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What Is The Duration Of Gold?

On Friday, the U.S. inflation market rallied further. Inflation-linked bonds were well-bid as zero-coupon inflation swaps rose 5-7bps across the board. Considering that nominal yields rose only 1-3bps, this is a noticeable outperformance: TIPS went up; nominal bonds went down.

Equities did their by-now-usual afternoon squirt higher. All of this occurred in a veritable news vacuum. Next week sees Retail Sales, Empire Manufacturing, Industrial Production, PPI, an updated Initial Claims number, and CPI. We will see how strong these hands are then.

There were some news stories out of Europe worth noting. In the "not a bad idea" category, the UK Telegraph reported some details from "senior sources" about the Basel rules that were scheduled to be agreed upon this weekend. They are said to require not only higher capitalization ratios from banks - many of whom, recall, are probably already undercapitalized if the PIIGS debt was properly scored - but also a special surcharge on banks deemed "too big to fail." This is bad news for shareholders of those banks, as return on equity will fall when the leverage falls, but good news for the debt holders. And it makes sense: make the costs of being big reflect the rewards of being big. That's a much better idea than just forcing them to break up (which probably wouldn't be legal in this country, anyway), although of course they will never set the price right. Either the price will be set too low and banks will strive to merge and become bigger so that they can get the safety net, or it will be too high and all of the big banks will split up. (Frankly, the latter outcome isn't that bad).

In news that is almost certainly related, Deutsche Bank reportedly plans to raise $11.4bln worth in a sale of new shares. Given how overvalued these banks are in an environment where the gross margins are shrinking, turnover is declining, and leverage is declining, it makes sense to be selling shares even if the Basel rules do not come to fruition as the Telegraph article suggests. I doubt Deutsche Bank will be the last. I imagine banks will keep selling to people who want to own banks at these prices until investors are sated.

I would target Norway as a potential investor. They announced recently a plan to buy more Greek debt, because after all it's a great idea to take limited upside and unlimited downside, especially when there is a pretty decent chance that downside will be realized. But if investors in Norway like Greek debt, then they probably will also like European bank debt because those institutions have lots of Greek debt too.


You might think I would learn my lesson about writing columns concerning the barbarous relic[1] after my experience last month. But today I wanted to consider briefly not the level of gold nor the value of gold, but rather a secondary investment characteristic of the metal: its duration.

An aside is warranted for those readers who are not familiar with the concept of duration. There are two types of duration we who focus on fixed-income markets are typically concerned with. One, called Macaulay Duration, is a measure of the average time to cash flows. Specifically, it is a weighted average time to receipt of the cash flows, where the weight is given by the present value of the cash flow in question.[2] Gold obviously has no cash flows, and an effectively unlimited life, so it makes no sense to talk about its Macaulay Duration.

The other sort of duration is called "modified" duration. The formula happens to look very similar to the one for Macaulay Duration, which is comforting, but the significance of the calculation is that it gives the percentage change in the bond's price for a percentage change in yield (and as traders, we multiply by the full price and divide by 100 to get the dollar value of an .01). In other words, it is the answer to the question "if this bond's yield declines by 1%, by about how much should the price of the security rise?" It is a measure of the sensitivity of the bond's price to changes in yield, which is useful for calculating the portfolio's sensitivity and for matching the sensitivity of the portfolio to the sensitivity of the liability mix (or the risk tolerance of the entity).

Of course, the sensitivity of the price of the bond to its yield-to-maturity turns out to be importantly related to the structure and timing of the cash flows, so it isn't surprising that the two formulas look similar. But when we look at the second concept, we can see other ways to achieve a reasonably-close answer. We can perturb the bond's yield slightly, recalculate the price, and observe the difference in price. Or, if we didn't have a bond calculator, in theory we could look at a series of observations of price and yield and run a regression to find out how price responds to a change in yield.

We could take this latter approach to evaluate gold's "duration" with respect to interest rates or inflation. Why might we want to do this? Well, a pension fund or endowment generally has a pretty good idea of how the present value of their obligations changes when interest rates and (depending on the obligation) inflation change. A corporate postretirement benefit plan, for example, reports (in footnotes to the annual 10-K, usually around footnote 14 or so) the effect of a 1% change in trend health care inflation rates. So plan sponsors who want to protect their plans from an unexpected increase in inflation need to know how much protection they need to buy. The answer, clearly, is related to how much protection is provided by each type of available asset in the "inflation protection" bucket.

Unfortunately, the historical data makes this approach difficult because (a) there is a lot of noise in the price of gold, which reflects a lot more than just the changing price level, and (b) over the last 35 years gold has had a positive bubble, a negative bubble, and ... well, I don't know where it is now. The result is that the relationships are sloppy. The chart below shows the percentage change in the front Gold futures contract as a function of the inflation rate.

Gold % Change as a Function of CPI Rate

If gold is a tight hedge, then these points should be arrayed in a generally straight-ish line from the origin to the upper-right of the chart. Clearly that's not the chart we are looking at. Indeed, if gold is a tight hedge, then it shouldn't have year-on-year declines unless the general level of prices in the economy is declining! The R2 here is 0.04, showing effectively no relationship between the inflation rate and the change in gold.[3]

This is surprising, is it not? Over the (very) long run, I would expect that gold keeps pace roughly 1:1 with the price level, since it is after all a real asset and should appreciate as the value of the currency (in real terms) depreciates (which is what CPI is measuring). And over the very long run that does seem to be the case. From December 1981 to December 2008, the total change in gold was 120% while the price level according to CPI rose 124%. Because gold is volatile, the exact ratio depends a lot on whether you're measuring peak-to-peak or trough-to-peak or peak-to-trough, but over the very long run it seems a decent 1:1 price hedge. This makes sense. However, I ought to point out that there have been several periods where gold has declined on net for more than 10 years before catching up, so you do need to realize that you're looking at a long-run price hedge, and the return after inflation is near zero. Both of these make sense.

But I started this exercise wanting to look at gold's duration, which is a little different. What I want to know is what happens to gold when inflation accelerates or decelerates. So I don't want the annual change in prices (the first difference); what I want to look at is the change in the rate of change in prices (the second difference).

The chart below shows the annual percentage change in gold versus the change in the CPI rate from year-to-year.

Gold % Change as a Function of CPI Rate Change

To the naked eye, this looks like there may be something of a positive relationship here, which is what we would expect: an acceleration in inflation tends to be associated with a rise in gold prices. The R2, however, still comes in at only 0.068.

It turns out that this is partly due to the fact that, because inflation has some persistence (that is, it tends to be pretty similar from year to year; it doesn't go from +10% to -2% to +6%), there are many small changes in inflation that are associated with a wide range of changes in gold. From the standpoint of our investor/hedger, this is white noise and it is reasonable to exclude some of these points from the relationship.

If we remove all of the points where the annual inflation rate changed by less than 0.5% (which are covered by the blue box in the chart), the R2 rises to 0.26. This isn't great, but it is at least respectable (equivalent to a correlation coefficient of about 0.5).

The slope of the line through those points is 8.3, which is to say that the expected response of the price of gold to a 1% change in CPI inflation is 8.3%.

This may seem disappointing, since most people who buy gold are doing so because they expect 20%, 30%, or larger gains. But that's a tactical investing decision. I'm looking at a deeper question, which is the role of gold as a portfolio hedge in the long run. I expected something a little longer, since as a zero-coupon perpetual investment the Macaulay duration is effectively infinite and the modified duration for a zero coupon bond in a period of low rates can get into the 20s, but 8% isn't bad. For hedging, I'd probably assume it is somewhat higher, maybe 10-15%. If you include 1979, the slope goes to 32, so something higher than 8% is defensible. As you can see from the chart, the relationship isn't exactly tight.

(What I would really like to know, actually, is the response of gold to changes in inflation expectations, but we have only a short history of inflation breakevens in the U.S. and that was mostly during a period of quiescent inflation. I could do the same analysis in sterling, since a longer history exists there, and perhaps if a client is interested I will do that).

The real investing problem, of course, is not the beta of gold with respect to inflation but its alpha. The total return of gold can be thought of then as something like this:

GoldReturn = 8.3% * (change in inflation) + alpha

In other words, alpha is the "unexplained" variance in this relationship. Visually, if you draw the 8% slope line on that chart, the alpha is the difference in the actual performance compared to the "predicted" performance from the line. And it can be 20% or more on both the positive and negative sides. For my money, I would set investment guidelines on that basis rather than on the basis of the "duration" of gold, and that - combined with the long-run expectation of a zero real yield - will limit my concentration in gold. But it does seem to have some value as a hedge against inflation accelerations, and obvious diversification benefits.


[1] Technically, I think Keynes was referring to the gold standard as a relic, not gold itself, but it seems many people apply the appellation to the yellow metal.
[2] In calculating Macaulay Duration, we use the bond's yield-to-maturity to discount all cash flows, even though
[3] If I start the chart a few years earlier, the R2 rises to 0.20 due to the single point of 1979, when 13.3% CPI inflation was matched with a 136% rise in gold. But without that outlier point, there is no relationship at least on a year-on-year sense.


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