grayfox wrote:
By the way, I made a startling discovery yesterday about bonds and YTM. You better be sitting down for this:
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?
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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?
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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.)
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.
Kevin
If I make a calculation error, #Cruncher probably will let me know.