TIPS-style inflation-linked bonds (more properly known as Canadian-style) pay a fixed coupon on a principal amount that varies with the price level. In this way, the real value of the principal is protected (you always get back an amount of principal that’s indexed to the price level, floored in the case of TIPS at the original nominal value), and the real value of the coupon is protected since a constant percentage of a principal that is varying with the price level is also varying with the price level. This clever construction means that “inflation-linked” bonds can be thought of as simply bonds that pay fixed amounts in real space.

I have illustrated this in the past with a picture of a hypothetical “cake bond,” which pays in units of pastry. The coupons are all constant-sized cupcakes (although the dollar value of those cupcakes will change over time), and you get a known-sized cake at the end (although the dollar value of that cake might be a lot higher). That’s exactly what a TIPS bond is essentially accomplishing, although instead of cupcakes you get a coupon called money, which you can exchange for a cupcake. This is a useful characteristic of money, that it can be exchanged for cupcakes.

The beauty of this construction is that these real values can be discounted using real yields, and all of the usual bond mathematics work just perfectly without having to assume any particular inflation rate. So you can always find the nominal price of a TIPS bond if you know the real price…but you don’t need the nominal price or a nominal yield to calculate its real value. In real space, it’s fully specified. The only thing which changes the real price of a real bond is the real yield.

All TIPS have coupons. Many of them have quite small coupons, just like Treasuries, but they all have coupons. So in the cake bond, they’re paying very small constant cupcakes, but still a stream of cupcakes. What if, though, the coupon was zero? Then you’d simply have a promise that at some future date, you’d get a certain amount of cake (or, equivalently, enough money to buy that certain amount of cake).

Of course, it doesn’t have to be cake. It can be anything whose price over a long period of time varies more or less in line with the price level. Such as, for example, gold. Over a very long period of time, the price of gold is pretty convincingly linked to the price level, and since there is miniscule variation in the industrial demand for gold or the production of new gold in response to price – it turns out to look very much like a long-duration zero-coupon real bond.

And that, mathematically, is where we start to run into problems with a zero-coupon perpetuity, especially with yields around zero.

[If you’re not a bond geek you might want to skip this section.] The definition of Macaulay duration is the present-value-weighted average time periods to maturity. But if there is only one “payment,” and it is received “never,” then the Macaulay duration is the uncomfortable ∞. That’s not particularly helpful. Nor is the mathematical definition of Modified duration, which is Macaulay Duration / (1+r), since we have infinity in the numerator. Note to self: a TIPS’ modified duration at a very low coupon and a negative real yield can actually be longer than the Macaulay duration, and in fact in theory can be longer than the maturity of the bond. Mind blown.  Anyway, this is why the concept of ‘value’ in commodities is elusive. With no cash flows, what is present value? How do you discount corn? Yield means something different in agriculture…

This means that we are more or less stuck evaluating the empirical duration of gold, but without a real strong mathematical intuition. But what we think we know is that gold acts like a real bond (a zero coupon TIPS bond that pays in units of gold), which means that the real price of gold ought to be closely related to real yields. And, in fact, we find this to be true. The chart below relates the real price of gold versus the level of 10-year real yields since TIPS were issued in 1997. The gold price is deflated by the CPI relative to the current CPI (so that the current price is the current price, and former prices seem higher than they were in nominal space).

When we run this as a regression, we get a coefficient that suggests a 1% change in real yields produces a 16.6% change in the real price of gold (a higher yield leads to a lower gold price), with a strong r-squared of 0.82. This is consistent with our intuition that gold should act as a fairly long-duration TIPS bond. Of course, this regression only covers a period of low inflation generally; when we do the same thing for different regimes we find that the real gold price is not quite as well-behaved – after all, consider that real gold prices were very high in the early 1980s, along with real yields. If gold is a real bond, then this doesn’t make a lot of sense; it implies the real yield of gold was very low at the same time that real yields of dollars were very high.

Although perhaps that isn’t as nonsensical as it seems. For, back in 1980, inflation-linked bonds didn’t exist and it may be that gold traded at a large premium because it was one of the few ways to get protection against price level changes. Would it be so surprising in that environment for gold to trade at a very low “gold real yield” when the alternative wasn’t investible? It turns out that during the period up until 1997, the real price of gold was also positively related to the trailing inflation rate. That sounds like it makes sense, but it really doesn’t. We are already deflating the price of gold by inflation – why would a bond that is already immunized (in theory) against price level changes also respond to inflation? It shouldn’t.

And yet, that too is less nonsensical as it seems. We see a similar effect in TIPS today. Big inflation numbers shouldn’t move TIPS higher; rather, they should move nominal bonds lower. TIPS are immunized against inflation! And yet, TIPS most definitely respond when the CPI prints surprise.

(This is a type of money illusion, by which I mean that we are all trained to think in nominal space and not real space. So we think of higher inflation leading to TIPS paying out “more money”, which means they should be worth more, right? Except that the additional amount of dollars they are paying out is exactly offset by the decline in the value of the unit of payment. So inflation does nothing to the real return of TIPS. Meanwhile, your fixed payment in nominal bonds is worth less, since the unit of payment is declining in value. Although this is obviously so, this ‘error’ and others like it – e.g. Modigliani’s insistence that equity multiples should not vary with inflation since they are paying a stream of real income – have been documented for a half century.)

For now, then, we can think of gold as having a very large real duration, along with a price-level duration of roughly one (that is just saying that the concept of a real price of gold is meaningful). Which means that higher inflation is actually potentially dangerous for gold, given low current real yields, if inflation causes yields (including real yields) to rise, and also means that gold bugs should cheer along with stock market bulls for yield curve control in that circumstance. Inflation indeed makes strange bedfellows.