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Consider a renewal insurance risk model with initial surplus u>0 and let A_u denote the deficit at the time of ruin. This paper investigates the asymptotic behavior of the moments of A_u as u tends to infinity. Under the assumption that the claim size is exponentially or subexponentially distributed, we obtain some asymptotic relationships for the φ-moments of A_u, where φ is a non-negative and non-decreasing function satisfying certain conditions.