up to about 6 years), and then is slightly higher for longer holding periods. The reason for the divergence is that in my CAGR for coupon bonds I kept it simple and did not assume coupons are reinvested, but the YTH measure is calculated as if they are. So you see a bit more divergence at higher coupon rates, but it still is quite small.

So, YTH is a more appropriate yield measure for yield-curve-riders who expect yield curve to remain static and plan to sell before maturity. It tells you exactly what your expected return is given your assumptions, assuming a zero-coupon bond is used or that interest from a coupon bond is reinvested at the YTH rate (and almost the same even without this assumption). I can go into more detail about this if there's any interest.

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Re: So do we care more about change in bond price or bond yi

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Kevin M wrote:Note that there are many more maturities for non-zeros (hence the gaps in the red curve), and note the little wiggles in the curve for non-zeros. I don't see any obvious reason for the wiggles, e.g., due to different coupons.
It looks like they are older bonds with very high coupons. I looked at the 6 to 7 year range and all three have prices in the 140 range. I understand that people don't like premium bonds although I don't know why.
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Re: So do we care more about change in bond price or bond yi

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Doc wrote:
Kevin M wrote:Note that there are many more maturities for non-zeros (hence the gaps in the red curve), and note the little wiggles in the curve for non-zeros. I don't see any obvious reason for the wiggles, e.g., due to different coupons.
It looks like they are older bonds with very high coupons. I looked at the 6 to 7 year range and all three have prices in the 140 range.
You're right! I didn't notice because of the way my data is lined up in the spreadsheet, and the wiggles are actually a bit understated in the graph since each point is an average of yields for that maturity date. For example, the dip in the non-zero curve for maturity date 2/15/2021 is an average of the yields for the two bonds below, but note the yield for the higher coupon is even lower.

Code: Select all

7.875  2/15/2021  AAA  --  136.859  137      1.69
3.625  2/15/2021  AAA  --  110.968  111.063  1.78
Doc wrote:I understand that people don't like premium bonds although I don't know why.
Isn't it just the opposite? The price is higher (which results in the lower YTM); doesn't that indicate higher demand? That's consistent with the lower price/higher YTM for the zeros (lower demand?).

Thinking in terms of discounted cash flows, the higher coupon bond will deliver more coupon payments at a lower discount rate (earlier), so it has a slightly lower duration, and therefore is slightly less risky, thus increasing demand. Indeed, the duration of the 7.875% coupon bond is 3.31 and for the 3.625% bond it's 3.37. Since the only cash flow of the zero is at maturity and therefore is discounted the most, it is riskier, as verified by the duration of 6.32 (= maturity) for the 2/15/2021 zero.

Does that all sound right?

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Re: So do we care more about change in bond price or bond yi

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Kevin M wrote: I didn't notice because of the way my data is lined up in the spreadsheet, and the wiggles are actually a bit understated in the graph since each point is an average of yields for that maturity date.
Use the "chart" display on Fido's site.
Doc wrote:I understand that people don't like premium bonds although I don't know why.
Kevin M wrote: Isn't it just the opposite? The price is higher (which results in the lower YTM); doesn't that indicate higher demand? That's consistent with the lower price/higher YTM for the zeros (lower demand?).
Yeh, like I said I don't understand. Maybe I've got it bass ackwards but I still wouldn't understand. But in any case the high coupon bonds are often outliers on the curve. The chart I looked at was after hours so the bid/ask should not be taken for a "truism".
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Re: So do we care more about change in bond price or bond yi

Post by stlutz »

Given two bonds with the same maturity date, the one with the higher coupon will have a lower duration. So, the bumps probably indicate that that bonds trade more on the basis of duration than maturity.
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Re: So do we care more about change in bond price or bond yi

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stlutz wrote:Given two bonds with the same maturity date, the one with the higher coupon will have a lower duration. So, the bumps probably indicate that that bonds trade more on the basis of duration than maturity.
Yes, that's pretty much what I said, and it also explains the slightly higher yields (and lower prices) for the zeros.

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Re: So do we care more about change in bond price or bond yi

Post by grayfox »

Nothing to add, but I just want to say keep up the good work. I'm reading all the posts and following bond fund analysis discussion. Very enlightening.
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Re: So do we care more about change in bond price or bond yi

Post by Doc »

stlutz & Kevin

You guys are quick. It took me 'til 04:52 local time Thursday to realize the "duration connection".

What's especially frustrating was that I went to the "zero" analogy several days ago to avoid the maturity/duration problem and then didn't realize it when it came back. :oops:
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Re: So do we care more about change in bond price or bond yi

Post by grayfox »

Kevin M wrote:
Doc wrote:
Kevin M wrote:Note that there are many more maturities for non-zeros (hence the gaps in the red curve), and note the little wiggles in the curve for non-zeros. I don't see any obvious reason for the wiggles, e.g., due to different coupons.
It looks like they are older bonds with very high coupons. I looked at the 6 to 7 year range and all three have prices in the 140 range.
You're right! I didn't notice because of the way my data is lined up in the spreadsheet, and the wiggles are actually a bit understated in the graph since each point is an average of yields for that maturity date. For example, the dip in the non-zero curve for maturity date 2/15/2021 is an average of the yields for the two bonds below, but note the yield for the higher coupon is even lower.

Code: Select all

7.875  2/15/2021  AAA  --  136.859  137      1.69
3.625  2/15/2021  AAA  --  110.968  111.063  1.78
Doc wrote:I understand that people don't like premium bonds although I don't know why.
Isn't it just the opposite? The price is higher (which results in the lower YTM); doesn't that indicate higher demand? That's consistent with the lower price/higher YTM for the zeros (lower demand?).

Thinking in terms of discounted cash flows, the higher coupon bond will deliver more coupon payments at a lower discount rate (earlier), so it has a slightly lower duration, and therefore is slightly less risky, thus increasing demand. Indeed, the duration of the 7.875% coupon bond is 3.31 and for the 3.625% bond it's 3.37. Since the only cash flow of the zero is at maturity and therefore is discounted the most, it is riskier, as verified by the duration of 6.32 (= maturity) for the 2/15/2021 zero.

Does that all sound right?

Kevin
So the higher coupon bonds at the same maturity have shorter duration, and would be considered less risky because you get some of your money back sooner. Therefore for the YTM is lower for higher coupons. Makes sense.

Then the highest YTM at any maturity should be for zero-coupon bonds. This chart seems to confirm that:

Image

By the way, I made a startling discovery yesterday about bonds and YTM. You better be sitting down for this:

:idea: The Yield-to-Maturity for Coupon Bonds is a fictitious number.

Now it may be a useful fiction, but is nevertheless fictitious.

The implication is that if you draw the Treasury Yield Curve using the YTM of coupon bonds, it's not the true yield curve. It is usually biased low. In the chart above, the actual true yield curve is the red line, i.e. the zero-coupon yield curve.

Anyone agree or disagree? Maybe everyone already knows this.
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Re: So do we care more about change in bond price or bond yi

Post by Kevin M »

grayfox wrote: By the way, I made a startling discovery yesterday about bonds and YTM. You better be sitting down for this:

:idea: The Yield-to-Maturity for Coupon Bonds is a fictitious number.

Now it may be a useful fiction, but is nevertheless fictitious.

The implication is that if you draw the Treasury Yield Curve using the YTM of coupon bonds, it's not the true yield curve. It is usually biased low. In the chart above, the actual true yield curve is the red line, i.e. the zero-coupon yield curve.

Anyone agree or disagree? Maybe everyone already knows this.
I think you're making two different statements; one about YTM and one about yield curves.

Yield to Maturity (YTM): fictitious?
------------------------------------------

Regarding YTM, it depends how you define "fictitious". YTM is defined as the single discount rate that equates bond price to discounted cash flows. Is a calculated number fictitious? It depends how you interpret it.

A common interpretation of YTM is the annualized rate of return you would earn if you reinvested all coupons at a rate equal to the YTM. If you interpret it this way, then yes, YTM is fictitious, since it is highly unlikely that you will actually reinvest the coupon payments at the YTM rate. This is the reinvestment-risk component of term risk (interest-rate risk), the other component being price risk. If held to maturity, there is no price risk (in nominal terms for nominal bonds), so the reinvestment risk is the only remaining risk (assuming default-free bond).

So your realized annualized return is unlikely to equal the original YTM of the bond.

For low rates, like now, the reinvestment risk does not introduce a huge uncertainty--at least on the downside. For example, even if you reinvest the coupon payments at 0%, the realized annualized return on a 10-year bond with YTM = 2.29% and coupon = 2% is 2.23%, so your maximum downside risk relative to YTM is only 6 basis points per year.

Or course one of the beauties of a zero coupon bond is that there is no interest-rate risk if the intention is to hold the bond to maturity. Price at maturity is certain, and there is no reinvestment risk since there are no coupons to reinvest.

Yield Curve: true?
----------------------

I'm realizing a problem is that we talk in terms of "the yield curve". We've seen now that there is not a single yield curve--even if we look just at treasuries. The yield curve depends not only on term to maturity, but also on coupon rate. So not only is the yield curve different for zero coupon bonds, but the yield curve is different for coupon bonds with large differences in coupon rates. That's why we see the wiggles in the (blue) yield curve in the chart.

The difference is even more dramatically visible if you chart price (instead of yield) vs. term to maturity. The chart below is not really the way I want to present the data, but it's OK for making this particular point clearly, so I'll show it for now.

(All bonds in this chart mature on 11/15 of each year. I used this date because the charts are from Fidelity quotes, and this is the closest date to today that had quotes for each type of bond for each year. I used only one date per year so the charts aren't so cluttered, and so further analysis (e.g., annual returns) is easier.)

Image

The red and green curves are YTM and price for our beautiful zeros. Wonderfully well-behaved because of the consistent coupon rate of 0%.

By contrast, the blue and orange curves are the yield and price for bonds maturing on the same dates but some with different coupon rates. So for this chart I did not average the yields (or prices) for bonds maturing on a given date, as in the chart you quoted in your reply.

First note the discontinuities in the blue yield curve at terms to maturity of 4, 7 and 8. Note that there are two yields at each of these terms, which are the yields for bonds with a big difference in coupon rate. Next note that the two prices at each of these terms show the difference much more dramatically.

The Fidelity quotes don't really provide enough data to draw a single yield curve for bonds with similar coupon rates. That's something I may work on next using bond quotes from the WSJ.

According to one of my investment textbooks, a yield curve should be constructed using bonds of similar characteristics, including coupon rate. According to another textbook, a yield curve constructed from zero coupon bonds sometimes is referred to as the "pure yield curve".

You can read about the methodology used to construct the yield curves provided on the treasury.gov site here: Treasury Yield Curve Methodology. Some highlights:
The Treasury's yield curve is derived using a quasi-cubic hermite spline function. Our inputs are the Close of Business (COB) bid yields for the on-the-run securities. <snip> Treasury reserves the option to input additional bid yields if there is no on-the-run security available for a given maturity range that we deem necessary for deriving a good fit for the quasi-cubic hermite spline curve.
<snip>
More specifically, the current inputs are the most recently auctioned 4-, 13-, 26-, and 52-week bills, plus the most recently auctioned 2-, 3-, 5-, 7-, and 10-year notes and the most recently auctioned 30-year bond, plus the composite rate in the 20-year maturity range.
So since Treasury is using only the most recently auctioned notes/bills in the 1-10 year range, there will be only one coupon rate for a given term to maturity in their yield curve.

However, there are significant differences in the coupon rates between say a 10-year note and a 2-year note, so that I don't know that we can use the treasury.gov yield curve to exactly forecast the yield/price of a 10-year bond 8 years from now based on the YTMs of the 10-year and 2-year terms on the assumption of the current yield curve remaining static. Here are the most recent auction results from TreasuryDirect: Institutional - Announcements, Data & Results

Code: Select all

Security                 Issue     Maturity    High   Interest    
 Term    CUSIP           Date        Date      Yield    Rate
10-Year 912828D56  Yes 10/15/2014  08/15/2024  2.381%  2.375%
3-Year  912828F54  No  10/15/2014  10/15/2017  0.994%  0.875%
7-Year  912828F21  No  09/30/2014  09/30/2021  2.235%  2.125%
5-Year  912828F39  No  09/30/2014  09/30/2019  1.800%  1.750%
2-Year  912828F47  No  09/30/2014  09/30/2016  0.589%  0.500%
So in 8 years the 10-year note will still have a coupon rate of 2.375%, but the current 2-Year YTM on the yield curve is based on a coupon rate of 0.500%. Here are the latest WSJ quotes (yesterday) for notes/bonds maturing on 9/30/2016 (2-year terms):

Code: Select all

Maturity   Coupon    Bid      Asked     Chg    Asked yield
9/30/2016  0.500  100.1875  100.2266  -0.0391  0.382
9/30/2016  1.000  101.1250  101.1328  -0.0391  0.411
9/30/2016  3.000  104.9375  104.9688  -0.0781  0.418
So the 3% coupon 2-year has an ask yield of about 0.42% vs.0.38% for the 0.5% coupon 2-year.

For comparison, treasury.gov reports 2-year bid yield at 0.41% as of yesterday. Computing bid yield from WSJ bid price gives 0.403% for the 0.5% coupon note and 0.437% for the 3% coupon note. Of course there will be some discrepancy because treasury.gov is using their quasi-cubic spline method to interpolate a 2-year rate as of today (or yesterday; so for a hypothetical bond maturing on 10/23/2016), whereas the quotes above are for bonds maturing about a month earlier; I think this plus rounding probably accounts for the difference between 0.403% and 0.41%.

At any rate, there's enough uncertainty about the yield curve remaining relatively static (i.e., highly uncertain) that the relatively small difference in yields for different coupons probably is a minor factor. On the other hand, the bid price for the 3% coupon note is 4.74% higher than for the 0.5% coupon note.

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Re: So do we care more about change in bond price or bond yi

Post by ogd »

Kevin M wrote:The red and green curves are YTM and price for our beautiful zeros. Wonderfully well-behaved because of the consistent coupon rate of 0%.

By contrast, the blue and orange curves are the yield and price for bonds maturing on the same dates but some with different coupon rates. So for this chart I did not average the yields (or prices) for bonds maturing on a given date, as in the chart you quoted in your reply.
I think this is simply by construction of the graph. If you fixed coupon rate to any value and were able to find samples at every point, you'd have a nice well-behaved curve.

Zero coupons do have a special position of popularity (no other single coupon rate will be as well represented at every maturity), and of course the reinvestment risk aspect.

For me, "the" yield curve is the one where coupon ~= YtM at every position, not only because it matches theTreasury data and its long history but also because other investments like CDs behave that way too, which is good for comparison.

Anyway, I wouldn't read too much into the differences between yield curves because they will be small and related mathematically (coupon bonds are a superposition of zeros, the exact shape of which depends on the coupon rates), plus some small tax effects and some random noise around the exact bids.
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Re: So do we care more about change in bond price or bond yi

Post by Doc »

Kevin M wrote:A common interpretation of YTM is the annualized rate of return you would earn if you reinvested all coupons at a rate equal to the YTM.
That is a common interpretation but it isn't actually true. If you look at the equation to determine YTM it is the present value equation which in non bond speak is the internal rate of return or IRR. But the key point is present value which doesn't require reinvestment of coupons. You can also calculate YTM using the future value equations which does make the assumption of reinvestment of coupons. But both equation are the same except for multiplying both sides by (1+i)^N. So why bother to make the reinvestment of coupons assumption when you don't have to do. Nevertheless as Kevin stated that is the "common" interpretation and I am not likely to make it uncommon even for Bogleheads.
Kevin M wrote:The Fidelity quotes don't really provide enough data to draw a single yield curve for bonds with similar coupon rates. That's something I may work on next using bond quotes from the WSJ.
Have you thought about constructing a yield curve with the x axis as duration instead of maturity. I think Fido has all the data needed in their quotes. I think all the "outliers" will disappear with this presentation. After thinking of this for a few days I believe that this is the reason that I stated earlier that "people" don't like buying premium priced bonds. Kevin pointed out that their viewpoint would force the price in the other direction. Yes but these "people" are in the "common interpretation" camp but the actual price of the bonds is set by the bond professionals that are using duration and not maturity in determining a particular bonds worth. :idea: :?:
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Yield vs. Duration

Post by Kevin M »

Doc wrote:Have you thought about constructing a yield curve with the x axis as duration instead of maturity. I think Fido has all the data needed in their quotes. I think all the "outliers" will disappear with this presentation.
I can do that. But I'll do it from the WSJ quotes, since there is more data.

Source: U.S. Treasury Quotes - Markets Data Center - WSJ.com

Image

Still looks pretty noisy.

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Re: So do we care more about change in bond price or bond yi

Post by grayfox »

Kevin M wrote: Yield to Maturity (YTM): fictitious?
------------------------------------------

Regarding YTM, it depends how you define "fictitious". YTM is defined as the single discount rate that equates bond price to discounted cash flows. Is a calculated number fictitious? It depends how you interpret it.
What I mean when I say it's "fictitious" is the way an average is fictitious or CAGR is fictitious.
With your feet in the oven and head in the freezer, you should happy because the average is room temperature? Average temperature is fiction.
E.g. if you have annual returns -0.30, 0.50, -0.20, 0.20, 0.15, then the annualized return 3% is what the return would have to be every year to get the same return. But the return was not the same every year. In fact it was never even close to 3%. That's the fiction.

Same idea with YTM. Like you wrote, it's the the single discount rate that you would have to discount every coupon and the final principle payment to get the actual bond price. The fiction is that the market is discounting all the cash flows are at the same discount rate. But the truth is each cash-flow is discounted at a different rate. And probably none are at the calculated YTM of the coupon bond. Fiction.

Now for the zero-coupon, the YTM is not fictitious at all. The YTM you calculate is the actual rate that the market discounts the single cash flow.
Kevin M wrote: Yield Curve: true?
----------------------
As far as the zero-coupon yield curve being the "true" yield curve, what I mean by that is, well, consider what a yield curve is suppose to show.
The Treasury yield curve should tell us how much, i.e. at what rate, the market us discounting a government-guaranteed cash-flow at some time in the future.
E.g. What is the discount rate at 5 years? Just look at the YTM of 5-year zero-coupon and that's the answer.
But is you look at the YTM of a 5-year coupon bond, it's polluted with all the discounted cash flows at 1, 2, 3, 4 years. So it's not purely 5 years.
According to another textbook, a yield curve constructed from zero coupon bonds sometimes is referred to as the "pure yield curve".
That's exactly how I would describe it, the pure yield curve.
Last edited by grayfox on Fri Oct 24, 2014 4:17 pm, edited 1 time in total.
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Re: So do we care more about change in bond price or bond yi

Post by Doc »

I think it looks better than the previous chart.
Kevin M wrote:Image
Maybe there is too much data. Is there anything that the "spikes" have in common? Are they random or is there something else in there that we can speculate about?
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Re: So do we care more about change in bond price or bond yi

Post by Kevin M »

Doc wrote:
Kevin M wrote:A common interpretation of YTM is the annualized rate of return you would earn if you reinvested all coupons at a rate equal to the YTM.
That is a common interpretation but it isn't actually true. If you look at the equation to determine YTM it is the present value equation which in non bond speak is the internal rate of return or IRR. But the key point is present value which doesn't require reinvestment of coupons. You can also calculate YTM using the future value equations which does make the assumption of reinvestment of coupons. But both equation are the same except for multiplying both sides by (1+i)^N. So why bother to make the reinvestment of coupons assumption when you don't have to do. Nevertheless as Kevin stated that is the "common" interpretation and I am not likely to make it uncommon even for Bogleheads.
I actually ran across a published academic paper a month or two ago that addresses this exact point. It was because of having read that paper that I carefully worded my statement with the phrase "an interpretation" of YTM.

www.economics-finance.org/jefe/econ/For ... lpaper.pdf

However, the criticism of the textbooks is somewhat overstated. Specifically, Bodie, Kane, Marcus (2004) was mentioned but not quoted. I have BKM textbooks published in 1996 and 2008, and they both define YTM correctly, but also state that YTM is "often viewed as a measure of the average rate of return that will be earned on a bond if it is bought now and held until maturity." (from the 2008 textbook). In the 1996 book, they make a similar statement, and also state that "Yield to maturity is widely accepted as a proxy for average return."

So although there is no assumption about reinvesting coupon payments in the precise definition of YTM, it obviously is a common way to think about it.

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Re: So do we care more about change in bond price or bond yi

Post by Kevin M »

Doc wrote: Maybe there is too much data. Is there anything that the "spikes" have in common? Are they random or is there something else in there that we can speculate about?
EDIT: I assume you're talking about the yield vs. duration chart. We already know the spikes in the yield vs. term to maturity chart are due to big differences in coupon rates.

I think it still comes down to differences in coupon rate, or perhaps more accurately the interplay of coupon rate and term to maturity. Here are the bonds with durations between 4.8 and 5.2, sorted by duration.

Image

Note for example the first two rows. Same duration, 13 basis point drop in yield, 7.75 point drop in coupon, and 0.87 year drop in term to maturity.

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Re: So do we care more about change in bond price or bond yi

Post by Doc »

Kevin M wrote:Note for example the first two rows. Same duration, 13 basis point drop in yield, 7.75 point drop in coupon, and 0.87 year drop in term to maturity.
We need more examples. I notice the spread is different. The older higher coupon may have some liquidity issues. Could there be a "bid duration" vs "ask duration" effect? At this pint do we really care anymore. (I'm leaving the typo and am going to go have one. :beer )
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Re: So do we care more about change in bond price or bond yi

Post by Kevin M »

Doc wrote:I think it looks better than the previous chart.
So we can compare the two plots using the same data, here is the YTM vs. Term to Maturity (TTM) for the data from the WSJ:

Image

And here again is the plot of Duration vs. TTM for that data:

Image
Doc wrote:We need more examples.
Anyone can peruse the data at the WSJ site using the link I provided earlier. I shared how to load the data into a spreadsheet in another thread in this reply: Importing yields from treasury.gov WSJ into Google spreadsheet. After that, just add a duration calculation, and if desired a bid YTM calculation, and analyze to your heart's content.
Doc wrote:At this pint do we really care anymore. (I'm leaving the typo and am going to go have one. :beer )
Sounds like a great idea. It is Friday night after all! :sharebeer

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Re: So do we care more about change in bond price or bond yi

Post by grayfox »

This idea of buying a 10-year STRIP and then selling it after one year sounds appealing.

It turns out I actually bought the 2024-08-15 Stripped-Coupon-Interest STRIPS, with settlement date 12/31/2013 for a price of 70.774.
My plan was to hold it to maturity as part of a LMP.

By the way, the are three kinds of STRIPS the WSJ shows:
1 Treasury Bond, Stripped Principal STRIPS
2 Treasury Note, Stripped Principal STRIPS
3 Stripped-Coupon-Interest STRIPS

The Stripped-Coupon-Interest STRIPS seem to have a slightly higher YTM than Stripped-Principle STRIPS. I have been told it's because of taxes.

From the WSJ U.S. Treasury Strips Friday, October 24, 2014

Treasury Bond, Stripped Principal

Maturity Bid Asked Chg Asked yield
2024 Aug 15 78.789 78.865 0.043 2.44
2025 Nov 15 75.107 75.189 0.025 2.60
2026 Nov 15 72.408 72.494 0.036 2.69
2027 Nov 15 69.794 69.884 0.060 2.76

So according to WSJ I am up about 11% in less than one year. :sharebeer
Vanguard shows me a slightly lower price of 78.588
:idea: I wonder if I should sell the 2024 and by the 2025-11-15 or maybe the 2027-11-15

It would be interesting to see what price I can actually sell it for. Maybe I get killed on the Bid-Ask spread when I actually try to sell.

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Re: Yield vs. Duration

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Kevin M wrote:
Doc wrote:Have you thought about constructing a yield curve with the x axis as duration instead of maturity. I think Fido has all the data needed in their quotes. I think all the "outliers" will disappear with this presentation.
I can do that. But I'll do it from the WSJ quotes, since there is more data.

Source: U.S. Treasury Quotes - Markets Data Center - WSJ.com

Image

Still looks pretty noisy.
What is going on here is that duration is also an abstraction. Duration is the time-weighted average of all future payments, but it doesn't correspond directly to risk. This is most noticeable with the distinction between long-term and short-term rates. $30K in a fund with a three-year duration will lose $900 if three-year rates rise by 1%. $10K in a fund with a nine-year duration and $20K in a money-market fund will lose $900 if nine-year rates rise by 1%, which is a lesser risk because long-term rates are less volatile.

Thus I would expect zeros to have slightly less risk than coupon bonds of equal duration, and it does appear from this chart that they have slightly lower yields.
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Re: So do we care more about change in bond price or bond yi

Post by Kevin M »

grayfox wrote:This idea of buying a 10-year STRIP and then selling it after one year sounds appealing.
Just remember that with this strategy you're making a bet on the 9-year rate/price one year after settlement. Your expected 1-year return is the the forecast Yield-to-Horizon (YTH) using your estimate of the 9-year price one year after settlement as the "redemption" price using the PRICE function. Your return will be higher or lower to the extent the forward 9-year price is higher or lower than your forecast price.
grayfox wrote:It turns out I actually bought the 2024-08-15 Stripped-Coupon-Interest STRIPS, with settlement date 12/31/2013 for a price of 70.774.
My plan was to hold it to maturity as part of a LMP.
Makes sense if your liabilities are in nominal dollars, or if you're building in an estimated inflation factor for the 10-year holding period. But there's still some risk of unexpected inflation.
grayfox wrote: By the way, the are three kinds of STRIPS the WSJ shows:
1 Treasury Bond, Stripped Principal STRIPS
2 Treasury Note, Stripped Principal STRIPS
3 Stripped-Coupon-Interest STRIPS

The Stripped-Coupon-Interest STRIPS seem to have a slightly higher YTM than Stripped-Principle STRIPS. I have been told it's because of taxes.

From the WSJ U.S. Treasury Strips Friday, October 24, 2014
Thanks for the education and the link. I added a WsjStrips sheet to my bond calculations spreadsheet, imported the quotes, worked some spreadsheet magic, and created the following charts.

First, the overview of all three types of STRIPS for the full 30-year period:

Image

This verifies your observation about the stripped coupon STRIPS having slightly higher yields than the stripped principle STRIPS. I also did not plot null values, so we can see that there are many more stripped-coupon STRIPS than stripped-principal bonds.

Here's the same chart, but just for the 0-10 year terms to maturity:

Image

We still see the slightly higher yields and availability of the stripped-coupons, and also notice the pricing is a bit noisier. One thing I noticed is that there are STRIPS with the exact same maturity date but different prices/yields. Explanation?

Finally, I charted just the stripped principal bonds and notes:

Image

Some gaps in availability, but pretty smooth and consistent pricing.
grayfox wrote:So according to WSJ I am up about 11% in less than one year. :sharebeer
Vanguard shows me a slightly lower price of 78.588
:idea: I wonder if I should sell the 2024 and by the 2025-11-15 or maybe the 2027-11-15
Congratulations! I recently realized 15.4% and 15.8% gains in about 15-month and 14-month periods (IRR = 12.4% and 13.3%) in a couple of muni-bond ETF trades. But I try to remember not to confuse strategy with outcome. It worked out over the last year or so, but it may not work out so well next year, or the year after that ...
grayfox wrote:It would be interesting to see what price I can actually sell it for. Maybe I get killed on the Bid-Ask spread when I actually try to sell.
But you can see the bid/ask spread online, can't you?
grayfox wrote:Zero-Coupon Bonds Rule!
They certainly do make the calculations easier! And I have to admit I'm tempted to play around with some yield-curve management strategies, perhaps using zeros, with some cash I have sitting in an IRA. I keep thinking about all the evidence that active-management is a loser's game, but if I limit it to a tiny portion of my portfolio, it's just entertainment and education--more enjoyable for me than going to Vegas.
:sharebeer

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Re: So do we care more about change in bond price or bond yi

Post by Doc »

Kevin M wrote:One thing I noticed is that there are STRIPS with the exact same maturity date but different prices/yields. Explanation?
From the announcement of the recent 30 year bond auction.
Minimum Amount Required for STRIPS $100
Corpus CUSIP Number 912803EJ8
Additional TINT(s) Due Date(s) and August 15, 2044
CUSIP Number(s) 912834NV6
You strip a particular issue into two parts therefore you always should have three types for the same original maturity. :idea:
grayfox wrote:By the way, the are three kinds of STRIPS the WSJ shows:
1 Treasury Bond, Stripped Principal STRIPS
2 Treasury Note, Stripped Principal STRIPS
3 Stripped-Coupon-Interest STRIPS
Once upon a time in America Treasury Bonds could be callable. The Treasury stopped issuing callable bonds in 1985 and I would suspect that all have matured making the bond/note distinction mute except for the initial maturity. Maybe the Treasury is maintaining the "bond" designation JIC. However I suspect issuing more longer TIPS would be the preferred method in a high interest rate environment.
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Re: So do we care more about change in bond price or bond yi

Post by Kevin M »

Doc wrote:
Kevin M wrote:One thing I noticed is that there are STRIPS with the exact same maturity date but different prices/yields. Explanation?
You strip a particular issue into two parts therefore you always should have three types for the same original maturity. :idea:
I should have been more specific, but it was in the context of the sentence before (not quoted): "... stripped-coupons, and also notice the pricing is a bit noisier".

So to be more specific, I noticed that there are STRIPS of the same type with same maturity date but different price/yield, and was using that as an obvious example that could introduce "noise" into the plot of price/yield vs. term to maturity. For example, from the recent quotes:

Code: Select all

Maturity  Bid      Asked  Chg    Asked
                                 yield
Stripped Coupon Interest
<snip>
5/15/2023  82.257  82.33  0.04  2.29
5/15/2023  82.868  82.94  0.06  2.20
So two stripped coupon interest zeros with same maturity date but difference price/yield. You see the same thing for the stripped principal bonds and notes.

Because of this I used the average price for each maturity date to generate the data shown in the charts; so there's actually a bit more variance than shown.

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Re: So do we care more about change in bond price or bond yi

Post by Kevin M »

For completeness, I added in the average bid yields of coupon bonds for comparison to the zeros. Here is the chart for maturities up to 10 years:

Image

So lower yields than stripped principal zeros (at least visible beyond six years to maturity), which in turn have lower yields than stripped coupon zeros.

What would be considered the "spot price"? I would guess yields on the stripped coupon securities (as opposed to the stripped principal securities).

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