The quantitative analysis of markets, and of the performance of fund managers, is in large part the mathematical treatment of various (often contested) metrics of risk.

Let us bring two points about risk into collision. First, over more than three decades scholars have debated the “equity premium puzzle.” Why do investors buy bonds, given the much higher returns available in the equities markets? The issue is a pressing one precisely because the metric of risk, that is, the relative safety of investment in bonds, especially the sovereign bonds of the United States, has never seemed enough to justify the size of the equity premium. Yet if it is not enough, what else is there?

Our second point: it is not the bonds properly speaking, but the US Treasury* bills* that have long been regarded as the paradigmatic risk-free asset. Is that the right choice? What if there were a better risk-free asset? How might calculations change?

**When the “Tea Party” Was New**

New research by David Blitz, of head of Quant Research at Robeco, looks at these issues nearly a decade after the 2011 near-default crisis in the US,* from an unusual angle.* Instead of wondering whether there is some non-arbitrary theoretical reason for taking one or another US Treasury as the best proxy of return free of risk, Blitz asks

*which*US Treasuries are the best as an empirical matter. Does the market itself treat the one-month bills as the benchmark? Blitz believes that it does not. The market’s behavior is implying, rather, that medium-term (five-year) bonds are best proxies.

One way to look at the issue: perhaps any investor’s risk-free asset depends on that investor’s investment horizon. An N-year zero coupon Treasury bond is the benchmark for an investor with an N-year horizon. A pension fund manager has then a benchmark defined by the long-term bonds.

One problem with that point of view though is that, with the longer horizons, the threat of inflation within the horizon becomes important. There are inflation-indexed T bonds, but as Blitz observes in a footnote these have limited availability and a short history, so they aren’t a great proxy.

Since CAPM requires a single risk-free asset, and its return (r), it is natural to press further.

**Finding the Market Implied Risk-Free Asset and Rate**

“What happens if the CAPM is tested assuming that 1-month Treasury bills are the risk-free asset, but the true risk-free asset is an N-year bond?” asks Blitz. A straightforward application of CAPM tells us that stocks with low equity beta would then exhibit a positive exposure to N-year bond returns. Stocks with high equity betas, on the other hand would exhibit a negative exposure to N-year bond returns. One more inferential step gets us to this: such a misidentified proxy for *r* would result in an inverse linear relationship between equity betas and bond betas.

Hedge funds learning of this finding will immediately start to think “Arbitrage!” Such inefficiencies are precisely where they live. But let us pull back from their trading floors for a moment and look to the epistemological point that this inverse linear relationship is a *testable* prediction.

That is a testable prediction, and in fact, the tests come back positive. The test can be run with various Ns, until one finds what N does not in fact lead to a false r. That answer tells you what benchmark asset, and what r, is implied by the markets. The answer, when the stocks involved in the testing are those of the US equity markets, is the five-year Treasury. When other equity markets are tested, the answer varies, but stays within the range of two- to 10-year bonds.

All this brings us back to our first data point above. For it is possible that Blitz has resolved the equity premium puzzle. Bonds have a higher average rate of return than bills. If five-year bonds rather than bills are risk-free asset; then the equity premium must be measured against that higher yield. So the premium is smaller than the world has assumed it is. And the risk/return trade-off may be sufficient to account for the smaller premium after all.

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