For many years now the “gold standard” in factor models has been the 1996 Fama-French 3-factor model:

Here r is the portfolio’s expected rate of return, *R*_{f} is the risk-free return rate, and *K*_{m} is the return of the market portfolio. The “three factor” *β* is analogous to the classical *β* but not equal to it, since there are now two additional factors to do some of the work. *SMB* stands for “**S**mall [market capitalization] **M**inus **B**ig” and *HML* for “**H**igh [book-to-market ratio] **M**inus **L**ow”; they measure the historic excess returns of small caps over big caps and of value stocks over growth stocks. These factors are calculated with combinations of portfolios composed by ranked stocks (BtM ranking, Cap ranking) and available historical market data. The Fama–French three-factor model explains over 90% of the diversified portfolios in-sample returns, compared with the average 70% given by the standard CAPM model.

The 3-factor model can also capture the reversal of long-term returns documented by DeBondt and Thaler (1985), who noted that extreme price movements over long formation periods were followed by movements in the opposite direction. (Alpha Architect has several interesting posts on the subject, including this one).

Fama and French say the 3-factor model can account for this. Long-term losers tend to have positive HML slopes and higher future average returns. Conversely, long-term winners tend to be strong stocks that have negative slopes on HML and low future returns. Fama and French argue that DeBondt and Thaler are just loading on the HML factor.

While many anomalies disappear under tests, shorter term momentum effects (formation periods ~1 year) appear robust. Carhart (1997) constructs his 4-factor model by using FF 3-factor model plus an additional momentum factor. He shows that his 4-factor model with MOM substantially improves the average pricing errors of the CAPM and the 3-factor model. After his work, the standard factors of asset pricing model are now commonly recognized as Value, Size and Momentum.

In a recent post, Alpha Architect looks as some possibilities for combining momentum and mean reversion strategies. They examine all firms above the NYSE 40th percentile for market-cap (currently around $1.8 billion) to avoid weird empirical effects associated with micro/small cap stocks. The portfolios are formed at a monthly frequency with the following 2 variables:

- Momentum = Total return over the past twelve months (ignoring the last month)
- Value = EBIT/(Total Enterprise Value)

They form the simple Value and Momentum portfolios as follows:

- EBIT VW = Highest decile of firms ranked on Value (EBIT/TEV). Portfolio is value-weighted.
- MOM VW = Highest decile of firms ranked on Momentum. Portfolio is value-weighted.
- Universe VW = Value-weight returns to the universe of firms.
- SP500 = S&P 500 Total return

The results show that the top decile of Value and Momentum outperformed the index over the past 50 years. The Momentum strategy has stronger returns than value, on average, but much higher volatility and drawdowns. On a risk-adjusted basis they perform similarly. The researchers then form the following four portfolios:

- EBIT VW = Highest decile of firms ranked on Value (EBIT/TEV). Portfolio is value-weighted.
- MOM VW = Highest decile of firms ranked on Momentum. Portfolio is value-weighted.
- COMBO VW = Rank firms independently on both Value and Momentum. Add the two rankings together. Select the highest decile of firms ranked on the combined rankings. Portfolio is value-weighted.
- 50% EBIT/ 50% MOM VW = Each month, invest 50% in the EBIT VW portfolio, and 50% in the MOM VW portfolio. Portfolio is value-weighted.

With the following results:

- The combined ranked portfolio outperforms the index over the same time period.
- However, the combination portfolio performs worse than a 50% allocation to Value and a 50% allocation to Momentum.

Yangru Wu of Rutgers has been doing interesting work in this area over the last 15 years, or more. His 2005 paper (with Ronald Balvers), *Momentum and mean reversion across national equity markets*, considers joint momentum and mean-reversion effects and allows for complex interactions between them. Their model is of the form where the excess return for country i (relative to the global equity portfolio) is represented by a combination of mean-reversion and autoregressive (momentum) terms. Balvers and Wu find that combination momentum-contrarian strategies, used to select from among 18 developed equity markets at a monthly frequency, outperform both pure momentum and pure mean-reversion strategies. The results continue to hold after corrections for factor sensitivities and transaction costs. The researchers confirm that momentum and mean reversion occur in the same assets. So in establishing the strength and duration of the momentum and mean reversion effects it becomes important to control for each factor’s effect on the other. The momentum and mean reversion effects exhibit a strong negative correlation of 35%. Accordingly, controlling for momentum accelerates the mean reversion process, and controlling for mean reversion may extend the momentum effect.

The presence of strong momentum and mean reversion in volatility processes provides a rationale for the kind of volatility strategy that we trade at Systematic Strategies. One sophisticated model is the Range Based EGARCH model of Alizadeh, Brandt, and Diebold (2002) . The model posits a two-factor volatility process in which a short term, transient volatility process mean-reverts to a stochastic long term mean process, which may exhibit momentum, or long memory effects (details here).

In our volatility strategy we model mean reversion and momentum effects derived from the level of short and long term volatility-of-volatility, as well as the forward volatility curve. These are applied to volatility ETFs, including levered ETF products, where convexity effects are also important. Mean reversion is a well understood phenomenon in volatility, as, too, is the yield roll in volatility futures (which also impacts ETF products like VXX and XIV).

Momentum effects are perhaps less well researched in this context, but our research shows them to be extremely important. By way of illustration, in the chart below I have isolated the (gross) returns generated by one of the momentum factors in our model.

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