There’s been a lot on Seeking Alpha about how TIPS pricing reveals inflationary, or, more to the point, deflationary expectations. I think it would be helpful to work through a relatively simple pricing model for TIPS to understand how that may not be so when there's a nip of deflation in the air.

**How TIPS Work.**

“TIPS” stands for “Treasury Inflation-Protected Securities.” A TIPS note is issued at a par value, say $1,000, and pays interest at 3% per annum, semiannually. The periodic interest is based, however, not on par, but on the par value adjusted for inflation/deflation since issuance. So, if inflation has been 10% since issue, the 1.5% semiannual coupon pays $16.50 instead of $15. If, on the other hand, there has been 10% *deflation*, the coupon pays $13.50. But, and this is important for pricing in today’s environment, at maturity, the TIPS note pays *the greater of* the inflation-adjusted amount *or* the face amount. So, even if there has been 10% deflation over the life of the bond, the holder still receives $1,000 at maturity of the bond.

One way to look at the maturity payment is to break it into two pieces: the inflation-adjusted principal of the bond and a “guaranty” payment, if required, necessary to bring the payment up to par. In inflationary times, the guaranty payment is essentially worthless. But in deflationary times, or even when the *possibility *of inflation is worth considering, the protection of principal against deflation has value, making the signal sent by the market price of TIPS much more difficult to decipher.

**The TIPS Spread**

The usual way to analyze TIPS as an indicator of inflation expectations is to compare what a TIPS bond would yield absent inflation to what traditional fixed payment bonds are yielding. Ordinarily, the TIPS bond will cost more and, therefore, yield less if no inflation occurs. But since the yields on the two instruments are presumably meant by their buyers to be the same (both having the same credit risk - zero), the lower nominal yield on the TIPS is expected to be compensated for by inflation adjustments. Thus, the spread between the yield on a fixed-rate note and the nominal yield on a TIPS note is an indication of the level of inflation expected over the life of the note.

But this analysis assigns no value to the TIPS' guaranty of repayment at par. When deflation becomes a serious possibility – *i.e., *when investors take that possibility into account in their decision making – the TIPS spread is distorted by an upward bias in the price of the TIPS, which makes the spread an unreliable measure of the mood of the market.

**Valuing the Par Guaranty**

I did some simplified math to try to understand the impact of the guaranteed principal payment in a deflationary scenario. My method follows the logic of the TIPS spread itself. I compare the likely price of a TIPS if there were no guaranty to the price determined with the guaranty, in both cases holding the target yield constant.

Assume a $1,000, 5-year inflation-adjusted, 3% coupon, inflation-indexed note, but with *no *guaranty at maturity, *i.e.,* with the payment at maturity equal to the inflation-adjusted value, even if it is less than par. A price of $866 will yield about 3% if -3% annual inflation occurs each year over the life of the note. That same price would yield about 6% if it were paid for a fixed 3% note. That’s a yield spread of –3%, which would match the deflationary expectation signaled by the $866 price.

Now assume an otherwise similar TIPS note with the guaranty of at least $1,000 at maturity. Thanks to the guaranty, a price of $988 will yield 3% at -3% annual inflation. But that price translates to a yield of 3.25% on a fixed bond. So, as it turns out, on a 5-year TIPS, a spread of -.25% results from an inflationary expectation of -3%. (Note that I say “results from,” not "signals.”)

The numbers change a bit, but not dramatically, if we lengthen the duration (which makes the size of the guaranty payment grow but increases the discount for the delay in receiving it) or if we adjust the nominal target return (3% in the preceding example) to reflect expected inflation, for example, by pricing a 5% return where 2% inflation is expected, a 3% return where no inflation is expected, and a 1% return where –2% inflation is expected.

**Refining the Model**

I promised a “relatively” simple pricing model but not, I hope, an overly simplistic one. Conceptually, our estimate of any variable is, in effect, the probability-weighted average of its possible values. (No, you haven't stumbled into Prof. Feynman's quantum mechanics seminar.) For example, when we say we believe that inflation will be x%, we really mean that if we were to take an average of the inflation rates we believe might occur, weighted by our assessment of their likelihood, x% would be the result. By the same logic, the value of a security – the price we will pay for it – is the probability-weighted average of the present values of its possible outcomes.

In many cases the effort of doing a weighted average of the values associated with each possible outcome isn’t justified. If the relationship between the price and a key variable is linear, we can figure the weighted average value of the variable and then calculate the price from the result. In the example above, the inflation-adjusted note without the guaranty was priced in that way, taking an assumed inflation level (*i.e., *the probability-weighted average of possible inflation rates) as the basis for our price. But once the guaranty is introduced, and deflation is regarded as a serious possibility, the relationship between the expected inflation rate and the price of the note ceases to be linear. In that case, we have to average the prices, not the inflation rates.

Let’s suppose that you assess the probabilities of annual inflation over the next five years as follows:

-2%__________25%

-1%__________25%

0%___________25%

1%___________25%

In other words, your “inflation expectation” is –.5%. But what would you pay/accept for a five year TIPS note? By my calculations, taking into account the guaranteed payment at par, a fair price for a 5-year TIPS note under these expectations is $1,009. That’s because the 1% inflation expectation would price out to $1,048, whereas, thanks to the guaranteed payment at par, the –1% and –2% expectations would price out to $996 and $992 respectively. (If you're checking my arithemetic, don't forget to include the $1,000 price for the 0% inflation assumption.) $1009 is what one would pay if one expected inflation to be a *positive .19%, *and that price would, accordingly, produce a TIPS spread of about .19%. Absent the guarantee, an inflation-indexed note would price out at $978, reflecting the –.5% average expectation and producing a similar spread against a fixed-rate bond. But the TIPS won't sell for $978; it'll sell for $1,009.

What this says to me is that, as deflation becomes a concern, the guaranty element of the TIPS instrument become increasingly relevant to pricing the paper, and the signal sent by the TIPS spread becomes harder and harder to decipher. And, as illustrated above, a small positive TIPS spread is entirely consistent with a deflationary expectation.

Nor is it possible simply to "adjust" for the value of the par guaranty. When the spread on a certain TIPS maturity is .19%, is that because inflation expectaions are clustered tightly around a small but positive inflation number, or are they spread widely as in the example above, perhaps with a deflationary bias that is masked by the pricing mechanism? We can't know. Which I think is something we need to know.

Do feel free to do your own math! I did some rounding and substituted annual coupons for semiannual ones. I don’t believe the point of this article is affected by those short-cuts.

**Disclosure: **No positions in TIPS. I have some ill-considered LQD puts expiring next week.