Real Return Bonds

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newguy
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Re: Real Return Bonds

Post by newguy »

Maybe someone can explain the weekly percent price changes in TIP vs IEF (bond fund etfs). I expected this difference but the other way around. :?
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Re: Real Return Bonds

Post by jiHymas »

newguy wrote:
BNP Paribas wrote:The DV01 is affected by the index ratio
Yes, because the DV01 give the absolute price change for a given absolute yield change.

DV01 = ModDur * Price
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Re: Real Return Bonds

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jiHymas wrote:
newguy wrote:
BNP Paribas wrote:The DV01 is affected by the index ratio
Yes, because the DV01 give the absolute price change for a given absolute yield change.

DV01 = ModDur * Price
Which is what I'd like to see for duration because I want to know how much the value of my portfolio will change depending on underlying yield changes of the same amount in both securities.

Except looking at my scatter plot that's kind of silly since they(each ones yield) probably don't change by the same amount, duh I don't know how I expected anything else.

I looked at regression between yield changes for Canada and the US. The Canada is close to 30 yr (only data available for rrb) and the US is 10 yr (not much 30 yr data and comparable to TIP and IEF etfs). They're all for close to the same period (Dec 5 2003, TIP inception).
rrbnom.jpg
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So the portfolio of TIPs changes around about .68 times IEF and the TIP's yield changes .73 times the nominal yield changes. Shouldn't duration, if it's to be useful, take these changes into account.

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Re: Real Return Bonds

Post by IdOp »

jiHymas wrote:I'm not quite sure I follow this. Modified duration multiplies the absolute change in yield to get the relative change in price.
Correct.
If the real yield changes by 1bp, then the nominal yield will also ... change by 1bp
This assumption is the the source of the confusion, and I feel certain it's incorrect in general. I'll try to explain this in two stages. First, I'll give a trivial counter-example that should convince you this is wrong. Second I'll elaborate a bit on the general framework. It's useful to do the example first, otherwise the generalities might sound like more mumbo-jumbo than they are. ;)

For the example, take a real return strip bond, which matures in one year and is sold to you today with an advertized real yield of 5% (annually). You pay $1 for the strip today, and in one year it matures and the broker gives you $1.10 in actual/nominal dollars. The difference between $1.05 and $1.10 was caused by inflation. This single financial situation can be looked at either through rose-coloured glasses (nominally), or blue-coloured glasses (in real terms). So let's examine everything both ways.

The nominal annual yield is 10%.

The real annual yield is 5%.

Both the real and nominal Macaulay durations are 1 year. (They happen to be the same because a strip is so simple. In general they will not be.)

Let's look at modified duration for annual yield.

The value of the nominal modified duration evaluated at the nominal yield, is 1/1.10.

The value of the real modified duration evaluated at the real yield, is 1/1.05.

For the given financial situation, the two values are not the same in this example, and therefore not in general either. (Although, as functions of their respective annual return factor they are the same, 1/R, again because the strip is so simple.)

Now to the general view of an RRB. It too can be viewed in nominal or real terms. The difference is in the $ amounts of the cash flows, which differ by inflation. These amounts are generally completely different. In real terms the RRB is like an normal bond, in nominal terms it isn't. Call the cash flows Cr and Cn for real and nominal, respectively. The (total) price -- yield relationship is, nominally:

Pn = f(Rn,Cn)

and in real terms

Pr = f(Rr,Cr).

[P is total price and R is return factor, and r and n should be read as subscripts.] For a given RRB, the functional form of f is the same in both cases. For a given cash flow C, the price--return relationship is given by f( . ,C), and it's a different function of R depending which cash flows C you use, real or nominal. It's like having two totally different bonds. The price as a function of return (or vice versa) is different. Any durations computed from f(.,C) will in general be different functions of R, again depending on the choice of C. And when such real and nominal durations are evaluated at the parameters corresponding to a given financial situation, there is again no reason the results (numbers) should be the same; the strip example shows this for modified duration.

I'm hoping the example provides the conviction, that if you flesh out the generalities it will make sense.

Regarding GlobeInvestor's data, I just want to clarify my view. There are basically two completely separate issues:

A) Compute real Macaulay duration.

B) Multiply it by the index ratio as your numerical observation indicates they did.

I'm fine with A), while B) makes no sense to me at all, nor to anyone else it seems.
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Re: Real Return Bonds

Post by IdOp »

ghariton wrote:The problem, I think, is with the definition of "duration". It is the change in the price of a bond in response to a 1% proportional change in the yield (or interest rate). [I know that it is really the response to an infinitesimal change (di/i), but I have been criticized here before for calculations in continuous time.)
Thanks for your comments. That's too bad about the criticism. My own background is in natural sciences, so my view is that finance takes place in the physical world, and so financial models are just specializations and simplifications of physical models. Viva la Newtonian time.

A thought about infinitesimal vs small changes in this discussion. One could take the approach of calculating some numerical examples with finite, but small, changes in the yield, etc. Trouble is, if this were to show a small difference in real and nominal modified duration, it could always be blamed on convexity, so it wouldn't prove anything. For that reason analytical results are better. We have calculus, so let's use it. Trouble is, they can be complicated to get in closed form. Maybe something quadratic could be done (a bond with 2 payments)? Maybe it's not necessary though, in view of the strip example.
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Re: Real Return Bonds

Post by ghariton »

I've reworked the algebra in continuous time, and I've come to the conclusion that James is right and Globe Investor is wrong. In continuous time (I will spare you the calculations), it is abundantly clear that the duration of a residual (or coupon) maturing in ten years is ten.

(In continuous time, Macauley duration and modified duration are the same, simplifying things so that even I can figure them out.)

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Re: Real Return Bonds

Post by IdOp »

ghariton wrote:In continuous time, Macauley duration and modified duration are the same, simplifying things so that even I can figure them out.
Now I'm puzzled as this suggests that what you meant by continuous time was not the same as what I called Newtonian time. I had thought we meant the same thing by those, but maybe not. By Newtonian time, I simply meant that time is a real variable, and so the payment times of cash flows didn't have to occur at, essentially, integer values.

FWIW a very pedagogical description of my understanding of duration (both Macaulay and modified) can be found in a note linked on my website, which is linked to this forum. (There is nothing about RRBs in it.)

In other news, it occurred to me overnight that it should be possible to directly calculate real and nominal modified durations for an example RRB with coupons using a spreadsheet. This would avoid convexity ambiguity. I'm off to try this now.
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Re: Real Return Bonds

Post by ghariton »

IdOp wrote:
ghariton wrote:In continuous time, Macauley duration and modified duration are the same, simplifying things so that even I can figure them out.
Now I'm puzzled as this suggests that what you meant by continuous time was not the same as what I called Newtonian time. I had thought we meant the same thing by those, but maybe not. By Newtonian time, I simply meant that time is a real variable, and so the payment times of cash flows didn't have to occur at, essentially, integer values.
By continuous time I mean that compounding happens at every instant. It is the limit of compounding once a year, twice a year, every month, every day, every minute, every second, and so on.

On one dollar, if the return is y, if compounding is yearly, the amount after a year is (1 + y). If compounding is k times a year, the amount after a year is [1 + (y / k)]^k. If the compounding is continuous, i.e. k becomes infinitely large, the amount after a year is exp (y). This last is what I refer to as continuous time. Interest is a continuous flow, not a periodic series of payments.

Modified duration is obtained by dividing duration by [1 + (y /k)]. As k becomes very large, (y / k) becomes very small, and so one is dividing by 1. Thus the two definitions of duration lead to the same result.
FWIW a very pedagogical description of my understanding of duration (both Macaulay and modified) can be found in a note linked on my website, which is linked to this forum. (There is nothing about RRBs in it.)
Thank you. I'll visit it when I get a bit more time.

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Re: Real Return Bonds

Post by jiHymas »

ghariton wrote:I've reworked the algebra in continuous time, and I've come to the conclusion that James is right and Globe Investor is wrong.
Yay! Less math for me to look at!
IdOp wrote:A thought about infinitesimal vs small changes in this discussion. One could take the approach of calculating some numerical examples with finite, but small, changes in the yield, etc. Trouble is, if this were to show a small difference in real and nominal modified duration, it could always be blamed on convexity, so it wouldn't prove anything. For that reason analytical results are better. We have calculus, so let's use it. Trouble is, they can be complicated to get in closed form.
There are closed-form equations for convexity effects as well, but as long as you keep yield changes genuinely small (like 1bp) convexity effects are very minor. You certainly couldn't blame anything close to the magnitude of the difference between GlobeInvestor Modified Duration and the Real Modified Duration on convexity, if you were to start with a bond at 0% Real and change the yield to 0.01% Real.

The other advantage (for me) of using 1bp changes is that it reflects the way I think about the markets, so it's easier to spot errors.
IdOp wrote:For the example, take a real return strip bond, which matures in one year and is sold to you today with an advertized real yield of 5% (annually). You pay $1 for the strip today, and in one year it matures and the broker gives you $1.10 in actual/nominal dollars. The difference between $1.05 and $1.10 was caused by inflation. This single financial situation can be looked at either through rose-coloured glasses (nominally), or blue-coloured glasses (in real terms).
...
Now to the general view of an RRB. It too can be viewed in nominal or real terms. The difference is in the $ amounts of the cash flows, which differ by inflation. These amounts are generally completely different. In real terms the RRB is like an normal bond, in nominal terms it isn't.
This example, where you are assuming that future inflation will be 5%, is an example of the backloading highlighted by Christiensen et al. in the BoC paper I linked above.

Yes, certainly, the Modified Duration will change when you change your assumptions about future inflation. But these are special cases - otherwise, any table of Modified Durations would have to be infinitely long, with each MD for each having a subscript indicating the inflation to maturity assumption. So by convention - and by convention only - you assume future inflation will be zero. Anything else is a special calculation - no less valid than the conventional calculation, just not conventional.
IdOp wrote:Regarding GlobeInvestor's data, I just want to clarify my view. There are basically two completely separate issues:

A) Compute real Macaulay duration.

B) Multiply it by the index ratio as your numerical observation indicates they did.

I'm fine with A), while B) makes no sense to me at all, nor to anyone else it seems.
Agreed (with the caveat that we have been discussing Modified, not Macaulay, duration). I just didn't want to be so emphatic with unlooked-at math still on the table.
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Re: Real Return Bonds

Post by IdOp »

Chronological note: Just saw the replies from ghariton and jiHymas. Thank you both! I'll reply later when they've been absorbed. :)

---------

I've put together a spreadsheet along the lines indicated above (it's attached below). It looks at a (hypothetical) 5-year annual-pay RRB with a real face value of $100. This is basically a more complex version of the strip example. If there are any errors in it (I just put it together this morning!) then any conclusions would be wrong, so it's important that anyone interested should check it independently. Please let me know of errors, concerns, etc.

Some comments on it:

* You can experiment by changing the green cells. These give the real coupon rate and price, and the assumed annual inflations.

* It calculates the return and Macaulay and modified durations, all both real and then nominal. The whole point is to compare the modified durations in the orange cells.

* The calculation avoids any convexity issues, in terms of comparing the orange cells. (There could be numerical or programming errors of course.)

* It does not use any sort of Excel-type function to calculate the durations. They're done by first principles.

* Many cells have a pedantic comment attached.

* The nominal view has been normalized by choosing the nominal price equal to the real price. This may seem strange at first, but it is not a mistake for the purposes of this calculation. That's because the entire return equation can always be divided by an overall constant, which can be chosen to make the two prices the same, and this has no effect on return or durations. (Of course when you buy an RRB at a broker the index ratio is very important to get the nominal dollars right!)

I've checked the results a bit by changing the price by a small amount and noting the change in the returns, which seem to agree with the correspoding duration (real or nominal). Yes, convexity is here.

As presently constructed, the spreadsheet gives generally different values for the real and modified duration values it calculates.

Enjoy. :)
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rrb_mod_durs.xls
RRB modified durations example.
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Re: Real Return Bonds

Post by IdOp »

ghariton wrote:By continuous time I mean that compounding happens at every instant. ...
Thanks for clarifying what you meant. I think I had heard of that before, but forgotten. So what you are describing is what in my note I called logarithmic return, eq.(4.7). (The name came from "somewhere on the Internet".) I guess the limiting process you describe a way of arriving at such a thing. OTOH you can just say that is how I want to parameterize the return factor, if it's convenient or desirable. That is how I look at it anyway. And I agree that with logarithmic return the two durations are "the same," eq.(5.9). So I'm glad we're in complete agreement on that. ;)
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Re: Real Return Bonds

Post by IdOp »

jiHymas wrote:There are closed-form equations for convexity effects as well, but as long as you keep yield changes genuinely small (like 1bp) convexity effects are very minor. You certainly couldn't blame anything close to the magnitude of the difference between GlobeInvestor Modified Duration and the Real Modified Duration on convexity, if you were to start with a bond at 0% Real and change the yield to 0.01% Real.
Agreed, and I wasn't suggesting convexity explained GlobeInvestor. What I meant was that for a given RRB in a given inflation scenario, you could try to calculate each of the real and nominal modified durations by making a small change in the price and noting the effect on the yield. If these two results were different it would be hard to conclude the difference was real, if you felt convexity was nagging in the background. Thought as you say it would be very small. Nevertheless, I wanted to side-step this nagging worry so in my spreadsheet I calculated the durations "exactly", i.e., using closed formulae rather than such a small-but-finite difference approach.
Yes, certainly, the Modified Duration will change when you change your assumptions about future inflation.
Awesome, that's all I was pointing out from the start. E.g., 0 inflation gives real modified duration, non-zero gives nominal modified duration. And the two are different. We are now in non-violent agreement. ;)
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Re: Real Return Bonds

Post by jiHymas »

IdOp wrote:
Yes, certainly, the Modified Duration will change when you change your assumptions about future inflation.
Awesome, that's all I was pointing out from the start. E.g., 0 inflation gives real modified duration, non-zero gives nominal modified duration. And the two are different. We are now in non-violent agreement. ;)
Reserving terminological problems. You will never hear me talking about "nominal modified duration" because it's a very poorly defined term - your calculated "nominal modified duration" will be very different if you assume future inflation of 1% as opposed to 100%.

I would prefer to refer to "Modified Duration with an inflation assumption of X"; and if I just say "Modified Duration" then it is understood that X = 0.
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Re: Real Return Bonds

Post by IdOp »

jiHymas wrote:Reserving terminological problems. You will never hear me talking about "nominal modified duration" because it's a very poorly defined term ...
Potentially, yes. What I would mean by it is modified duration in the presence of the nominal cash flows. If those cash flows have not been specified or are not clear from the context of the discussion, then the concept would be less specific than it could be.
- your calculated "nominal modified duration" will be very different if you assume future inflation of 1% as opposed to 100%.
Certainly. But in both the strip and spreadsheet examples, it is quite clear what the nominal cash flows are.
I would prefer to refer to "Modified Duration with an inflation assumption of X"; and if I just say "Modified Duration" then it is understood that X = 0.
A clear and reasonable approach.
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Re: Real Return Bonds

Post by ghariton »

The real yield on the Bank of Canada's benchmark is now north of 50 bp, and so of course I'm starting to fuss again.

Should I sell all my RRBs? Should I sell half? How can I just sit here and watch the price crater?

Well, the average duration of my RRBs is around 10 (I overweight the 2021s). So if the real return goes from 0.5% to 1.5% (a plausible scenario), the market value of the RRBs would decline by 10%. If the real return goes to 2.5% (which it did very briefly during the 2008 liquidity crunch), the market value would decline by 20%. I can live with that.

More importantly, I'm holding to maturity. I won't be forced to sell before 2017, when I have to convert to a RRIF. By then the duration of some of the RRBs should be below 4, and then I won't care.

On the other side, the problem is really that nominal rates are increasing while inflation is holding steady. How far can that go? As long as inflation is low, the Bank of Canada will try to keep nominal rates low too. They might not succeed, but at least they and I will be on the same side. So I suspect that chances are good that the increase we are seeing in real returns is unlikely to continue very long.

I think I'll go lie down.

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Re: Real Return Bonds

Post by ghariton »

ghariton wrote:Bank of Canada reporting yesterday's average yield as 1.24%.

I'm posting this here so that, in a year or two when I look back, I will be able to kick myself for not buying more at these prices. :shock:

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Re: Real Return Bonds

Post by BRIAN5000 »

IMHO an interesting comment in the TDW ETF fixed income list about XRB and ZRR, because of there duration if there is Inflation combined with rising interest rate increases the adjustment to principle due to inflation may be offset.

So no inflation protection
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Re: Real Return Bonds

Post by Shakespeare »

My 2021 RRB has a real return that is essentially zero. As long as I hold to maturity, the principal is protected.
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Re: Real Return Bonds

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Shakespeare wrote:My 2021 RRB has a real return that is essentially zero. As long as I hold to maturity, the principal is protected.
That seems like a pretty good deal in the current environment. I've been trying to stay relatively high quality, short duration, and manage a break even return on a net, net, net basis. For some time, I've been frustrated that a seemingly modest requirement has proven much more difficult to satiate than I would have anticipated.

And you get protection from unanticipated inflation to boot....
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Re: Real Return Bonds

Post by zeno »

Do I need RRBs as an inflation hedge if I own a house?
This may seem like a provocative (or obtuse) statement, but it's an honest question. I've never owned real return bonds. I have a conservative portfolio with a generous fixed income allocation (GICs for short term and Provinicial strips for long term). My home represents about a third of my net worth, and it's mortgage free. I've always considered it to by my hedge against inflation.
I can think of some obvious flaws in this strategy: housing prices don't always move in lockstep with overall inflation, houses aren't liquid, tastes and markets change and while I'm in a desirable, walkable downtown neighbourhood today, I may find that future home buyers don't find it as desirable.
Even with all those risks, I can't justify an RRB allocation to myself. To really protect myself against inflation I'd need a significant allocation to RRBs. After all what's the point of protecting 5% of your wealth?
I can see that once I exit the accumulation phase and downsize, RRBs would be important to protect my nest egg, but do I need them now?
Enlighten me.
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Re: Real Return Bonds

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zeno wrote:My home represents about a third of my net worth, and it's mortgage free. I've always considered it to by my hedge against inflation.
I can think of some obvious flaws in this strategy: housing prices don't always move in lockstep with overall inflation, houses aren't liquid, tastes and markets change and while I'm in a desirable, walkable downtown neighbourhood today, I may find that future home buyers don't find it as desirable.
The obvious flaw, IMHO, is why do you care about inflation in the paper value of your house, and how does it help you in the event your monthly expenses go up?
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Re: Real Return Bonds

Post by ghariton »

adrian2 wrote:
zeno wrote:My home represents about a third of my net worth, and it's mortgage free. I've always considered it to by my hedge against inflation.
I can think of some obvious flaws in this strategy: housing prices don't always move in lockstep with overall inflation, houses aren't liquid, tastes and markets change and while I'm in a desirable, walkable downtown neighbourhood today, I may find that future home buyers don't find it as desirable.
The obvious flaw, IMHO, is why do you care about inflation in the paper value of your house, and how does it help you in the event your monthly expenses go up?
Perhaps zeno is thinking of his house as his retirement plan, i.e. sell the house when he reaches 65 and live off the proceeds. To the extent that house prices equal or exceed inflation, that will give inflation protection in the accumulation stage. But your comment about operating expenses during this time is still valid.

Why RRBs? The main reason to my mind is protection against unanticipated inflation. I've also found that RRBs offer diversification, and offset to some degree the ups and downs of equity markets. But for me that is a secondary reason.

It's not clear to me that any other asset class will provide equivalent protection against unanticipated inflation. Certainly housing would be unlikely to, in the short and medium terms.

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Re: Real Return Bonds

Post by patriot1 »

zeno wrote:Do I need RRBs as an inflation hedge if I own a house?
Owning a house is a hedge against rising rents. That's all. Just as owning an oil well is a hedge against rising oil prices or what have you.

You cannot take it for granted that you can sell your house at some expected price in the future. That includes expecting that the current market price will continue to track consumer price inflation.
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Re: Real Return Bonds

Post by leoc2 »

Inflation = high interest rates.
High interest rates = lower house prices.
I know I bought a house in 1982.Had a 19% interest rate mortgage.
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Re: Real Return Bonds

Post by zeno »

[quote="ghariton"]
Perhaps zeno is thinking of his house as his retirement plan, i.e. sell the house when he reaches 65 and live off the proceeds. To the extent that house prices equal or exceed inflation, that will give inflation protection in the accumulation stage. But your comment about operating expenses during this time is still valid.

Why RRBs? The main reason to my mind is protection against unanticipated inflation. I've also found that RRBs offer diversification, and offset to some degree the ups and downs of equity markets. But for me that is a secondary reason.
[/quote]

Yes, I suppose I am thinking of it, if not as my retirement plan, at least as part of the mix. I didn't buy my house as an investment. I bought it to live in and because it suits my lifestyle. I have teenaged kids living at home, so I need the space and it's within walking distance of a good university, so they may be at home for many years yet. Even so, I'm conscious that the house is a significant portion of my net worth, and until the kids move out and I downsize, I am in effect overweighting my allocation to housing. The only way for it not to be so would be to remortgage and invest the proceeds elsewhere.

I think I understand RRBS as a protection against [u]unanticipated [/u]inflation and as diversification (they don't correlate to many of the other asset classes). I guess what I struggle with is allocation. Most asset allocation models allocate no more than 5% to inflation protected securities. That's a tiny fraction of your wealth that's protected. It hardly seems worth it.
Would I be right in saying that during the accumulation stage, this really is to provide some balance to your portfolio, so that when everything is going down, something is going up - a psychological (but not to be dismissed) benefit that keeps you from panic selling, allowing you to recover over time? Then, in the redemption stage, with a larger allocation, it's there to protect your capital when you've less runway to earn it back?

Again, thanks to all upthread for their insights.
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