### 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考| A 10-Year P&C ALM Problem

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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|A 10-Year P&C ALM Problem

We present results for a test problem modelling a large $P \& C$ company assumed to manage a portfolio worth 10 billion $€$ (bln) at $t_{0}$. The firm’s management has set a first-year operating profit of 400 million $€(\mathrm{mln})$ and wishes to maximize in expectation the company realized and unrealized profits over 10 years. Company figures have been disguised for confidentiality reasons, though preserving the key elements of the original ALM problem. We name the P\&C company Danni Group.
The following tree structure is assumed in this case study. The current implementable decision, corresponding to the root node, is set at 2 January 2010 (Table 5.4).

Within the model the $P \& C$ management will revise its strategy quarterly during the first year to minimize the expected shortfall with respect to the target.

Initial conditions in the problem formulation include average first-year insurance premiums $R(t)$ estimated at $4.2$ bln $€$, liability reserves $\Lambda(0)$ equal to $6.5$ bln $€$, and expected insurance claims $L(t)$ in the first year of $2.2$ bln $€$. The optimal investment policy, furthermore, is constrained by the following upper and lower bounds relative to the current portfolio value:

• Bond portfolio upper bound: $85 \%$
• Equity portfolio upper bound: $20 \%$
• Corporate portfolio upper bound: $30 \%$
• Real estate portfolio upper bound: $25 \%$
• Desired turnover at rebalanced dates: $\leq 30 \%$
• Cash lower bound: $5 \%$
The results that follow are generated through a set of software modules combining MATLAB $7.4 \mathrm{~b}$ as the main development tool, GAMS $21.5$ as the model generator and solution method provider, and Excel 2007 as the input and output data collector running under a Windows XP operating system with $1 \mathrm{~GB}$ of RAM and a dual processor.

The objective function (5.1) of the P\&C ALM problem considers a tradeoff between short-, medium-, and long-term goals through the coefficients $\lambda_{1}=$ $0.5, \lambda_{2}=0.2$, and $\lambda_{3}=0.3$ at the 10-year horizon. The first coefficient determines a penalty on profit target shortfalls. The second and the third coefficients are associated, respectively, with a medium-term portfolio value decision criterion and long-term terminal wealth. Rebalancing decisions can be taken at decision epochs from time 0 up to the beginning of the last stage; no decisions are allowed at the horizon.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Optimal Investment Policy Under P&C Liability Constraints

The optimal portfolio composition at time 0 , current time, the only one under full uncertainty regarding future financial scenarios, is displayed in Fig. 5.5.

Relying on a good liquidity buffer at time 0 the insurance manager will allocate wealth evenly across the asset classes with a similar portion of long Treasury and corporate bonds. As shown in Fig. $5.6$ the optimal strategy will then progressively

reduce holdings in equity and corporate bonds to increase the investment in Treasuries. A non-negligible real estate investment is kept throughout the 10 years. The allocation in the real estate index remains relatively stable as a result of the persistent company liquidity surplus. Real estate investments are overall preferred to equity investments due to higher expected returns per unit volatility estimated from the 1999 to 2009 historical sample. The optimal investment policy is characterized by quarterly rebalancing decisions during the first year with limited profit-taking operations allowed beyond the first business year. The 3 -year objective is primarily associated with the medium-term maximization of the portfolio asset value; the strategy will maximize unrealized portfolio gains specifically from long Treasury bonds, real estate, and corporate bonds. Over the 10 years the portfolio value moves along the average scenario from the initial 10 bln $€$ to roughly $14.3$ bln $€$.

The optimal portfolio strategy is not affected by the liquidity constraints from the technical side as shown in Table $5.5$. Danni Group has a strong liquidity buffer generated by the core insurance activity, but only sufficient on its own to reach the target profit set at the end of the first business year.

At the year I horizon the investment manager seeks to minimize the profit target shortfall while keeping all the upside. Indeed a $1.1$ billion $€$ gross profit is achieved prior to the corporate tax payment corresponding to roughly $750 \mathrm{mln}$ $€$ net profit. Thanks to the safe operational environment, no pressure is put on the investment side in terms of income generation and the investment manager is free to focus on the maximization of portfolio value at the 3-year horizon. Over the final 7-year stage the portfolio manager can be expected to concentrate on both realized and unrealized profit maximization, contributing to overall firm business growth, as witnessed in Table $5.5$ and Fig. 5.6. Empirical evidence suggests that P\&C optimal portfolio strategies, matching liabilities average life time, tend to concentrate on assets with limited duration (e.g. 1-3 years). We show below that such a strategy without an explicit risk capital constraint would penalize the portfolio terminal value.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Dynamic Asset Allocation with λ1 = 1

Remaining within a dynamic framework in the objective function (5.1) of the $P \& C$ ALM problem we set $\lambda_{1}=1$, with thus $\lambda_{2}=\lambda_{3}=0$ over the 10 -year horizon. The optimal policy will in this case be driven by the year 1 operating profit target, carrying on until the end of the decision horizon.

Diving a different view from Fig. 5.6, we display in Figs. $5.7$ and $5.8$ the optimal strategies in terms of the portfolio’s time-to-maturity structure in each stage. Relative to the portfolio composition in Fig. $5.8$ which corresponds to giving more weight to medium- and long-term objectives, the portfolio in Fig. $5.7$ concentrates from stage 1 on fixed income assets with lower duration. During the first year it shows an active rebalancing strategy and a more diversified portfolio. At the 9-month horizon part of the portfolio is liquidated and the resulting profit will minimize at the year 1 horizon the expected shortfall with respect to the target. Thereafter the strategy, suggesting a buy and hold management approach, will tend to concentrate on those assets that would not expire within the 10 -year horizon.

Consider now Fig. 5.8, recalling that no risk capital constraints are included in the model. The strategy remains relatively concentrated on long bonds and assets without a contractual maturity. Nevertheless the portfolio strategy is able to achieve the first target and heavily overperform over the 10 years. At $T=10$ years the first portfolio in this representative scenario is worth roughly $12.1$ billions $€$, while the second achieves a value of $14.2$ billions $€$.The year 1 horizon is the current standard for insurance companies seeking an optimal risk-reward trade-off typically within a static, one-period, framework. The above evidence suggests that over the same short-term horizon, a dynamic setting would in any case induce a more active strategy and, furthermore, a 10 -year extension of the decision horizon would not jeopardize the short-medium-term profitability of the P\&C shareholder while achieving superior returns in the long term.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|A 10-Year P&C ALM Problem

• 债券组合上限：85%
• 股票投资组合上限：20%
• 企业投资组合上限：30%
• 房地产投资组合上限：25%
• 重新平衡日期的期望营业额：≤30%
• 现金下限：5%
下面的结果是通过一组结合MATLAB的软件模块生成的7.4 b作为主要的开发工具，GAMS21.5作为模型生成器和求解方法提供者，Excel 2007 作为输入和输出数据收集器，在 Windows XP 操作系统下运行1 G乙RAM 和双处理器。

P\&C ALM 问题的目标函数 (5.1) 通过系数考虑了短期、中期和长期目标之间的权衡λ1= 0.5,λ2=0.2， 和λ3=0.3在 10 年的范围内。第一个系数确定对利润目标不足的惩罚。第二和第三个系数分别与中期投资组合价值决策标准和长期终端财富相关联。可以在从时间 0 到最后阶段开始的决策时期进行再平衡决策；不允许在地平线上做出任何决定。

## 广义线性模型代考

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。