Limits To Arbitrage And The Case For Minimum Volatility

by: Dan Morillo

As I’ve noted before on this blog, no matter the strength of historical observation, I am often asked by clients whether minimum volatility (MV) portfolios can continue to perform as they have in the past. In my last blog I answered that question by looking at a “behavioral” explanation - the “favorite/long-shot bias” - as a potential source for the performance of minimum volatility portfolios.

But it’s worth noting that the case for MV portfolios does not rely purely on arguments about behavioral biases. There’s also an interesting argument around what is known as “limits to arbitrage” that may lead to an outperformance of low-beta stocks.

“Limits to arbitrage” refers to situations where investors operate under restrictions that shape their investment behavior - even if they are not subject to behavioral biases. One such “limits to arbitrage” explanation for the outperformance of low-beta stocks is linked to the widespread use of index benchmarks as a mechanism to evaluate active managers [1].

When an active manager is given the mandate to beat a benchmark, he or she is effectively told to care only about the tradeoff between returns in excess of the benchmark and similarly to care only about the active or relative risk against that benchmark. This means that the manager is unconcerned with the tradeoff between total returns and total risk because they will only get paid on the basis of relative performance against the benchmark.

Why does this matter?

Imagine an active manager that operates under such a mandate who has two stocks that he believes are similarly undervalued and thus have the same expected “alpha." This manager expects to obtain total returns equal to each stocks’ exposure to the market (or the “beta” component of the return) plus an additional positive alpha arising from his or her expertise in stock-selection (or the “alpha” component of the return).

Let’s say these two stocks have a beta of 0.75 and 1.25. The manager expects higher total returns for the high-beta stock because it has the same alpha as the low-beta stock, but it is levered to the positive expected returns of the market. But he also understands that this stock has higher total risk since its high beta “amplifies” the return variation of the market. The two stocks end up with comparable levels of total return per unit of total risk. If the manager cared about total return and total risk, he or she would therefore want to hold a similar, if not larger, amount of the low-beta stock compared with the high beta stock.

In this case, however, the active manager cares only about active risk so he ends up with a significant preference for the high-beta stock. That is because it induces the same amount of active risk as the low-beta stock (both stocks deviate equally from a beta of 1) but it also offers the benefit of being levered to the positive returns of the market and offers higher expected active returns. In short, the ratio of active return to active risk is much better for the high-beta stock.

The upshot is that if there is a sufficiently large number of managers who operate under this sort of active mandate, high-beta stocks can end up being “overbought” - leading to their future underperformance - while low beta stocks can end up being “oversold” - leading to their future outperformance.

What does this mean for minimum volatility portfolios? They are able to take advantage of this stock picking behavior by overweighting low-beta stocks and underweighting high-beta stocks.

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