Summary: For a multivariate random walk with independent and identically distributed jumps satisfying the Cramér moment condition and having mean vector with at least one negative component, we derive the exact asymptotics of the probability of ever hitting the positive orthant that is being translated to infinity along a fixed vector with positive components. This problem is motivated by and extends results of F. Avram et al. [Ann. Appl. Probab. 18, No. 6, 2421–2449 (2008; Zbl 1163.60010)] on a two-dimensional risk process. Our approach combines the large deviation techniques from a series of papers by A. A. Borovkov and A. A. Mogul’skij [Sib. Adv. Math. 2, No. 3, 1 (1992; Zbl 0847.62012); translation from Tr. Inst. Mat. 19, 1–63 (1992); Sib. Math. J. 37, No. 4, 647–682 (1996; Zbl 0878.60023); translation from Sib. Mat. Zh. 37, No. 4, 745–782 (1996); Theory Probab. Appl. 43, No. 1, 1–12 (1998; Zbl 0927.60040); translation from Teor. Veroyatn. Primen. 43, No. 1, 3–17 (1998); ibid. 45, No. 1, 3–22 (2000; Zbl 0980.60030); translation from Teor. Veroyatn. Primen. 45, No. 1, 5–29 (2000); Sib. Mat. Zh. 42, No. 2, 289–317 (2001; Zbl 0983.60011); translation in Sib. Math. J. 42, No. 2, 245–270 (2001)] with new auxiliary constructions, enabling us to extend their results on hitting remote sets with smooth boundaries to the case of boundaries with a ‘corner’ at the ‘most probable hitting point’. We also discuss how our results can be extended to the case of more general target sets.