I document a new empirical puzzle in initial public offerings (IPOs): trading volume over the 600 business days following an IPO exhibits a pronounced U-shaped pattern. After the well-known post-IPO drop in turnover, trading activity rises steadily by about 22% per year. This increase cannot be explained by standard drivers of high turnover and is robust to aggregate market trends and to firm size at issuance. I further show that the U-shape is closely linked to lockup agreements, but not through their direct consequences such as insider sales or changes in free float. Instead, the results suggest that lockups shape long-run liquidity dynamics, pointing to persistent effects from investor anticipation, underwriter reputation, or shifts in trading clientele. These findings link IPO contract design to secondary market liquidity and open new questions for asset pricing and market microstructure.

This paper shows that probability weighting can help explain three retirement-saving puzzles at the same time: people save too late, invest too little in stocks, and rarely buy annuities. I study this in a calibrated life-cycle model with labor income risk, stock market risk, mortality risk, and an annuitization choice at retirement. Probability weighting makes distant retirement consumption feel less valuable, makes labor income and equity risk look larger, and makes actuarially fair annuities less attractive. As a result, annuitization falls sharply, from 78% without probability weighting to 4.2% with probability weighting. The model also helps reconcile why people can be optimistic about their life expectancy while still avoiding annuities: LISS survey beliefs show patterns that are consistent with probability weighting. Overall, the results point to probability distortions as an important driver of retirement-saving behavior.