Un-diversifying during crises: Is it a good idea?

High levels of correlation among financial assets and extreme losses are typical during crises. In such situations, investing in few assets might be a better choice than holding diversified portfolios. We show that constraining the sparse -norm of portfolio weights automatically controls diversification and selects portfolios with a small number of active weights and low risk, in presence of high correlation and volatility. We highlight the diversification relationships between the minimum variance portfolio, risk budgeting strategies and diversification-constrained portfolios. Finally, we show empirically that the lq- strategy can successfully cope with bear markets by shrinking portfolio weights and total amount of shorting.