We show that including a term which accounts for finite liquidity in portfolio optimization naturally mitigates the instabilities that arise in the estimation of coherent risk measures on finite samples. This is because taking into account the impact of trading in the market is mathematically equivalent to introducing a regularisation on the risk measure. We show here that the impact function determines which regularizer is to be used. We also show that any regularizer based on the norm p with

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... > 1 makes the sensitivity of coherent risk measures to estimation error disappear, while regularizers with p < 1 do not. The 1 norm represents a border case: its "soft" implementation does not remove the instability, but rather shifts its locus, whereas its "hard"