This paper studies the impact of an unfunded social security system on the distribution of altruistic transfers in a framework where savings are due to both life cycle and random altruistic motivations. We show that the effect of social security on the distribution of these transfers depends crucially on the strength of the bequest motive in explaining savings behaviour. We measure this strength by the expected weight that individuals attach to the utility of future generations. On the one hand, if the bequest motive is strong, then an increase in the social security tax raises the bequests left by altruistic parents. On the other hand, when the importance of altruism in motivating savings is sufficiently low, the increase in the social security tax could result in a reduction of the bequests left by altruistic parents under some conditions on the attitude of individuals toward risk and on the relative returns associated with private saving and social security. Some implications concerning the transitional effects of introducing an unfunded social security scheme are also discussed.