### 数学代考|金融数学代考Financial Mathematics代写|The Effect of the Assumed Interest Rate and Smoothing

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• 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 expectation AIR

One may be interested to choose the AIR $\mathrm{a}{0}(\mathrm{~h})$ in such a way that the expected pension payments are constant with respect to $\mathrm{h}$, that is such that $\mathrm{E}{0}\left[\mathrm{~W}_{\mathrm{h}}(\mathrm{h})\right]=\mathrm{W} 0(0)$ (recall that the first pension payment $\mathrm{W} 0(0)$ is without investment risk).

Proposition 2.1. The AIR $A I R \bar{a}{0}(h \mid w)$ that leads to constant expected payments for variable annuities equals $$\bar{a}{0}(h \mid w)=r+w \lambda \sigma .$$
Proof. From (2.11), we find that $\mathrm{E}{0}\left[\mathrm{~W}{\mathrm{h}}(\mathrm{h})\right]=\mathrm{W}{0}(0)$ implies $\frac{W{0}(h)}{W_{0}(0)}=\exp (-h(r+w \lambda \sigma))$,
or (2.14), using (2.8).
This constant AIR leads to nominally constant expected pension payments. In case our financial market would exhibit interest rate risk (that is, a horizondependent risk-free term structure) and/or stock market predictability, we would need horizon-dependent AIRs to obtain expected constant pension payments. We will see that, even in the present financial market,also smoothing financial market returns leads to a horizon-dependent AIR if annuity payments are required to be constant in expectation.

## 数学代考|金融数学代考Financial Mathematics代写|Optimal AIR

For given preferences, we derive the optimal AIR that maximizes the expected utility from all the pension payments subject to the budget constraint of the total available pension wealth. The optimal withdrawal is determined by the optimal allocation strategy $a_{0}^{*}(h \mid w)$. The retiree has to determine how much of his wealth he allocates to each horizon for a given investment strategy w. Thus, a retiree who maximizes the expected utility subject to the budget constraint solves the following optimization problem:
Problem 2.1.
$\max {\left|W{0}(h)\right\rangle} \mathbb{E}{0}\left[\sum{h=0}^{H-1} \exp (-\beta h) u\left(W_{h}(h)\right)\right]$
(2.16)

s.t. $\quad W_{0}=\sum_{h=0}^{H-1} W_{0}(h)$,
(2.17)
where $\beta$ is the subjective discount rate that reflects time preferences, that is, impatience.

Proposition 2.2. The optimal AIR that solves Problem $2.1$ with utility function $(2.4)$ is
$$a_{0}^{*}(h \mid w)= \begin{cases}r+\frac{1}{\gamma}(\beta-r)-\left(\frac{1}{\gamma}-1\right) w \sigma\left(\lambda-\frac{1}{2} \gamma w \sigma\right) & \text { if } \gamma>0, \gamma \neq 1 \ \beta & \text { if } \gamma=1\end{cases}$$
Proof. Using Itô’s lemma and (2.10), we find
$$\mathrm{d} W_{t}(h)^{1-\gamma}=\left(r+w \lambda \sigma-\frac{1}{2} \gamma w^{2} \sigma^{2}\right)(1-\gamma) W_{t}(h)^{1-\gamma} \mathrm{d} t+w \sigma(1-\gamma) W_{t}(h)^{1-\gamma} \mathrm{d} Z_{t} .$$
This leads to the optimization problem
$$\max {\left(W{0}(h)\right)} \sum_{h=0}^{H-1} \exp (-\beta h) \frac{W_{0}(h)^{1-\gamma}}{1-\gamma} \exp \left(\left(r+w \lambda \sigma-\frac{1}{2} \gamma w^{2} \sigma^{2}\right)(1-\gamma) h\right)$$
s.t. $W_{0}=\sum_{h=0}^{H-1} W_{0}(h)$.
The Lagrangian is
\begin{aligned} \mathcal{L}\left(W_{0}(h)\right)=& \sum_{h=0}^{H-1} \exp (-\beta h) \frac{W_{0}(h)^{1-\gamma}}{1-\gamma} \exp \left(\left(r+w \lambda \sigma-\frac{1}{2} \gamma w^{2} \sigma^{2}\right)(1-\gamma) h\right) \ &-\kappa\left(\sum_{h=0}^{H-1} W_{0}(h)-W_{0}\right) \end{aligned}

## 数学代考|金融数学代考Financial Mathematics代写|SMOOTHING FINANCIAL RETURNS

If agents have habit-formation preferences, they may want to reduce yeartoyear volatility in the pension payments. Utility functions of this type capture individuals who receive utility from consumption relative to a habit. It rationalizes the demand for smooth consumption as shown by Abel (1990), Constantinides (1990), Fuhrer (2000), Carroll (2000), Crawford (2010) and Davidoff et al. (2005).

The traditional view to achieve smooth consumption, that is lower yearto-year volatility, is to “smooth” financial market returns. That is, in case portfolio returns are $-20 \%$, instead of reducing the pension payment immediately by $20 \%$, it is only reduced by a fraction, say, $20 \% / 5=4 \%$. This implies that pension payments later in the retirement phase need to be cut by more than $20 \%$ to fulfill the budget constraint. Smoothing thus leads to a smaller year-to-year volatility, but the long-term volatility is larger. We derive the conceptual implications of smoothing in the framework of the discrete pension buckets and show the change in the design via the AIR that generates constant expectations.

The reduced year-to-year volatility can be achieved as follows. Recall that the initial pension payment at time 0 is given by $W_{0}(0)$. In order to have limited risk in the pension payment $\mathrm{W} t(1)$, we do not invest it according to a stock exposure w, as in Section 2 , but with a stock exposure w0(1) = $\mathrm{w} / \mathrm{N}$, where $\mathrm{N}$ denotes the smoothing period, say, $\mathrm{N}=5$ years. Subsequently, the pension wealth $\mathrm{W}{0}(2)$ for the pension payment $\mathrm{W}{2}(2)$ is invested with exposure $\mathrm{w}{0}(2)=2 \mathrm{w} / \mathrm{N}$ the first year and $\mathrm{w}{1}(2)=\mathrm{w} / \mathrm{N}$ the second year. Different smoothing mechanisms can be chosen as long as the exposure is decreased. All results, that is, formulas, below hold for general wj-1(h), which is the stock exposure from year $\mathrm{j}-1$ to $\mathrm{j}$ for the pension wealth that generates the payment in year h. For illustration, we provide figures based on the exposures
$$w_{j-1}(h)=w \min \left{1, \frac{1+h-j}{N}\right}, \quad j=1, \ldots, h,$$

for given smoothing period $\mathrm{N}$ and long-term stock exposure $\mathrm{w}$. Figure 12 shows these stock exposure w
(h) as a function of $j$ for $h=17$.

## 数学代考|金融数学代考Financial Mathematics代写|Optimal AIR

(2.16)

(2.17)

d在吨(H)1−C=(r+在λσ−12C在2σ2)(1−C)在吨(H)1−Cd吨+在σ(1−C)在吨(H)1−Cd从吨.

## 数学代考|金融数学代考Financial Mathematics代写|SMOOTHING FINANCIAL RETURNS

w_{j-1}(h)=w \min \left{1, \frac{1+hj}{N}\right}, \quad j=1, \ldots, h,w_{j-1}(h)=w \min \left{1, \frac{1+hj}{N}\right}, \quad j=1, \ldots, h,

(h) 作为函数j为了H=17.

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