Primer: Indexation Calculations For Inflation-Linked Bonds

Although it is squeamish to recall of inflation-linked bonds equally providing a furnish inwards existent terms, whatever detailed analysis needs to empathize how the indexation calculations work. For Canadian model linkers (as opposed the early on United Kingdom design), nosotros tin recall of the bond equally beingness priced inwards some other currency, land all cash flows hand off inwards the local currency. The CPI index that is used inwards the indexation calculation is the telephone substitution charge per unit of measurement betwixt the currencies, amongst a daily interpolation betwixt monthly values.

Daily Interpolation

If i wants a high degree agreement of an inflation-linked bond, i tin say that all cash flows associated past times the bond are multiplied past times an indexation factor, which converts the "real" cost to "nominal." These cash flows include:

• the amount paid for the bond;
• accrued interest;
• coupon cash flows;
• proceeds of sale;
• redemption value if held to maturity.

At a high level, nosotros tin recall of the indexation factor equally the ratio of the value of the CPI index at the valuation appointment divided past times the value of the CPI index at the appointment of issue. (I am referring to the cost index used inwards the bond calculations equally the CPI equally a shorthand; inwards some countries, it may endure some other index.) Please banknote that the next statements are alone roughly correct; the balance of the article explains the added factors to have got into account.

Let us have got a 30-year Canadian Real Return Bond issued inwards Dec 2018. Example (simplified!) indexation values include the following.

• At issuance, the indexation factor is the CPI(December 2018) divided past times CPI(December 2018) which ever equals 1. This agency that the nominal cost for the bond on the twenty-four hours of issuance (at auction) would equal the existent price.
• The coupon payment inwards Dec 2019 would equal the coupon charge per unit of measurement times the chief amount held times the indexation factor, which equals CPI(December 2019) divided past times CPI(December 2018). This factor is equal to 1 summation the one-year inflation rate.
• The chief payment on Dec 2048 would usage the indexation factor CPI(December 2048) divided past times CPI(December 2018). This factor is equal to (1+i)^30, where i is the annualised 30-year inflation rate.

As tin endure seen, the chief value is (roughly!) rising inwards delineate of piece of job amongst the inflation rate, equally expected.

However, at that spot are 2 problems amongst this simplified approach.

1. Cash flows tin hand off on whatever working day, together with nosotros take away to avoid pathological deportment that results from this. We cannot associate the indexation factor for the month to the whole month, equally that would imply that the indexation factor would endure unchanged during the month, together with boundary from the in conclusion working  twenty-four hours to the outset working twenty-four hours of the month. Large known one-day cost jumps create a real high coin marketplace equivalent yield (since a 1% boundary inwards working twenty-four hours tin correspond to over 365% annualised, depending on the coin marketplace convention). We take away to shine the indexation factor changes over the month.
2. CPI information are non available inwards existent time; nosotros alone have a monthly figure inwards a afterwards month. Since it would endure impossible to back-date payment amounts, nosotros take away to piece of job amongst information from an before month (introduce a calculation lag). For Canadian bonds, the value is based on linearly interpolating the values from 2 together with 3 months ago.* For example, values inwards Dec are based on interpolating the monthly CPI values for September together with October.

If nosotros desire to recall almost monthly inflation rates, they exhibit upward amongst a 2-month lag nether this system. For example, imagine that the CPI index is unchanged at 100 for all months upward until September, together with hence the index jumps to 101 inwards October. The indexation factor would apartment at 100 for all days until the terminate of November. However, after Dec 1, the indexation factor starts rising toward 101 (which it reaches on Jan 1). This agency that the 1% monthly increment shows upward equally (1% divided past times 31) steps daily during the month.

This calculation lag implies that these bonds cannot protect y'all from a hyperinflation. The park technical Definition of a hyperinflation is an inflation charge per unit of measurement of greater than 50% per month. H5N1 two-month calculation lag inwards a hyperinflation would withal imply a devastating loss of purchasing power. For this reason, nosotros take away to endure careful almost assertions that index-linked providing a guaranteed existent charge per unit of measurement of return. That said, deviations from an ideal real-time CPI index would non endure pregnant nether normal weather condition (modulo the number of seasonality inwards the CPI).

In practice, i annoying belongings of these bonds is that each uses a index ratio that is rebased to 1 at the appointment of issuance of the bond. Although that makes the development of that especial bond's valuation easier to understand, it has the side number that if i has a portfolio of these bonds, each i has a dissimilar indexation ratio (absent the special instance of 2 such bonds issued on the same day). In effect, each bond has its ain unique telephone substitution charge per unit of measurement for calculations.

Example

Assume that the CPI index evolves equally follows.

• September 2017: 100.0
• October 2017: 100.0
• November 2017: 101.0
• December 2017: 102.0

The daily CPI index evolves equally follows. (Note that the index ratio of a especial bond at a appointment is the ratio of the daily CPI at that appointment divided past times the daily CPI index at the appointment of number of the bond. For example, for whatever bond issued inwards Dec 2017, the index ratio is the daily CPI value shown divided past times 100.0.)

We consider that the daily CPI is apartment at 100 during Dec 2017, equally it based on interpolating the September together with Oct CPI figures. (Note that the appointment axis are floating indicate numbers, rather than pretty month labels. The charts inwards this article was generated past times the script ex20180126_indexation.py in the simplepricers module; described inwards the appendix to this article.)

The rising inwards the CPI index inwards Nov 2017 results inwards the daily index rising inwards Jan 2018.

The nautical chart inwards a higher house shows the annualised charge per unit of measurement of alter of the daily index. It zooms off to almost 12.6% inwards January, together with hence 26.5% inwards February. If nosotros desire to recall inwards terms of inflation rates, nosotros consider that the boundary inwards the 1-month inflation charge per unit of measurement inwards the Nov monthly figure (when it went from 100.0 to 101.0) translates to positive acquit inwards the month of January. This agency that if nosotros desire to recall almost the number on carry, it is alone a two-month lag -- non three. It takes 3 months to fully achieve the novel degree of the CPI, but y'all alternative upward the number of the acquit inwards the month ahead.

The real high annualised inflation rates inwards this instance are non that far-fetched; the CPI index used for indexation is non seasonally adjusted, together with is headline inflation (including energy). Having a boundary of almost 1% on monthly ground is non that extreme, yet the annualised inflation charge per unit of measurement of 12.6% dwarfs modern coin marketplace rates. This highly variable inflation compensation inwards the brusque term makes index-linked bonds an extremely unreliable coin marketplace instrument, together with explains why at that spot is no appetite to number index-linked newspaper at coin marketplace maturities.

One may likewise banknote that the annualised inflation charge per unit of measurement during the months of Jan together with Feb are non perfectly flat. This is because the daily index is linearly interpolated, whereas a apartment annualised inflation charge per unit of measurement would imply an exponential correspond betwixt the endpoints. This number is small, but nosotros would likewise take away to add together inwards the fact that coin marketplace quote conventions differ from uncomplicated mathematical quote conventions. This agency nosotros could non just calculate nominal equivalent returns from inflation rates together with the quoted indexed yield on the linker.

Appendix: Simple Pricing Module

One difficulty amongst fixed income calculations is that if nosotros desire to piece of job amongst traded instruments together with quoted data, nosotros take away to follow the diverse quote conventions that are used inwards the associated market. Having fixed quote conventions is important: it allows traders to negotiate inwards yield terms, together with the agreed yield tin endure just translated to a in conclusion village price. If at that spot were ambiguity inwards the Definition of the bond yield, the 2 dorsum offices could terminate upward disagreeing on the transaction price. However, these quote conventions larn inwards hard to do illustrative calculations when writing tutorial material.

Textbooks together with fiscal theory quite ofttimes larn approximately this past times using a simplified yield calculation formalism. Real ground pricers volition interpret marketplace conventions to the theoretical equivalent, using them nether the hood. I decided to reinvent the wheel, together with railroad train a modest library of simplified functions that I volition usage for illustrative pricers - the opened upward source Python SimplePricers module, available at https://github.com/brianr747/SimplePricers. The code is designed to endure straightforward, hence that interested readers could replicate them if necessary inwards whatever programming linguistic communication they use. The objective is non to construct a fiscal software library, but but have got a unmarried location where I set all my instance code. This way, I do non have got to waste matter fourth dimension writing out equations: if the reader wants the details, they are available inwards the code.

I select this upward because inwards my writings almost inflation-linked bonds, I desire to illustrate calculations using the SimplePricers module. The number is that nosotros encounter the ugliness associated amongst the calendar. The number of days inwards the yr is variable, together with bond cash flows are diverted from weekends together with holidays depending on the calendar convention used inwards each jurisdiction. For uncomplicated bond calculations, nosotros tin ignore the calendar. For example, nosotros tin aspect at a hypothetical bond that matures inwards just 10 years, without worrying whether that is a weekend or not. Unfortunately, nosotros take away some notion of calendar dates to relate to inflation indexation.

In gild to maintain things simple, the bundle uses a simplified calendar of 12 months, each amongst xxx days (and no notion of weekends or holidays) -- a 360 twenty-four hours year. The appointment itself is but a existent (decimal) number, amongst the integer element representing the year, together with the fractional component corresponding to the twenty-four hours of the year. (In most reckoner representations of dates, a existent number is likewise used, but the integer component represents the number of days from a fixed start date. If nosotros were modelling loan shark fiscal calculations that are day-based, such a representation mightiness endure useful.)

"January 1" (the outset twenty-four hours of the outset month) corresponds to an offset of 0.0, together with hence each twenty-four hours increases the offset past times 1/360, amongst "December 30" (the in conclusion twenty-four hours of the year) is 359/360.

Footnote:

* The index calculation is described as:

The reference CPI for the outset twenty-four hours of whatever month is the CPI for the tertiary preceding calendar month. For example, the reference CPI for Dec 1 inwards whatever yr volition endure the CPI for September inwards that yr (which volition endure published inwards Oct of that year).

 Bank of Canada Review, referenced at:  http://www.collectionscanada.gc.ca/eppp-archive/100/201/301/bank_can_review/2006/spring/cover/en/pdf/real_return_eng.pdf


(c) Brian Romanchuk 2018