Primer: Agreement Inflation-Linked Bonds
Canadian Model Linkers
In this article, I am trying to decease on my comments fairly generic, but at that spot are marketplace position convention differences betwixt linkers issued past times dissimilar countries. I am focusing on what is called the "Canadian model" for index-linked bonds, which is the construction of the Government of Canada's Real Return Bond programme.The UK of Britain in addition to Northern Republic of Ireland started regularly issuing index-linked debt earlier, using a dissimilar model. The former U.K. model was designed to bargain amongst sure technical issues that were concerns at that time, but the bond pricing is extremely hard to understand. (When I tried edifice a pricer for these bonds, fifty-fifty the U.K. Treasury did non lay the pricing calculation for non-coupon dates inward their introductory documentation equally it was ridiculous.) The U.K. finally gave upwards in addition to at nowadays issues Canadian model bonds.
Understanding the Bonds
I experience that the best way to sympathise these bonds is to intermission downwards their construction into iii logical steps, which are described below.The alternative is to await at the pricing formula, and/or play amongst a fixed income pricer to run into how they act. The work amongst memorisation of the formula is that it does non explicate why the bonds are structured the way that they are. I volition practise roughly pricing examples hither in ane lawsuit I extend my "simplepricers" Python package to handgrip index-linked calculations. (That is non a high priority; I desire to extend the functionality of my SFC model packet first.)
The logic of these bonds piece of work inward iii steps.
- The bonds human activeness similar manifestly vanilla bonds inward a "real prices" globe (or prices expressed relative to the cost degree of a fixed date). If yous sympathise conventional pricing of bonds, it is just the same.
- We in addition to thence imagine a globe where the solely fourth dimension units are monthly, in addition to nosotros exercise the monthly inflation fourth dimension serial to interpret prices from the "real prices" globe to the existent world, where prices are nominal. This translation is just similar translating prices from ane currency to another.
- We in addition to thence have got to bring the fact that nosotros involve to cost bonds on a daily basis, in addition to bring into concern human relationship publication delays inward inflation data. We interpolate the monthly "currency conversion" serial to a daily footing for calculations.
The residual of this article expands upon the inward a higher house summary description.
Real Prices World
For the purposes of this article, I am largely assuming that the reader understands how conventional bonds work.However, equally a quick recapitulation, volition exercise equally an instance a hypothetical $100 facial expression upwards value 10-year bond amongst a 2% coupon. We assume that "now" is Dec 2016, in addition to the bond matures inward Dec 2026. (I am ignoring the twenty-four hours of the month for now.)
If nosotros are inward a province where bonds pay semi-annually, the cash flows received are:
- $1 coupon payment on every June in addition to Dec upwards to in addition to including Dec 2026. (We assume that the $1 coupon for Dec 2016 was already paid.)
- Additionally, nosotros are paid $100 principal at maturity (December 2026). This agency that the lastly cash flow is $101.
If nosotros alive inward a province where bonds pay annual coupons, the cash flows would be:
- $2 every December, upwards to in addition to including Dec 2026.
- $100 principal payment on Dec 2026. (Making the full lastly payment $102.)
I am non going to explicate bond pricing conventions, but they are based on what is called a clean price, which has the next property.
- If the build clean cost is less than $100, the quoted yield-to-maturity is greater than the coupon charge per unit of measurement (2% inward this case).
- If the build clean cost is greater than $100, the quoted yield-to-maturity is less than the coupon charge per unit of measurement (2%).
In other words: cost up, yield down, in addition to vice-versa.
However, this build clean cost is theoretical. What nosotros genuinely pay (the invoice price, too known equally the dirty price) too includes accrued interest. When involvement rates are positive, nosotros desire to larn payments to a greater extent than quickly, in addition to thence the invoice cost volition live higher only before a coupon payment than after the coupon payment, if the charge per unit of measurement of furnish is unchanged. (The yield on a bond tin sack roughly live idea of equally the internal charge per unit of measurement of furnish on the bond, albeit expressed inward a somewhat funky quote convention.)
For our purposes here, nosotros involve to realise that when nosotros verbalise most a bond's value, nosotros genuinely have got iii dissimilar concepts:
- the facial expression upwards value of the bond;
- the build clean cost of the bond;
- the invoice cost of the bond.
We in addition to thence have got roughly other pace -- roughly other dark box -- to convert the build clean cost to what nosotros genuinely pay (the invoice price).
The solely divergence betwixt an index-linked bond in addition to a conventional bond is that the instant dark box (that converts the build clean cost to invoice price) is to a greater extent than complicated.
The Monthly World
The side past times side phase is to imagine a globe which conveniently solely has months equally the fourth dimension axis (and inflation serial are forthwith available).All actual cash flows inward this globe are equal to: the cash flows inward the "real prices" globe times the monthly indexation factor. The monthly indexation factor is a monthly fourth dimension series, in addition to tin sack live idea of equally an telephone substitution charge per unit of measurement betwixt the "real prices" globe in addition to the cash flows seen past times investors.
(In fact, selection models on inflation-linked products are based on cross-currency options. That topic is good out of reach of this essay.)
For a given bond, the indexation factor inward a month equals the CPI degree for that month, divided past times the degree of the CPI at the month of bond issuance.
Example
To decease on this instance equally uncomplicated equally possible, I volition await at a bond amongst ane cash flow.- The electrical flow fourth dimension is Dec 2016, in addition to the CPI index degree for Dec 2016 is 200.
- The bond inward enquiry is a $100 one-year bond, amongst an annual coupon of 2%, that was issued inward Dec 2016.
- We purchase the bond at auction (in Dec 2016) amongst a build clean cost of $100 (which too equals the muddied price).
- Therefore, at that spot is solely ane cash flow received: $102 inward Dec 2017 inward "real" dollars.
- The degree of CPI inward Dec 2017 is 204.
There are 2 cash flows inward this example:
- The initial purchase (at auction) inward Dec 2016.
- The lastly coupon/principal payment inward Dec 2017.
We examine these inward turn.
Initial payment. The monthly indexation factor for Dec 2016 is equal to the CPI degree at that appointment (December 2016) divided past times the CPI degree at issuance (also Dec 2016). Therefore, the indexation factor = 200/200 = 1. Therefore, the invoice cost (paid inward actual dollars) does equal the build clean cost of $100.
(This is why nosotros cook the indexation factor to stand upwards for that of the bond number date. In theory, nosotros could only exercise the degree of the CPI equally the translation factor, but nosotros would in addition to thence have got oddball invoice prices for auction buyers. The implication is that each bond volition have got its ain soul indexation factor, which somewhat mucks upwards portfolio calculations.)
Final Payment. The lastly payment inward "real" dollars is $102. To interpret that into actual dollars, nosotros involve to multiply past times the indexation factor. In this case, the indexation factor is equal to the CPI degree for Dec 2017 divided past times that of the number date, or 204/200 = 1.02. Therefore, the full dollar amount received is 1.02*($102) = $104.04. [Update: corrected typo spotted past times commenter, thanks!]
This agency that the annual involvement charge per unit of measurement is 4.04% (using an annual quote convention), piece the charge per unit of measurement of inflation was 2% (204/200=1.02). Although the quoted existent yield (or indexed yield) of the bond was 2%, nosotros genuinely received 2.04% to a greater extent than than the inflation rate. This is because of the interest-on-interest effect; subtracting the inflation charge per unit of measurement from nominal returns is solely an approximation of the truthful existent return.
The Final Step: The "Actual" World
In reality, nosotros cost bonds on the footing of the transaction settling on a exceptional calendar date, in addition to non only the month of the year.
This is dealt amongst past times interpolating the monthly indexation factor serial described inward a higher house to the twenty-four hours of the month.
We define the reference value for a month to live the indexation value on the start twenty-four hours of the month. For the instant or after twenty-four hours inward a calendar month, the indexation factor is calculated past times linearly interpolating betwixt the reference value on the start of the month, in addition to the reference value on the start twenty-four hours of the side past times side month.
Since the CPI index for a month is solely available amongst a fourth dimension delay, the reference value for a month is the CPI index for the 3rd preceding month (for Canada at least; I retrieve most countries follow the iii month lag).
For example, if nosotros are inward June (30 twenty-four hours month), the value of the indexation factor on the 16th is equal the average of the reference value on June 1st in addition to July 1st (since at that spot is an equal number of days betwixt the 2 reference dates). Those reference values are equal to the monthly CPI index values on March in addition to April.
CPI Index Used
In all the cases I am aware of, the CPI index used inward calculations is all-items, in addition to is not seasonally adjusted. People who follow economical commentary volition complaint that economists mainly verbalise most inwardness (ex-food in addition to energy) seasonally adjusted inflation series. This tin sack practise a wedge betwixt economist chatter in addition to inflation strategy inward the close term.
The non-seasonally adjusted serial bring roughly getting used to. There is a fairly stable ripple inward the CPI series, in addition to this creates menstruum of high in addition to depression acquit for index-linked bonds (link to article that discusses this). In turn, this tin sack hateful that the standard breakeven inflation rate can live misleading for short-term bonds, if ane is thinking inward terms of the seasonally adjusted series.
(The argue why seasonal adjustments cannot live used is that seasonal adjustment is solely an approximation, in addition to could modify equally a outcome of algorithm changes, or if the seasonal adjustment factor is re-calibrated inward reply to novel data. The authorities does non desire to facial expression upwards off against investors inward courtroom equally a outcome of duelling seasonal adjustment algorithms.)
Principal Put
In roughly markets (notably the United States), the bond construction has an added characteristic that benefits investors: a principal put.
With this feature, the indexation factor on the principal payment is guaranteed to live no lower than 1; that is yous volition e'er larn at to the lowest degree $100 for every $100 facial expression upwards value of bonds yous own. This guarantee does not extend to coupon payments, including the lastly one.
That is, the indexation factor for the principal = max(1, coupon indexation factor).
For those of yous interested inward the wild globe of fixed income derivatives, yous volition recognise the max performance equally a shape of selection (a put). This agency that if nosotros desire to live really careful most bond pricing, nosotros involve to bring into concern human relationship the value of the principal put.
That said, the principal lay is solely in-the-money if the CPI index falls over the life of the bond. We have got non seen that behavior inward the U.S.A. inward the post-World War II world, although Japan's cost index has been largely stable in addition to would have got historically triggered a hypothetical principal lay at roughly points inward fourth dimension if such bonds existed. (Working from memory, Japanese linkers practise non have got a principal lay inward their construction anyway.)
Tax Treatment
The taxation handling of index-linked bonds is somewhat brutal; inward the cases that I am aware of, the authorities imposes a taxation based on the inflation uplift on the principal. This agency that yous are paying taxes on a marketplace position value change, but non on actual cash flows. (Otherwise, the taxes would live back-weighted, equally the inflation compensation on the principal is a major element of the bond's full return.)
I sure enough cannot give taxation advice, but it is security to tell that yous desire to concord index-linked bonds inward tax-deferred vehicles (retirement savings plans, similar RRSPs inward Canada). Otherwise, yous in all probability involve a expert accountant.
The Hyperinflation Disclaimer
If yous are paranoid, ane should complaint a disclaimer most such bonds protecting investors from inflation. (And all expert fixed income analysts are paranoid.) In a hyperinflation, the delay inward the indexation calculation agency that payments volition live based on what are highly out-of-date index values, in addition to the existent value of the bond cash flows would notwithstanding live largely inflated away.
Only real-time indexation into a hard currency or commodity could protect against a hyperinflation.
Concluding Remarks
The calculations are somewhat complex, but nosotros run into that they let index-linked bonds to give investors a guaranteed rate-of-return inward a higher house inflation (under normal circumstances). They hit this via creating an "exchange rate" betwixt "real cash flows" in addition to actual cash flows.
Calculation References:
(c) Brian Romanchuk 2016
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