### 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考| The ALM Model

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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

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|The ALM Model

The ALM problem is formulated as a linear MSP recourse problem (Birge and Louveaux, 1997; Consigli 2007; Consigli and Dempster, 1998) over six stages. The optimal root node decision is taken at time $t_{0}=0$ and, from a practical viewpoint, represents the key implementable decision. Recourse decisions occur at $t_{1}=0.25$ (after a quarter), $t_{2}=0.5, t_{3}=0.75, t_{4}=1$ year, and $t_{5}=3$ years. The objective function includes a first-year profit shortfall with respect to a target (at $\left.t_{4}\right)$, a 3-year expected excess portfolio value relative to the insurance reserves (at $\left.t_{5}\right)$, and a 10-year wealth objective (at $t_{6}=H$ ):
$$\max {x \in X}\left{-\lambda{1} E\left[\tilde{\Pi}{t 4}-\Pi{t 4} \mid \Pi_{t 4}<\tilde{\Pi}{t 4}\right]+\lambda{2} E\left(X_{t 5}-\Lambda_{t 5}\right)+\lambda_{3} E\left[W_{H}\right]\right}$$
with $\lambda_{1}+\lambda_{2}+\lambda_{3}=1,0 \leq \lambda_{1}, \lambda_{2}, \lambda_{3} \leq 1$.
The first-year horizon reflects the short-term objective to be achieved at the end of the current year: a target net profit is included and for given liabilities and random cash flows over the initial four quarters the model will tend to minimize the expected shortfall with respect to the target (Artzner et al., 1999; Rockafellar and Uryasev, 2002). The intermediate 3 -year objective reflects the maximization of the investment portfolio value above the $\mathrm{P} \& \mathrm{C}$ liabilities. The 10-year expected terminal wealth objective, finally, reflects management’s need to maximize on average the long-term realized and unrealized profits.

The prospective optimal strategy is determined as a set of scenario-dependent decisions to maximize this objective function subject to several constraints. Decision variables include holding, selling, and buying indices in a representative strategic investment universe. We distinguish the set $i \in I_{1}$, including all interest-bearing assets with a specified maturity, from $i \in I_{2}$, which includes real estate and equity assets without an expiry date. The asset universe is $I=I_{1} \cup I_{2}$ :

$x^{+}(i, t, s)$ imestment in stage $t$, scenario $s$, of asset $i$ (with maturity $T_{i}$ for $\left.i \in I_{1}\right)$;
$x^{-}(i, h, t, s)$ selling in stage $t$, scenario $s$, of asset $i$ (with maturity $T_{i}$ for $i \in I_{1}$ ) that was bought in $h$;
$x^{\exp }(i, h, t, s)$ maturity in stage $t$, scenario $s$, of asset $i \in I_{1}$ that was bought in $h$;
$x(i, h, t, s)$ holding in stage $t$, scenario $s$, of asset $i$ (with maturity $T_{i}$ for $i \in I_{1}$ ) that was bought in $h$;
$z(t, s)=z^{+}(t, s)-z^{-}(t, s)$ cash account in stage $t$, scenario $s$.
The investmént epoochs $h=t_{0}, l_{1}, \ldots, t_{n-1} \mathrm{~ a ̈ r e ̀ ~ a ̊ l s o ~ i n t r o ̛ d u c e ́ d ~ i n ~ o ̛ r d e ́ r ~ t o}$ mate the capital gains on specific investments. An extended set of linear constraints will determine the feasibility region of the SP problem (Birge and Louveaux, 1997; Consigli and Dempster, 1998).

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Inventory Balance Constraints

The inventory balance constraints affect the time evolution of the asset portfolio. Given an initial input portfolio the constraints take care of the stage evolution of the optimal strategy. Starting at time 0 we model the market value of the investment at each node along a scenario, for each asset class $i$. Unlike real estate and equity investments, fixed income portfolio holdings are assumed to carry a maturity date. At time 0 an initial input portfolio $x_{i}$, prior to any rebalancing decision, is assumed. We distinguish between the initial time 0 constraints and those for the later stages:
\begin{aligned} x_{i, 0} &=\stackrel{\circ}{x}{i}+x{i, 0}^{+}-x_{i, 0}^{-} \quad \forall i \in I \ X_{0} &=\sum_{i \in I} x_{i, 0} \end{aligned}
For $t=t_{1}, t_{2}, \ldots, H$, all scenarios $s=1,2, \ldots, S$
\begin{aligned} x_{i, h, t_{j}}(s) &=x_{i, h, t_{j-1}}(s)\left(1+\rho_{i, t_{j}}(s)\right)-x_{i, h, t_{j}}^{-}(s)-x_{i, h, t_{j}}(s) & \forall i, h<t_{j}, \ x_{i, t_{j}}(s) &=\sum_{h<t_{j}} x_{i, h, t_{j}}(s)+x_{i, t_{j}}^{+}(s) & \forall i, \ X_{t}(s) &=\sum_{i} x_{i, t}(s) & \end{aligned}
At each stage the market returns $\rho_{i, t, j_{j}}(s)$ for asset $i$ realized in scenario $s$ at time $t_{j}$ will determine the portfolio revaluation paths: previous stage holdings plus buying decisions minus selling and expiry will define the new portfolio position for the following period. Each rebalancing decision, jointly with all cash flows induced by $\mathrm{P} \& \mathrm{C}$ premium renewals and claims, will determine the cash balance evolution up to the horizon.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Cash Balance Constraints

We consider cash outflows due to liability payments, negative interest on cash account deficits, dividend payments to the holding company or to shareholders in
5 Dynamic Portfolio Management for Property and Casualty Insurance
107
the second quarter of each year, corporate taxes, buying decisions, and operating and human resource costs. Cash inflows are due to insurance premiums, equity dividends, fixed income coupon payments, asset expiry (fixed income benchmarks), selling decisions, and interest on cash account surpluses. For $t=0$, given an initial cash balance 2
$$\stackrel{\circ}{z}+\sum_{i} x_{i, 0}^{-}-\sum_{i} x_{i, 0}^{+}+z_{0}^{+}-z_{0}^{-}=0 \quad \forall i \in I .$$
For $t=t_{1}, t_{2}, \ldots, t_{n}$ and $s=1,2, \ldots, s$
$$\begin{gathered} z_{t_{j-1}}^{+}(s)\left(1+\zeta_{t_{j}}^{+}(s)\right)-z_{t_{j-1}}^{-}(s)\left(1+\zeta_{t_{j}}^{-}(s)\right)-z_{t_{j}}^{+}(s)+z_{t_{j}}^{-}(s) \ +\sum_{i \in I_{1}} \sum_{h<t_{j}} x_{i, h, t_{j}}^{-}(s)\left(1+\eta_{i, h, t_{j}}(s)\right)+\sum_{i \in I_{2}} \sum_{h<t_{j}} x_{i, h, t_{j}}^{-}(s) \ +\sum_{i \in I_{1}} \sum_{h \in t_{j}} x_{i_{i, h, t_{j}}}(s)-\sum_{i \in 1} x_{i, t_{j}}^{+}(s)+\sum_{i \in I_{2}} x_{i, t_{j-1}}(s) \delta_{i, t_{j}}(s) \ +R_{t_{j}}(s)-L_{t_{j}}(s)-C_{t_{j}}(s)-D_{t_{j}}^{-}(s)-T_{t_{j}}(s)=0 \end{gathered}$$
Along each scenario, consistent with the assumed tree structure, cash surpluses and deficits will be passed forward to the following stage together with the accrual interest. Very low positive interest rates $\zeta_{t_{j}}^{+}(s)$ and penalty negative interest rates $\zeta_{t_{j}}^{-}(s)$ will force the investment manager to minimize cash holdings over time. The cash surplus at the end of the horizon is part of company terminal wealth.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|The ALM Model

ALM 问题被表述为六个阶段的线性 MSP 追索问题（Birge 和 Louveaux，1997；Consigli 2007；Consigli 和 Dempster，1998）。最优根节点决策是在时间吨0=0并且，从实际的角度来看，它代表了关键的可实施决策。追索决定发生在吨1=0.25（四分之一之后），吨2=0.5,吨3=0.75,吨4=1年，和吨5=3年。目标函数包括相对于目标的第一年利润缺口（在吨4)，相对于保险准备金的 3 年预期超额投资组合价值（在吨5)，以及 10 年的财富目标（在吨6=H):
\max {x \in X}\left{-\lambda{1} E\left[\tilde{\Pi}{t 4}-\Pi{t 4} \mid \Pi_{t 4}<\tilde{ \Pi}{t 4}\right]+\lambda{2} E\left(X_{t 5}-\Lambda_{t 5}\right)+\lambda_{3} E\left[W_{H}\是的是的}\max {x \in X}\left{-\lambda{1} E\left[\tilde{\Pi}{t 4}-\Pi{t 4} \mid \Pi_{t 4}<\tilde{ \Pi}{t 4}\right]+\lambda{2} E\left(X_{t 5}-\Lambda_{t 5}\right)+\lambda_{3} E\left[W_{H}\是的是的}

X+(一世,吨,s)阶段性投入吨， 设想s, 资产一世（随着成熟吨一世为了一世∈一世1);
X−(一世,H,吨,s)阶段性销售吨， 设想s, 资产一世（随着成熟吨一世为了一世∈一世1) 买的H;
X经验(一世,H,吨,s)阶段性成熟吨， 设想s, 资产一世∈一世1那是买的H;
X(一世,H,吨,s)在舞台上举行吨， 设想s, 资产一世（随着成熟吨一世为了一世∈一世1) 买的H;

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Inventory Balance Constraints

\begin{aligned} x_{i, 0} &=\stackrel{\circ}{x} {i}+x {i, 0}^ {+}-x_{i, 0}^{-} \quad \forall i \in I \ X_{0} &=\sum_{i \in I} x_{i, 0} \end{aligned} F这r吨=吨1,吨2,…,H,一种一世一世sC和n一种r一世这ss=1,2,…,小号 X一世,H,吨j(s)=X一世,H,吨j−1(s)(1+ρ一世,吨j(s))−X一世,H,吨j−(s)−X一世,H,吨j(s)∀一世,H<吨j, X一世,吨j(s)=∑H<吨jX一世,H,吨j(s)+X一世,吨j+(s)∀一世, X吨(s)=∑一世X一世,吨(s)

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Cash Balance Constraints

5 财产和意外伤害保险动态投资组合管理
107中考虑了由于负债支付、现金账户赤字的负利息、向控股公司或股东支付的股息、公司税、购买决策和

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