### 数学代考|金融数学代考Financial Mathematics代写|Constant expectations with smoothing

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• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代考|金融数学代考Financial Mathematics代写|Constant expectations with smoothing

The present setting allows for an exact derivation of the AIR that leads to pension payments that have a constant expectation. The AIR $\tilde{a}_{0}(h \mid w)$ leads to a pension payment that is constant in expected nominal terms and is given by the next proposition.

Proposition 3.1. The AIR $\tilde{a}{0}(h \mid w)$ that leads to constant expected payments in nominal terms when financial shocks are smoothed according to (3.1) equals $\tilde{a}{0}(h \mid w)=r+\lambda \sigma \frac{1}{h} \sum_{j=1}^{h} w_{j-1}(h) .$
Proof. With smoothing, the expected nominal pension payment at time $\mathrm{h}$ is given by

$\mathbb{E}{0}\left[W{\sharp}(h)\right]=W_{0}(h) \exp \left(\sum_{i=1}^{h}\left(r+w_{j-1}(h) \lambda \sigma\right)\right) .$
In order to have a constant expected nominal pension payment, we must choose the $\operatorname{AIR} \tilde{a}{0}(h \mid w){\text {such }}$ that this expectation equals $\mathrm{W}{0}(0)$ for all $\mathrm{h}$. Recall, see (2.8), $\frac{W{0}(h)}{W_{0}(0)}=\exp \left(-h a_{0}(h)\right)$.
Thus, together with $\mathbb{E}{0}\left[W{h}(h)\right]=W_{0}(0)$, it follows that a $0(\mathrm{~h} \mid \mathrm{w})$ is implied by $\tilde{a}{0}(h \mid w)$ is implied by the $\mathrm{a}{0}(\mathrm{~h})$ that solves exp $\left(-h a_{0}(h)\right) \exp \left(\sum_{j=1}^{h}\left(r+w_{j-1}(h) \lambda \sigma\right)\right)=1$ for a given exposure w.

The dash-dotted gray line in Figure 13 shows the expected pension payment and the dashed gray lines show the $5 \%$ and $95 \%$ quantiles with smoothing period $\mathrm{N}=5$ years for a stock exposure $\mathrm{w}=35 \%$ and the AIR equal to $\tilde{a}_{0}(h \mid w)$ of Proposition 3.1. The black dotted and black solid lines are obtained without smoothing, similar to Figure 3 .

## 数学代考|金融数学代考Financial Mathematics代写|CONCLUSION

\text { This paper provides analytical expressions for the risk-return trade-off of } variable annuities, with a special focus on the explicit allocation of initial wealth across the “pension buckets” reserved for future payments. The latter is completely determined by the AIR and relates to the consumption portfolio problem. This conceptual division provides useful insights by analyzing different AIRs. We derive the AIR that leads to constant expected pension payments as well as the optimal AIR for an investor with CRRA preferences. The utility loss between these two is small when the risk exposure is optimal, that is for the Merton fraction. We also consider the situation where financial market returns may be smoothed over the remaining retirement period. In order to obtain, in a contract with smoothing, a constant expected nominal pension, the AIR has to be horizon dependent. Other insights that we obtain from investigating the effect of the AIR on the variable annuity is that the allocation’s sensitivity on the payments toward the end of life is large while early or intermediate expected payments are hardly affected by the AIR. Moreover, we show that variable annuities do not solve conversion risk since we find that shocks in the risk-free rate have the same effect for variable and fixed annuities.

## 数学代考|金融数学代考Financial Mathematics代写|ABSTRACT

We study the effects on the macroeconomic equilibrium, the wealth distribution, and welfare of adverse selection in private annuity markets in a closed economy inhabited by overlapping generations of heterogeneous agents who are distinguished by their health status. If an agent’s health type is private information there will be a pooling equilibrium in the private

annuity market. We also study the implications for the macro-economy and welfare of a social security system with mandatory contributions that are constant across health types. These social annuities are immune to adverse selection and therefore offer a higher rate of return than private annuities do. However, they have a negative effect on the steady-state capital intensity and welfare. The positive effect of a fair pooled rate of return on a fixed part of savings and a higher return on capital in equilibrium is outweighed by the negative consequences of increased adverse selection in the private annuity market and a lower wage rate.

Keywords:-Annuity markets, adverse selection, overlapping generations, Demography.

## 数学代考|金融数学代考Financial Mathematics代写|CONCLUSION

\text { 本文提供了 } 可变年金的风险回报权衡的分析表达式，特别关注初始财富在为未来支付保留的“养老金桶”中的显式分配。后者完全由 AIR 决定，与消费组合问题有关。这个概念划分通过分析不同的 AIR 提供了有用的见解。我们推导出导致持续预期养老金支付的 AIR 以及具有 CRRA 偏好的投资者的最佳 AIR。当风险敞口最佳时，即默顿分数，这两者之间的效用损失很小。我们还考虑了金融市场回报可能在剩余退休期间平滑的情况。为了在平滑合同中获得恒定的预期名义养老金，AIR 必须依赖于地平线。我们从调查 AIR 对可变年金的影响中获得的其他见解是，分配对临终付款的敏感性很大，而早期或中期预期付款几乎不受 AIR 影响。此外，我们表明可变年金并不能解决转换风险，因为我们发现无风险利率的冲击对可变年金和固定年金具有相同的影响。

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