Real Return Bonds

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Re: Real Return Bonds

Postby newguy » 24 Dec 2011 17:16

canopus wrote:I'm just betting that it's politically easier to print money than to default. Unless you're Greece, or Italy, or Argentina, or Brazil, etc.

Some countries can't print money, ie. euro or gold backed. Some countries owed money in foreign currencies, ie. Argentina and much of Europe's periphery.

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Re: Real Return Bonds

Postby ghariton » 12 Jan 2012 14:50

Bank of Canada shows 0.38% real return, as of yesterday.

Time to sell? Time to buy? Time to go lie down?

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Re: Real Return Bonds

Postby Pickles » 12 Jan 2012 18:10

ghariton wrote:Bank of Canada shows 0.38% real return, as of yesterday.

Time to sell? Time to buy? Time to go lie down?

George


Over in the Portfolio reutns for 2011 thread, Dan H posted this:

For your comparisons as you continue to post figures, here are selected index returns for calendar 2011 (all are total return figures unless otherwise noted).

CANADIAN STOCKS
S&P/TSX Composite: -8.74%
Cdn Value (Dow Jones): -4.53
Cdn Growth (Dow Jones): -11.41%
Small Caps (S&P/Citigroup): -13.35%
...
CANADIAN BONDS
RRBs: 18.36%
Broad Market: 9.68%
Short Term: 4.97%


I confess, George, I don't understand RRBs no matter how patiently you and Bylo explain them. But I was surprised that they did so well last year and wonder how this jibes with your own experience as a bond holder.
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Re: Real Return Bonds

Postby Shakespeare » 12 Jan 2012 20:35

Time to go lie down?
My vote.
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Re: Real Return Bonds

Postby ockham » 12 Jan 2012 21:00

ghariton wrote:Bank of Canada shows 0.38% real return, as of yesterday.

Time to sell? Time to buy? Time to go lie down?

George


Today, ishares shows XRB (sporting its .37 MER) with a weighted average ytm of .17%; and XBB (MER of .32) with a weighted average ytm of 2.33. Lying down looks good. :beer:
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Re: Real Return Bonds

Postby ghariton » 12 Jan 2012 21:13

Pickles wrote:I confess, George, I don't understand RRBs no matter how patiently you and Bylo explain them. But I was surprised that they did so well last year and wonder how this jibes with your own experience as a bond holder.

Gosh, I didn't realize that they had done that well. I figured it was more like 12% (plus 3% coupon cash flow, and I'm heavily weighted toward the 2021's, so I guess Dan is right after all). The RRBs saved my portfolio. The rest, all in equities, was down a lot, but the portfolio as a whole was up 8%.

I dunno what to say. I'm holding these things for capital preservation, not expecting gains. I guess there's a flight to safety or something. Will it reverse any time soon? Who knows? (I do know that the Bank of Canada and the U.S. Fed have both committed to keeping interest rates low for a couple of years at least. But they really control only short term rates.)

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Re: Real Return Bonds

Postby northbeach » 12 Jan 2012 21:31

If core inflation rises to 5% in 12 months, what will the RRBs do?

If core inflation remains around 2.0% which way will RRBs go in 12 months?

Or does the price depend solely on long term rates?
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Re: Real Return Bonds

Postby newguy » 12 Jan 2012 21:35

ghariton wrote:Gosh, I didn't realize that they had done that well. I figured it was more like 12% (plus 3% coupon cash flow, and I'm heavily weighted toward the 2021's, so I guess Dan is right after all).
It's all just duration, he probably quoted longer ones. The long nominal bonds did about the same.

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Re: Real Return Bonds

Postby scomac » 12 Jan 2012 21:52

Pickles wrote:I was surprised that they did so well last year and wonder how this jibes with your own experience as a bond holder.


My return experience with the 2021 Strip has been 7.99% CAGR over the last 4 1/4 years since purchase. The calendar year return for 2011 was 13.35%
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Re: Real Return Bonds

Postby Shakespeare » 12 Jan 2012 21:56

http://www.globeinvestor.com/servlet/Pa ... ype=fedgov gives 143.91 on the 2021; my spreadsheet gives -.145% - i.e. 10-year RRBs are now negative.
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Re: Real Return Bonds

Postby ghariton » 13 Jan 2012 00:22

northbeach wrote:If core inflation rises to 5% in 12 months, what will the RRBs do?

If core inflation remains around 2.0% which way will RRBs go in 12 months?

RRBs track total inflation, as measured by the CPI. They do not track core inflation, which can be higher or lower.

Or does the price depend solely on long term rates?

In theory, the yield on a RRB equals the yield on a nominal bond of equal maturity (or better, duration), minus expected inflation, minus a "premium" for unanticipated inflation, plus a liquidity premium (there are more nominal bonds actively traded than RRBs). In practice this holds over the long run, but may not hold at any given point in time.

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Re: Real Return Bonds

Postby ghariton » 13 Jan 2012 00:28

Shakespeare wrote:http://www.globeinvestor.com/servlet/Page/document/v5/data/bonds?type=fedgov gives 143.91 on the 2021; my spreadsheet gives -.145% - i.e. 10-year RRBs are now negative.

You mean the ten-year RRB shown with a yield of 0.00% and a duration of 12.33 years? :wink:

That's why I don't trust globeinvestor's numbers.

I note in passing that, by interpolation, a nominal bond of comparable maturity yields about 2.00%. Coincidentally, the Bank of Canada targets an inflation rate of 2%. If expectations are the same (i.e. if the Bank succeeds in "anchoring" inflation), we are getting the insurance against unanticipated inflation for free.

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Re: Real Return Bonds

Postby jiHymas » 13 Jan 2012 02:20

ghariton wrote:
Shakespeare wrote:http://www.globeinvestor.com/servlet/Page/document/v5/data/bonds?type=fedgov gives 143.91 on the 2021; my spreadsheet gives -.145% - i.e. 10-year RRBs are now negative.

You mean the ten-year RRB shown with a yield of 0.00% and a duration of 12.33 years? :wink:

That's why I don't trust globeinvestor's numbers.

This is interesting. As far as I can tell, they are first calculating the Macaulay Duration for a normal bond with a coupon of 4.25% and a yield of 0%, which (according to MS-Excel) is about 8.47.

Then, I think, they are multiplying this by the Index Ratio, which is 120.9 / 83.07713 and coming up with their answer of 12.33 years (actually, I make it 12.32 years, using the actual/365 Excel convention and settlement today, but I get 12.33 using US or European 30/360 and settlement today. Canadian Convention is actual/365).

This may have been discussed somewhere already (can somebody give me a link?), but I can't think of a rationale for multiplying duration by the Index Ratio. Programmers gone wild? Note that using normal bond calculations for the duration of RRB ignores the back-loading effect which can be significant unless the assumed inflation rate is zero.
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Re: Real Return Bonds

Postby ghariton » 13 Jan 2012 02:27

jiHymas wrote:This may have been discussed somewhere already (can somebody give me a link?), but I can't think of a rationale for multiplying duration by the Index Ratio. Programmers gone wild? Note that using normal bond calculations for the duration of RRB ignores the back-loading effect which can be significant unless the assumed inflation rate is zero.

This has been discussed elsewhere on this forum, but none of us could figure out how a reported duration can exceed the maturity of a bond. Even for a strip, which is completely back-end-loaded, the duration equals the maturity (or maturity divided by gross return).

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Re: Real Return Bonds

Postby newguy » 13 Jan 2012 08:53

jiHymas wrote:This may have been discussed somewhere already (can somebody give me a link?), but I can't think of a rationale for multiplying duration by the Index Ratio. Programmers gone wild? Note that using normal bond calculations for the duration of RRB ignores the back-loading effect which can be significant unless the assumed inflation rate is zero.
What if they use DV01 methods for calculating duration. If the RRB yield changes from 0.2% to 0.21% it will result in a bigger change than a nominal going from 2.2% to 2.21%

If inflation expectations stay the same and you want to go short nominal bonds and long RRBs, what ratio would you use to hedge interest rate changes. I don't think it's 1 to 1 but it should be the ratio of DV01s, not the index ratio - unless someone can show they're the same.

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Re: Real Return Bonds

Postby ghariton » 13 Jan 2012 13:10

jiHymas wrote: but I can't think of a rationale for multiplying duration by the Index Ratio.


Okay, I think I understand.

For simplicity, consider a residual, i.e. a bond with only one payment, at maturity, T years from now.

If this is a nominal bond, the value of the bond is

V(nominal) = P / [(1 + i) ^ T]

where P is the principal and i is the nominal rate of return.

If this is a real return bond, the value of the bond is

V(real) = [P * (1 + f) ^ T] / [(1 + i) ^ T]

where f is the rate of inflation.

But since 1 + i = (! + f) * (1 + r), where r is the real rate of return, then

V(real) = P / (1 + r) ^ T

Now duration is the first derivative of the present value with respect to the rate of return. But to make comparisons among bonds, that reference rate must be the same across different kinds of bonds. Since most bonds are nominal bonds, use a nominal rate, even for real bonds. So

d[V(real)] / di = {P * (1 + f) ^ T} * d{1/[(1 + i) ^ T] } / di = {P * (1 + f) ^ T)} * (-T) / [(1 + r) ^ (T + 1)]

Since duration is defined as d[V(real)] / di divided by V(real), then duration is

(-T) * (1 + f) ^ T divided by (1 + r)

and modified duration is

(-T) * (1 + f) ^ T

which is the formula James is using.


My apologies for doubting Globe Investor.

So the reason for multiplying by the inflation index is to have the duration measure the impact of changes in the nominal rate of interest on the value of the real bond -- not changes in the real rate.

To explain it in more intuitive (to me) words, Globe Investor treats all duration calculations as if they were the result of a small change in the nominal rate. Since the discount rate for a real return bond is less than for a nominal bond, the impact of a given change in the nominal interest rate is larger, and so the reported duration is bigger. (In even simpler words, the denominator is smaller.)

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Re: Real Return Bonds

Postby jiHymas » 13 Jan 2012 15:27

ghariton wrote:So the reason for multiplying by the inflation index is to have the duration measure the impact of changes in the nominal rate of interest on the value of the real bond -- not changes in the real rate.

I feel sure there's an error here, but won't have time to take apart the math until Monday.

Firstly:
ghariton wrote:This has been discussed elsewhere on this forum, but none of us could figure out how a reported duration can exceed the maturity of a bond. Even for a strip, which is completely back-end-loaded, the duration equals the maturity (or maturity divided by gross return).


By back-loading, I mean (following the example of the linked paper by Christiensen et al.) the importance of the changes in the expected value of the coupons and principal, i.e., for a nominal 5% bond (assumed to pay annually for simplicity) the cash flows are $5, $5, ..., $105, but the coupons recieved from a real bond will increase with inflation (call it 2%), $5, $5.10, $5.20, ..., $105*factor. This had importance consequences in their calculation of the Break-Even Inflation Rate - they had to solve for this factor reiteratively until they found a BEIR that worked for bonds of the same duration when the modified duration of the RRB is determined by changing the cash-flows by the BEIR.

ghariton wrote:which is the formula James is using.

Well, yeah, but only because when I looked at the figures I thought 'gee, it looks like the geometric difference between the duration of the RRB expressed in real terms and that reported by Globe Investor is the Index Factor!'. When I performed the calculations, it worked! I am by no means convinced that this method makes sense.

I look at it like this: You calculate the Real Price using the real coupon and the real yield, according to convention:
IIAC wrote:Aside from the indexing provision, RRBs are identical to conventional semi-annual pay bullet bonds that repay 100 per cent of principal at maturity. Thus, the clean price, given yield to maturity for a RRB, is calculated in a similar fashion. The price resulting from this calculation is known as the “real price.”

A change in inflation will not affect the real price; only a change in real yield will.

You get the Nominal Price by multiplying the Real Price by the Index Ratio:
IIAC wrote:Settlement amounts for transactions in RRBs are based on the nominal price and nominal
accrued interest, which are calculated as follows:
Nominal.PriceDate = Real.Price * Index.RatioDate


Now, since you're getting the Real Price by treating the thing as a normal bond, you can also get the Real Macaulay Duration and the Real Modified Duration using the same assumptions (which doesn't mean it's right - it just means you can). It appears from my reconstruction of the GlobeInvestor numbers that this is exactly what they have done.

But the Real Modified Duration of the RRB expresses the percentage change in Real Price that will result from a change in the Real Yield. And further, the percentage change in the Nominal Price will be exactly the same as the percentage change in the Real Price, no matter what the value of the Index Ratio. That is to say, the Nominal Modified Duration should be equal to the Real Modified Duration.

So I don't get understand the Globe Investor calculation. And, to my chagrin, I'll be too busy to look at your math carefully until Monday.
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Re: Real Return Bonds

Postby jiHymas » 13 Jan 2012 15:33

It may be that they are considering the effect of an anticipated change in inflation on the final value received; and it may be that this is what your math shows. Changes in realized inflation will, of course, affect the Nominal calculation without affect the Real calculation. But at this point I don't know.
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Re: Real Return Bonds

Postby IdOp » 13 Jan 2012 15:41

A couple of quick comments about ghariton's calculation:

1) Totally minor point: the definition of duration is missing a minus sign ... evidenced by the negative result.

2) Main point: the factor (1+f)^T surely represents future inflation. Thus it cannot be the index ratio, can it?

In general it seems to me for RRB's you have real returns and real duration, quite analogous to nominal returns and nominal duration for nominal bonds. Since future inflation is unknown, and will not be the same every period, you simply cannot compute nominal yield or nominal duration for an RRB that hasn't matured. Any attempt to do so is speculation. If that is what GlobeInvestor are doing it seems like a slippery slope. Why don't they similarly try to report nominal YTM for RRBs?
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Re: Real Return Bonds

Postby ghariton » 13 Jan 2012 18:04

I should make clear that I'm not agreeing with what Globe Investor does. I'm just trying to understand why. My conclusion is that they are trying to "standardize" durations across various bonds. I personally don't find that helpful (I never buy nominal bonds anyway). I would rather do the calculation on the RRB's own terms, as James says. But it's interesting to figure out what Globe Investor is up to. (There's only so much excitement I can get from walking the dogs.)

As to IdOp's point about using future inflation rather than past inflation, I agree. Perhaps they are using past inflation to forecast future inflation. Or perhaps I haven't understood Globe Investor after all.

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Re: Real Return Bonds

Postby IdOp » 13 Jan 2012 18:54

ghariton wrote:I should make clear that I'm not agreeing with what Globe Investor does. I'm just trying to understand why. My conclusion is that they are trying to "standardize" durations across various bonds. I personally don't find that helpful (I never buy nominal bonds anyway). I would rather do the calculation on the RRB's own terms, as James says. But it's interesting to figure out what Globe Investor is up to.

I understand and it's great that you post your thoughts on it. :thumbsup:

There's only so much excitement I can get from walking the dogs.

Maybe time to switch to squirrel-feeding for a while? ;)

As to IdOp's point about using future inflation rather than past inflation, I agree. Perhaps they are using past inflation to forecast future inflation.

I was mainly trying to point out that (1+f)^T couldn't be what jiHymas multiplied by. The BoC I'm sure has smart people working there, but they don't yet publish statistics on future inflation, AFAIK. ;)

Or perhaps I haven't understood Globe Investor after all.

Neither have I, and I never yet made an effort to.
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Re: Real Return Bonds

Postby IdOp » 14 Jan 2012 22:57

hi,

jiHymas wrote:But the Real Modified Duration of the RRB expresses the percentage change in Real Price that will result from a change in the Real Yield.

Correct, and similarly with all 3 Real --> Nominal.

And further, the percentage change in the Nominal Price will be exactly the same as the percentage change in the Real Price, no matter what the value of the Index Ratio

Correct, but note that this is the relative change in the nominal price due to the given (small) change in the Real yield. The corresponding change in the Nominal yield will in general not be the same as the change in the Real yield. Therefore ...

That is to say, the Nominal Modified Duration should be equal to the Real Modified Duration.

... does not follow. Just like the real and nominal yields are different, so to the values of real and nominal durations will typically be different, because they arise from different (non-proportional) cash flows (if inflation isn't zero).

Probably you'll have already sorted this out by Monday, but if not, hope this helps a bit.
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Re: Real Return Bonds

Postby ghariton » 14 Jan 2012 23:16

IdOp wrote:Just like the real and nominal yields are different, so to the values of real and nominal durations will typically be different, because they arise from different (non-proportional) cash flows (if inflation isn't zero).

Yes. 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.)

So are we talking of a 1% change in the real rate, or a 1% change in the nominal rate? The two won't reflect the same phenomenon. For simplicity, assuming Fisher's relationship between real and nominal rates,

(1 + i) = (1 + r) * (1 + f)

so that, if inflation is 2% and real returns are 2%, then nominal returns are 4%. If real rates increase by 1%, then the proportional increase in real rates is 50%. But if inflation is unchanged, nominal rates rise to 5%, i.e. a proportional increase of 25%. The problem, I think, is that I calculate durations on RRBs using the 50%, but Globe Investor calculates using the 25%. That's why they have to multiply by a larger number than I do, to get the same change in the price of the bond.

Or maybe not. I will wait for James' analysis.

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Re: Real Return Bonds

Postby jiHymas » 15 Jan 2012 00:04

IdOp wrote:I was mainly trying to point out that (1+f)^T couldn't be what jiHymas multiplied by.


Quite right, it wasn't. I multiplied by the Index Ratio, which is simply the cumulative inflation since issuance. For any given RRB with five years term remaining, it could be 1.00000000000000000000000000000 (for a five year new issue) or 20-bazillion (for a RRB issue 495 years ago with an original term of 500 years).

The Index Ratio is simply what you multiply the Real Price by to get the Nominal Price.

IdOp wrote:
That is to say, the Nominal Modified Duration should be equal to the Real Modified Duration.

... does not follow. Just like the real and nominal yields are different, so to the values of real and nominal durations will typically be different, because they arise from different (non-proportional) cash flows (if inflation isn't zero).


I'm not quite sure I follow this.

Modified duration multiplies the absolute change in yield to get the relative change in price.

If the real yield changes by 1bp, then the nominal yield will also change by 1bp and (assuming that the Modified Duration of the instrument in question is 8.0) then the price of the bond - whether expresses as the real price or the nominal price - will change by 0.08% of its prior value, e.g. from 100.00 to 100.08.

If the original 100.00 was the Real Price (e.g., a RRB with a real coupon of 2% trading to yield 2% real), and the yield decreases by 1bp (to 1.99% real), and the Modified Duration (real) is 8, then the new Real Price is 100.08 (ignoring convexity, which will be negligible for a 1bp yield change). If the index ratio is 1.5, then the Nominal Price will change from 150.00 to 150.12.

ghariton wrote:so that, if inflation is 2% and real returns are 2%, then nominal returns are 4%. If real rates increase by 1%, then the proportional increase in real rates is 50%. But if inflation is unchanged, nominal rates rise to 5%, i.e. a proportional increase of 25%. The problem, I think, is that I calculate durations on RRBs using the 50%, but Globe Investor calculates using the 25%. That's why they have to multiply by a larger number than I do, to get the same change in the price of the bond.

This cannot be right, for the same reason as above. Modified Duration does not address the proportional change in yield, it addresses the absolute change in yield, which in your example is 100bp.

I'm leaning towards the idea the Globe Investor is just plain wrong. But I still haven't looked at your math properly.
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Re: Real Return Bonds

Postby newguy » 15 Jan 2012 08:53

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