In this paper, we consider the risk–return trade-off for variable annuities in a Black–Scholes setting. Our analysis is based on a novel explicit allocation of initial wealth over the payments at various horizons. We investigate the relationship between the optimal consumption problem and the design of variable annuities by deriving the optimal so-called assumed interest rate for an investor with constant relative risk aversion preferences. We investigate the utility loss due to deviations from this. Finally, we show analytically how habit-formation-type smoothing of financial market shocks over the remaining lifetime leads to smaller year-to-year volatility in pension payouts, but to increases in the longer-term volatility.

中文翻译：

---

假定利率和平滑度对可变年金的影响

在本文中，我们考虑了布莱克-斯科尔斯环境下可变年金的风险-收益权衡。我们的分析是基于在各个阶段对付款进行的新颖的初始财富显式分配。我们通过推导具有相对风险厌恶偏好不变的投资者的最优所谓假设利率，来研究最优消费问题与可变年金设计之间的关系。我们调查了由于此而导致的效用损失。最后，我们分析性地显示了在剩余生命周期中金融市场冲击的习惯形成型平滑如何导致养老金支出的年度波动性较小，但长期波动性增大。