## How TIPS Work

Since 1981, bonds linked to a broad price index have become increasingly common. In that year, the UK issued the first sovereign bond linked to a broad price index. Australia issued bonds in 1985. Canada followed in 1991, and introduced the model used by most countries since then (we will discuss this below). In 1997, the U.S. issued TIPS and, because of the country’s insatiable need for financing the inflation-linked bond market began to dramatically increase in size. The list of countries that now issue inflation-linked bonds (ILBs) is now quite long: Sweden (1995), France (1998), Italy (2003), Japan (2004), Germany (2006), Korea (2007), and so on. Many if not most Latin American countries issue ILBs as does the Czech Republic, Israel, India, New Zealand, Kazakhstan…

The global ILB market is huge, with more than $4.5 trillion in market value outstanding, in just the developed countries, as of mid-2021. Roughly 39% of that is in US TIPS, 48% is in developed markets ex-U.S., and 13% is in Emerging Markets as the chart below (Source: Bloomberg) illustrates.

In countries that issue inflation-linked bonds, the choice of a broad price index is almost always one that measures the average price change of a basket of consumer goods. In the U.S., this is the Consumer Price Index, or CPI.

Today there are two common structures traded in the U.S. inflation-linked bond market: Treasury Inflation-Indexed Securities (colloquially known as TIPS) and Corporate Inflation-Protected Securities (collectively known as CIPS). These two types of securities have significantly different structures, although the degree of this difference is obscured by the similarity of the names. In this lesson, we are focusing on TIPS.

## Now, How TIPS Work:

The U.S. Treasury’s version of inflation-linked bonds is based on what is often called the Canadian model.[1] A TIPS bond has a **stated coupon rate**, which does not change over the life of the bond and is paid semiannually. However, the **principal amount on which the coupon is paid changes over time**, so that the stated coupon rate is paid on a different principal amount each period. The bond’s final redemption amount is the greater of the original par amount or the inflation-adjusted principal amount.

Specifically, the principal amount changes each period based on the change in the Consumer Price All Urban Non-Seasonally Adjusted Index (NSA CPI, series CPURNSA<Index> on Bloomberg), which is released monthly as part of the Bureau of Labor Statistics’ CPI report. Note that we are talking here about the *index level, *not the monthly or annual rate of change that is usually the most widely-reported part of the release.The NSA CPI for a given day is called the *Reference CPI* (Ref CPI). In order to create a *daily* series from a *monthly* data point,[2] the Treasury defines the daily Ref CPI as follows:

- The Ref CPI for the first day of a month is equal to the NSA CPI for the month three months prior. For example, the Ref CPI for June 1
^{st}is the March NSA CPI, which was reported in April.

- The Ref CPI for any other day of the month is straight-line interpolated based on the number of calendar days in that month. For example, the Ref CPI for June 10
^{th}would be (9/30 * NSACPI_{Apr}) + (21/30 * NSACPI_{Mar}). The chart below shows the Ref CPIs for February and March 2009. The February 1st Ref CPI is equal to the November 2008 NSA CPI value of 212.425; the March 1st Ref CPI is equal to the December 2008 NSA CPI value of 210.228; and the April 1st Ref CPI is equal to the January 2009 NSA CPI value of 211.143. All other Ref CPIs are interpolated, so that we have one value for every date even though we only have one release per month.

- The current principal value of a TIPS bond is equal to the original principal times the
*Index Ratio*for the settlement date. The Index Ratio for a particular bond on a given day is defined as the Ref CPI for that day divided by the Ref CPI that applied to the day the bond was issued. This latter value is referred to as the*Base CPI*of the bond.[3]

Treasury Direct, part of the U.S. Department of the Treasury Bureau of the Fiscal Service, reports the Ref CPI (which is the same on any given day for all bonds) for each day and the Index Ratio (which differs for each bond, since each one has a different Base CPI) for each bond on each day as they become calculable, on its website at https://www.treasurydirect.gov/instit/annceresult/tipscpi/tipscpi.htm. But although there are quite a few terms involved, the actual calculation is quite simple. The important thing to remember is this: *the Index Ratio is used to adjust the principal of the bond, and the coupon payment is a fixed percentage of that principal.*

To illustrate how TIPS work, consider the example of a bond in its final pay period. Suppose that when it was originally issued, the Ref CPI for the bond’s original issue date (that is, its Base CPI) was 188.49677. The Ref CPI for its maturity date, we will suppose, is 2, is 296.64400. The bond pays a stated 2.375% coupon, half of which is paid semiannually. The two components to the final payment are as follows:

1. Principal Value = Stated Par * Index Ratio

= $1000 * (296.64400/188.49677)

= $1,573.74

2. Coupon Payment = Principal Value * Coupon Rate * Fraction of a Year

= $1,573.74 * 2.375% * ½

= $18.688

Notice that it is fairly easy to see how the construction of TIPS protects the real return of the asset. The Index Ratio of 296.64400/188.49677, or 1.57374, means that since this bond was issued, the total rise in the NSA CPI – that is, the aggregate rise in the price level – has been 57.374%. The coupon received has risen from 2.375% to an effective 3.7376% of the original notional (18.688/1000, times two because there are two coupons per year), a rise of 57.373%, and the bondholder has received a redemption of principal that is 57.374% higher than the original investment. In short, the investment produced a return stream that adjusted upwards (and downwards) with inflation, and then redeemed an amount of money that has the same purchasing power as the original investment. Clearly, this represents a real return very close to the original “real” coupon of 2.375%.[4]

Note that with most TIPS-style bonds, there is an added proviso that *at maturity*, the redemption value cannot be below the original par amount, although for any coupon period it can be. So, if in the above example the Ref CPI had been 180.00000, then the coupon value would have been:

1. Principal Value = $1000 * 180.00000 / 188.49677 = $954.92

2. Coupon Payment = $954.92 * 2.375% * ½ = $11.34

However, if that had been the *final* coupon period then the bondholder would have received $1011.34 – the coupon payment plus the **greater** of the calculated principal value or the original par value, $1,000 in this case.

Now, when you speak to your broker to buy a TIPS bond, the quote you will get back will be what is known as the *real, clean price* of the bond. To see what we actually have to pay for the bond, we then have to adjust that price by the accrued interest…just as with a normal bond…and by the inflation “uplift”, as we just did. So, for example, if a bond was quoted a price of 108-00, there was 0.27106 points of accrued interest, and there had been 43.881% inflation since the bond was issued, the invoice price would actually be:

Bloomberg makes this calculation easy, if you have access to Bloomberg. Here is the “YA” screen for the bond we just calculated the invoice price of, with the relevant numbers circled:

Now, if you are an individual investor the fine mechanics of this computation are probably not particularly important to you, but I want you to understand why, when you are quoted $108 by your broker, the actual price you pay ends up being $155.78. The difference in the quoted price, due to the convention of quoting a real, clean price, and the actual price you pay can become quite large as the amount of inflation accretion increases – as this example illustrates.

So why, then, do we quote the real, clean price rather than the actual price you pay? The reason is that using the real price allows us to calculate the real yield of the bond using normal bond mathematics. Avert your eyes if you are allergic to math, but it is really a beautiful fact that because of the way TIPS bonds are constructed, all of the “inflation uplifts” drop out when we are calculating real yields. That is, you *don**’t need to know* how much inflation has already been experienced by a bond in order to calculate its real yield to maturity, nor do you have to make any assumptions about future inflation. Again, this is only cool in a geeky kind of way, but if this is the price of a normal, “nominal” bond:

Then it turns out that this is the price of a “real” bond:

So, we use *exactly the same equation*, but we use real quantities- that is, we use the stated (real, unaccreted) price, the stated (real, unaccreted) coupon, and the stated (real, unaccreted) par value of 100. But now we are solving for r, the real yield, rather than n, the nominal yield. We are doing the calculation “in real space,” so we can ignore the effect of inflation since the effect is the same on each piece of the puzzle. And that is why the “Canada model” is such a popular model for inflation-linked bonds. Because geeks are in charge.

And that is how TIPS work.

[1] Most other countries also use the Canadian model now, with the biggest remaining exception being Australia.

[2] Necessary to, for example, properly account for accrued interest when a TIPS is bought or sold.

[3] The CPURNSA is very rarely revised, but it does happen. The issue specifications for TIPS, however, dictate that once the Base CPI is set for a bond, it becomes a permanent characteristic of that bond no matter what future revisions to the CPI occur.

[4] It isn’t *exactly* 2.375% in real terms because the inflation index “looks back” three months and this lag introduces a subtle mismatch between the inflation period covered by the bond and the actual inflation rate experienced by the holder during the holding period.