A popular proposal for reforming social security is to supplement or replace traditional publicly financed benefits with a new system of mandatory defined-contribution private pensions. Proponents claim that private plans offer better returns than traditional social security. To achieve higher returns, however, contributors are exposed to extra risks associated with financial market fluctuations. This paper offers evidence on the extent of these risks by considering the hypothetical pensions U.S. workers would have obtained between 1911 and 1999 if they had accumulated retirement savings in individual accounts. The 89 hypothetical contributors are assumed to have identical careers and to contribute a fixed percentage of their wages to private investment accounts. Contributors differ only with respect to the stock market returns, bond interest rates, and price inflation they face over their careers. These differences occur because of the differing start and end dates of workers' careers. The analysis demonstrates that returns under private plans would usually have been good, but that financial market risks in a private account system are empirically quite large. Some of these risks are also present in certain types of public retirement system, but a public system has one important advantage over private pensions. Because public social security is backed by the taxing and borrowing authority of the state, it can spread risks over a much larger population of potential contributors and beneficiaries, including contributors and beneficiaries in several generations.