Axioma to Quants: Beware of Cherry Picking by Optimizers

Three executives of the risk and analytics company Axioma have jointly written an article on “Factor Alignment Problems and Quantitative Portfolio Management.” The gist of it is that optimization through certain standard techniques may lead to a sub-optimal result.

The paper begins by distinguishing three components of any active portfolio management strategy: the modeling of expected return, the modeling of risk, and application of the appropriate restraints. Drawing from statistics lingo, the authors – Sebastián Ceria (Axioma’s CEO); Anureet Saxena (senior associate of research) and Robert A. Stubbs (vice president of research) – call expected return the “first moment of the equity returns process,” whereas risk is the “second moment.”

There is an inevitable tension between those two moments in portfolio management, so we already have the material for the “factor alignment problems” (FAP) of the article title. Then there are constraints: A manager will want to follow best practices with regard to turnover and tax efficiency, for example, and may be obliged by contract to avoid investing in companies that sell alcohol, weapons, or other products his clients consider unethical. The graver the constraints, the more serious the threat of FAP.


Consider, as a simple example of the first and second moment, the possibility of a restatement of the earnings of large cap computer manufacturer Dell. In the first quarter of 2011, Dell took a $100 million charge to settle an investigation of possible accounting fraud. A portfolio manager might create a misalignment by reworking her expected returns model in accord with this new information, but leaving her risk model unchanged.

Why would our manager modify the one and not the other? Perhaps because “the risk model is not developed in-house but purchased from a third-party vendor, making it prohibitively expensive to recalibrate the risk model after modifying the factor exposures.”

In this case, Ceria at al don’t expect that the theoretical misalignment of the two moments will do much harm or, in their language, they think it “innocuous and unlikely to damage the ex post performance of the optimal portfolio.”  Why? Because the “orthogonal component” of alpha is asset specific rather than systemic. The problem is specific that is to Dell stock and seems unlikely to infect the other assets of the portfolio.

A more troublesome mismatch would occur, though, if an active manager were seeking alpha on the basis of the liquidity based model devised by Lubos Pástor and Robert F. Stambaugh (Liquidity Risk and Expected Stock Returns 2003) while using the Fama-French model on the risk side. This could produce “deterioration in the quality of risk production” in the portfolio as a whole.


Reliance on optimization tools that in turn rely on standard “user risk factors” will make such misalignment worse, caution Ceria et al. An optimizer will cherry pick “the aspects of the model of expected returns that it deems desirable when gauged on the yardstick of marginal contribution to systemic risk.” This amounts to making, and betting on, the erroneous assumption that a lack of correlation with the used risk factors is a lack of systemic risk altogether.

In giving specific empirical content to that theoretical argument, Ceria et al work from “a test-bed of 25 backtests derived from real alphas” and the decisions of real portfolio managers. The authors quantified misalignment of the first and second moments as the misalignment coefficient (MC) and they found, as their theoretical analysis predicts, that the value of MC is higher “for the optimal holdings as compared to that of the alphas.” This is true both when the back testing is run with a fundamental, and when it is run with a statistical, model, as you can see in the above graph.

The authors weren’t surprised by that result, but they were somewhat surprised by the size of MC. “For some of the backtests,” they write, “the MC of the optimal holdings was 100% greater than that of the alphas for backtests that used the fundamental risk model.”

They propose that active managers limit the damage from FAP with what they call an “augmented risk model approach.” What they have in mind by such an approach rather resists my powers of summarization, but they do indicate that if a portfolio manager can master it, she can “[push] the ex post efficient frontier upward … [accessing] portfolios that lie above the traditional risk-return frontier defined by the user risk model.”

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