New results on precautionary saving under two risks

•We find a new sufficient condition for precautionary saving under two risks.•We provide three economic interpretations for this condition.•An interpretation focuses on the marginal effect of saving on total income variance.•An interpretation focuses on the covariance between total income and saving returns.•An interpretation focuses on the effect of saving on the utility premium.

This paper identifies a new sufficient condition for a prudent agent to have positive precautionary saving in the presence of labor income and interest rate risks of any size. We also provide three economic interpretations for this condition focusing respectively on the marginal effect of saving on total income variance, on the sign of the covariance between total income and the return of saving, and on the effect of saving on the utility premium.