Currently, strategies based on premium factors are everywhere: from funds or ETFs built on ratios or statistics perfectly specified, trying to exploit specific factor premia, to boutique instruments more or less opaque that following one or more risk premia.

In any case, one of the questions we may pose is what the best way to combine several risk premia is and what interaction can expect. In this post, we are going to examine this question with an example. Concretely, we will evaluate **how a strategy, that is guided by only a factor, could be completely disrupted by limiting the universe of stocks** by means of a different factor.

## Factors

In this post we are going to consider four factors/anomalies: Value, Quality, Momentum, and Low Vol. Each of them are build as **an aggregation of several ratios and statistics**, to make them more robust and less dependent on a specific statement.

- Value factor considers 9 ratios, among which you can find Price to Book, EV to EBITDA, or Price to Earnings.

- The quality factor takes into consideration mostly profitability ratios like ROE, ROA, or ROIC.

- Momentum and Volatility follow classic implementations with several calculation windows between 6 months and 12 months.

Backtesting of the four factors based on the S&P 500 components in the period 1/1/2000 – 17/4/2020, selecting every week the decile with the highest ranking according to each factor, **generate quite different statistics depending on the factor used**:

- While Value shows tremendous performance, it suffers dramatic drawdowns and amazing runups, with huge volatility.

- Momentum presents modest performance, with notorious drawdowns.

- Quality and Low Volatility produce prominent performance with moderate drawdowns (mostly Low Volatility) and with poor runups in the case of Low Volatility.

Seen these results, an immediate reaction would be** to join several factors to obtain strategies with better properties**, keeping the best of each factor and smoothing the worse effect of them.

## Cascade Selection

A very straightforward way to combine several factors and analyse the effect between them – apart from the classic techniques of combining scores or techniques of rank aggregation – is by a **cascade selection** doing consecutive selections based on the factors, limiting step by step the universe.

Let us examine what would happen if, in a Pure Low Vol strategy, that selects the stocks of the S&P 500 with the lowest volatility, we would do a **previous selection of 3 deciles** (150 stocks) based on the other 3 factors

respectively.

In the boxplot graphs representing the backtests from 1/1/2000 to 17/4/2020 in different days of portfolio revision, you can see **how the behaviour of LowVol changes tremendously with each previous selection**, taking each backtest some of the characteristics of the first factor selected rather than the one of the second and final selections (LowVol).

As you can imagine, the more stocks selected in the first “cascade”, the softer bias to this factor we get.

For example, the cascade Value-Low Vol gets closer to the behaviour of the Pure Value, mainly in terms of risk: higher runups, a large increment of volatility, deeper drawdowns, and an increment of annualized return.

### Factor Exposure

If you analyse the factor exposure to Low Vol factor and Value factor according to the method described in Factor Exposure: The Turn of The Screw, we can** obtain the evolution of the exposures** **along the period **with regards to the S&P 500 index, where the value 0.5 would represent the exposure of the index to a specific factor, and 1 the component of the index with maximum factor.

In the analysis of the pure Low Vol selection, the exposure of the portfolio backtested is near 1 regarding the Low Vol factor, while the Value exposure is close to 0.5 (index exposure) most of the time, but at the beginning.

When we implement a cascade selection based on Value factor previous to the Low Vol Selection, **the factor exposure changes enormously**, with a clear bias to Value Factor but also with some periods where the exposure to Low Volatility factor is quite high (sure higher than the one of a Pure Value Selection).

## Conclusions

Cascade factor selection would allow us to incorporate some nuances to a stand-alone factor selection, correcting some lacks that appear in every factor. It is possible to use several “factor falls”, generating more complex interactions among the factors.

A drawback in this kind of implementations is the need of broad universes, as in each step of the cascade the universe gets smaller. Besides, the order of the factors and number of selected stocks play an important role to be controlled.