Greg Mankiw has a cool New York Times article and blog post, "On Au" analyzing the case to be made for gold in a portfolio, including a cute problem set. (Picture at left from Greg's website. I need to get Sally painting some gold pictures!)

I think Greg made two basic mistakes in analysis.

First, he assumed that returns (gold, bonds, stocks) are independent over time, so that one-period mean-variance analysis is the appropriate way to look at investments. Such analysis already makes it hard to understand why people hold so many long-term bonds. They don't earn much more than short term bonds, and have a lot more variance. But long-term bonds have a magic property: When the price goes down -- bad return today -- the yield goes up -- better returns tomorrow. Thus, because of their dynamic property (negative autocorrelation), long term bonds are risk free to long term investors even though their short-term mean-variance properties look awful.

Gold likely has a similar profile. Gold prices go up and down in the short run. But relative prices mean-revert in the long run, so the long run risk and short run risk are likely quite different.

Second, deeper, Greg forgot the average investor theorem. The average investor holds the value-weighted portfolio of all assets. And all deviations from market weights are a zero sum game. I can only earn positive alpha if someone else earns negative alpha. That's not a theorem, it's an identity. You should only hold something different than market weights if you are identifiably different than the market average investor. If, for example, you are a tenured professor, then your income stream is less sensitive to stock market fluctuations than other people, and that might bias you toward more stocks.

So, how does Greg analyze the demand for gold, and decide if he should hold more or less than market average weights? With mean-variance analysis. That's an instance of the answer, "I diverge from market weights because I'm smarter and better informed than the average investor." Now Greg surely is smarter than the average investor. But everyone else thinks they're smarter than average, and half of them are deluded.

In any case, Greg isn't smarter because he knows mean-variance analysis. In fact, sadly, the opposite is true. The first problem set you do in any MBA class (well, mine!) makes clear that plugging historical means and variance into a mean-variance optimizer and implementing its portfolio advice is a

In any case, Greg shouldn't have phrased the question, "how much gold should I hold according to mean variance analysis, presuming I'm smarter than everyone else and can profit at their expense by looking in this crystal ball?" He should have phrased the question, "how much

That's especially true of a New York Times op-ed, which offers investment advice to everyone. By definition, we can't

I don't come down to a substantially different answer though. As Greg points out, gold is a tiny fraction of wealth. So it should be at most a tiny fraction of a portfolio.

There is all this bit about gold, guns, ammo and cans of beans. If you think about gold that way, you're thinking about gold as an out of the money put option on calamitous social disruption, including destruction of the entire financial and monetary system. That might justify a different answer. And it makes a bit of sense why gold prices are up while TIPS indicate little expected inflation. But you don't value such options by one-period means and variances. And you still have to think why this option is more valuable to you than it is to everyone else.

I think Greg made two basic mistakes in analysis.

First, he assumed that returns (gold, bonds, stocks) are independent over time, so that one-period mean-variance analysis is the appropriate way to look at investments. Such analysis already makes it hard to understand why people hold so many long-term bonds. They don't earn much more than short term bonds, and have a lot more variance. But long-term bonds have a magic property: When the price goes down -- bad return today -- the yield goes up -- better returns tomorrow. Thus, because of their dynamic property (negative autocorrelation), long term bonds are risk free to long term investors even though their short-term mean-variance properties look awful.

Gold likely has a similar profile. Gold prices go up and down in the short run. But relative prices mean-revert in the long run, so the long run risk and short run risk are likely quite different.

Second, deeper, Greg forgot the average investor theorem. The average investor holds the value-weighted portfolio of all assets. And all deviations from market weights are a zero sum game. I can only earn positive alpha if someone else earns negative alpha. That's not a theorem, it's an identity. You should only hold something different than market weights if you are identifiably different than the market average investor. If, for example, you are a tenured professor, then your income stream is less sensitive to stock market fluctuations than other people, and that might bias you toward more stocks.

So, how does Greg analyze the demand for gold, and decide if he should hold more or less than market average weights? With mean-variance analysis. That's an instance of the answer, "I diverge from market weights because I'm smarter and better informed than the average investor." Now Greg surely is smarter than the average investor. But everyone else thinks they're smarter than average, and half of them are deluded.

In any case, Greg isn't smarter because he knows mean-variance analysis. In fact, sadly, the opposite is true. The first problem set you do in any MBA class (well, mine!) makes clear that plugging historical means and variance into a mean-variance optimizer and implementing its portfolio advice is a

*terrible*guide to investing. Practically anything does better. 1/N does better. Means and variances are poorly estimated (Greg, how about a standard error?) and the calculation is quite unstable to inputs.In any case, Greg shouldn't have phrased the question, "how much gold should I hold according to mean variance analysis, presuming I'm smarter than everyone else and can profit at their expense by looking in this crystal ball?" He should have phrased the question, "how much

*more or less than the market average should I hold*?" And "what makes me different from average to do it?"That's especially true of a New York Times op-ed, which offers investment advice to everyone. By definition, we can't

*all*hold more or less gold than average! If you offer advice that A should buy, and hold more than average, you need to offer advice that B should sell, and hold less than average.I don't come down to a substantially different answer though. As Greg points out, gold is a tiny fraction of wealth. So it should be at most a tiny fraction of a portfolio.

There is all this bit about gold, guns, ammo and cans of beans. If you think about gold that way, you're thinking about gold as an out of the money put option on calamitous social disruption, including destruction of the entire financial and monetary system. That might justify a different answer. And it makes a bit of sense why gold prices are up while TIPS indicate little expected inflation. But you don't value such options by one-period means and variances. And you still have to think why this option is more valuable to you than it is to everyone else.

## Post a Comment