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

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

Post by LadyGeek »

Are you now delving into a discussion of convexity?

(Background info In the wiki: Convexity, also see: Advanced Bond Concepts: Convexity on Investopedia. I'm using Frank J. Fabozzi's Fixed Income Mathematics which treats this at a more advanced level. I don't understand everything as it applies here, but I think this is where the discussion is going.)
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Re: So do we care more about change in bond price or bond yi

Post by Kevin M »

LadyGeek wrote:Are you now delving into a discussion of convexity?
Not really, but it's a good question, and good to keep in mind.

(Edited below to correct for missing minus sign on the MD formula, and reword slightly)

Modified duration (MD) is the opposite of the first derivative of the price yield relationship divided by price at a given point on the curve, so MD = -(dP/dY)/P, and convexity is the second derivative (rate of change of the slope). So MD gives us the slope of the curve at point P (with sign reversed), and convexity tells us the rate at which the slope is changing at point P. So the price vs. yield relationship is a non-linear curve, duration is the opposite of the slope of the curve at point P1 divided by P1 (a linear estimate of duration), and convexity indicates how curved the curve is.

The implication is that using the MD approximation formula to estimate change in bond price given change in bond yield applies most accurately for small changes in yield. For larger changes, convexity will result in the slope (duration) changing enough between Y1,P1 and Y2,P2 to decrease the accuracy of the estimated price change, P2 - P1, estimated at the duration/slope at P1.

More relevant here is that duration decreases as maturity decreases, so the duration for a given bond now is smaller than it was six months ago. That's why I used the average of the durations at T2 and T1 to scale the yields in the chart. It's all good enough to see visually that the relationship holds roughly (with some spikiness in there that I can't really explain). The main reason for scaling yield with duration is to see the change in yield and price in the same range, so we can see more clearly that they are inversely related.

EDIT: I realize this can be a little confusing, because you might be visualizing YTM decreasing on a price/yield chart, but in that chart we're looking at a fixed term to maturity. Here what we're looking at is maturity that has decreased by six months, which is the dominant factor in the decrease in duration. Convexity could have a secondary effect. Since I know you know spreadsheets, a good way to develop an understanding of this is to play with the PRICE, YIELD and DURATION or MDURATION functions, and create some charts using different values for the inputs to those functions. That's what I did to create many of the charts in this thread (mainly PRICE and YIELD, but I used MDURATION to calculate the durations to use as scaling factors for the most recent chart).

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

Post by Kevin M »

One thing I"ve been working toward is trying to see if I could verify or debunk the notion that there was a positive contribution to total return due to price increase related to declining maturity (aka, roll yield) over a recent one-year period. The easiest period to use is 8/1/2013 through 7/31/2014, since there are semiannual reports that I can use to see prices and yields for bonds held by the fund at the start and end of this holding period. So I loaded the data from the reports into my spreadsheet (replacing the 1/31/2014 data used to generate the most recent chart I posted with 7/31/2013 data).

Here is the result:

Image

First an explanation about the chart. I only have the data to compute change in price and yield for bonds that were owned by the fund at both start and end of holding period. Since most of the bonds that were in the 3-yr to 4-yr range at start of holding period were sold, I don't have price and yield data for the end of holding period, so can't compute change in price and yield for these. To do so, I'll need to figure out how to look up historical treasury bond quotes (any tips?), and at least fill in the blanks for the big sales. Or, I can just estimate price and yield at and of period based on the yield curve; I work through an example of that below.

At any rate, this is why there is no price/yield data in the chart for the bonds maturing on or before 7/31/2017.

The first thing in this chart that jumped out at me was the significant lack of inverse correlation between change in price and change in yield. After some investigation I figured out that this is because of the impact of changing maturity on the relationship between price and yield at different coupon rates (It's ironic that in my reply to LadyGeek above that I mentioned that the price/yield curves we look at assume fixed maturity, and yet I ignored this in expecting to see the inverse relationship in these charts where we're looking at the change in values between two different dates). This became much more apparent when looking at change over a year than in change over six months.

This explains the "spikiness", which was even somewhat evident in the previous chart. There can be a significant price decrease for little to no increase in YTM for a bond with a coupon that is large relative to YTM, and here can be both price increase and YTM increase for a bond with a relatively small coupon.

For example, for the first big sale (-$154,000 K) for which we see price/yield change data, the yield was just a smidge higher at the end of the period, but price declined 0.82%. The coupon rate on this bond is 1.88%, which is high relative to the YTM of 1.08% at end of period.

Conversely, we see a few sales and purchases in the 2017-2018 range for which change in both YTM and price were positive. The coupon rates on these bonds range from 0.63% to 0.88%, low relative to the YTM range of 1.25% to 1.44% (at end of period) for these bonds.

All of this is pretty much just a verification of something I said in an earlier post, and that someone else pointed out in one of the "riding the yield curve" posts (to which there was no reply); i.e., whether a bond increases or decreases in price as it approaches maturity is dependent not only on declining YTM due to declining maturity (i.e., rolling down the yield curve), but also on the coupon rate (not to mention any shift in the yield curve).

Although I don't have the quotes, one thing I can say is that many of the bonds with large sale amounts were high coupon bonds, e.g., 2.75% - 3.25%. Even without looking up the quotes or estimating yields from the yield curve (which I may do next), I think it's safe to assume that these bonds lost value during the holding period. I'll work through one representative example.

The single largest sale was $199M face value of a 2.75%-coupon bond maturing 5/31/2017, which would have had a term of 3.8 years at start of period, so perhaps one that would have been sold toward end of period as it neared 3-yr maturity. At T1 YTM was 0.95% and price was 106.75. Assuming it was sold at end of period with no change in YTM, price would have been 105.01 for a decline of 1.63%. Assuming same YTM is reasonable, since 3-yr YTM on 7/31/2014 was about the same as 4-yr YTM on 7/31/2013.

Even if yield curve had remained static, so YTM dropped about 40 basis points, price still would have declined by about 50 basis points. The relatively high coupon makes all the difference.

So my conclusion is that there was no benefit from "riding the yield curve" as bonds were sold in the 3-yr to 4-yr range.

As a final observation, I note that total return for the period was 1.53%, of which 1.59% was from dividends, 0.64% was from capital gain distributions, and -0.71% was from price change. At the start of the period distribution yield was 1.50% and SEC yield was 1.30%.

The return did not come from riding the yield curve, the return came from dividend distributions. Capital gain distributions simply partially offset price decline. The fund did not return more than expected based on SEC yield because of riding the yield curve, it returned more than expected based on SEC yield because distribution yield was higher than SEC yield at the start and throughout the holding period.

Of course not too much should be read into this in terms of decision making, other than not to have unrealistic expectations for a benefit from riding the yield curve. For this fund SEC yield and distribution yield now are almost the same anyway, so pick whichever one you want to make your predictions. I don't own the fund. I prefer my CDs and bond funds that are compensating me better for the extra risk with higher yields.

EDIT: This is an example of a fund where for me it's a no-brainer to prefer a good direct CD to this fund if one has the flexibility to choose. The best nationally-available direct CD now is a 2.52% 5-year CD, so much higher yield than both SEC yield of 1.74% (Admiral shares) and distribution yield of 1.79%. Maybe hold some of each if you want the liquidity of treasuries but don't want to take any credit risk.

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

Post by ogd »

Kevin M wrote: One thing I"ve been working toward is trying to see if I could verify or debunk the notion that there was a positive contribution to total return due to price increase related to declining maturity (aka, roll yield) over a recent one-year period.
...
All of this is pretty much just a verification of something I said in an earlier post, and that someone else pointed out in one of the "riding the yield curve" posts (to which there was no reply); i.e., whether a bond increases or decreases in price as it approaches maturity is dependent not only on declining YTM due to declining maturity (i.e., rolling down the yield curve), but also on the coupon rate (not to mention any shift in the yield curve).
Kevin: this is wrong and you are misleading yourself with the focus on price. The roll-down return is independent of the coupon rate. Exactly how much of the return is in price is dependent, but who cares.

The roll-down return is the portion of total return that is attributable to the steepness of the yield curve, i.e. it wouldn't have occured had the yield curve been flat. Alternatively, you can think of it as the excess return earned from a roll transaction vs holding the bond to maturity. Nowhere do we require that the roll-down return is in price alone.

Let's take two extreme examples: 5 year Treasuries held between your dates, 8/1/2013 and 7/31/2014, with coupons of 1% and 5% respectively.

On 8/1/2013, the 5 year yield was 1.49%. The four year yield is not something I have available, but we can interpolate it from the 3 year yield to 1.07%.
On 7/31/2014, the 5 year yield was 1.76%; the four year yield (interpolated) was 1.39%. It's not great that the four year yield is almost the same as the previous 5 year for this argument, but we can try to make do with that difference.

The 1% Treasury appreciated from $97.64 to $98.48, or $0.84. It also paid $1 in coupons during the year, for a total return of $1.84, which is 1.89% of purchase price.
The 5% Treasury depreciated from $116.88 to $114.022, or -$2.86. It paid $5 in coupons, for a total return of $2.14, which is 1.83% of purchase price.

So the return is pretty much identical, the small difference accounted for by the fact that the 5% gave you the coupons earlier, which could have been reinvested.

Both Treasuries exceeded their start and end YtM's of 1.49% and 1.39% respectively, despite an adverse move in yields. To gauge exactly how much return is attributable to the steep yield curve, the best thing to do is imagine that the end curve had been flat (the steepness of the beginning curve doesn't actually come into play; if you want to take that hypothetical road, the total return will be equal to the YtM).

Under the hypothesis that all yields had flattened to the 5 year yield 1.76%, total return is:
1% Treasury: -$0.57 price, $1 coupons results in 0.44% return. This matches the duration equation reasonably well: -0.25% x 4.86 = -1.25%, plus a year of average yield = 0.41%.
5% Treasury: -$4.39 price, $5 coupons results in 0.52% return. Duration equation: -0.25% x 4.48 plus the 1.62% average yield for the year = 0.505%

Note the smaller duration of the 5% Treasury making it hurt less. If you get hung up on the smaller duration, feel free to redo with a Treasury of the exact same duration; for now I want to get moving with this rather long message.

In conclusion, as much as 1.4% of total return is explained by roll-down return (i.e. the fact that the four year yield was smaller than the 5 year), compared with what you would have expected to get based on 5 year yield moves and duration. This varies very little between the two bonds, and consequently has very little to do with distribution yield.

We can also redo the numbers under the hypothesis that the 5 and 4 year yields had stayed constant at the 8/1/2013 numbers (1.49% and 1.07% respectively):
1% Treasury: total return 3.16%
5% Treasury: total return 2.84% (as above, delivered sooner, so more valuable)

So I stand by the "riding the yield curve" thread http://www.bogleheads.org/forum/viewtop ... 32601&f=10. The returns in that period cannot be explained by anything other than roll-down return.

And distribution yield has nothing to do with it, unless you think the fund managers use it to indicate in advance the roll yield they will be able to extract, views that are likely to prove correct. In which case, maybe, but I maintain that focusing on distribution yield is harmful. Case in point, this recent post http://www.bogleheads.org/forum/viewtop ... 8#p2222558.

You can argue that something above is incorrect, e.g. the flatness hypothesis or the interpolated yield, but the basic point will stand regardless, and it's this: for any period in the past 1.5 years of steepness, an intermediate Treasury fund, active or passive, will have had much more total return than can be explained by the Treasury yields of the period, once you account for the appreciation / depreciation that should have occured due to yield moves at its duration. If you find me a counterexample, I will print out this post and eat it. Well, maybe not. I could handwrite it and type "wrong" on top, because I hate handwriting and it would be punishment enough.

Edit: well, scratch that too. I almost lost this post because I got logged out before hitting preview, but I eventually figured out how to recover it in Chrome and I'll make a "forum issues" post about it. This was enough annoyance for one post. :annoyed
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Re: So do we care more about change in bond price or bond yi

Post by Kevin M »

ogd wrote: Kevin: this is wrong and you are misleading yourself with the focus on price. The roll-down return is independent of the coupon rate. Exactly how much of the return is in price is dependent, but who cares.
It seems like you're redefining the "roll-down return" relative to what stlutz originally defined it in the thread that started this discussion.
stlutz wrote: Looking only at the SEC Yield underestimates your expected bond return as it does not include the capital appreciation that one can expect from yield curve riding, which is relatively significant currently for an intermediate term treasury fund/etf.
The major focus of stlutz's posts was the additional capital appreciation due to riding down a steep portion of the yield curve. Capital appreciation is the appreciation in price.

You also are now redefining profiting from riding the yield curve relative to it's standard interpretation. Here's what Investopedia says:
A trading strategy that is based upon the yield curve and used for interest rate futures. Investors hope to achieve capital gains by employing this strategy.
Again, capital gains depend only on change in price.

I don't know how to make it any simpler. So what exactly is wrong with what I'm saying?

Most of my challenge to stlutz's original post was that it was unreasonable to assume you would get a much higher return due to yield-curve-riding. I showed the actual one-year return numbers for the int-term treasury fund, which is the fund stlutz was kind of defending a little over a year ago. The return for the year I looked at wasn't that great, and the realized capital appreciation didn't even offset the unrealized capital depreciation. The total return was about 20 basis points higher than the YTM at the beginning of the period, and pretty close to the distribution yield at the beginning of the period, in the ballpark of 1.5%. Did you look at those numbers? Do they show anything close to the 2.8% stlutz was throwing out in his example?

You accuse me of making "dangerous" statements, yet you post things like this:
ogd wrote: So everyone riding the curve in the intermediate range made quite a bit more than they were supposed to. Say what you will about uncertain bond times, but I like making 60% more than what I was promised
Consider what I'm posting now as a sobering counterpoint to statements like that.

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

Post by ogd »

Kevin M wrote:
ogd wrote: Kevin: this is wrong and you are misleading yourself with the focus on price. The roll-down return is independent of the coupon rate. Exactly how much of the return is in price is dependent, but who cares.
It seems like you're redefining the "roll-down return" relative to what stlutz originally defined it in the thread that started this discussion.
stlutz wrote: Looking only at the SEC Yield underestimates your expected bond return as it does not include the capital appreciation that one can expect from yield curve riding, which is relatively significant currently for an intermediate term treasury fund/etf.
The major focus of stlutz's posts was the additional capital appreciation due to riding down a steep portion of the yield curve. Capital appreciation is the appreciation in price.
A-ha! We finally come to the crux of our misunderstanding!

It is not up to stlutz to define roll-down return; but if it were, I'm sure he'd agree to the following:

Roll down return is total return earned by a bond portfolio from the steepness of the yield curve using a strategy of selling bonds before maturity. This return is in excess of bond YtM and of price changes caused by changes in yields.

The return does not have to be due to capital gains alone. stlutz considered bonds with YtM close to coupons, for simplicity or maybe realism. But even he said:
stlutz wrote:Right now, this fund holds the 9/30/2018 bond yielding 1.54% and priced at 99.188. It has a coupon rate of 1.375%. The common assumption is that the "expected" return on this fund is therefore 1.54%.
, proceeding then to show return in excess of the 1.54% yield. Not the 1.375% coupon. The YtM is the baseline and it already includes some amount of price changes, positive (in this example) or negative.

We can take more extreme examples. If I bought today a Treasury maturing in February 2016 with coupon 9.25% at a YtM of 0.213% it is unreasonable for me to expect 9.25% returns for, say, the next year; in fact, that would be downright impossible. My expectation is 0.213%. If I decide to take advantage of 0.1% of yield curve steepness in a year and roll back to 1.5 years, I can exceed that by a bit and get some money that I wouldn't have gotten had I kept the Treasury to maturity. That's roll-down return. Even if I'm moving from a big price decline to a slightly smaller price decline and as such getting to keep a little more of the gigantic coupons than if the yield curve had been flat, it's still roll return. Your logic says, I believe, that because the 0.1% extra came from the 9.25% coupons, it doesn't qualify as roll return. I beg to differ. To realize it, I needed my maneuver and I needed some steepness, so it's roll return.

The investopedia definition you quoted probably considers bonds with coupon = YtM for simplicity, or perhaps it only refers to interest rate futures which don't have to deal with bonds of varying coupons. Other sources don't do this. For example, this Forbes article directly addresses your point:
Forbes wrote:Why do some bonds, including treasuries, not increase in value as they roll down the yield curve?

When bonds are bought at a premium to their face value, their price may not rise as a result of moving towards maturity. However, the effect is still the same. Essentially, these bonds have coupon rates which are higher than their yield. You can think of the coupon payments as providing a market interest rate + extra interest.
Nevertheless, if you still don't agree on the definition, how about we leave aside the version with capital gains only (you can keep it) and I define my own "ogd roll yield" which doesn't care how the extra return is delivered. All that it cares about is that it's return that would have not occured had the yield curve been flat. In my posts and understanding, this was always about yields and never about coupons.
ogd wrote: So everyone riding the curve in the intermediate range made quite a bit more than they were supposed to. Say what you will about uncertain bond times, but I like making 60% more than what I was promised
Consider what I'm posting now as a sobering counterpoint to statements like that.[/quote]
Not at all. The funds greatly exceeded what we expected from them, and they did so because of the yield curve, in the case of Treasuries entirely, in the case of riskier funds on top of favorable credit risk moves. I fully stand by that statement.
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Re: So do we care more about change in bond price or bond yi

Post by stlutz »

It is not up to stlutz to define roll-down return; but if it were, I'm sure he'd agree to the following:

Roll down return is total return earned by a bond portfolio from the steepness of the yield curve using a strategy of selling bonds before maturity. This return is in excess of bond YtM and of price changes caused by changes in yields.
Yes. I agree.

Whether returns come from distributions or capital appreciation often depends upon random factors I can't really control as an investor. For example, the VG Intermediate term treasury fund has a TTM yield of 1.61% while Int. Term Gov't Bond is 1.34%. Yet, their portfolios are very similar. Before tax, I would expect the returns from both funds to be very close to one another. Actually, their prior one year returns are exactly the same currently (1.58%).

In a taxable account, I would prefer to use the fund with the lower distribution yield if the duration and SEC yield were the same, as returns from capital gains are better than returns from income in such as case. I don't care either way in an IRA.
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Re: So do we care more about change in bond price or bond yi

Post by Kevin M »

I thought it would be useful to simplify this by starting from bond basics. This will show that "riding the yield curve" is all about change in price, and that the shape of the yield curve has no impact on the component of return due to the coupon rate. This can be demonstrated with simple bond algebra, and we can do so without the complexities of considering discount rates, time value of money, etc.

The holding period return (HPR; i.e., how much money you make) of a bond over a given holding period, HP, is completely determined by the sum of the coupon payments and the change in bond price (P2 - P1). Hopefully there's no disagreement about that.

For simplicity I'll assume a single coupon payment, C, at the end of the holding period.

HPR = C + P2 - P1

The HP yield (HPY; i.e., rate of return) for a one-year HP is the HPR divided by P1:

HPY = HPR/P1 = (C + P2 - P1)/P1 = C/P1 + (P2 - P1)/P1
HPY = C/P1 + P2/P1 - 1

This shows clearly that the HPY consists of the coupon yield, CY = C/P1 and the price yield, P2/P1 - 1.

The coupon yield, CY = C/P1, is known with certainty at the beginning of the holding period, since C and P1 are both fixed and known.So Price at the end of the holding period has no impact on the coupon yield, and therefore no impact on the component of return due to the coupon rate (C/100).

So the only unknown at the beginning of HP is P2, the price at the end of the holding period.

Bond price and bond yield (YTM) are two ways of expressing the same thing; they are directly related by a formula. Therefore P2 will depend directly on Y2, the YTM at the end of the holding period. We know Y1, YTM at the beginning of the holding period since we know P1 at the beginning of HP.

If we want to think of YTM as the independent variable, then P2 is directly determined by Y2. All other variables used to determine P2 are fixed and known at the beginning of the HP (e.g., coupon, maturity). Y2 is the only unknown variable at the beginning of HP that affects P2.

Now, to bring this back to the topic at hand.

As time elapses, the time to maturity, M, of the bond decreases. At all times we know the value of M; e.g., M1 at T1 and M2 at T2 (in this example M2 = M1 - 1 year). For example, if M1 is 4 years then M2 is 3 years. M is a parameter used in the bond PRICE and YIELD functions.

So the YTM, Y2, at the end of HP is the YTM we get from the yield curve at T2 for M2. We now have all the parameters to calculate P2, e.g., using the spreadsheet PRICE function.

Using this terminology, "riding the yield curve" means to make a profit (i.e., P2 - P1 > 0) because the yield curve has a positive slope between M1 and M2, so Y2 < Y1. A steeper yield curve will result in a larger difference between Y1 and Y2, therefore a larger profit (P2 - P1).

Although the coupon rate, CR, is one of the parameters used to calculate P from Y or Y from P, coupon rate is fixed and known: CR = C/100. Since C and CR are fixed and known, the slope of the yield curve has no effect on the coupon component of return. This is just a restatement to emphasize that we know C/P1 with certainty at the beginning of the holding period.

To summarize, the only impact of the "shape of the yield curve" or "riding the yield curve" on HPR or HPY is the component of return due to change in price, P2 - P1 (which is directly determined by the difference between Y1 and Y2).

Any flaws in the basic bond algebra here?

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

Post by ogd »

Kevin M wrote: To summarize, the only impact of the "shape of the yield curve" or "riding the yield curve" on HPR or HPY is the component of return due to change in price, P2 - P1 (which is directly determined by the difference between Y1 and Y2).

Any flaws in the basic bond algebra here?
No, completely agreed. Because coupons are fixed for the holding period, roll return manifests as deltas to the price.

Where we differ is this: I count any positive delta, to the difference in price, that arises from the steepness of the yield curve, as roll return. If the price difference is negative, but it is less negative than if the yield curve had been flat, I call that "less negative" amount, i.e. a positive to my bottom line, a roll return. It doesn't have to be a positive impact to a positive or zero price difference for it to count.

At the beginning of a period, I might expect a decline in price based on the YtM; even a huge one like in my 9.25% example. YtM and SEC yield both assume a flat yield curve at the end of the period. If instead the yield curve is not flat, I can exceed those yields; if in fact the yield curve stays exactly the same for a while, I can keep exceeding them; that is, I buy a 0.2% YtM, hold it for a year, and make 0.3%. This is not because I had a 9.25% coupon, which was never mine to begin with, although the excess does end up being paid as part of the coupon. Simply put, I get to keep a little more of those coupons that weren't mine.

The nice part about this view is we don't care which type of bonds we have, high or low coupon. Which is right, because both types make the exact same total returns, equally affected by a flattening of the yield curve for example. If we attribute a component of returns to the yield curve shape, this kind of robustness needs to be a feature of the delimitation.
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Re: So do we care more about change in bond price or bond yi

Post by ogd »

stlutz wrote:
It is not up to stlutz to define roll-down return; but if it were, I'm sure he'd agree to the following:

Roll down return is total return earned by a bond portfolio from the steepness of the yield curve using a strategy of selling bonds before maturity. This return is in excess of bond YtM and of price changes caused by changes in yields.
Yes. I agree.

Whether returns come from distributions or capital appreciation often depends upon random factors I can't really control as an investor. For example, the VG Intermediate term treasury fund has a TTM yield of 1.61% while Int. Term Gov't Bond is 1.34%. Yet, their portfolios are very similar. Before tax, I would expect the returns from both funds to be very close to one another. Actually, their prior one year returns are exactly the same currently (1.58%).
Thanks for chiming in, stlutz! I was afraid I'd overreached with the "I'm sure he'd agree".

Kevin: another way to put it is this: suppose we have Stlutz Fund I , which does the original thing from the original thread. It uses mostly 2012 vintage bonds (because it was a good year), i.e. lower coupons, and always sees price appreciation for every bond.

In the same firm but a fiercely competing department, we have Slutz Fund II. These guys don't like low coupon bonds at all and they only buy 2006-2007 vintage bonds with 5% coupons, at premium, but still keeping duration about the same, so they buy'em and roll'em a little closer to maturity than SF1. They never see anything but price declines, day in and day out, but are otherwise satisfied with the high coupons.

In the original 2013 conditions, SF1 and SF2 make about double their (equal) YtM. Both will have their returns cut to half if the yield curve flattens. Similarly, both will have their returns cut to half if they stop rolling bonds and switch to holding to maturity.

Now what I'm saying is this: if a definition tells me that SF1 and SF2 are making different roll returns, or that SF2 isn't making any, I don't like that definition and it's not the one I'll be using (or I was using). It's clear to me that they should be in the same bucket.

By the way, I don't think that the IRS will actually let SF1 get away with it, consistently delivering returns as capital gains by buying low coupon bonds. Something akin to the discount rules for individuals must be stopping the practice, probably the same rule that makes VFIUX amortize all discounts like it says in their prospectus.
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Re: So do we care more about change in bond price or bond yi

Post by stlutz »

By the way, I don't think that the IRS will actually let SF1 get away with it, consistently delivering returns as capital gains by buying low coupon bonds. Something akin to the discount rules for individuals must be stopping the practice, probably the same rule that makes VFIUX amortize all discounts like it says in their prospectus.
The one [small] advantage that the market discount bond offers is that you can pay the tax on the discount at maturity or when you sell the bond and not each year.
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Re: So do we care more about change in bond price or bond yi

Post by Doc »

Question only. Most of this discussion is above my pay grade and I don't want to take sides.

My understanding of "riding the yield curve" was that the benefit was always there if the yield curve had a positive slope but that the benefit might not be enough to cover the trading costs.

We have two definitions that have may have a significant difference besides any total return/price difference.
A trading strategy that is based upon the yield curve and used for interest rate futures. Investors hope to achieve capital gains by employing this strategy.
Roll down return is total return earned by a bond portfolio from the steepness of the yield curve using a strategy of selling bonds before maturity.
It seems to me that a big difference is the use of derivatives instead of actually buying and selling the underlying security. That's a cost difference.

:?: Question: Is some of the problem with the data one of looking at funds which do not make a large use of derivatives? :?:
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Re: So do we care more about change in bond price or bond yi

Post by ogd »

Doc wrote: We have two definitions that have may have a significant difference besides any total return/price difference.
A trading strategy that is based upon the yield curve and used for interest rate futures. Investors hope to achieve capital gains by employing this strategy.
Roll down return is total return earned by a bond portfolio from the steepness of the yield curve using a strategy of selling bonds before maturity.
It seems to me that a big difference is the use of derivatives instead of actually buying and selling the underlying security. That's a cost difference.

:?: Question: Is some of the problem with the data one of looking at funds which do not make a large use of derivatives? :?:
I think the investopedia definition (first quote) refers to interest rate futures specifically. A consequence is it doesn't have to deal with bonds of various coupons and this might be why it only talks about capital gains. We're not interested in futures over here (well, not me / Kevin / stlutz as far as I can tell)
Doc wrote:My understanding of "riding the yield curve" was that the benefit was always there if the yield curve had a positive slope but that the benefit might not be enough to cover the trading costs.
Yes, same understanding here. Good point about the trading costs -- for Treasuries they're negligible but I think you see this expressed in the much lower turnover of muni funds, for example. With munis you might know your bond is making too little YtM, but it would be too expensive to sell so you hold it to maturity anyway. Or roll less frequently, which is what I think VWITX does.
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Re: So do we care more about change in bond price or bond yi

Post by Doc »

ogd wrote:I think the investopedia definition (first quote) refers to interest rate futures specifically. A consequence is it doesn't have to deal with bonds of various coupons and this might be why it only talks about capital gains. We're not interested in futures over here (well, not me / Kevin / stlutz as far as I can tell)
ogd wrote:... about the trading costs -- for Treasuries they're negligible but I think you see this expressed in the much lower turnover of muni funds, for example.
Trading costs for Treasuries might by 12 bps for a round trip (from memory of quotes). Trading costs to accomplish the same thing with futures maybe 2 bps (WAG).

Is the theoretical benefit from "riding the curve" about 10 bps? If so an intermediate term bond ETF or even Vanguards "actively" managed fund can't take advantage of it but a PIMCO managed fund might be able to. All I am trying to say is that we might not see the effect in the fund data because that fund doesn't use derivatives to take advantage of the phenomena.
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Re: So do we care more about change in bond price or bond yi

Post by ogd »

Doc wrote: Trading costs for Treasuries might by 12 bps for a round trip (from memory of quotes). Trading costs to accomplish the same thing with futures maybe 2 bps (WAG).

Is the theoretical benefit from "riding the curve" about 10 bps? If so an intermediate term bond ETF or even Vanguards "actively" managed fund can't take advantage of it but a PIMCO managed fund might be able to. All I am trying to say is that we might not see the effect in the fund data because that fund doesn't use derivatives to take advantage of the phenomena.
Oh no, the theoretical bonus is huge presently.

From WSJ's treasury quotes on Thursday (last time it worked, for some reason), I can take these two Treasuries:
9/30/2018 1.375 100.1406 100.1875 0.0156 1.326
9/30/2019 1.750 100.8438 100.8906 0.0234 1.563

Suppose I buy the latter and suppose a year from now I can still sell it at 1.326% YtM, which is where a bond shorter by one year is now trading (ignoring the tiny spreads). From the price difference plus coupons I get 2.49%, which is almost 1% higher than the YtM I am promised. Yup, it's a big deal. Last year it was even steeper; since then the curve has flattened, but still remained steep enough for the bonus to have worked out in practice. Next year it might be different of course and four year-ish bonds might trade at 1.56% or above by then.
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Re: So do we care more about change in bond price or bond yi

Post by ogd »

ogd wrote: From WSJ's treasury quotes on Thursday (last time it worked, for some reason), I can take these two Treasuries:
9/30/2018 1.375 100.1406 100.1875 0.0156 1.326
9/30/2019 1.750 100.8438 100.8906 0.0234 1.563

Suppose I buy the latter and suppose a year from now I can still sell it at 1.326% YtM, which is where a bond shorter by one year is now trading (ignoring the tiny spreads). From the price difference plus coupons I get 2.49%, which is almost 1% higher than the YtM I am promised. Yup, it's a big deal. Last year it was even steeper; since then the curve has flattened, but still remained steep enough for the bonus to have worked out in practice. Next year it might be different of course and four year-ish bonds might trade at 1.56% or above by then.
This does raise a question having to do with intepretation, btw. I'm definitely open to opinions here.

Suppose a year from now, the curve looks like this: 1.56% at four year and 2% at five year, so it's even steeper than right now. The 1.56% bond I bought today is not giving me the 2.49% return I was (unreliably) forecasting above, but merely its 1.56% YtM.

On one hand, my bond hasn't gained anything by this exercise. On the other hand, if I look at the portfolio / fund as a whole, I'm seeing 0.44% higher yields at its duration, so I would have expected to lose about 2.2% value to the higher yields. But because the yield curve is steep and I'm able to sell bonds for the same YtM I bought them, I haven't actually lost anything and what I got for the year was the 1.56% that I would have (conservatively) expected from the SEC yield / YtMs.

Should I say I got a 2.2% roll return, which moved to counteract the value loss? Personally I think we should, because whenever we talk about bond funds here this 2.2% loss is front and center in every discussion, and if there's a factor that erases it we shouldn't discount it. So I would break down the return as such: 1.56% YtM or SEC yield, -2.2% yield increases, 2.2% roll return. But I can see why someone would just say 1.56% YtM and 0% for the others, but if we talk about it in those terms we should also say that the 2.2% loss might simply not happen despite the yields going up, if the yield curve stays steep enough. Or something.

Edit: the other reason to label them separately is that they might occur on entirely different timelines: the -2.2% value loss could be immediate like in summer 2013, the 2.2% roll return gradually over the course of the next year. So it would be hard to get away with "the 2.2% loss might not happen", like I suggested above, when people see it clearly on a graph.

(Note: It's a 2.2% roll return because that's the difference that the steep yield curve made at the end of my period. Had it gone completely flat at 2%, the new yield for my duration, I would have been 2.2% poorer)

(Note 2: the roll return is equal to the price decline in my example because I chose the steepness to be such that the four year yield is precisely where the five year yield used to be. This choice was arbitrary, but perhaps a likely occurence if the bond market is right about things)

(Note 3: this is not the issue Kevin was talking about, which had to do with requiring positive price movements. I can still tweak the type of bonds that I'm buying, i.e. their coupons, so as to show either a capital gain or a capital loss at the end of the period)
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Re: So do we care more about change in bond price or bond yi

Post by Kevin M »

ogd wrote:
Kevin M wrote: To summarize, the only impact of the "shape of the yield curve" or "riding the yield curve" on HPR or HPY is the component of return due to change in price, P2 - P1 (which is directly determined by the difference between Y1 and Y2).

Any flaws in the basic bond algebra here?
No, completely agreed. Because coupons are fixed for the holding period, roll return manifests as deltas to the price.
Perfect. I think in terms of the fundamental bond algebra when thinking about about all of this, and I think it can be used as the basis to explain everything we're discussing. I plan to show some examples. Hopefully sticking with the fundamental bond return equation and showing some examples will help others better understand what we're discussing, and avoid the apparent disagreements because of different terminology and mental models.
ogd wrote:
Where we differ is this: I count any positive delta, to the difference in price, that arises from the steepness of the yield curve, as roll return. If the price difference is negative, but it is less negative than if the yield curve had been flat, I call that "less negative" amount, i.e. a positive to my bottom line, a roll return. It doesn't have to be a positive impact to a positive or zero price difference for it to count.
I do not disagree with this, so we don't differ on this either.

You may have gotten that impression because of the history of this discussion, starting with stlutz's post on June 25, 2013. I have consistently been cautioning against expecting much higher returns due to riding the yield curve. So I have put some emphasis on, "see, it didn't work out--you didn't get 2.8% after all". But I also agree that the effect of some of the bonds increasing in price due to the yield-curve-ride effect works to offset the decline in price due to the yield curve shifting up in the range of interest. I've already acknowledge that several times in this thread, so based on that, you should see that we don't disagree on the point as you've stated in the quote above.

I was thinking last night that this really deserves an stlutz parody thread, as it is quite humorous and ironic if you step back and look at the positions taken over time, and the evolution of the back and forth. I'll provide a rough draft for stlutz in my next post.

But ogd, now that we've agreed on the basic bond algebra (thank goodness!), why did you say I was wrong to focus on price change in evaluating the benefit of riding the yield curve? Please explain how the comments I made focusing on bond price were wrong, and explain how they were wrong with some simple algebra if possible.

Yes, you can get a larger potential benefit by moving P1/Y1 further up the yield curve; i.e., buying a bond with higher YTM and more risk. But that's just taking more term risk, which you should be rewarded for with higher YTM. Once you've placed your bet, the only unknown is P2/Y2; i.e., price and yield at the end of the holding period.

Of course part of your riding the yield curve strategy could be selecting a bond at the top of the steepest portion of the current yield curve, so you're not just looking at term risk (since the steepest portion of the yield curve is not necessarily at longer maturities), but also potential benefit from maximum price appreciation assuming the yield curve does not flatten over that segment during your holding period. But for that strategy it comes down even more to price appreciation as the key parameter.

I keeping thinking we agree on a particular aspect, but then you say, "no, we don't agree" or "no, you're thinking about it wrong". Color me puzzled. :confused

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Riding the yield curve recap: Part 1

Post by Kevin M »

EDIT: I'm correcting minor errors and making minor changes that might make it more entertaining; e.g., I forgot to mention the announcement of "The Stlutz Intermediate Term Treasury Fund".

Well, it's probably a waste of time, but I'm retired, so I can waste time however I choose, so here's part 1 of the recap of the discussion about riding the yield curve. It would be much funnier in the form of an stlutz parody thread, but perhaps some of you will find it mildly entertaining and even helpful in getting more oriented to this discussion.

(For those who don't know what I mean by an "stlutz parody thread", you should read this, The Final, Definitive Thead on Value, especially if you have followed any of the debates about tilting to small-cap and value stocks. I've read it twice, and laughed out loud both times.)

I don't have the stlutz humor chops, but here's the recap. Aside from some simplifications, omissions, and some very minor attempts at humor, this is basically how the conversation actually went.

stlutz (6/25/13): People have been panicking about bonds, and CDs earning 2% are considered a free lunch. I've crunched some numbers, and think that treasuries are getting the short end of the stick. We could earn 2.8% if the yield curve doesn't change. So the intermediate-term treasury fund is better than CDs, which are not a free lunch after all; the treasury fund may even be better after tax than a muni fund.

ogd: hold on now, the treasury fund has a yield of only 1.07%. I think you're double counting the yield.

stlutz: You're looking at SEC yield, which is an average. The YTM probably is closer to 1.5%, and the rest of the return comes from price appreciation by selling a 5-year bond after one year.

ogd: I still don't see how you can get 2.8%.

Kevin M: You're making it too complicated. a 5-year CD yields 2%, and 5-year treasury yields 1.5%. There's your free lunch. The mythical free lunch is squeezing 2.8% out of a bond that yields 1.5%. And by the way, most people don't see CDs as a free lunch; I'm in the minority (editor: "Kevin's unloved CDs" vs. "Mel's unloved midcaps"), and most people want to "stay the course" with their lousy bond funds.

Simplegift: [asks Kevin M a question about CD strategy, which gets temporarily lost in the shuffle]

richard: He's not getting a 2.8% yield, he's getting 1.4% in capital appreciation by "riding the yield curve". (editor's note: first time the term is introduced into the discussion). But it's not a free lunch -- Google it to find out more.

stlutz: [chimes in with more explanation of riding the yield curve, and points out that in normal times it may not be a huge deal, but now the capital gain portion is a large part of the returns on a fund like VG int-term treasury.]

ogd: OK, now that it's morning and I've had my coffee, I get it. SEC yield assumes bonds are held to maturity, but they're not in an intermediate-term fund. "The distribution yield is more like it, but it has other reliability problems. Will somebody please tell me what my fund will be making if nothing changes???

Doc: If the yield curve and fund strategy remains constant, you'll get the SEC yield for the duration.

ogd: That's what I used to think, but then how do we reconcile the low SEC yield with the much higher results of stlutz's scenario.

Doc: [Tries to explain it, but nobody understands what he's saying. He says he's going to shut up because he's off topic anyway.]

ogd: No,you're not off topic. We've been comparing bonds to CDs based on SEC yield, and I want to know if this is wrong.

stlutz Yes, ogd, you've got it! SEC yield understates expected return for the bond fund, because it does not include the capital appreciation from riding the yield curve.

stlutz: I just looked at some more numbers, and now I'm confused about what SEC yield is.

Kevin M: SEC yield is wacky, but it's still a better than distribution yield to estimate total return.

Kevin M: [Finally gets around to answering SimpleGift's question, and launches into a dissertation on his CD strategy. (editor: Oh boy, he thinks he's so smart).

[Someone says CDs are dangerous because of what happened in Cyprus. ogd asks if we can please start another thread to discuss that, and Kevin M gives ogd's comment a thumbs up. (editor: they actually do agree from time to time)]

ogd: I'll consider the SEC the conservative lower bound, without expecting much more.

stultz: SEC yield is fine for yield, but don't forget the capital appreciation from riding the yield curve, which is higher than usual now due to a steeper than usual yield curve.

[A side discussion is launched about how SEC yield is calculated, and why price and SEC yield of the fund changed in a particular way over a short time.]

Kevin M: I still think there's something fishy in Denmark with this yield curve riding stuff. Here are a bunch of actual numbers from the fund for a one year period ending 6/28/13. [a bunch of numbers with calculations follow]. The fund had a total return of -1.62%. The drop in fund price looks even worse than expected, so how did riding the yield curve help?

stlutz: [tries to answer Kevin M's question with a bunch of numbers and calculations]

Kevin M: I don't understand enough about bonds to evaluate your numbers, but I don't think they represent the bond fund returns. [Launches into another dissertation, in which he points out that Vanguard has said that SEC yield is a reliable predictor of subsequent 10-year total return]. SEC yield is a reasonable indicator of long-term returns, and the higher distribution yields aren't worth the extra interest-rate risk to me.

ogd: [responds to some of the points in Kevin M's dissertation (editor: well at least someone read it).] You might not get the benefit of riding the yield curve, because the yield curve could flatten out. SEC yield is a conservative lower bound.

[Kevin M and ogd launch into a debate about how bond funds work, and how the performance of individual bonds can or can't explain the bond fund returns. Nobody else cares.]

stlutz [provides an example using an individual bond that he says explains the fund results Kevin M had presented earlier, and announces the launch of "The Stlutz Intermediate Term Treasury Fund".]

ogd: Bummer! I made some decisions comparing FDIC yields to SEC yields. I should have bought treasuries instead. I better move it into the Stlutz treasury fund; what's the ticker symbol?.

Kevin M: OK stlutz, you won the bond math contest, but I'm still not going to invest in your fund. I'm sticking with my CDs.

Kevin M: You should be comparing a 5-year CD to a 5-year treasury, not to a bond fund. If the bond fund has a higher expected return, then it must have higher risk than the 5-year treasury. [Kevin provides a quote from a book on bonds about riding the yield curve, and highlights the part that says "if interest rates rise, it may backfire"]. SEC yield may be too low, so bump it up a bit, but the risk of the bond fund still is much greater than that of the CD. Your yield curve assumptions could come back to bite you.

ogd: You're thinking about it wrong. I eat interest-rate risk for breakfast, so I get a free lunch (uh, breakfast that is) by taking more interest-rate risk than I can get with 5-year CDs. Since I dig interest-rate risk, I like taking more of it in the treasury fund. I'm more worried about stock risk anyway.

[Some back and forth between ogd and Kevin M on "the right way to look at it"; as usual, nobody else cares]

[a little more conversation about various aspects of the topic, then thread peters out]

-------------------------------------- END of PART ONE ----------------------------

Preview of chapter two:

ogd: Well, the numbers are in, and we've benefited nicely from riding the yield curve.

Kevin M: The returns look lousy--I don't know what you're so excited about.
Last edited by Kevin M on Tue Oct 14, 2014 5:11 pm, edited 2 times in total.
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Re: So do we care more about change in bond price or bond yi

Post by peppers »

Kevin M

Nice job. See, some of us are paying attention.
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Re: So do we care more about change in bond price or bond yi

Post by ogd »

Kevin: funny! Not exactly stlutz-style but a very passable summarody. You do have a lot of time on your hands :sharebeer
Kevin M wrote:ogd: Bummer! I used made some decision comparing FDIC yields to SEC yields. I should have bought treasuries instead. I better move it back into treasuries.
For part two, please consider this new information:

ogd, as he rebalances into stocks for the first time in a while: cha-ching!
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Re: So do we care more about change in bond price or bond yi

Post by Doc »

Let me set up a trivial example in order to try to understand what this is all about.

We have two Treasury funds that invest only in a minimum number of five year zero coupon Treasuries. The yield curve does not change during our analysis.

Fund 0-5 invests in five zero coupon securities and holds them until they mature and then reinvests in another five year note at auction (a traditional 5 year ladder). Average maturity/duration of the fund is ~2.5 years.

Fund 1-5 invests in only four zero coupon securities and sells them when they have one year remaining and reinvests the proceeds in another five at auction (a 1-5 year ladder). Average maturity/duration of ~3.0 years.

Questions:

1) Fund 1-5 has a higher return because it is making use of "rolling the curve". But isn't the extra return the same as the higher duration? If not the bond traders would arbitrage it away.

2) Wouldn't Fund 1-5 report some of the income as capital gains not interest? Shouldn't we be able to see get this number from annual reports or 1099's?

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

Post by ogd »

Doc: for your example, my spreadsheet says:

10/15/2019 0% 1.45% 93.01876765 10/15/2018 0.09% 99.9100506 7.41% 1.80%

1.45% and 0.09% are today's yields at five and one year respectively. The last number, 1.80%, is the annualized return for this strategy. Note that it's much greater than the five year and approaching a seven year yield, so duration is not a good explanation. It's easy to understand why: five year's worth of 1.45% was delivered almost entirely within the first four years, boosting the yield by 5/4 in those first four years. The fund that holds to maturity spends another year earning only 0.09%, very much not worth it.

For kicks, look at what happens for a 5 to 3 rolling strategy, where the curve is even steeper:

10/15/2019 0% 1.45% 93.01876765 10/15/2016 0.80% 97.63091097 4.96% 2.45%

This approaches 20 year yield territory, so again a four year duration can't possibly account for it.

The "bond traders" will probably increase the yield curve so that it's not 0.80% @ 3 years in 2016. If I change that number to the same 1.45% I started with, I get the expected 1.45% annual return (spreadsheet verification, you might call it). However, if they don't, like they didn't since summer 2013 because things remain uncertain, I (the hypothetical bond roller) am making a lot of money. This is what we've been seeing in practice for the past 1.5 years.

As for reporting, I am unsure how it works for funds, but in this example the OIDs would make this strategy income-generating (as opposed to capital gains -generating) for individuals so it probably does for the fund too.
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Re: So do we care more about change in bond price or bond yi

Post by ogd »

Kevin M wrote:But ogd, now that we've agreed on the basic bond algebra (thank goodness!), why did you say I was wrong to focus on price change in evaluating the benefit of riding the yield curve? Please explain how the comments I made focusing on bond price were wrong, and explain how they were wrong with some simple algebra if possible.
Kevin: there were a bunch of posts in the middle-to-late thread where you were making the argument that if a bond was not sold for a higher price than bought, then there is no roll return. Digging them up:
Kevin M wrote: All of this is pretty much just a verification of something I said in an earlier post, and that someone else pointed out in one of the "riding the yield curve" posts (to which there was no reply); i.e., whether a bond increases or decreases in price as it approaches maturity is dependent not only on declining YTM due to declining maturity (i.e., rolling down the yield curve), but also on the coupon rate (not to mention any shift in the yield curve).

Although I don't have the quotes, one thing I can say is that many of the bonds with large sale amounts were high coupon bonds, e.g., 2.75% - 3.25%. Even without looking up the quotes or estimating yields from the yield curve (which I may do next), I think it's safe to assume that these bonds lost value during the holding period. I'll work through one representative example.

...example can be found with Ctrl-F above...
So my conclusion is that there was no benefit from "riding the yield curve" as bonds were sold in the 3-yr to 4-yr range.

As a final observation, I note that total return for the period was 1.53%, of which 1.59% was from dividends, 0.64% was from capital gain distributions, and -0.71% was from price change. At the start of the period distribution yield was 1.50% and SEC yield was 1.30%.

The return did not come from riding the yield curve, the return came from dividend distributions. Capital gain distributions simply partially offset price decline. The fund did not return more than expected based on SEC yield because of riding the yield curve, it returned more than expected based on SEC yield because distribution yield was higher than SEC yield at the start and throughout the holding period.
This contradicts what I think we're saying in the last couple of posts, which is this: if you bought a high coupon bond at a lower YtM, thus signing up for high dividends but a capital loss, and the steep yield curve gives you a smaller capital loss than initially expected, the difference should be counted as positive roll return, even if it was delivered as coupons that you kept more of than you were expecting.

In your next post, you seemed to argue that only capital appreciation can count as roll return and "less depreciation" shouldn't:
Kevin M wrote: The major focus of stlutz's posts was the additional capital appreciation due to riding down a steep portion of the yield curve. Capital appreciation is the appreciation in price.

You also are now redefining profiting from riding the yield curve relative to it's standard interpretation. Here's what Investopedia says:

A trading strategy that is based upon the yield curve and used for interest rate futures. Investors hope to achieve capital gains by employing this strategy.
If all of the above allow for roll return from the "less depreciation" effect, then this was a misunderstanding. But I still don't know why then you'd be discussing higher and lower coupon bonds and how their price behaviour differed -- and why we're discussing distribution yield at all. For example, here's what Doc's requested scenario looks like with high coupon bonds as opposed to zero coupons.

10/15/2019 5% 1.45% 117.0919826 10/15/2018 0.09% 104.9072394 6.67% 1.63%

So hardly a difference, once you account for those coupons delivering returns earlier for reinvestment. Coupons and the direction of "natural" price movements (before roll return comes into play) just don't matter much for this.

Btw, I would welcome your input on this post above: http://www.bogleheads.org/forum/viewtop ... 0#p2224926 , which is where I could be swayed either way. Should we call it roll return when the steepness of the yield curve lessened the losses not only from the expected price declines of high coupon bonds, but also from an increase in yields at the fund's duration point? I think we should. This relates to one other point of yours:
Kevin M wrote:The total return was about 20 basis points higher than the YTM at the beginning of the period, and pretty close to the distribution yield at the beginning of the period, in the ballpark of 1.5%. Did you look at those numbers? Do they show anything close to the 2.8% stlutz was throwing out in his example?
, which I didn't address specifically because I don't think it's fair to expect 2.8% returns within those specific dates, with all the yield increases that occured. I have much better behaved ranges in my "Riding the yield curve" thread where I was specifically trying to find dates with similar-ish yield curves on both ends so I could get a clearer picture of this effect on its own.
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Re: So do we care more about change in bond price or bond yi

Post by Kevin M »

Doc wrote: 1) Fund 1-5 has a higher return because it is making use of "rolling the curve". But isn't the extra return the same as the higher duration? If not the bond traders would arbitrage it away.
I think simple bond math/algebra answers question #1a, and I know you know bond math--probably better than I do. I'm developing a preference to stay away from made-up terms or misused terms, but if you price a bond at two points on your static yield curve, you see exactly how much return you get from price change, and since your example bonds have no coupon payments, all of your return came from price change (not exactly true due to net amortization of premiums and discounts, but I address that below). The price change will depend on the steepness of the yield curve between the two points, or more simply, on the YTMs of the bond at the two points.

I showed this graphically up-thread, but here it is again (I'm only showing the price change chart, but I also included the yield-curve chart earlier):

Image

It's clear that the slope of price change for the 0% coupon bond is steeper in the 1-5 year range than in the 0-1 year range, so the 1-5 year fund will have a higher return with a static yield curve. A 2-5 or 3-5 year fund would do even better, assuming a static yield curve that looks something like today's yield curve (but note the chart above was from the yield curve on the day I first posted it).

With respect to question #1b about duration, then the answer is at least partially "yes". In a riding-the-yield-curve strategy you are rewarded both for taking more term risk, and for selling the bond before it enters a flatter portion of the yield curve.

Some of my comments quoted just above by ogd were attempting to explain this. With a zero coupon bond, essentially all return is reflected in price change. For coupon bonds, some of the return is from price change and some from coupon; the coupon component is known in advance--there is no risk to it, and the steepness of the yield curve does not affect it. In the real world, the risky part is the price change based on YTM at the end of the holding period, which is unknown in advance. In your static-yield-curve scenario world, there is no risk to the price-change component either, and we can all get rich by simply rolling over zero coupon bonds wherever the price change slope is the steepest.

I believe the answer to question #1c is at least partially "yes". If this were a free lunch, it would be arbitraged away. The key is that your premise of a static yield curve is not reality. If I knew the yield curve would be static, I most definitely would buy bonds only at the top of the steepest section of the yield curve and sell them at the bottom of the steepest section, after factoring in transaction costs and taxes.

Again, some of my comments that ogd quoted just above (and many others in our discussions about this since the introduction of the Stlutz treasury fund) were attempts to emphasize this point. In late June 2013, the chairman of the Stlutz Intermediate Term Treasury Fund (finally--a fund you can understand; it owns only one bond!) was forecasting a 1-year return of 2.8%, despite the much lower SEC yield (with a caveat in the fine print that a static yield curve was assumed ). In the fund's annual report at the end of its first fiscal year, 7/31/2014, the chairman explained that although the change in the yield curve had hurt their results, their yield-curve-riding strategy had kept it from being even worse, and after all, they did beat the SEC yield from one year earlier by about 20 basis points.
Doc wrote: 2) Wouldn't Fund 1-5 report some of the income as capital gains not interest? Shouldn't we be able to see get this number from annual reports or 1099's?
Based on my reading of the annual and semiannual reports for the Vanguard treasury funds, the net amortization of discount and premium (based on purchase price) would be reported as net investment income, and would have been delivered to the shareholders in the form of dividend distributions. After accounting for discount/premium, the remainder of the return would have been delivered as capital gains, through capital gain distributions for the realized capital gains (from selling bonds as they hit 1-year maturity), and as NAV price appreciation for the unrealized capital gains (or any retained realized capital gains).

Your 1099s will tell you the form in which the fund distributed the returns, either as dividends or capital gains. A perusal of the Statement of "Changes in Net Assets", "Financial Highlights", and "Statement of Operations" in the Annual or SemiAnnual report will show you that the dividend distributions came from "Net Investment Income", and will also show the the distributions from capital gains, as well as the unrealized capital gains, and on how all of this resulted in the ending NAV (relative to the NAV at start of period).

However, to see that Net Investment Income included the net amortization of premiums and discounts, you'll have to read the Notes to Financial Statements, as I've shared in a previous post (and on which you commented).

Bottom line: it's mostly about the bond math and risk. The quantitative questions can be mostly answered by plugging numbers into a spreadsheet and using the PRICE and YIELD functions (as you know), along with the simple arithmetic required to compute any coupon return component. Creating some graphs from the resulting numbers, like the one above, helps add intuitive understanding (at least for me). There's really not much to debate if we rely on the basic bond algebra I posted earlier.

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

Post by Doc »

Kevin, I have not spent enough time analyzing your response but intend to later.

What I am attempting to do is to simplify the problem as much as possible so that we can get away from what Vanguard does or what the shape of the yield curve is now or what it will be tomorrow. This is based on the crawl, before walk, before run idea.

I purposely used the zero coupon analogy so we can dispense with any market premium/discount amortization/accretion complications. This also eliminates any maturity versus duration complications. I further specified that the yield curve doesn't change over the time frame of our analysis. I am now going to simplify further by saying that the yield curve has no curvature over the time frame in question. (I think this means there is zero convexity.) I am not sure this makes later any difference.

First principle of bonds (which is not dependent on any of the above): They are fixed income. The purchaser enters into a contract with the lender to trade a certain cash flow for an initial payment. Barring default there is nothing the purchaser can do for the "duration" of the contract to increase his initial yield unless he makes some kind of trade that increases his risk. The whole idea of "riding the yield curve" can have no impact at all at least before taxes, on the original agreed upon terms unless the purchaser reinvests in more risky securities. In the basic "riding the curve" tactic the investor sells before maturity, realizes a capital gain now instead of waiting for the interest payments and reinvest the proceeds in a longer term note thus increasing his risk. Therefore if all the bond traders could know for certainty what the future curve would look like there is no way that any gain from riding the curve would not be arbitraged away except for differences in trading costs or tax implications (ordinary income versus capital gains).

If you and I and all the other posters are aware of this idea then all of the professional bond traders also know it and any advantage that one has over another has to disappear and there is no advantage to the technique other than trading costs or tax differences. Of course one manager may have more skill than another but that is a whole different discussion. (Think PIMCO Total Return maybe?)

Now if the yield curve changes during the course of the investment time frame selling before maturity and reinvesting could have some real benefit that those managers required to or choosing to hold to maturity cannot take advantage of.
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Re: So do we care more about change in bond price or bond yi

Post by Kevin M »

Doc, I think I agree with everything you say, but like you, I'll take a closer look later and see if there are any discrepancies between your understanding and mine. I am interested to see if you think there are any discrepancies after you review my post more carefully.

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

Post by gerrym51 »

if you own individual bonds-or brokered CD's-or term limited bond fund-then an increase in nav is a good thing if you want to sell. i posees a term limited bond fund and it has been going up like crazy latelly. when i bought it last year and bond navs were going down it gave me agita. who knows
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Re: So do we care more about change in bond price or bond yi

Post by Doc »

In reply to Kevin, we are in agreement except maybe for some of the tax aspects of amortization/accretion accounting and that not worth worrying about.
The price change will depend on the steepness of the yield curve between the two points, or more simply, on the YTMs of the bond at the two points.
Right
It's clear that the slope of price change for the 0% coupon bond is steeper ...
OK. The steeper the curve the bigger the effect.
With respect to question #1b about duration, then the answer is at least partially "yes". In a riding-the-yield-curve strategy you are rewarded both for taking more term risk, and for selling the bond before it enters a flatter portion of the yield curve.
Ok but the before it enters a flatter portion is just a nuance of the "rolling" aspect.
The key is that your premise of a static yield curve is not reality.
Right it is just an artificial mechanism to try to isolate different effects.
... the chairman explained that although the change in the yield curve had hurt their results, their yield-curve-riding strategy had kept it from being even worse, and after all, they did beat the SEC yield from one year earlier by about 20 basis points.
Chicken and egg argument. Was the benefit from yield-curve-riding or by lengthening the duration using yield-curve-riding.
After accounting for discount/premium, the remainder of the return would have been delivered as capital gains, ...
OK. I haven't looked at the distributions and don't intend to. My point was that if a fund is making good use of yield-curve-riding we should see some capital gain distributions.
However, to see that Net Investment Income included the net amortization of premiums and discounts, you'll have to read the Notes to Financial Statements, as I've shared in a previous post (and on which you commented).
Amortization/accretion of market premium/discount is determined at the time of purchase. Any price change on the sale that is different from the original cost after adjusting for amortization/accretion is a capital item and should show up in the realized capital gain (loss) line item.
Bottom line: it's mostly about the bond math and risk. The quantitative questions can be mostly answered by plugging numbers into a spreadsheet and using the PRICE and YIELD functions (as you know), along with the simple arithmetic required to compute any coupon return component. Creating some graphs from the resulting numbers, like the one above, helps add intuitive understanding (at least for me). There's really not much to debate if we rely on the basic bond algebra I posted earlier
Yes but what is the question? If two similar funds have slightly different SEC yields one should look at their duration. Take the difference in duration and a "standard" current yield curve and eyeball a correction to the SEC difference. Are they now the same? Do you want to take the risk of the higher duration for the higher yield or not? If they are not close at least one of the two is doing something you might not like or maybe you will depending on your manager risk tolerance.

FWIW I have personally had the opportunity to use the riding-the-yield-curve strategy on seven occasions in the last couple of years. Twice I sold for capital gain/ordinary income trade off. Four times I sold for for duration lengthening. And once I didn't do it at all because the transaction costs were greater than the benefit. Since then I have moved much of our (Treasury) bond portfolio from individual notes to "actively" managed Vanguard funds. Ask me in five or ten years whether or not that was a good decision.
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Re: So do we care more about change in bond price or bond yi

Post by Kevin M »

Doc, I think we are basically on the same page. The complexities arise when we try to use words and phrases to subjectively describe what is happening. I think we'll all end up agreeing if we simply stick to the bond math descriptions of what is happening. Perhaps we also have to agree that there is at least a general relationship between risk and expected return, and although there may be some exceptions and anomalies, I don't think any of them apply to this particular discussion.

Here's my point by point reply to your post.
Doc wrote: I purposely used the zero coupon analogy so we can dispense with any market premium/discount amortization/accretion complications.
OK, by definition a zero is purchased at a discount to par. My understanding is that the discount is amortized over the life of the bond, and the annual imputed interest is taxed as ordinary income. If the bond is sold before maturity, then there also could be a capital gain or loss. So by itself, this doesn't simplify things, but when you add on your assumption of a static yield curve, it does, since then there would be no capital gain, and all return would be from interest income. For a bond fund, expenses would be subtracted to arrive at net investment income, which would be distributed to shareholders (as dividend distributions in Vanguard bond fund terms).
Doc wrote: This also eliminates any maturity versus duration complications.
Yes, duration = maturity for a zero.
Doc wrote: I further specified that the yield curve doesn't change over the time frame of our analysis.
I think this is one of things that causes confusion. If we know the yield curve is static for our holding period, there is no risk. If we hold to this stipulation for the rest of the discussion, we can eliminate any discussion of risk. We simply purchase a zero at the top and sell at the bottom of the steepest portion of the yield curve.

But this introduces a paradox, because if there were no risk, the yield curve would not have a positive slope--it would be flat. A positively sloped yield curve is the motivation to take more term risk.

So if we want to assume a static yield curve simply for purposes of illustrating the concept of riding the yield curve, OK, but then we have to be careful to remember the logical inconsistency of the assumption and conclude that this somehow provides a risk-free return.
Doc wrote: I am now going to simplify further by saying that the yield curve has no curvature over the time frame in question. (I think this means there is zero convexity.) I am not sure this makes later any difference.
We can look at the impact of this assumption, but better not to mention convexity, since that is related to the price/yield relationship for an instantaneous change in yield. To look at how price changes over time as a bond "rolls down the yield curve" (in this case, under the assumption of a static yield curve with constant slope), I find it most useful to simply plug some numbers into a spreadsheet and look at the resulting graphs.

Image

So with a constant-slope yield curve the slope of the price/maturity relationship still is steeper at longer maturities, and there would be more return for riding the yield curve with longer-maturity bonds. This seems puzzling unless we remember the logical inconsistency of the assumption of a static yield curve, for which we get the logically-inconsistent result that investors are rewarded for investing in longer-term bonds without taking any additional risk. Or to put it succinctly, garbage (assumptions) in, garbage (conclusions) out.
Doc wrote: First principle of bonds (which is not dependent on any of the above): They are fixed income. The purchaser enters into a contract with the lender to trade a certain cash flow for an initial payment. Barring default there is nothing the purchaser can do for the "duration" of the contract to increase his initial yield unless he makes some kind of trade that increases his risk.
I find the last sentence confusing given all of the assumptions. No comment unless you want to clarify this with a more detailed scenario given your assumptions.
Doc wrote: The whole idea of "riding the yield curve" can have no impact at all at least before taxes, on the original agreed upon terms unless the purchaser reinvests in more risky securities.
Same comment as the immediately preceding one.
Doc wrote: In the basic "riding the curve" tactic the investor sells before maturity, realizes a capital gain now instead of waiting for the interest payments and reinvest the proceeds in a longer term note thus increasing his risk.
Yes, except under the stated assumptions there is no risk, and none of this makes any sense. If we stick with the assumptions, and simply state the resulting facts mathematically, we can see from the graphs above that by selling before maturity and reinvesting at the original price/maturity point, the investor reaps the gains of the steeper price/maturity slope at longer maturities.

Now, if you want to change the simplifying assumption from constant-slope yield curve to constant-slope price/maturity relationship, then there is no extra return from riding the yield curve over any particular range of maturities. But again, this is a nonsensical assumption and result.
Doc wrote: Therefore if all the bond traders could know for certainty what the future curve would look like there is no way that any gain from riding the curve would not be arbitraged away except for differences in trading costs or tax implications (ordinary income versus capital gains).
Exactly! Except for such practicalities as you mention, the yield curve would flatten to a horizontal line. Without risk there is no risk premium, and therefore no reason for longer-term bonds to be priced higher than shorter-term bonds. An investor could simply pick a maturity that suited her preference or matched her liability and pay the same price for a bond of any maturity.
Doc wrote: If you and I and all the other posters are aware of this idea then all of the professional bond traders also know it and any advantage that one has over another has to disappear and there is no advantage to the technique other than trading costs or tax differences. Of course one manager may have more skill than another but that is a whole different discussion. (Think PIMCO Total Return maybe?)
To the extent this is a restatement of the previous statement, yes, exactly. Some confusion and debate may result from the extra words like "no advantage to the technique", because someone probably then will start throwing out more words and numbers to try and show that there is an advantage to the technique, which I think is exactly what has been happening with this entire discussion. If bond traders believed there was no advantage to the technique, we wouldn't get any hits when we Google "riding the yield curve". But we can quickly verify that this is not the case, and find many articles that suggest that there is an advantage to this technique.

Again, if we stick with bond math, there is no debate (unless we make mistakes with the math, which can be objectively demonstrated/corrected).

The bond math shows us that declining maturity has a positive effect on the price of a zero coupon bond (more so at longer maturities where the yield curve is steeper, or more accurately, where the price/maturity curve is steeper), and a shift in the yield curve can either have a positive or negative effect on the price. The net capital gain is the net of the two effects. If we stick with the bond math, that's the end of the story.

I think articles that promote riding the yield curve are likely to focus on the fact that the positive declining maturity effect on price can offset the negative price effect of an increase in rates, with the conclusion that taking more term risk is not as risky as it seems.

Or if we reframe the original stlutz post in terms of a zero coupon bond (that always will increase in value with declining maturity for a positively-sloped yield curve), selling a bond before maturity could result in a higher return than indicated by the YTM at time of purchase, if the yield curve did not shift up or flatten (or more simply, if YTM2 at M2 did not change), because the bond was sold before it entered the less-steep portion of the yield curve (or more accurately, the less steep portion of the price/maturity curve). This is because it's the less-steep portion of the original yield curve that lowers the calculated YTM value (at least that's one way to think about it).
Doc wrote: Now if the yield curve changes during the course of the investment time frame selling before maturity and reinvesting could have some real benefit that those managers required to or choosing to hold to maturity cannot take advantage of.
Of course, but we can restate this simply as a mathematical fact. The capital gain will either be higher, lower, or the same as it would have been if the yield curve had remained static, depending on whether the YTM at the end of the holding period is lower, higher or the same as YTM as at that maturity at the beginning of the holding period.

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

Post by Doc »

Kevin I don't have the time to go through your detailed post but let me clarify a couple of points.

By a static yield curve I mean that it doesn't shift during the time period under consideration but only for purposes of the discussion to try separate the different pieces of the "puzzle". You still have term risk as long the slope is positive in the real world. (Of course if you didn't have term risk the curve would be flat so my simplification does not exist in the real world.)
Kevin M wrote:Doc wrote:
First principle of bonds (which is not dependent on any of the above): They are fixed income. The purchaser enters into a contract with the lender to trade a certain cash flow for an initial payment. Barring default there is nothing the purchaser can do for the "duration" of the contract to increase his initial yield unless he makes some kind of trade that increases his risk.

I find the last sentence confusing given all of the assumptions. No comment unless you want to clarify this with a more detailed scenario given your assumptions.


Fixed income: I give the Guv $1000 and the Guv agrees to give me back $1 every six months for five years and then gives me back my $1000. That contract doesn't change no matter what the market interest rate does over those five years. (I'm ignoring duration/maturity considerations and linearizing the bond equations for simplicity.) Say after four years the market interest on a one year note is only $.50 so my original note is going to have a market value $1001. The five year 2% with 1 year remaining has the same value as a new one year note. I can sell my note and buy the new note for $1000, put the extra $1 in the bank, receive 2 x $.50 interest payment and when the note matures I have a total of $1002 which is what I would have if I hadn't ever sold the original five year note. I cannot get ahead by selling with one year remaining because of my original contract which is still fixed if I buy a new one year note with a different coupon. What I can do and what your curve riders do is sell that note for $1001 and invest $1000 in a new five year note that is paying $2 a year. But by doing that I have increased my risk because I am not guaranteed to get back my $1000 for an extra four years.

There is nothing inheritance better or worse for a fund selling before maturity. It is just another way of increasing the duration of the fund. If you want to have two five year funds (my Fund 0-5 and Fund 1-5) to have the same maturity the Fund 0-5 would have to become a Fund 0-6. An active fund manager may get an advantage because he use the technique to position himself slightly differently on the yield curve as in "I think the yield on the five is going to go down in the next 12 months so I better buy it now instead of waiting for my current five to mature in 12 months" but that is a different story.
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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 I don't have the time to go through your detailed post but let me clarify a couple of points.

By a static yield curve I mean that it doesn't shift during the time period under consideration but only for purposes of the discussion to try separate the different pieces of the "puzzle". You still have term risk as long the slope is positive in the real world. (Of course if you didn't have term risk the curve would be flat so my simplification does not exist in the real world.)
Yes, I understand, and when you do get a chance to read my detailed reply you'll see that I understand, but that an important point is that the simplifying assumption leads to a paradox that can cause confusion.

I read your more detailed example, and I don't think it introduces anything new or that isn't addressed in my previous reply. I think you and I can each plug the same numbers into a spreadsheet and get the same results, and I think we both agree that higher expected return only comes from taking more risk--at least if we restrict our domain to marketable bonds.

To summarize, I think we are in violent agreement with respect to this topic!

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

Post by Sunny Sarkar »

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

They all (stocks and bonds) come and go, and all they leave behind are the dividends.
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Re: So do we care more about change in bond price or bond yi

Post by ogd »

Doc: the simple risk story that I believe you're looking for is this: the 5-to-zero fund has a fifth of its assets in Treasuries in their final year making 0.09% yield; these Treasuries are, for the purposes of the individual investor with 1% bank accounts, deadbeats. The 5-to-1 fund doesn't, and it has more risk as a consequence, therefore it's entitled to make 20% more yield, which it does. It's as if the 5-to-zero fund were the 5-to-1 fund diluted with cash.

A-ha! -- I hear you say -- more risk, I knew it! Well, yeah. Thing is, as one of said individual investors, I simply don't want the deadbeat Treasuries at all. I can dilute the 5-to-1 (or rather, the 5-to-3) fund myself, much more beneficially. So because I'm able to take more risk in the market, I'm getting rewarded more in the market. Now one day I might have to pay, but even then I'm 1% x #years better off than the fund that held 0.09% Treasuries, for the cash portion.

The main point of the riding the yield curve discussion here is that yes it's more risky in isolation, yes it might go away soon, but while the bond market is in its present state of uncertainty and "interest rates can only go up", I am reaping the fruit of that uncertainty, not just the risks.

The rest is bond mathematics and wrapping your mind around that dual time axis that shows up in a proper discussion of the yield curve, which is not easy from my own experience.
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Re: So do we care more about change in bond price or bond yi

Post by Kevin M »

ogd wrote: The main point of the riding the yield curve discussion here is that yes it's more risky in isolation, yes it might go away soon, but while the bond market is in its present state of uncertainty and "interest rates can only go up", I am reaping the fruit of that uncertainty, not just the risks.
This is an example of the kind of wordy abstraction, and not enough bond math that leads to endless debate. Can you quantify "risky in isolation", "present state of uncertainty", and "reaping the fruit of that uncertainty, not just the risks"?

Financial markets are always in a state of uncertainty. One of the most common, textbook definitions of risk is uncertainty of return over the investor's holding period. This makes a phrase like "reaping the fruit of that uncertainty, not just the risks" sound like nonsense to me.

It also seems nonsensical to say something like that in the present tense. It's the future that is uncertain, not the present. You don't know whether or not you'll reap any fruit from the uncertainty until the end of your holding period. Riding the yield curve is not some magical thing that has somehow revoked the fundamental relationship between risk and expected return.

Can we please stick with the fundamentals of risk/uncertainty, expected return, and the applicable bond math?

Most if not all of the rest of what you said does exactly that. No one can argue with the fact that a retail investor can earn more in an FDIC-insured savings account than on a 1-year, 2-year, and as of today, even a 3-year treasury if held to maturity. No one can argue with the fact that a retail investor can earn more over the next five years in a 5-year CD than a 5-year treasury held to maturity.

It might be a fact that you might earn more than a 5-year CD over the next two years by buying a 5-year treasury and selling it after 2 years; this can easily be shown with a spreadsheet calculation. One of the variables is yield (YTM2) of the 3-year treasury in 2 years. But it's also a fact that this is not a certainty. You can guess at YTM2 based on the steepness of the yield curve between the current 5-year and 3-year maturities, but it's just a guess. Maybe you're really smart, and your guess is better than my guess, and your bet will pay off. But it's still a bet, and the payoff is uncertain. If it were otherwise, you would have found the magic formula to guaranteed investment success, and you would not be sharing it with us.

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

Post by ogd »

Kevin M wrote:This is an example of the kind of wordy abstraction, and not enough bond math that leads to endless debate. Can you quantify "risky in isolation", "present state of uncertainty", and "reaping the fruit of that uncertainty, not just the risks"?
Sure.

"risky in isolation" -- I am taking more risk in the bond market but balancing it out with cash.
"present state of uncertainty" -- steep yield curve. We know what it means, the market is forecasting higher rates. This is well established.
"reaping the fruit of that uncertainty, not just the risks" -- roll return from said yield curve.

I had just presented all of these before that conclusion paragraph.

All bond math and no conclusions [would] make this a very dull thread.
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Re: So do we care more about change in bond price or bond yi

Post by stlutz »

There is nothing inheritance better or worse for a fund selling before maturity. It is just another way of increasing the duration of the fund. If you want to have two five year funds (my Fund 0-5 and Fund 1-5) to have the same maturity the Fund 0-5 would have to become a Fund 0-6. An active fund manager may get an advantage because he use the technique to position himself slightly differently on the yield curve as in "I think the yield on the five is going to go down in the next 12 months so I better buy it now instead of waiting for my current five to mature in 12 months" but that is a different story.
I've gone back and forth on this in my own mind a few times over the past year on this. Here is my [current] thinking.

On one hand, I'm comfortable saying that the Treasury market is ruthlessly efficient. As such, a rolling ladder with a 5 year duration and a "bullet" strategy with a 5 year duration have very similar risks and should expect the same returns. With this thinking, there is nothing magicial about yield curve riding--it's really just giving you the peformance of a ladder but you only have to manage once security instead of a bunch of them.

However, I have to temper this conclusion with another fact: Not all investors experience risk in the same way. People buy short-term Treasury instruments because they cannot take any risk. Such instruments are therefore priced accordingly. As such, the investor who holds all the way to maturity has their returns diluted in the last year of the ladder by these "risk-free" investors. As such, if I am willing/able to take some term risk, I actually will most likely increase my returns by selling a year or two before maturity.

Sometime I may sit down with some historical rate data and try to prove/disprove that.
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Re: So do we care more about change in bond price or bond yi

Post by Doc »

ogd wrote:Doc: the simple risk story that I believe you're looking for is this: the 5-to-zero fund has a fifth of its assets in Treasuries in their final year making 0.09% yield; these Treasuries are, for the purposes of the individual investor with 1% bank accounts, deadbeats. The 5-to-1 fund doesn't, and it has more risk as a consequence, therefore it's entitled to make 20% more yield, which it does. It's as if the 5-to-zero fund were the 5-to-1 fund diluted with cash.
This statement is at best misleading and at worst just wrong. It's the word problem that Kevin keeps talking about.

That one fifth of the assets in the 0-5 may have a yield to maturity of only 0.09% but it was likely bought at par four years ago with a 2% coupon and therefore it's return on investment is 2%. Selling the original five year note at one year only benefits you if you reinvest the proceeds in a note that is yielding more than the 0.09% of the current one year note. Like maybe buying another five instead of that new one year note.

This problem is part of the originate YTM/current return discussion which we never resolved "ages" ago.
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Re: So do we care more about change in bond price or bond yi

Post by Kevin M »

I thought it would be worthwhile to expand a bit on the part of my earlier reply to Doc that discussed and showed graphs of the price/maturity curve for a yield curve with linear slope. In that reply, I mentioned that the assumption of a static yield curve with a positive slope leads to a paradox (under equilibrium conditions). The resolution of the paradox is that market forces would push longer-term yields down and/or shorter-term yields up until the yield curve was flat.

So I thought it would be interesting to explore the price/maturity curve of a zero-coupon bond for a flat yield curve.

First, kudos to Doc for suggesting using zero-coupon bonds to explore this topic--it really does simply things and allow us to focus only on price-change effects without the added complexities introduced by coupon payments. The more I work through it the more I appreciate the simplification.

The yield curve and price/maturity graphs are shown below with the assumption of a flat yield curve with constant YTM of 2%:

Image

This seems to make sense. The price/maturity curve looks very linear, so investors receive approximately the same price return by buying and selling at any two points on a static, flat yield curve. But the reality is that the price/maturity curve is not quite linear.

To explore this, I start with the observation that although the formula used by the spreadsheet PRICE function is somewhat complex, for a zero-coupon bond, with a couple of simplifying assumptions the spreadsheet PRICE formula can be simplified to the simple version of the present value formula. Since I suspect most people are more familiar with the future value formula, as it's often used to explain how interest compounding works, I'll start with that.

FV = PV x (1 + i)^n

For those not familiar with spreadsheet operators, ^ is the exponentiation operator. So assuming i is the annual interest rate rate and n is the number of years, this is read as:

Future Value = Present Value times (1 + annual interest rate) raised to the nth power. Solving for PV, we get the simple version of the present value formula:

PV = FV / (1 + i)^n

Now I'll substitute variable names that are more consistent with my previous posts:

P1 = P2 / (1 + Y)^n

where P2 is price of the bond at maturity, P1 is price on the date we are pricing the bond, and Y is yield (specifically, yield to maturity or YTM) for the term to maturity of the bond (we get Y from the yield curve). Bond prices are quoted as a percent of par (face value), so in the equation above P2 = 100.

This formula gives exactly the same result for a zero-coupon bond price as the spreadsheet PRICE function if we assume Frequency = 1. Frequency is the number of coupon payments per year, and although zero-coupon bonds don't have coupon payments, quoted prices assume Frequency = 2 so that prices can be compared to coupon treasury notes/bonds , which have two coupon payments per year. Using Frequency = 2 in the PRICE function will give slightly different results than the present value formula above if we assume Y is the annualized YTM.

The important point is that we can see from the formula above that it is not a linear equation., but an equation of the form y = 1/(x^n). However, for constant values of x that are only slightly greater than 1, or even if x increases by small enough increments, this equation approaches a straight line with a negative slope as seen in the graph above. For that graph x = Y+ 1 = 1.02, just slightly greater than 1, and is a constant.

To see the non-linear effect, we can increase the yield to a value quite a bit larger than 2%. We can start to see some curvature at 10%, but at 20% it becomes quite obvious:

Image

Note that the curvature if this price/maturity graph is the opposite of that of the graph in the earlier reply about a yield curve with a constant positive slope. We can discuss this if anyone is interested, but this post is long enough for now.

Hope this is interesting or helpful to someone.

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

Post by ogd »

Doc wrote:
ogd wrote:Doc: the simple risk story that I believe you're looking for is this: the 5-to-zero fund has a fifth of its assets in Treasuries in their final year making 0.09% yield; these Treasuries are, for the purposes of the individual investor with 1% bank accounts, deadbeats. The 5-to-1 fund doesn't, and it has more risk as a consequence, therefore it's entitled to make 20% more yield, which it does. It's as if the 5-to-zero fund were the 5-to-1 fund diluted with cash.
This statement is at best misleading and at worst just wrong. It's the word problem that Kevin keeps talking about.

That one fifth of the assets in the 0-5 may have a yield to maturity of only 0.09% but it was likely bought at par four years ago with a 2% coupon and therefore it's return on investment is 2%. Selling the original five year note at one year only benefits you if you reinvest the proceeds in a note that is yielding more than the 0.09% of the current one year note. Like maybe buying another five instead of that new one year note.

This problem is part of the originate YTM/current return discussion which we never resolved "ages" ago.
Doc: it was hard reading that first paragraph from you. Maybe I should treat it as my cue to be done with this thread.

The statement is absolutely right and it's easy to prove. First, with a single bond: if I invested $10K four years ago in a zero coupon at about the current 1.45% yield (example works the same with 2% or any other number, edit: and it works the same with coupon bonds currently sitting at a premium), what I have now is a bond worth $10740. If I hold it for another year, I will have $10750; it is impossible, no matter what happens in the bond market, for me to get a single penny more. If, instead, I sell it now and put the money in my 1% bank account for a year, I will have $10847. The latter position also has slightly smaller risk.

What's there to argue about in the statement "for the purposes of individual investors with 1% bank accounts, this Treasury is a deadbeat"? It was once a 1.45% investment, but it has delivered almost the entirety of that in its first four years and it's now a 0.09% investment, plain and simple. You don't get to define your way around $100 more in my pocket.

Then with the fund: a position that consists of 80% bond fund rolling from 5 years to 1 and 20% bank savings account on the present yield curve is slightly less risky and yields 1.64%, vs the 1.45% yield of a fund that holds 5 year bonds to maturity. This may one day change if the one year Treasuries climb above bank accounts, but it's a circumstance I can adapt to when the time comes; with that provision (that I will switch) there is no possible way the second position outperforms the former. Again, no matter what happens in the bond market. It makes complete sense for me to avoid the low end of the maturity curve while it's making next to nothing.

Having 1% bank accounts available was a qualifier for my statement and it's easily arranged, switching fixed income to taxable if needed and booking a tax penalty to the 1%; still plenty left. But what if I really didn't? Then indeed I would have a tradeoff to make. However, personally I don't see myself ever taking 0.09% for anything but a negligible amount of my money.
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Re: So do we care more about change in bond price or bond yi

Post by Doc »

ogd wrote:The statement is absolutely right and it's easy to prove. First, with a single bond: if I invested $10K four years ago in a zero coupon at about the current 1.45% yield (example works the same with 2% or any other number, edit: and it works the same with coupon bonds currently sitting at a premium), what I have now is a bond worth $10740. If I hold it for another year, I will have $10750; it is impossible, no matter what happens in the bond market, for me to get a single penny more. If, instead, I sell it now and put the money in my 1% bank account for a year, I will have $10847. The latter position also has slightly smaller risk.
We were addressing funds not your personal portfolio. The fund can't sell that 4 year old five year note and put the proceeds in a bank. That five year note had a return on investment of 1 to 2 % at the start and it has the same return on investment for the last year. The return on market price is of course much lower as you note. If putting the money in a bank is your idea of avoiding the "deadbeat" your statement is misleading or at least irrelevant to the discussion.

Money in the bank is not the same as Treasuries. I can't sell the bank deposit and buy Apple stock in the time it takes to punch a few keys. It can't be used as collateral for margin. It doesn't do anything to help me get to flagship status at Vanguard. It has only a limited guarantee if the bank goes under and what guarantee it has may take some time. If your only concern for short term investments is how much interest they pay you can call the short bond portion of a bond fund "deadbeat" if you want. But if your sole concern is about how much interest it pays why a bank - why not five or ten year Treasuries or even (gasp) high yield bonds.
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Re: So do we care more about change in bond price or bond yi

Post by ogd »

Doc: in your first message you asked what is the risk difference that produces extra yield. I answered that there is indeed one, and I also made an argument why it's a good tradeoff for the typical investor who is not using margin and who is not on the cusp of Flagship. This investor can match the less risky fund with a better yield, covered by what is a guarantee as good as Treasury, if not slightly stronger for political reasons. External bank accounts are every bit as fast as Prime MM at Vanguard starting last year.

If I'm sitting on a one year Treasury that once had a 2% yield, then back then it was a good buy. Good for me. But now it's an even better sell. I don't want to hold it and I don't want to hold a fund holding it; neither risk nor reward justifies it. Despite my use of "I", this applies to an awful lot of investors.

That is all.
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Re: So do we care more about change in bond price or bond yi

Post by Kevin M »

stlutz wrote: On one hand, I'm comfortable saying that the Treasury market is ruthlessly efficient. As such, a rolling ladder with a 5 year duration and a "bullet" strategy with a 5 year duration have very similar risks and should expect the same returns. With this thinking, there is nothing magicial about yield curve riding--it's really just giving you the peformance of a ladder but you only have to manage once security instead of a bunch of them.
Are you interested in what the managers of the VG int-term treasury fund think? Or at least what they were thinking on 7/31/2013 and 7/31/2014? Now that I have their holdings on those dates loaded into a spreadsheet we can view them at a finer resolution than reported in the SemiAnnual reports.

Image

So clearly neither a ladder nor what you're referring to as a bullet strategy. It looks to me like there's some very active "yield-curve management" going on. As of 7/31/2014 it looked something like a roller-coaster strategy (trace the peaks of the red bars), and as of 7/31/2013 it looked like more of a water-slide-with-a-bump or ski-slope-with-a-jump strategy (trace peaks of blue bars from left to right).

And for what it's worth, here are the yield curves for the two dates based on actual holdings, with the yield curve numbers from Treasury.gov shown below:

Image

(There were a few holdings with maturities of less than three years as of 7/31/2014, but they didn't amount to enough to show in the holdings-by-maturity graph above).
stlutz wrote:However, I have to temper this conclusion with another fact: Not all investors experience risk in the same way. People buy short-term Treasury instruments because they cannot take any risk. Such instruments are therefore priced accordingly. As such, the investor who holds all the way to maturity has their returns diluted in the last year of the ladder by these "risk-free" investors. As such, if I am willing/able to take some term risk, I actually will most likely increase my returns by selling a year or two before maturity.
Mostly makes sense, except for the following.

Some investors stay short not because they "cannot take any risk", but because they don't think the risk/expected-return tradeoff justifies it. William Bernstein and Warren Buffet appear to be in this camp. Clearly returns are diluted in the last year of the ladder given the current shape of the yield curve, assuming the yield curve does not move too far against you. Once again, "most likely increase my returns by selling a year or two before maturity" depends on the yield-curve not shifting too much against you.

I keep stressing the fact that the yield curve is not static because that is the big fly in the ointment of a yield-curve-riding strategy.

I take some term risk too. I also take some credit risk. It's hard to describe my strategy with a catchy phrase, since I'm earning short-term returns with no risk (FDIC-insured savings and reward checking accounts), intermediate-term returns with short-term risk (5-year to 7-year CDs with cheap early withdrawal options), entirely skipping any short-term bonds, and boosting expected returns a bit by taking term risk (and credit risk) with intermediate-term (and even a dollop of long-term) investment-grade or tax-exempt bonds instead of treasuries. Kind of a barbell strategy I guess, but with higher-return/lower-risk on the left side of the barbell, and higher-return/higher-risk/return on the right side.

I really appreciate you starting the whole yield-curve riding discussion back in June 2013. I found it a bit confusing and counter-intuitive at first, but now think I understand it completely. I didn't quite appreciate it as much, nor could I articulate my concerns as well back then as now.

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

Post by Doc »

My summary of the simplified case used for illustration.

Five year zero with 2% YTM. Original cost ~$900.

Straight yield curve - slope is 0.4% per year.

Two funds with different strategies.

"F0-5" fund: A zero to five year fund - buy at auction and hold to maturity. Duration 2.5 years. YTM of fund 2.5 x 0.4% = 1.0%.

"F1-5" fund: Buys five year zero at auction and sells with one year remaining. Duration 3.0 year. YTM of fund 3.0 x 0.4% = 1.2%.

Theory 1: F1-5 has a 20 bps higher YTM because of the longer duration.

"Yield-curve-roll" effect:

F0-1 earns $100 over five years for a return on investment of 100/5/900=0.0222 2.22%

F1-0 earns only $96 (selling price is 1000*(1-.004)=996 ) but over only four years for a return on investment of 96/4/900=0.0267 2.67%

This is a 47 basis point advantage over F0-1 of which 27 bps cannot be accounted for by the duration difference. (This difference is probably arbitraged away by the bond traders which is why the yield curve is not straight but I don't have a good argument for this.)

Theory 2: F1-5 has another 27 bps points advantage because of yield roll effect.

My conclusion: The analysis above is flawed. Yield to maturity is a calculation that uses as inputs cash flows if the security is held to maturity. F1-5 does not hold securities to maturity.

My take away which was developed before this discussion started. I prefer a 1-3, 1-5 or 1-7 fund over their 0-3, 0-5 or 0-7 siblings. I get a little higher yield perhaps and I don't have to worry about putting 20% of my cash in ogd's bank account.
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Re: So do we care more about change in bond price or bond yi

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I thought it would be interesting to look at some returns for a fairly recent 2-3 year period when the yield curve went from moderately steep to flat. I'll continue to use zero-coupon bonds since all return is from price-change, so it's easier to see the offsetting effects on price of maturation and yield-curve change.

The graph below shows the yield curves for the first trading day of each year, 2004-2007 (just look at the graph for now). Note that in 2004 (blue curve) the yield curve had a fairly typical, positively sloped shape, with the steepest portion of the curve between 1 and 5 years. By 2005 (red curve) shorter-term rates had increased, the 1-5 year portion of the curve was much less steep, and the steepest portion was 6 months to 1 month. By 2006 (orange line) shorter-term rates had increased enough that the yield curve was basically flat between 6 months and 10 years. By 2007 (green curve), the entire curve had shifted up a bit, with the shortest terms increasing the most.

Image

The table above the graph shows the return numbers for three zero-coupon bonds as they traveled toward maturity, starting on 1/2/2004. The first two rows show the maturity date and the initial term to maturity in years for the three bonds. I'll walk through the 5-year bond to explain the other rows.

The 5-year bond had an initial YTM of 3.36% (5-year point on the blue curve) and price of 84.65 on 1/2/2004 . Since no 4-year rate is reported by treasury.gov, there is no 4-year YTM to read off the red curve one year later, so YTM and price are N/A for 2005. At the two year mark, 1/3/2006, YTM had increased to 4.3% (3-year point on orange curve), and price had increased to 88.03. This was a total 2-year return of 3.99%, and an annualized return (CAGR) of 1.97%. One year later, on 1/2/2007, yield had increased to 4.80% (2-year point on the green curve) and price had increased to 90.95, for a 1-year return of 3.32%, and a CAGR of 2.42% over the 3-year period.

This illustrates some of the key points we've been discussing:
  • You cannot depend on the yield curve remaining static.
  • Due to the yield curve flattening, with shorter term rates moving up, the annualized return over the 3-year period (2.42%) was less than the YTM at the start of the period (3.36%).
  • Due to the maturation effect (price increase as term-to-maturity decreases), return was positive even though YTM was significantly higher at end of period than beginning of period.
I find it interesting to note that the return as maturity declined from 3 years to 2 years (3.32% between 2006 and 2007) was significantly greater than the annualized return as maturity declined from 5 years to 3 years (1.97% between 2004 and 2006), especially since the yield curve was basically flat in the 2-3 year range in 2006, and the 2-year rate was even higher in 2007. This is not what one would have expected if one put too much weight on yield-curve steepness as the primary source of return.

It's also interesting to compare the performances of the different bonds. Note that the 2-year bond had almost the same annualized return as the 5-year bond over the first two years of the holding period, despite having a significantly lower expected return (YTM) at the beginning of the holding period. Similarly, at the end of three years, the 3-year bond had delivered it's promised initial YTM of 2.49% (accounting for rounding error), slightly more than the annualized 3-year return of 2.42% for the 5-year bond.

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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'll continue to use zero-coupon bonds since all return is from price-change, so it's easier to see the offsetting effects on price of maturation and yield-curve change.
Kevin, first what are you doing up at 3 O'clock in the AM posting about bond yields. I have also been awake at that time but have been doing the calc's in my head which is why I keep linearizing the bond equations.

Anyway I reworked my numbers on the straight line yield curve with the 0-5 year fund and the 1-5 year fund using the actual discounted cash flow equations and got results very close to what I had yesterday. An analysis I decided was flawed.

I also reworked the number using Kevin's 2004 yield curve which is "typical" as he noted. The YTM points at five and one year are the only numbers needed for the calc.

I got a 53 bps difference from the calculation and eyeballing the chart at 2.5 and 3 years got ~25 bps from the duration difference.

This is telling me that yield-curve-riding has an effect that cannot be explained by duration and active management can have a positive effect as long as the yield curve shifting that Kevin is investigating doesn't somehow negate it.

When I have time I'm going to try to look at some 0-5 or 1-5 bond funds that are both active and indexed and see if there is some persistent advantage of the active before expenses. The problem is that I don't trust the yield curve data to be typical for the past five years because of the large scale Fed action. (See Kevin's charts for 2006 and 2007.)
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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, first what are you doing up at 3 O'clock in the AM posting about bond yields.
It was only 11:30pm my time.
Doc wrote: This is telling me that yield-curve-riding has an effect that cannot be explained by duration and active management can have a positive effect as long as the yield curve shifting that Kevin is investigating doesn't somehow negate it.
I don't know what this statement means. We can use bond math to determine exactly what the return will be given a certain set of conditions and assumptions. I remember seeing a statement in a BH post about not trying to do bond math in one's head--the poster's member name rhymes with "Spock". The more I work through this stuff, the more I think that was good advice.

Duration is just one of the numbers we can calculate for a bond at a given point in time, and it gives us an approximation of how much the price of the bond will change for an instantaneous (or very quick) change in the market YTM for the bond. Once the bond has traveled through time for awhile, the duration number will be different because the term to maturity will be different and the yield will be different, and those are both inputs to the duration calculation.

I think perhaps a source of confusion is trying to think about this topic in time-static terms. That leads to conflating things like duration and convexity, which are related to price vs. yield for a fixed term to maturity, with calculations that are related to change in term to maturity over time.

Or maybe a problem is trying to visualize things in two dimensions. To visualize bond price change over time, we need three dimensions; something like price on the y axis, YTM on the x axis, and term to maturity on the z axis. This is clear from looking at the PRICE function, since both yield and term to maturity (maturity - settlement) are parameters. We know how term to maturity will change over time, but we don't know how YTM will change over time, since it is a function of market forces (resulting in the changing yield curve), in addition to the changing term to maturity for our bond.

Rather than try to visualize it, now I'm just doing the calculations, and letting the resulting graphs show me what happens.
Doc wrote:When I have time I'm going to try to look at some 0-5 or 1-5 bond funds that are both active and indexed and see if there is some persistent advantage of the active before expenses. The problem is that I don't trust the yield curve data to be typical for the past five years because of the large scale Fed action. (See Kevin's charts for 2006 and 2007.)
I find that the easiest thing is just to do the calculations based on whatever assumptions you want to make. None of us knows what the yield curve will look like one year from now. "The market" may expect something, but the market may be wrong. As stated in the VG treasury funds semiannual report on 7/31/2014:
Vanguard wrote:Bond yields have confounded expectations so far this year. The markets had anticipated that interest rates would continue rising across the board once the Federal Reserve began scaling back its stimulative bond-buying program. Although yields of some shorter-dated bonds did keep drifting higher, yields of their longer-dated counterparts reversed course as demand for those securities revived.
The managers of the Vanguard int-term treasury bond fund obviously have been placing their bets, as can be seen by the lopsided holding distributions and the changes in distributions between 7/31/2013 and 7/31/2014, per my earlier posts on this. They admitted that their bet on the shorter end of the yield curve did not pay off as expected:
Vanguard wrote:Falling bond prices clipped the performances of the Short-Term and Intermediate-Term Treasury Funds, given that yields rose at the shorter end of the yield curve.
We can't say what will happen, we can only speculate on what might happen given different scenarios. The bond math will tell us exactly what will happen for a particular bond given a particular scenario. We can then extend that for a particular bond ladder or any other collection of bonds given a specified trading strategy.

Or we can do the same calculations for various past holding periods to determine exactly what happened for a specified collection of bonds and a specified trading strategy. For example, one could extend the analysis I did for the 2004-2007 period to see what would have happened for different trading strategies involving the 5-year, 3-year and 2-year zero-coupon bonds (or bonds of different maturities and coupons, like a 1-5 or 0-5 bond ladder).

Most of my bond fund holdings are with actively-managed Vanguard bond funds, so I hope they do a good job of yield-curve management and whatever other strategies they follow.

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

Post by Kevin M »

Continuing with my attempts to graphically illustrate the relationship between price change, time-to-maturity change, yield change, and return, below is a chart built as follows.
  • Use the latest yield curve rates from treasury.gov, and use linear interpolation to populate values for the missing terms to maturity (i.e., 4,6,8,9 years). The resulting yield curve is shown as the blue YTM curve on the chart, reading values from the left-vertical axis.
  • Assume the yield curve is static throughout the holding period. (But try and remember that this leads to a paradox in which extra return can be earned without taking extra risk).
  • Calculate price using the spreadsheet PRICE function. The changing inputs are time to maturity and YTM from the yield curve for time to maturity. All other parameters are fixed (e.g., rate, redemption). The resulting prices are shown on the red curve on the chart, reading values from the right-vertical axis.
  • Calculate return for each year and annualized return (CAGR) for holding period to date; the CAGR values assume a 10-year zero-coupon bond is purchased at the beginning of the holding period. The resulting returns are shown on the orange and green curves respectively, reading values from the left-vertical axis.
The chart below shows the results. The chart is read from right to left as the bond rolls down the (assumed static) yield curve. The yield and return figures use the left vertical axis, and price uses the right vertical axis. I'll walk through a read of the chart, and make some observations, but since small changes can't be seen on the chart, the table of data used to create the chart is shown below the chart. I also added a few extra columns to make a point or two.

Image
Image


Starting at the far-right of the chart, YTM for a 10-year term to maturity is 2.20% (keep this number in mind). This is read from the far-right point on the blue curve using the left-vertical scale, as well as from the first row of the table. At the same term, price is 80.35, the far-right point on the red curve using the right-vertical axis, as well as the first row of the table. There are no returns at term = 10 years, since it is the beginning of the holding period for a 10-year bond.

Moving one year to the left (term = 9 years), yield has dropped to 2.08% and price has risen to 82.98 (blue and red curves, second row of table). The return for the year is 3.28%, and since compounding period is one year CAGR is the same.

Note that although YTM is 2.20% for a 10-year bond at beginning of holding period, return is more than 100 basis points higher than that for a 1-year holding period. Note that this is not the steepest part of the yield curve, yet this is the highest one-year return.

So one lesson is that the highest return is not necessarily earned by holding a bond over the steepest portion of the yield curve. In this case the reward for taking the most term risk is a bigger factor. Term risk reward dominates yield-curve riding reward. And we have to keep remembering that we're assuming a static yield curve, and not get confused by real-world results where the yield curve changes over time.

The annual return falls over the next two years (as the bond hits 8 and then 7-years to maturity). We can see from the table that the annual change in yield (dY) is the same in each of these years (as it must be, since linear interpolation is used to generate these values). Falling annual return for the same incremental decrease in yield is consistent with the principle of more reward for more term risk.
Another way to interpret it is that term-risk dominates any yield-curve riding effect between years 7 and 10. A higher return would have been generated by rolling the bond over between years 10 and 9; i.e., selling the bond when it hits 9-year maturity and buying another 10-year bond. This actually would be the way to generate the highest return in this scenario.

A quick note that CAGR return is smoother, as expected, since it is a geometric average of the annual returns to date. Also note that if the 10-year bond is held to maturity, CAGR = original YTM (allowing for rounding error) as expected.

Year 7 to year 6 is interesting, since return increases. This is because the yield curve is steeper between years 5 and 7 (22 basis points per year) than between years 7 and 10 (12 bp per year). One way to interpret this is that during this 1-year period the yield-curve riding effect dominates the declining term risk effect. So return for this year is 3.20%--higher than the previous two years, but still not quite as high as the first year.

So, the investor who took the greater term risk by rolling over a 10-year bond each year earned a higher return than the investor who rolled over a 7-year bond each year (term risk reward dominates yield-curve riding reward), but the latter investor earned a higher return than an investor who rolled over a 7-year or 8-year bond each year (yield-curve riding reward dominates term risk reward).

After the 6-year term to maturity, annual return resumes its downward decline as declining time to maturity earns a lower return due to less term risk, but with enough upward pressure from yield-curve riding between terms of 4 and 5 years to hold annual return flat over that one year.

Note that the steepest 1-year section of the yield curve is between terms to maturity of 3 to 2 years, yet the return for this 1-year period is lower than for any 1-year period with longer term to maturity. So in this scenario, taking more term risk is more highly rewarded than curve riding over the steepest portion of the yield curve. Term risk reward dominates yield-curve riding reward.

In all of the above, I've characterized the yield-curve riding reward as reward for holding a bond over a steeper portion of the yield curve, as opposed to just the reward for taking more term risk. But I think maybe this isn't the way some folks might be thinking about it, which is why the term might be best avoided.

At any rate, an observation that might be more relevant to the riding-the-yield-curve enthusiasts is that the annualized return for a 10-year bond over any holding period shorter than 10 years is higher than the YTM at the beginning of the holding period. This is visualized as drawing a vector from the 10-year point on the (blue) yield curve to any point on the (green) CAGR curve.

We could construct a CAGR curve for a bond with any initial term and see a similar result, but even without doing that we can get a sense of things by visualizing a vector from any point on the (blue) yield curve to a point up and to the left on the (orange) annual return curve. Even a 3-year bond with an initial YTM of only 0.76% earns a return of 1.55% if held for one year. Perhaps the yield-curve-riding enthusiast will seek to optimize the balance between original YTM, anticipated CAGR, and term risk.

I added the dY column and the columns to the right of it to help clarify that you can't think in terms of the static price/yield relationship when observing price change over time. First note that Duration (Dur) = term to maturity for a zero coupon bond, but that Modified Duration (MD) is close to this value but slightly less. Modified duration is the duration used in the duration rule of thumb.

Note that percent price change, dP/P, is much higher over any one-year period than predicted by the duration rule of thumb. To help clarify this I added the -dYxMD column, which multiplies the change in yield for each year by the modified duration at the beginning of the year, and then switches sign (the duration rule of thumb). This would be predictive of price change for an instantaneous (or very quick) change in YTM; i.e., for a fixed term to maturity.

To verify this, I've shown the price change assuming a drop in yield of 12 basis points for a fixed maturity in the bottom two rows of the table. The return is 1.16% vs. the 1.15% predicted by the duration rule of thumb--much lower than the 3.28% return generated due to declining maturity. The relevant cells are highlighted with gray background.

A big take-away for me is that you can't do bond math in your head (at least I can't), and that the math doesn't always generate results you might expect based on the shape of the yield curve. I think this also does a pretty good job of quantifying an illustration of the impact of taking more term risk vs. riding the yield curve, but of course that depends on how you define your terms (Doc, does this help?).

One more time, it's important to emphasize a huge caveat to all of the above: the assumption of a static yield curve leads to a paradox in which extra return is earned without taking extra risk. So every time the word "risk" was used above was really a misuse of the term, but people seem to like the simplifying assumption of a static yield curve. The next step would be to do some what-if scenarios in which the yield curve changes in various ways.

Kevin
If I make a calculation error, #Cruncher probably will let me know.
stlutz
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Re: So do we care more about change in bond price or bond yi

Post by stlutz »

Hi Kevin,

Thanks for the post.

I don't think I agree with your inputs into the PRICE function.

When calculating the price gain from holding the bond, I think you need to use a settlement date one year in the future (i.e. 10/20/15) as that is what the price one-year from now would be based on. So, looking at your 10 year bond, if I plug in a settlement date of 10/20/15, maturity of 10/20/24, a rate of 2.2%, a yield of 2.08%, and redemption of 100, I end up with a price of 100.98. So, I'm calculating the return on the 10 year bond as 2.2% (the coupon) + .98% (roll return) = 3.18%.

Using actual rates for the 4 year bond gives some more realistic numbers as well.

I tried this using the rates from a real 5 year bond and a real 4 year bond from fidelity.com (i.e. 1.437% for the 5 year and 1.201% for the 4 year). Running PRICE the same way as above, I get a roll return of .91% + 1.44% yield = 2.35%, which is relatively close to the 2.2% yield that you would get from from the 10 year bond, which is what I would expect based on the assumption that a bullet strategy and a rolling ladder approach should average out to be relatively equivalent over time, assuming the same duration.
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Re: So do we care more about change in bond price or bond yi

Post by Kevin M »

stlutz wrote: I don't think I agree with your inputs into the PRICE function.

When calculating the price gain from holding the bond, I think you need to use a settlement date one year in the future (i.e. 10/20/15) as that is what the price one-year from now would be based on.
I don't think so. The assumption of a static yield curve means that the price of a 9-year bond one year from now will be the same as the price of a 9-year bond today. Therefore I calculate the price of a 9-year bond today using today as the settlement date, and that will be the price of a 9-year bond one year from now. One year from now the 10-year bond I buy today will be a 9-year bond at today's 9-year bond price. Ditto for a bond of every maturity.
stlutz wrote: So, looking at your 10 year bond, if I plug in a settlement date of 10/20/15, maturity of 10/20/24, a rate of 2.2%, a yield of 2.08%, and redemption of 100, I end up with a price of 100.98. So, I'm calculating the return on the 10 year bond as 2.2% (the coupon) + .98% (roll return) = 3.18%.
You are not looking at my 10-year bond. My analysis uses a zero-coupon bond, per Doc's brilliant suggestion. The "rate" parameter is the coupon rate, so this is 0% for pricing all bonds at all maturities. The coupon return is 0%, so all return is price return. Using a zero-coupon bond eliminates the need for made-up/misused terms like "roll return", although I still use made-up terms like "yield-curve riding reward", to try and characterize return differences due to steepness of yield curve.

I expect that the results will be similar for a coupon bond, except that some of the return will be from the coupon and some from the change in price as term to maturity declines. But since I'm learning that it's dangerous to make too many assumptions without running the numbers, maybe I'll do the same analysis for a coupon bond next.
stlutz wrote:Using actual rates for the 4 year bond gives some more realistic numbers as well.
A check on the "realism" of the numbers in my analysis is as follows. The CAGR for the 10-year bond held to maturity equals the initial yield of 2.2% (allowing for small rounding error). The 10-year bond CAGR for each year is calculated exactly the same way: (Pn/P10)^(1/(10-n)) - 1, where n is term to maturity, Pn is the price at term to maturity n, and P10 is the initial price of the 10-year bond. Next, just eyeballing the numbers you can see that the CAGR is a gradually changing average of the annual returns for the 10-year bond. Another check on the CAGR formula is that it gives the same number as the annual return for the first year (3.28%). Finally, just for grins I calculated the geometric average of the 9 annual returns using the GEOMEAN function, and of course get the exact same CAGR value of 2.21% (which is all I really had to do).
stlutz wrote:I tried this using the rates from a real 5 year bond and a real 4 year bond from fidelity.com (i.e. 1.437% for the 5 year and 1.201% for the 4 year). Running PRICE the same way as above, I get a roll return of .91% + 1.44% yield = 2.35%, which is relatively close to the 2.2% yield that you would get from from the 10 year bond, which is what I would expect based on the assumption that a bullet strategy and a rolling ladder approach should average out to be relatively equivalent over time, assuming the same duration.
I don't know what you're trying to do, but here are calculations using a "real" 5-year bond and 3-year bond from Fidelity. Maturity date and YTM are taken from Fidelity, settlement is today, and the Fidelity quoted price is shown for comparison to my calculated price.

Image

Note that my calculated prices are essentially the same as the prices displayed by Fidelity. Consistent with my analysis, the CAGR over the 2-year period assuming static yield curve (2.52%) is quite a bit higher than the current YTM of the 5-year zero (1.52%). You can estimate this in your head by looking at the prices: 97 - 92 = 5, so a 2-year return of about 5% (97 is pretty close to 100), which is an annualized return of about 2.5%. As a yield-curve-riding enthusiast I would think that you'd be pleased with these numbers.

So, per the static yield curve assumption, we price the 5-year and 3-year bonds today, and assume the prices for 5-year and 3-year bonds are the same at all future times. Two years from now the 5-year bond priced/bought/sold today can be priced/bought/sold at the price of a 3-year bond priced/bought/sold today (again, since we're assuming static yield curve). Of course we are ignoring bid/ask spread and any other transaction costs.

Whether you pull YTM and term to maturity from Fidelity or from treasury.gov, the yield curve will be similar, and the math is the same.

Kevin
If I make a calculation error, #Cruncher probably will let me know.
stlutz
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Re: So do we care more about change in bond price or bond yi

Post by stlutz »

Got it--I see what you're doing now. I've never actually messed much with zero-coupon bonds before.

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%.
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