Do factors widely used to model cross-sectional returns of U.S. stocks exhibit reward-for-risk behaviors? In other words, are expected factor returns higher (lower) when factor return volatility is high (low)? In their January 2017 paper entitled “The Risk-Return Tradeoff Among Equity Factors”, Pedro Barroso and Paulo Maio examine reward-for-risk behaviors of the size (small minus big market capitalizations), value (high minus low book-to-market ratios), momentum (winners minus losers from 12 months ago to one month ago), profitability (robust minus weak gross profit) and investment (conservative minus aggressive) risk factors. They compute risk as realized variance based on the last 21 daily factor returns and predict factor returns via regressions that relate monthly returns to respective factor variances. They test out-of-sample predictive power by comparing forecast errors from inception-to-date regressions (minimum 10 years of data) to those of the historical average. They test out-of-sample economic significance of findings via a strategy that each month holds a broad stock market index plus a 150% or 200% long (short) position in a factor portfolio when the regression-predicted factor risk premium is positive (negative). They compare the performance of this strategy to buying and holding the broad market index. They also consider a long-only factor exposure strategy. Finally, they perform an ancillary test of the ability of realized factor variances to predict the equity risk premium. Using daily and monthly factor returns for the U.S. equity market during January 1964 through December 2015, *they find that:*

- Regarding factor gross performance statistics over the sample period:
- Momentum has the largest average monthly return (1.35%), followed by market (0.49%), value (0.34%), investment (0.31%), profitability (0.25%) and size (0.24%).
- Momentum exhibits the highest monthly volatility (7.00%), followed by market (4.46%), size (3.11%), value (2.87%), profitability (2.13%) and investment (2.01%).
- Momentum has the highest annualized Sharpe ratio (0.67), followed by investment (0.53), value (0.41), profitability (0.40), market (0.38) and size (0.27).
- Momentum has the deepest maximum drawdown (-80%), led by market (-56%), size (-53%), value (-45%), profitability (-41%) and investment (-18%).
- Momentum has the highest average monthly realized variance as specified above (0.39), following by market (0.22), size (0.06), value (0.05), profitability (0.03) and investment (0.03).

- In-sample regressions over the full sample period indicate a positive relationship between reward (monthly return) and risk (monthly realized variance) for profitability and investment, but a negative relationship for market and momentum.
- Inception-to-date out-of-sample regressions indicate that realized factor variance predicts only profitability and investment returns.
- The out-of-sample forecasting power is economically significant and robust only for the profitability and investment.
- Buying and holding the broad market index generates average gross monthly return 0.98% and gross annual Sharpe ratio 0.74.
- Applying the 1.5X/-1.5X factor boost strategy as described above for profitability (investment) produces average gross monthly return 1.60% (1.48%) and gross annual Sharpe ratio 1.12 (1.18), with maximum drawdown -40% (-51%).
- These improvements translate to annual utility gains 5.3% to 6.8% (6.4% to 7.2%) for profitability (investment), depending on assumed level of investor risk aversion.
- Starting the out-of-sample test in 1984 rather than 1974 yields similar results.
- Applying 2X/-2X factor leverage also yields similar results.
- A “long-only” 1.5X/0 version of the strategy that exploits only positive expected factor returns for profitability (investment) produces annual utility gains 1.8% to 3.6% (0%), depending on assumed level of investor risk aversion. However, this version of the strategy applied to momentum produces annual utility gains 2.4% to 9.4%, suggesting upside momentum forecasts are exploitable even though overall momentum forecasts are not.

- Factor variances exhibit only weak out-of-sample power to predict the equity risk premium.

In summary, *evidence indicates that investors may be able to exploit the predictability of: (1) profitability and investment factors via long (short) positions when forecasted returns are good (bad); and, the momentum factor via a long position when forecasted returns are good.*

Cautions regarding findings include:

- Factor returns are gross, not net. Analyses do not account for trading frictions due to periodic portfolio rebalancing, or shorting costs and constraints. Accounting for these costs/constraints, which vary considerably over time and across factors, may affect findings.
- Using an index to represent the market ignores the costs of maintaining a liquid fund. These costs may be substantial early in the sample period.
- The economic significance tests are based on a strategy employing 1.5x and -1.5x (or just 1.5X) leverage for factor returns, but they do not account for any cost of leverage. Leverage costs would reduce performance of the active strategy.
- Testing a strategy or model on different factors introduces snooping bias (luck extracted by trying different sets of data), such that the factor providing the best performance overstates expectations.