Understanding IL clean/dirty prices and redemption yield

[hairgoneJohn]

Hi

I new to linkers and am trying to get my head around valuing them.

Clean vs dirty price

If an IL bond with a zero coupon has been issued some time ago and its clean price is say 60 and inflation over the relevant period since issue has been 50% what will the dirty price be?
a] 60 x 1.5 = 90
b] 60 +100*0.5 = 110
c] another value

Calculating redemption yield

If the bond has say 10 years to redemption is the GRY?

10√(100/60) + inflation to maturity
That is easy to calculate but if the dirty price on purchase is not based on the first assumption in my previous question then it seems that the GRY above/below inflation will vary depending on future inflation and an assumption of future inflation is necessary to get a GRY.

Is there a calculator anwhere on the web that will give GRY based on various future inflation rates?

[hairgoneJohn]

And as an aside, an explanation of each of the columns of the IL page on yieldgimp would be great.

[helfordpirate]

If an IL bond with a zero coupon has been issued some time ago and its clean price is say 60 and inflation over the relevant period since issue has been 50% what will the dirty price be?
a] 60 x 1.5 = 90
b] 60 +100*0.5 = 110
c] another value

DirtyPrice = CleanPrice * IndexRatio + AccruedInterest
IndexRatio = ReferenceRPI/BondsBaseRPI

The reference RPI is the interpolated RPI 90 days ago.
The BondsBaseAPI is the interpolated RPI 90 days before the bond was issued.

So (a)!

If the bond has say 10 years to redemption is the GRY?

10√(100/60) + inflation to maturity

The first term would be a real yield to maturity in this case. But the "official" calculations are different and involve working out the future cash flows under assumptions of inflation, and assuming coupons are reinvested, to give a gross redemption yield (GRY) and then discounting it back using current inflation to give a real redemption yield.

See https://www.lseg.com/content/dam/ftse-r ... o-calc.pdf

Is there a calculator anwhere on the web that will give GRY based on various future inflation rates?[

The DMO website gives all the calculation details in excruciating detail including when and what to round. There is also a calculator which gives redemption yields based on certain assumptions about inflation. Here...https://www.dmo.gov.uk/data/pdfdatarepo ... rtCode=D9C

[Yieldy]

"DirtyPrice = CleanPrice * IndexRatio + AccruedInterest
IndexRatio = ReferenceRPI/BondsBaseRPI"

Is not quite correct. In your example, I think it would be more appropriate to write:

DirtyPrice = (CleanPrice + AccruedInterest) * IndexRatio

Accrued is multiplied by the IndexRatio too.

[helfordpirate]

Is not quite correct. In your example, I think it would be more appropriate to write:

DirtyPrice = (CleanPrice + AccruedInterest) * IndexRatio

Accrued is multiplied by the IndexRatio too.

I dont think so. This from the DMO web site - definition of Dirty Price

"The total price payable on the purchase of a bond calculated as the clean price plus accrued interest for all gilts except index-linked gilts with a 3-month indexation lag. In the case of the latter, the dirty price is calculated as the product of the real clean price and the relevant index ratio, plus the accrued interest."

See here https://www.dmo.gov.uk/help/glossary/#: ... tion%20lag.

The AccruedInterest is only the interest accrued on the bond since the last coupon payment. Whereas the Indexratio index adjusts from the issue of the bond to the current day.

[BondSquared]

Is not quite correct. In your example, I think it would be more appropriate to write:

DirtyPrice = (CleanPrice + AccruedInterest) * IndexRatio

Accrued is multiplied by the IndexRatio too.

I dont think so. This from the DMO web site - definition of Dirty Price

"The total price payable on the purchase of a bond calculated as the clean price plus accrued interest for all gilts except index-linked gilts with a 3-month indexation lag. In the case of the latter, the dirty price is calculated as the product of the real clean price and the relevant index ratio, plus the accrued interest."

See here https://www.dmo.gov.uk/help/glossary/#: ... tion%20lag.

The AccruedInterest is only the interest accrued on the bond since the last coupon payment. Whereas the Indexratio index adjusts from the issue of the bond to the current day.

The accrued interest does get adjusted, but not by the IndexRatio but its own AccruedRatioGrowth (or AccruedUplift).

A fully worked example can be found on page https://www.ons.gov.uk/economy/governme ... nmentgilts in the spreadsheet https://www.ons.gov.uk/file?uri=/econom ... excel.xlsx under tabs Table 2 / Table 2A. A UK government agency providing quality information - Christmas is coming ...

[helfordpirate]

The accrued interest does get adjusted, but not by the IndexRatio but its own AccruedRatioGrowth (or AccruedUplift).

I am not saying the accrued interest is not adjusted for inflation. But my understanding is that it is reported already adjusted such that it is just added to the product of the clean price and index ratio as per the DMO definition.

[Yieldy]

Hi, I think I am correct, but happy to be proven wrong.

@BondSquared - That's an interesting explanation from the ONS about how they account for gilt coupons (lumpy) throughout the year. However it's not appropriate for trading gilts.

@Helford - yes, I agree that they say:

"The total price payable on the purchase of a bond calculated as the clean price plus accrued interest for all gilts except index-linked gilts with a 3-month indexation lag. In the case of the latter, the dirty price is calculated as the product of the real clean price and the relevant index ratio, plus the accrued interest."

However the definition of Accrued Interest further up the page points to this link: https://www.dmo.gov.uk/media/1sljygul/yldeqns.pdf

Section 3 (page 25) deals with Accrued Interest. Part (ii) deals with 3-month IL gilts. "Real Accrued Interest" is calculated in the same way as a conventional gilt (as you would expect - no RPI uplift for "Real"). "Inflation-adjusted accrued interest" = RAI x IndexRatio[SettlementDate]... and this is the real world accrued we all pay when we buy an index linked gilt.

As IndexRatio[SettlementDate] is applied to both the clean price and the real accrued, you might write that:

Dirty Price = IndexRatio x (Clean Price + Real Accrued).

It is easily checked on a settlement ticket from HL (or similar platform, I would guess).

[Yieldy]

Ah @Helford, sorry, I missed your reply. Apologies.

[helfordpirate]

Ah @Helford, sorry, I missed your reply. Apologies.

Yes I did follow that link to. In order to make the DMO definition of DirtyPrice work, the AccruedInterest has to be already inflation adjusted. I agree. The question I have now is whether it is reported e.g. by TradeWeb, adjusted or not. I just assumed it was as TradeWeb are "DMO approved" so we can directly add it to the clean*indexratio as per DMO definition.

I dug out the TradeWeb/FTSE Actuaries definition of Accrued Interest and it is not helpful. So I will take a dataset and see if the accrued interest is consistent with being reported already adjusted or not.

In terms of practical consequence, it is of course pretty small! But worth getting it right.

[BondSquared]

Hi, I think I am correct, but happy to be proven wrong.

@BondSquared - That's an interesting explanation from the ONS about how they account for gilt coupons (lumpy) throughout the year. However it's not appropriate for trading gilts.

@Helford - yes, I agree that they say:

"The total price payable on the purchase of a bond calculated as the clean price plus accrued interest for all gilts except index-linked gilts with a 3-month indexation lag. In the case of the latter, the dirty price is calculated as the product of the real clean price and the relevant index ratio, plus the accrued interest."

However the definition of Accrued Interest further up the page points to this link: https://www.dmo.gov.uk/media/1sljygul/yldeqns.pdf

Section 3 (page 25) deals with Accrued Interest. Part (ii) deals with 3-month IL gilts. "Real Accrued Interest" is calculated in the same way as a conventional gilt (as you would expect - no RPI uplift for "Real"). "Inflation-adjusted accrued interest" = RAI x IndexRatio[SettlementDate]... and this is the real world accrued we all pay when we buy an index linked gilt.

As IndexRatio[SettlementDate] is applied to both the clean price and the real accrued, you might write that:

Dirty Price = IndexRatio x (Clean Price + Real Accrued).

It is easily checked on a settlement ticket from HL (or similar platform, I would guess).

We're talking about the same thing - I was referring to how to calculate accrued interest bottom up from the nominal interest rate of the linker.
More details (including spreadsheets and a powerpoint presentation which explains the difference in calc when the DMO moved to a 3month lag) under https://www.dmo.gov.uk/publications/gil ... -examples/ (which are examples on the basis of the document quoted above).

[helfordpirate]

So I will take a dataset and see if the accrued interest is consistent with being reported already adjusted or not.

So this is all getting in the weeds for me now! But I took a Tradeweb dataset and looked at the reported accrued interest and sanity checked it against an estimate of the interest.

Here is one clear example...

Gilt Name Close of Business Date ISIN Type Coupon Maturity Clean Price Dirty Price Yield Mod Duration Accrued Interest
UKGI 1.25 11/27 10/11/2023 GB00B128DH60 Index-linked 1.25 22/11/2027 103.138 201.684888 0.462086 3.90757 1.155737

Accrued Interest is reported as 1.155737. The dataset is from 10/11/23 and the bond issue date is 22/11 so it is close to paying its half coupon. The coupon is 1.25 so 0.625 real. The accrued interest must clearly be already inflation adjusted as otherwise it would be less than the half coupon - eyeballing the dirtyprice/clean a bit less than doubled.

So, at least with Accrued Interest reported by TradeWeb/FTSE the DMO definition is correct - Dirty = Clean * IndexRatio + Accrued
(because it is already inflation adjusted)

[hairgoneJohn]

Thanks. This is fantastic. The DMO calculator will give me hours of Christmas entertainment!

Anyone up for explaining the column headers for the yieldgimp table please?

[mc2fool]

Anyone up for explaining the column headers for the yieldgimp table please?

Which ones don't you understand from the explanations at the bottom of the index linkers page there?

BTW, there have already been some long and detailed threads fairly recently about index linkers:

viewtopic.php?p=604783#p604783
viewtopic.php?p=604678#p604678
viewtopic.php?p=597837#p597837

[hairgoneJohn]

Sorry I was a numpty and didn't see the explanations beneath the chart. All is clear now.