## 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

statistics-lab™ 为您的留学生涯保驾护航 在代写金融中的随机方法Stochastic Methods in Finance方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融中的随机方法Stochastic Methods in Finance方面经验极为丰富，各种代写金融中的随机方法Stochastic Methods in Finance相关的作业也就用不着说。

• 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

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写金融中的随机方法Stochastic Methods in Finance方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融中的随机方法Stochastic Methods in Finance方面经验极为丰富，各种代写金融中的随机方法Stochastic Methods in Finance相关的作业也就用不着说。

• 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|Scenario Generation

The optimal ALM strategy in the model depends on an extended set of asset and liability processes. We assume a multivariate normal distribution for monthly price returns of a representative set of market benchmarks including fixed income, equity, and real estate indices. The following $I=12$ investment opportunities are considered: for $i=1$ : the EURIBOR 3 months; for $i=2,3,4,5$, 6: the Barclays Treasury indices for maturity buckets $1-3,3-5,5-7,7-10$, and $10+$ years, respectively; for $i=7,8,9,10$ : the Barclays Corporate indices, again spanning maturities $1-3,3-5,5-7$, and $7-10$ years; finally for $i=11$ : the GPR Real Estate Europe index and $i=12$ : the MSCI EMU equity index. Recall that in our notation $I_{1}$ includes all fixed income assets and $I_{2}$ the real estate and equity investments.

The benchmarks represent total return indices incorporating over time the securities’ cash payments. In the definition of the strategic asset allocation problem, unlike
5 Dynamic Portfolio Management for Property and Casualty Insuranee
109
real estate and equity investments, money market and fixed income benchmarks are assumed to have a fixed maturity equal to their average duration ( 3 months, 2 years, 4,6 , and $8.5$ years for the corresponding Treasury and Corporate indices and 12 years for the $10+$ Treasury index). Income cashflows due to coupon payments will be estimated ex post through an approximation described below upon selling or expiry of fixed income benchmarks and disentangled from price returns. Equity investments will instead generate annual dividends through a price-adjusted random dividend yield. Finally, for real estate investments a simple model focusing only on price returns is considered. Scenario generation translates the indices return trajectories for the above market benchmarks into a tree process (see, for instance, the tree structure in Fig. 5.3) for the ALM coefficients (Consigli et al., 2010; Chapters 15 and 16 ): this is referred to in the sequel as the tree coefficient process. We rely on the process nodal realizations (Chapter 4 ) to identify the coefficients to be associated with each decision variable in the ALM model. We distinguish in the decision model between asset and liability coefficients.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Asset Return Scenarios

The following random coefficients must be derived from the data process simulations along the specified scenario tree. For each scenario $s$, at $t=t_{0}, t_{1}, \ldots, t_{n}=H$, we define
$\rho_{i, t}(s)$ : the return of asset $i$ at time $t$ in scenario $s$;
$\delta_{i, t}(s)$ : the dividend yield of asset $i$ at time $t$ in scenario $s$;
$\eta_{i, h, t}(s)$ : a positive interest at time $t$ in scenario $s$ per unit investment in asset $i \in I_{1}$, in epoch $h$;
$\gamma_{i, h, t}(s)$ : the capital gain at time $t$ in scenario $s$ per unit investment in asset $i \in I_{2}$ in epoch $h$;
$\zeta_{t}^{+/-}(s):$ the (positive and negative) interest rates on the cash account at time $t$ in scenario $s$.

Price refurns $\rho_{i, t}(s)$ under scenario s are directly computed from the associated benchmarks $V_{i, t}(s)$ assuming a multivariate normal distribution, with $\rho:=$ $\left{\rho_{i}(\omega)\right}, \rho \sim N(\mu, \Sigma)$, where $\omega$ denotes a generic random element, and $\mu$ and $\Sigma$ denote the return mean vector and variance-covariance matrix, respectively. In the statistical model we consider for $i \in I 12$ investment opportunities and the inflation rate. Denoting by $\Delta t$ a monthly time increment and given the initial values $V_{i, 0}$ for $i=1,2, \ldots, 12$, we assume a stochastic difference equation for $V_{i, t}:$
\begin{aligned} \rho_{i, t}(s) &=\frac{V_{i, t}(s)-V_{i, t-\Delta t}(s)}{V_{i, t}-\Delta t(s)}, \ \frac{\Delta V_{i, t}(s)}{V_{i, t}(s)}=\mu \Delta t+\Sigma \Delta W_{t}(s) . \end{aligned}

In $(5.12), \Delta W_{t} \sim N(0, \Delta t)$. We show in Tables $5.2$ and $5.3$ the estimated statistical parameters adopted to generate through Monte Carlo simulation (Consigli et al., 2010 ; Glasserman, 2003 ) the correlated monthly returns in (5.12) for each benchmark $i$ and scenario $s$. Monthly returns are then compounded according to the horizon partition (see Fig. 5.1) following a prespecified scenario tree structure. The return scenarios in tree form are then passed on to an algebraic language deterministic model generator to produce the stochastic program deterministic equivalent instance (Consigli and Dempster, 1998).

Equity dividends are determined independently in terms of the equity position at the beginning of a stage as $\sum_{i \in l_{2}} x_{i, f_{j-1}}(s) \delta_{i, t_{j}}(s)$, where $\delta_{i, t_{j}}(s) \sim N(0.02,0.005)$ is the dividend yield.

Interest margin $M_{t_{j}}(s)=I_{t_{j}}^{+}(s)-I_{t_{j}}^{-}(s)$ is computed by subtracting negative from positive interest cash flows. Negative interest $I_{t_{j}}^{-}(s)=z_{t_{j-1}}^{-}(s) \zeta_{t_{j}}^{-}(s)$ is generated by short positions on the current account according to a fixed $2 \%$ spread over the Euribor 3 -month rate for the current period. The positive interest rate for cash surplus is fixed at $\zeta^{+}(t, s)=0.5 \%$ for all $t$ and $s$.

Positive interest $I_{t_{j}}^{+}(s)$ is calculated from return scenarios of fixed income investments $i \in I_{2}$ using buying-selling price differences assuming initial unit investments; this may be regarded as a suitable first approximation. In the case of negative price differences, the loss is entirely attributed to price movements and the interest accrual is set to 0 .
We recognize that this is a significant simplifying assumption. It is adopted here to avoid the need for an explicit yield curve model, whose introduction would go beyond the scope of a case study. Nevertheless we believe that this simplification allows one to recognize the advantages of a dynamic version of the portfolio management problem.

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

Annual premium renewals by P\&C policyholders represent the key technical income on which the company relies for its ongoing business requirements. Premiums have associated random insurance claims over the years and, depending on the business cycle and the claim class, they might induce different reserve requirements. The latter constitute the insurance company’s most relevant liability. In our analysis premiums are assumed to be written annually and will be stationary over the 10-year horizon with limited volatility. Insurance claims are also assumed to remain stable over time in nominal terms, but with an inflation adjustment that in the long term may affect the company’s short-term liquidity. For a given estimate of insurance premiums $R_{0}$ collected in the first year, along scenario $s$, for $t=t_{1}, t_{2}, \ldots, H$, with $e_{r} \sim N(1,0.03)$, we assume that
$$R_{t j}(s)=\left[R_{t j-1}(s) \cdot e_{r}\right] .$$
Insurance claims are assumed to grow annually at the inflation rate and in nominal terms are constant in expectation. For given initial liability $L_{0}$, with $e_{l} \sim$ $N(1,0.01)$
$$L_{t_{j}}(s)=\left(L_{t_{j-1}}(s) \cdot e t\right)\left(1+\pi_{t_{j}}(s)\right) .$$
Every year the liability stream is assumed to vary in the following year according to a normal distribution with the previous year’s mean and a $1 \%$ volatility per year with a further inflation adjustment. Given $\pi(0)$ at time 0 , the inflation process $\pi_{t}(s)$ is assumed to follow a mean-reverting process

$$\Delta \pi_{t}(s)=\alpha_{\pi}\left(\mu-\pi_{t}(s)\right) \Delta t+\sigma_{\pi} \Delta W_{t}(s)$$
with $\mu$ set in our case study to the $2 \%$ European central bank target and $\Delta W_{t} \cdots$ $N(0, \Delta t)$. As for the other liability variables technical resenves are computed as a linear function of the current liability as $\Lambda_{t}(s)=L_{t}(s) \cdot \lambda$, where $\lambda$ in our case study is approximated by $1 / 0.3$ as estimated by practitioner opinion.

Operational costs $C_{t_{j}}(s)$ include staff and back-office and are assumed to increase at the inflation rate. For a given initial estimate $C_{0}$, along each scenario and stage, we have
$$C_{t_{j}}(s)=C_{t_{j-1}}(s) \cdot\left(1+\pi_{t_{j}}(s)\right)$$
Bad liability scenarios will be induced by years of premium reductions associated with unexpected insurance claim increases, leading to higher reserve requirements that, in turn, would require a higher capital base. A stressed situation is discussed in Section $5.4$ to emphasize the flexibility of dynamic strategies to adapt to bad technical scenarios and compensate for such losses. The application also shows that within a dynamic framework the insurance manager is able to achieve an optimal trade-off between investment and technical profit generation across scenarios and over time, and between risky and less risky portfolio positions.

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

5 财产险和意外险的动态投资组合管理
109

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Asset Return Scenarios

ρ一世,吨(s): 资产回报一世有时吨在情景中s;
d一世,吨(s): 资产的股息收益率一世有时吨在情景中s;

C一世,H,吨(s)：当时的资本收益吨在情景中s单位资产投资一世∈一世2在时代H;
G吨+/−(s):现金账户当时的（正和负）利率吨在情景中s.

ρ一世,吨(s)=五一世,吨(s)−五一世,吨−Δ吨(s)五一世,吨−Δ吨(s), Δ五一世,吨(s)五一世,吨(s)=μΔ吨+ΣΔ在吨(s).

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

R吨j(s)=[R吨j−1(s)⋅和r].

C吨j(s)=C吨j−1(s)⋅(1+圆周率吨j(s))

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 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

statistics-lab™ 为您的留学生涯保驾护航 在代写金融中的随机方法Stochastic Methods in Finance方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融中的随机方法Stochastic Methods in Finance方面经验极为丰富，各种代写金融中的随机方法Stochastic Methods in Finance相关的作业也就用不着说。

• 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

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考| Dynamic Portfolio Management for Property

statistics-lab™ 为您的留学生涯保驾护航 在代写金融中的随机方法Stochastic Methods in Finance方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融中的随机方法Stochastic Methods in Finance方面经验极为丰富，各种代写金融中的随机方法Stochastic Methods in Finance相关的作业也就用不着说。

• 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|Casualty Insurance

Increasing competition in the insurance sector in developed countries, and more recently record property and casualty $(\mathrm{P \& C}$ ) insurance claims reported by global players (CEA, 2010; Europe Economics, 2010), has gencrated remarkable

pressures on the financial stability of $\mathrm{P} \& \mathrm{C}$ divisions within insurance firms, leading to increased technical reserves and requiring a higher capital base (European Parliament, 2009). At the same time we have witnessed a remarkable expansion of investment management divisions, reinforcing the role of insurers as institutional investors competing in fixed income and equity markets with other players such as pension and mutual funds. Increasing market volatility in the last few years has, as a result, affected large insurers’ market risk exposure. Regulatory bodies, through the Solvency II regulation (European Parliament, 2009; ISVAP, 2010), have supported risk-based capital allocation measures for insurance firms as a whole. As a response large insurance companies have pursued restructuring aimed from an operational perspective at integrating the historical insurance business with the investment management business. Such an integration is also motivated by the perceived role of the $\mathrm{P} \& \mathrm{C}$ division as a liquidity buffer for the cash deficits generated by fixed income portfolios typically held by risk-averse investment divisions. A trade-off between safe liquidity conditions, key to shareholders’ short-mediumterm returns, and long-term business sustainability has emerged and lead to strategies based on long-term horizons. This motivates the adoption of a multistage stochastic programming (MSP) problem formulation (Birge and Louveaux, 1997; Cariño et al., 1994; Consigli and Dempster, 1998; Mulvey and Erkan, 2005; Zenios and Ziemba, 2007a) able to capture both short- and long-term goals. Contrary to current standards in insurance-based investment divisions which largely rely on one-period static approaches (de Lange et al., 2003; Mulvey and Erkan, 2003; Zenios and Ziemba, 2007b), the adoption of dynamic approaches allows both the extension of the decision horizon and a more accurate short-term modelling of $P \& C$ variables.

In this chapter we present an asset-liability management (ALM) problem integrating the definition of an optimal asset allocation policy over a 10-year planning horizon with the inclusion of liability constraints generated by an ongoing P\&C business (de Lange et al., 2003; Dempster et al., 2003; Mulvey and Erkan, 2005; Mulvey et al., 2007). Relying on a simplified P\&C income statement we clarify the interaction between the investment and the classical insurance business and introduce an objective function capturing short-, medium-, and long-term goals within a multistage model. The planning horizon is divided in six time periods: four quarters in the first year and then 2 and 7 years to reach the 10 -year horizon. Over this period alternative insurance and financial scenarios will affect the insurance management optimal forward policy. We show that integrated management of the insurance liability and asset portfolios is required to protect firm solvency in the presence of unexpectedly increased P\&C claims. A relatively simple multivariate Gaussian return model is adopted to generate return scenarios (Dupačová et al., 2001) for a realistic investment universe including fixed income, equity, and real estate investment opportunities.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|P&C Income Generation and Asset Allocation

We consider an insurance company holding a property and casualty liability portfolio with annual issuance of insurance contracts and a diversified asset portfolio spanning fixed income, real estate, and equity markets. The insurance management sets a target for the end-of-the-year operating profit on an annual basis and seeks an optimal trade-off between this short-term goal, medium-term industrial plan targets, and a (10-year) long-term sustainability goal. The company’s financial soundness will depend on the solidity of both the $\mathrm{P} \& \mathrm{C}$ business division, hereafter also referred to as the technical division, and the imvestment division. In a simplified framework, with stable insurance provisions and reserves, we assume that technical profitability primarily depends on collected annual premiums, operating and human resource costs, and, crucially, recorded insurance claims. We assume throughout that the company will fix insurance premiums so as to maintain its market position within a highly competitive market and that the operational efficiency with minimal operating, staff and administrative costs is maintained over time exogenously to the optimal financial management problem. In this setting alternative insurance claim scenarios are likely from 1 year to the next to heavily affect the company technical profitability. Increasing technical provisions may, on the other hand, weaken the capital base. This is indeed the risk faced currently by several global players in the insurance sector (Europe Economics, 2010), with record claims in the automotive and increasingly the real estate sector, and regarding catastrophic events (CAT risk events). Under such conditions the management will search for an asset portfolio strategy able to preserve the firm’s liquidity in the short term and its overall sustainability in the long term. Only the first goal ean be aecommodated within

a static 1-year decision problem. The investment profit depends on realized price gains from trading, dividends, and interest cash flows generated by the asset portfolio; see Table 5.1. Risky investment strategies may very well generate sufficient liquidity and financial profit over I year but then lead to heavy long-term losses, jeopardizing the company’s market position and solvency. Revenues and costs, as shown in Table $5.1$, will affect annually the firm’s profitability and are considered in the ALM model formulation.

The P\&C annual operating profit does not consider the portfolio’s gain and losses, which, if unrealized, reflect the asset portfolio’s revaluation over time (and can translate into current realized gains or losses if required). If actually realized portfolio profits and losses are nevertheless considered outside the core insurance activity. Net of future liabilities, the maximization of the investment portfolio expected value, can thus be considered a medium-term goal to be pursued by the management. Realized investment and technical profits, together with unrealized gains, will eventually jointly determine the business long-term sustainability reflected in the 10-year target. We assume a 10-year decision horizon with the first year divided into four quarters, a subsequent 2 -year stage and a final 7 -year stage as shown in Fig. 5.1.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Asset–Liability Management for P&C Companies

A generic P\&C ALM problem relies on the specification by the insurance firm’s actuarial division of the reserves and expected cash flows from premiums and insurance claims. In a multistage approach such inputs provide a first-year average scenario to depart from in order to assess, assuming ongoing business, the effects of these departures on longer term technical scenarios and to find an optimal asset management strategy under exogenous liquidity constraints. This set of scenario-dependent variables reflects the uncertainty faced by management in a dynamic framework. As shown in Fig. $5.1$ the initial model decision at time $t=0$ is the only one taken under full uncertainty, while subsequent decisions will all depend on previous decisions and future residual uncertainty. Such uncertainty is modelled through a scenario tree (Consigli et al., 2010; Dupačová et al., 2001; Kouwenberg, 2001; Pflug and Römisch, 2007) such as the one shown schematically in Fig. 5.3: every path, from the root node to a leaf node at the 10 -year horizon represents a scenario.

The stochastic programming formulation relies on the specification of the underlying random factors as tree processes endowed with a given branching structure. In the formulation of the ALM problem we denote by $R(t, s)$ the insurance premiums collected in stage $t$ under scenario $s$. Premiums represent the fundamental cash flows generated every year by the technical business. For given P\&C contract renewals, the insurance actuarial division will periodically revise its estimate on forthcoming claims $L(t, s)$ and required statutory and discretional reserves $\Lambda(t, s)$. These quantities for given human resources and administrative costs $C(t, s)$ will determine the behaviour of the company’s combined ratio: this is the ratio of all insurance

claims and technical costs to the earned premiums. A bad liability scenario will thus be associated with increasing insurance claims at a time of reducing premium renewals and operational inefficiency. Consequently the ex post combined ratio will increase and may go over the $100 \%$ threshold. The firm’s investment profit, instead, is generated by realized capital gains $G(t, s)$ in stage $t$, scenario $s$; the interest margin, defined as the difference between positive and negative interest cash flows, $M(t, s)=I^{+}(t, s)-I^{-}(t, s)$; and by dividends $D(t, s)$ and other financial revenues $Y(t, s)$.

The annual income $\Pi(t, s)$ is determined by the sum of technical and investment profit. The net income, here assumed entirely distributed, is derived by subtracting the corporate tax $T$ from the profit $D^{-}(t, s)=\Pi(t, s)(1-T)$.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|P&C Income Generation and Asset Allocation

P\&C 年度营业利润不考虑投资组合的损益，如果未实现，则反映资产组合随时间的重估（如果需要，可以转化为当前已实现的损益）。如果实际实现的投资组合利润和损失仍然被视为核心保险活动之外。因此，扣除未来负债后，投资组合预期价值的最大化可以被视为管理层追求的中期目标。已实现的投资和技术利润，以及未实现的收益，最终将共同决定10年目标所体现的业务长期可持续性。我们假设一个 10 年的决策期限，第一年分为四个季度，随后是 2 年阶段和最后 7 年阶段，如图 5.1 所示。

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写金融中的随机方法Stochastic Methods in Finance方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融中的随机方法Stochastic Methods in Finance方面经验极为丰富，各种代写金融中的随机方法Stochastic Methods in Finance相关的作业也就用不着说。

• 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|Constraints

The portfolio composition is optimized under various restrictions. The model includes the classical inventory balance constraints on the nominal amount invested in each bond, for each node of the scenario tree:
$$\begin{gathered} x_{i 0}=\bar{x}{i}+x{i 0}^{+}-x_{i 0}^{-} \quad \forall i \in I \ x_{i n}=x_{i a_{n}}+x_{i n}^{+}-x_{i n}^{-} \quad \forall i \in I, \quad \forall n \in N \end{gathered}$$
The cash balance constraint is imposed for the first and later stages:
\begin{aligned} &\sum_{i \in I} v_{i 0} x_{i 0}^{+}\left(1+\chi^{+}\right)+z_{0}-g_{0} \ &=C_{0}-\bar{g}+\sum_{i \in I} v_{i 0} x_{i 0}^{-}\left(1-\chi^{-}\right) \ &\sum_{i \in I} v_{i n} x_{i n}^{+}\left(1+x^{+}\right)+z_{n}-g_{n} \ &=\sum_{i \in I} v_{i n} x_{i n}^{-}\left(1-x^{-}\right)+z_{a_{n}} e^{r_{e_{n}}} \ &-g_{a_{n}} e^{b_{a_{n}}}+\sum_{i \in I} f_{i n} x_{i a_{n}} \quad \forall n \in \mathcal{N} \end{aligned}
P. Bcraldi et al.
84
The model also includes constraints bounding the amount invested in each rating class (4.19) as well as in investment grade (4.20) and speculative grade (4.21) classes, respectively, as fractions of the current portfolio value:
$$\begin{gathered} \sum_{i \in I_{k}} v_{i n} x_{i n} \leq v_{k} \sum_{i \in I} v_{i n} x_{i n} \quad \forall k \in K, \quad \forall n \in \mathcal{N}, \ \sum_{k=0}^{4} \sum_{i \in I_{k}} v_{i n} x_{i n} \leq \phi \sum_{i \in I} v_{i n} x_{i n} \quad \forall n \in \mathcal{N}, \ \sum_{k=5}^{7} \sum_{i \in I_{k}} v_{i n} x_{i n} \leq \zeta \sum_{i \in I} v_{i n} x_{i n} \quad \forall n \in \mathcal{N} \end{gathered}$$
Finally, a limit on the debt level for each node of the scenario tree is imposed:
$$g_{n} \leq \gamma\left(C_{0}+\sum_{i \in I} v_{i 0} \bar{x}_{i}-\bar{g}\right) \quad \forall n \in \mathcal{N}$$

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

The goal of financial planning is twofold: maximize the expected wealth generated by the investment strategy while controlling the market and credit risk exposure of the portfolio. This trade-off can be mathematically represented by adopting a risk-reward objective function:
$$\max (1-\alpha) E\left[\mathcal{W}{n}\right]-\alpha \theta{c}\left[\mathcal{W}{n}\right]$$ where $\alpha$ is a user-defined parameter accounting for the risk aversion attitude and $n$ are the leaf nodes $\left(n \in \mathcal{N}{T}\right)$ of the scenario tree. The higher the $\alpha$ the more conservative, but also the less profitable, the suggested financial plan. The first term of (4.23) denotes the expected value of terminal wealth, computed as
$$E\left[W_{n}\right]=\sum_{n \in N_{T}} p_{n} W_{n}$$
where the wealth at each node $n$ is
$$\mathcal{W}{n}=\sum{i \in l} v_{i n} x_{i n}+z_{n}-g_{n}$$
The second term in (4.23) accounts for risk. In particular, we have considered the conditional value at risk (CVaR) at a given confidence level $\epsilon$ (usually $95 \%$ ). CVaR measures the expected value of losses exceeding the value at risk (VaR). It is a “coherent” risk measure, suitable for asymmetric distributions and thus able to control the downside risk exposure. In addition, it enjoys nice computational properties (Andersson et al. 2001; Artzner et al. 1999; Rockafellar and Uryasev 2002) and

admits a simple linear reformulation. In tree notation the CVaR of the portfolio terminal wealth can be defined as
$$\theta_{c}=\xi_{c}+\frac{1}{1-\epsilon} \sum_{n \in \mathcal{N}{T}} p{n}\left[L_{n}-\xi_{c}\right]{+\uparrow}$$ where $\xi{e}$ denotes the VaR at the same confidence level. Here $L_{n}$ represents the loss at node $n$, measured as the negative deviation from a given target value of the portfolio terminal wealth:
$$L_{n}=\max \left[0, \tilde{W}{n}-\mathcal{W}{n}\right]$$
where $\tilde{W}{n}$ represents a reference value, computed on the initial wealth l-year compounded value for given current (known) risk-free rate: $$\begin{gathered} \mathcal{W}{0}=\left(C_{0}+\sum_{i \in l} v_{i} \bar{x}{i}-\bar{g}\right), \ \tilde{W}{n}=\mathcal{W}{0} e^{r{0} f_{n}} \quad \forall n \neq 0 . \end{gathered}$$
The overall objective function can be linearized through a set of auxiliary variables $\left(\zeta_{n}\right)$ and the constraints as follows:
$$\begin{gathered} \theta_{c}=\xi_{c}+\frac{1}{1-\epsilon} \sum_{n \in \mathcal{N}{T}} p{n} \zeta_{n} \ \zeta_{n} \geq L_{n}-\xi_{c} \quad \forall n \in \mathcal{N}{T} \ \zeta{n} \geq 0 \quad \forall n \in \mathcal{N} T \end{gathered}$$
This model specification leads to an easily solvable large-scale linear multi-stage stochastic programming problem. Decisions at any node explicitly depend on the corresponding postulated realization for the random variables and have a direct impact on the decisions at descendant nodes. Depending on the number of nodes in the scenario tree, the model can become very large calling for the use of specialized solution methods.

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

The bond portfolio model implementation requires the definition of an extended set of random coefficients (Dupačová et al. 2001; Heitsch and Rōmisch 2003; Pflug 2001). We focus in this section on the relationship between the statistical model

actually implemented to develop the case study and the set of scenario-dependent coefficients included in the stochastic programming formulation.

The statistical model drives the bond returns over the planning horizon $0, \ldots, T$. At time 0 all coefficients in $(4.2),(4.3),(4.4),(4.5),(4.7)$, and (4.8) have been estimated and, for a given initial portfolio, Monte Carlo simulation can be used to estimate the credit risk exposure of the current portfolio. In this application a simple random sampling algorithm for a fixed, pre-defined, tree structure is adopted to define the event tree structure underlying the stochastic programming formulation.

A method of moments (Campbell et al. 1997) estimation is first performed on $(4.2)$, (4.3), and (4.4) from historical data, then Moody’s statistics (Moody’s Investors Service 2009 ) are used to estimate the natural default probability and the recovery rates, which are calibrated following $(4.7)$, to account for recent market evidence and economic activity, and (4.8), allowing a limited dispersion from the average class-specific recovery rates.

In the case study implementation the risk-free rate and the credit spread processes are modeled as correlated square root processes according to the tree specification, for $s \in S, t=1, \ldots, T, n \in \mathcal{N}{t}$, where $h{n}$ denotes the child node in the given scenario $s$. For each $k \in K$ and initial states $r_{0}$ and $\pi_{0}^{k}$, we have
$$\begin{gathered} \Delta r_{n}=\mu^{r}\left(t_{h_{n}}-t_{n}\right)+\sigma^{r} \sqrt{r_{n}} \sqrt{t_{h_{n}}-t_{n}} e_{n} \ \Delta \pi_{n}^{k}=\mu^{k}\left(t_{h_{n}}-t_{n}\right)+\sigma^{k} \sqrt{\pi_{n}^{k}} \sqrt{t_{h_{n}}-t_{n}} \sum_{l \in K} q_{n}^{k l} e_{n}^{l} \end{gathered}$$
In $(4.33), e_{n}^{l} \sim N(0,1)$, for $l=0,1, . ., 7$, independently and $q^{k l}$ denote the Choleski coefficients in the lower triangular decomposition of the correlation matrix. The nodal realizations of the risk-free rate and the credit spreads are also used to identify the investor’s borrowing rate $b_{n}=r_{n}+\pi_{n}^{k}$ in the dynamic model implementation, where $\bar{k}$ denotes the investors specific rating class.

The incremental spread $\eta_{n}^{i}$ for security $i$ has been implemented in the case study as a pure jump-to-default process with null mean and volatility. The associated idlosyncratic tree processes, all independent from each other, will in this case for all $i$ follow the dynamic
$$d \eta_{n}^{i}=\beta_{n}^{i} d \Psi^{i}\left(\lambda^{i}, n\right)$$

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

X一世0=X¯一世+X一世0+−X一世0−∀一世∈一世 X一世n=X一世一种n+X一世n+−X一世n−∀一世∈一世,∀n∈ñ

∑一世∈一世v一世0X一世0+(1+χ+)+和0−G0 =C0−G¯+∑一世∈一世v一世0X一世0−(1−χ−) ∑一世∈一世v一世nX一世n+(1+X+)+和n−Gn =∑一世∈一世v一世nX一世n−(1−X−)+和一种n和r和n −G一种n和b一种n+∑一世∈一世F一世nX一世一种n∀n∈ñ
P. Bcraldi 等人。
84

∑一世∈一世到v一世nX一世n≤v到∑一世∈一世v一世nX一世n∀到∈到,∀n∈ñ, ∑到=04∑一世∈一世到v一世nX一世n≤φ∑一世∈一世v一世nX一世n∀n∈ñ, ∑到=57∑一世∈一世到v一世nX一世n≤G∑一世∈一世v一世nX一世n∀n∈ñ

Gn≤C(C0+∑一世∈一世v一世0X¯一世−G¯)∀n∈ñ

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

(4.23) 中的第二项说明了风险。特别是，我们考虑了给定置信水平下的条件风险值 (CVaR)ε（通常95%）。CVaR 衡量损失的预期价值超过风险价值 (VaR)。它是一种“连贯”的风险度量，适用于非对称分布，因此能够控制下行风险敞口。此外，它还具有很好的计算特性（Andersson et al. 2001; Artzner et al. 1999; Rockafellar and Uryasev 2002）和

θC=XC+11−ε∑n∈ñ吨pn[大号n−XC]+↑在哪里X和表示相同置信水平下的 VaR。这里大号n表示节点的损失n，测量为与投资组合终端财富给定目标值的负偏差：

θC=XC+11−ε∑n∈ñ吨pnGn Gn≥大号n−XC∀n∈ñ吨 Gn≥0∀n∈ñ吨

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

Δrn=μr(吨Hn−吨n)+σrrn吨Hn−吨n和n Δ圆周率n到=μ到(吨Hn−吨n)+σ到圆周率n到吨Hn−吨n∑一世∈到qn到一世和n一世

d这n一世=bn一世dΨ一世(λ一世,n)

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考| Price–Yield Relationship

statistics-lab™ 为您的留学生涯保驾护航 在代写金融中的随机方法Stochastic Methods in Finance方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融中的随机方法Stochastic Methods in Finance方面经验极为丰富，各种代写金融中的随机方法Stochastic Methods in Finance相关的作业也就用不着说。

• 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|Price–Yield Relationship

Following the intensity specification in (4.7) an individual security can at any time be affected by a jump-lo-default event attributed by the arrival of new information that will drive the security into a default state. The relationship between the incremental spread behavior and the security price is as follows. Let the value of the default process $d \Psi_{\tau}^{i}=1$ at a random time $\tau$. Then, given a recovery value $\beta^{i}$, we have

$$v_{\tau, T_{i}}^{i}=\sum_{\tau<m \leq T_{i}} c_{m}^{i} e^{-\beta_{i}^{i}} e^{-\left(r_{z}+\pi_{t}^{k_{i}}\right)(m-\tau)}$$
At the default announcement, the bond price will reflect the values of all payments over the bond residual life discounted at a flat risky discount rate including the risk-free interest rate and the class $k_{i}$ credit spread after default. Widening credit spreads across all the rating classes will push default rates upward without discriminating between corporate bonds within each class. On the other hand, once default is triggered at $\tau$ then the marginal spread upward movement will be absorbed by a bond price drop which is consistent with the postulated recovery rate.

Following $(4.5)$ the price movement will depend on the yield jump induced by $\eta_{t}^{i}$ This is assumed to be consistent in the mean with rating agencies recovery estimates (Moody’s Investors Service 2009 ) and carries a user-specified uncertainty reflected in a lognormal distribution with stochastic jump size $\beta^{i} \sim \ln N\left(\mu_{\beta^{i}}, \sigma_{\beta^{i}}\right)$. Then $\ln \left(\beta^{i}\right) \sim N\left(\mu_{\beta^{i}}, \sigma_{\beta^{i}}\right)$ so that $\beta_{t}^{i} \in(0,+\infty)$ and both $e^{-\beta^{i}}$ (the recovery rate) and $1-e^{-\beta^{i}}$ (the loss rate) belong to the interval $(0,1)$.

In summary the modeling framework has the following characteristic features:

• default arrivals and loss upon default are defined taking into account ratingspecific estimates by Moody’s $(2009)$ adjusted to account for a borrower’s sector of activity and market-default premium;
• over a l-year investment horizon default intensities are assumed constant and will determine the random arrival of default information to the market; upon default the marginal spread will suffer a positive shock immediately absorbed by a pricenegative movement reflecting the constant partial payments of the security over its residual life;
• the Poisson process and the Wiener processes drive the behavior of an independent ideosyncratic spread process which is assumed to determine the security price dynamics jointly with a short rate and a credit spread process;
• the credit spreads for each rating class and the short interest rate are correlated.
The model accommodates the assumption of correlated risk factors whose behavior will drive all securities across the different rating categories, but a specific default risk factor is included to differentiate securities behavior within each class. No contagion phenomena are considered and the exogenous spread dynamics impact specific bond returns depending on their duration and convexity structure.

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

We now extend the statistical model to describe the key elements affecting the portfolio value dynamics over a finite time horizon $T$. Assuming an investment universe of fixed income securities with annual coupon frequency we will have only one possible coupon payment and capital reimbursement over a l-year horizon. Securities may default and generate a credit loss only at those times. We assume

a wealth equation defined at time 0 by an investment portfolio value and a cash account $\mathcal{W}{0}=\sum{k \in K} \sum_{i \in I_{k}} X_{i 0}+C_{0}$, where $X_{i 0}$ is the time-0 market value of the position in security $i$, generated by a given nominal investment $x_{i 0}$ and by the current security price $v_{0, T_{i}}^{i}$. For $t=0, \Delta t, \ldots, T-\Delta t$, we have
$$\mathcal{W}{t+\Delta t, \omega t}=\sum{k \in K} \sum_{i \in h_{h}} x_{i t} v_{i, T_{i}}^{i}\left(1+\rho_{t+\Delta t}^{i}(\omega)\right)+C(t+\Delta t, \omega)$$
In (4.9) the price return $\rho_{t+\Delta t}^{i}$ is generated for each security according to (4.5) as $\frac{v^{i}(t+\Delta t, \omega)}{v^{i}(t)}-1=\left[-\delta_{t}^{i} d y_{t}^{i}+y_{t}^{i} d t+0.5 \gamma_{t}^{i}\left(d y_{t}^{i}\right)^{2}\right]$, where $y_{t}^{i}$ represents the yield return at time $t$ for security $i$. For $k=0$ we have the default-free class and the yield will coincide with the risk-free yield. As time evolves an evaluation of credit losses is made. Over the annual time span as before, we have
\begin{aligned} C(t&+\Delta t, \omega)=C(t) e^{r_{i}(\omega) \Delta t}+\sum_{k \in K} \sum_{i \in I_{k}} c_{i, t+\Delta t}+\ &-\sum_{k \in K} \sum_{i \in l_{k}} c_{i, t+\Delta t}\left(1-e^{\left[-\beta_{t+\Delta t}^{i}(\omega) d W_{i+\Delta t}^{i}(\omega)\right]}\right) . \end{aligned}
The cash balance $C(t)$ evolution will depend on the expected cash inflows $c_{i, t}$ and the associated credit losses for each position $i$, the third factor of (4.10). Coupon and capital payments in (4.10) are generated for given nominal investment by the security-specific coupon rates and expiry date. If a default occurs then, for a given recovery rate, over the planning horizon the investor will face both a cash loss in (4.10) and the market value loss of (4.9) reflecting an assumption of constant partial payments over the security residual time span within the default state.

Equation (4.10) considers payment defaults explicitly. However, the mathematical model introduced below considers an implicit default loss definition. This is obtained by introducing the coefficient specification
$$f_{i, t+\Delta t}(\omega)=c_{i}\left(e^{-\left[\beta_{i+\Delta t}^{i}(\omega) d \Psi_{l+\Delta i^{i}}^{i}(\omega)\right]}\right)$$
and, thus, $c_{i, t+\Delta t}=x_{i t} f_{i, s+\Delta t}$, where $c_{i}$ denotes the coupon rate on security $i$.
In this framework the corporate bond manager will seek an optimal dynamic strategy looking for a maximum price and cash return on his portfolio and at the same time trying to avnid possihle defanlt losses. The model is quite general and alternative specifications of the stochastic differential equations (4.2), (4.3), and (4.4) will lead to different market and loss scenarios accommodating a wide range of possible wealth distributions 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|Dynamic Portfolio Model

In this section we present a multistage portfolio model adapted to formulate and solve the market and credit risk control problem. It is widely accepted that stochastic programming provides a very powerful paradigm in modeling all those applications characterized by the joint presence of uncertainty and dynamics. In the area of portfolio management, stochastic programming has been successfully applied as witnessed by the large number of scientific contributions appearing in recent decades (Bertocchi et al. 2007; Dempster et al. 2003; Hochreiter and Pflug 2007; Zenios and Ziemba 2007). However, few contributions deal with the credit and market risk portfolio optimization problem within a stochastic programming framework. Here we mention (Jobst et al. 2006) (see also the references therein) where the authors present a stochastic programming index tracking model for fixed income securities.
Our model is along the same lines, however, introducing a risk-reward trade-off in the objective function to explicitly account for the downside generated by corporate defaults.

We consider a 1 -year planning horizon divided in the periods $t=0, \ldots, T$ corresponding to the trading dates. For each time $t$, we denote by $\mathcal{N} t$ the set of nodes at stage $t$. The root node is labeled with 0 and corresponds to the initial state. For $t \geq 1$ every $n \in \mathcal{N}{t}$ has a unique ancestor $a{n} \in \mathcal{N}{t-1}$, and for $t \leq T-1$ a non-empty set of child nodes $H{n} \in \mathcal{N}{t+1}$. We denote by $\mathcal{N}$ the whole set of nodes in the scenario tree. In addition, we refer by $t{n}$ the time stage of node $n$. A scenario is a path from the root to a leaf node and represents a joint realization of the random variables along the path to the planning horizon. We shall denote by $S$ the set of scenarios. Figure $4.1$ depicts an example scenario tree.

The scenario probability distribution $\mathcal{P}$ is defined on the leaf nodes of the scenario tree so that $\sum_{n \in N_{r}} p_{n}=1$ and for each non-terminal node $p_{n}=$ $\sum_{m \in H_{n}} p_{m}, \forall n \in N_{t}, t=T-1, \ldots, 0$, that is each node receives a conditional probability mass equal to the combined mass of all its descendant nodes. In our case study portfolio revisions imply a transition from the previous time $t-1$ portfolio allocation at the ancestor node to a new allocation through holding, buying, and selling decisions on individual securities, for $t=1, \ldots, T-1$. The last possible revision is at stage $T-1$ with one period to go. Consistent with the fixed-income portfolio problem, decisions are described by nominal, face-value, positions in the individual security $i \in I$ of rating class $k_{i} \in K$.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Price–Yield Relationship

• 违约到达和违约损失的定义考虑了穆迪对特定评级的估计(2009)调整以考虑借款人的活动部门和市场违约溢价；
• 在 1 年的投资期限内，违约强度假设为常数，并将决定违约信息随机进入市场；违约时，边际价差将立即受到正向冲击，价格负向变动反映了证券在其剩余期限内不断部分支付；
• Poisson 过程和 Wiener 过程驱动独立的异质价差过程的行为，该过程假设与短期利率和信用价差过程共同确定证券价格动态；
• 每个评级等级的信用利差和短期利率是相关的。
该模型包含相关风险因素的假设，其行为将推动不同评级类别的所有证券，但包含特定的违约风险因素以区分每个类别中的证券行为。没有考虑传染现象，外生的价差动态会根据债券的久期和凸度结构影响特定的债券回报。

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

C(吨+Δ吨,ω)=C(吨)和r一世(ω)Δ吨+∑到∈到∑一世∈一世到C一世,吨+Δ吨+ −∑到∈到∑一世∈一世到C一世,吨+Δ吨(1−和[−b吨+Δ吨一世(ω)d在一世+Δ吨一世(ω)]).

F一世,吨+Δ吨(ω)=C一世(和−[b一世+Δ吨一世(ω)dΨ一世+Δ一世一世一世(ω)])

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考| Hedging Market and Credit Risk in Corporate Bond Portfolios

statistics-lab™ 为您的留学生涯保驾护航 在代写金融中的随机方法Stochastic Methods in Finance方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融中的随机方法Stochastic Methods in Finance方面经验极为丰富，各种代写金融中的随机方法Stochastic Methods in Finance相关的作业也就用不着说。

• 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|Hedging Market and Credit Risk in Corporate Bond Portfolios

The 2007-2009 financial turmoil witnessed an unprecedented market downturn for financial instruments carrying credit risk (Abaffy et al. 2007; Berndt 2004; Crouhy et al. 2000; Duffie and Singleton 1999) spanning both the secondary markets for corporate and sovereign securities and the markets for derivatives based on such instruments. The crisis propagated in US markets from mortgage-backed securities (MBS) and collateralized debt obligations (CDO) to credit instruments traded overthe-counter and thus into international portfolios. Widespread lack of liquidity in the secondary market induced first the Federal Reserve, then the European Central Bank, to adopt an expansive monetary policy through a sequence of base rate

reductions. Such policy partially limited the fall of bond prices but could do very little against instability in the corporate equity and fixed income markets.

The debate over the causes and possible remedies of such a prolonged financial crisis involved, from different perspectives, policy makers, financial intermediaries, and economists (European Central Bank 2010), all interested in analyzing the equilibrium recovery conditions, possibly within a quite different market architecture.
Financial investors, on the other hand, suffered dramatic portfolio losses within increasingly illiquid money, secondary stock and bond and credit derivative markets. In this chapter we present a stochastic model for interest rate and credit risk applied to a portfolio of corporate bonds traded in the Eurobond market with portfolio strategies tested over the 2008-2009 crisis. The portfolio management problem is formulated as a dynamic stochastic program with recourse (Consigli and Dempster 1998; Pflug and Römisch 2007 ; Zenios and Ziemba 2007). This chapter provides evidence of the potential offered by dynamic policies during a dramatic market crisis. Key to the results presented are the definitions of

• a statistical model capturing common and bond-specific credit risk factors that will determine jointly with the yield curve the defaultable bonds price behavior (Dai and Singleton 2003; Das and Tufano 1996; Duffie and Singleton 2000; Jarrow and Turnbull 2000; Kijima and Muromachi 2000; Longstaff et al. 2005) and
• a multistage strategy determined by a risk-reward objective function explicitly considering an extreme risk measure (Bertocchi et al. 2007; Consigli et al. 2010; Dempster et al. 2003; Jobst et al. 2006; Jobst and Zenios 2005; Rockafellar and Uryasev 2002). The problem considers a constant investment universe excluding credit derivatives and exogenous hedging strategies.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Corporate Bonds Risk Exposure

The recent credit crisis witnessed an unprecedented credit spread increase across all maturities in the Eurobond market while, first in the USD monetary area and then in the UK and Europe, interest rates were rapidly, though ineffectively, driven down to try to facilitate a liquidity recovery in the markets. In this work we assume a risk model incorporating both common and specific risk factors and allow the investor to determine her optimal policy by exploiting a scenario representation of the credit spreads evolution which may be affected at random times by severe market shocks. We consider a universe of Euro-denominated bonds with ratings from $A A A$ to $C C C-C$ (see (Abaffy et al. 2007)) plus one default-free government bond. Ratings are alphanumeric indicators specified by international rating agencies sueh as standard and poor, moody and fiteh, defining the eredit merit of the bond issuer and thus the likelihood of possible defaults over a given risk horizon. Rating revisions can be induced by market pressures or expert opinions and only occur infrequently. Bonds trading in the secondary market on the other hand generate a continuous information flow on expected interest rates and yield movements.

Bonds belonging to the same rating class may be distinguished according to their maturity and more importantly the activity sector to which the issuer belongs. Bond issuers within the same rating class and industry sector may finally be distinguished according to their market position and financial strength: the riskier the issuer, according to the above classification, the higher the credit spread that will be requested by investors to include a specific fixed income security in the portfolio.
Corporate prices are thus assumed to be driven by

• a common factor affecting every interest-sensitive security in the market (Cox et al. 1985), related to movements of the yield curve,
• a credit risk factor, related to movements of the credit curves (Abaffy et al. 2007; Duffie and Singleton 1999; Jobst and Zenios 2005), one for each rating class, and
• a specific bond factor, related to the issuer’s economic sector and its general financial health.

A degree of complexity may be added to this framework when trying to take into account the possible transition over the portfolio lifetime of bond issuers across different rating classes (Jarrow et al. 1997) and bond-specific liquidity features resulting in the bid-ask spread widening in the secondary market. In this work we will focus on the solution of a l-year corporate bond portfolio management problem with monthly portfolio revision, not considering either transition risk or liquidity risk explicitly.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Market and Credit Risk Model

We consider a credit-risky bond market with a set $I$ of securities. For each bond $i \in I$, we denote by $T_{i}$ its maturity. Let $K$ be the set of rating classes. Assuming

a canonical $\mathrm{S} \& \mathrm{P}$ risk partition from AAA to $\mathrm{CCC}-\mathrm{C}$ and $\mathrm{D}$ (the default state), we will denote by $k=1,2, \ldots, 8$ the credit risk indicator associated, respectively, with default-free sovereign and corporate AAA, AA, A, BBB, BB, B, CCC-C, and D ratings. A difference is thus postulated between a sovereign and a corporate AAA rating: the first one will be associated with the prevailing market yield for defaultfree securities with null credit spread over its entire market life, while the second may be associated with a positive, though negligible, credit spread. The security $i$ credit risk class is denoted by $k_{i}$, while $I_{k}$ is the set of bonds belonging to rating class $k$. The price of security $i$ at time $t$ will be denoted by $v_{t, T_{i}}^{i}$. For given security price and payment structure the yield $y_{t, T_{i}}^{i}$ can be inferred from the classical price-yield relationship:
$$v_{t, T_{i}}^{i}=\sum_{t<m \leq T_{i}} c_{m}^{i} e^{-y_{t, T_{i}}(m-t)}$$
where the cash payments over the security residual life are denoted by $c_{m}^{i}$ up to and including the maturity date. In (4.1) the yield $y_{t, T_{i}}^{i}$ will in general reflect the current term structure of risky interest rates for securities belonging to class $k_{i}$. We assume $y_{t, T_{i}}^{i}$ generated by a bond-relevant credit spread $\pi_{t, T_{i}}^{i}$ and a defaul $t$-free interest rate $r_{t, T_{i}}$, i.e. $y_{t, T_{i}}^{i}=r_{t, T_{i}}+\pi_{t, T_{i}}^{i}$

A one-factor model is considered for both the default-free interest rate curve and the security-specific credit spread. The latter is assumed to be generated by a rating-specific factor $\pi_{t}^{k}$ and an idyosincratic factor $\eta_{t}^{i}$.
The three state variables of the model are assumed to follow the s.d.e.’s:
\begin{aligned} d r_{t}(\omega) &=\mu_{r} d t+\sigma_{r}(t, r) d W_{t}^{r}(\omega) \ d \pi_{t}^{k}(\omega) &=\mu_{k} d t+\sigma_{k}\left(t, \pi^{k}\right) \sum_{l \in K} q^{k l} d W_{t}^{l}(\omega) \quad \forall k \ d \eta_{t}^{i}(\omega) &-\mu_{\eta} d t+\sigma_{i} d W_{t}^{i}(\omega)+\beta^{i}(\omega) d \Psi_{t}^{i}\left(\lambda^{i}\right) \quad \forall i \end{aligned}
where $\omega$ is used to identify a generic random variable. The first equation describes the short interest rate evolution as a diffusion process with constant drift and possibly state- and time-dependent volatility. $d W_{t}^{r} \sim N(0, d t)$ is a normal Wiener increment. The $k=1,2, \ldots, 7$ credit spread processes in $(4.3)$ are also modeled as diffusion processes with time- and state-varying volatilities. Credit spreads and the risk-free rate are correlated: the coefficients $q^{k i}$ denote the elements of the correlation matrix lower triangular Choleski factor linking the eight risk factors together, given the independent Wiener increments $d W_{t}^{k}, k=1,2, \ldots, 7$.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Hedging Market and Credit Risk in Corporate Bond Portfolios

2007-2009 年的金融风暴见证了带有信用风险的金融工具的前所未有的市场低迷（Abaffy 等人 2007；Berndt 2004；Crouhy 等人 2000；Duffie 和 Singleton 1999），横跨企业和主权证券的二级市场以及基于此类工具的衍生品市场。危机在美国市场从抵押贷款支持证券 (MBS) 和债务抵押债券 (CDO) 蔓延到场外交易的信贷工具，进而传播到国际投资组合。二级市场普遍缺乏流动性，首先是美联储，然后是欧洲央行，通过一系列基准利率采取扩张性货币政策

• 一个统计模型，捕捉共同的和特定于债券的信用风险因素，这些因素将与收益率曲线共同决定违约债券的价格行为（Dai 和 Singleton 2003；Das 和 Tufano 1996；Duffie 和 Singleton 2000；Jarrow 和 Turnbull 2000；Kijima 和 Muromachi 2000 ; Longstaff 等人，2005 年）和
• 一种多阶段策略，由明确考虑极端风险度量的风险回报目标函数确定（Bertocchi et al. 2007; Consigli et al. 2010; Dempster et al. 2003; Jobst et al. 2006; Jobst and Zenios 2005; Rockafellar and Uryasev 2002）。该问题考虑了一个不包括信用衍生品和外生对冲策略的恒定投资范围。

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Corporate Bonds Risk Exposure

• 影响市场上每一种对利息敏感的证券的共同因素（Cox 等，1985），与收益率曲线的变动有关，
• 一个信用风险因素，与信用曲线的移动有关（Abaffy 等人 2007；Duffie 和 Singleton 1999；Jobst 和 Zenios 2005），每个评级类别一个，以及
• 与发行人的经济部门及其总体财务状况相关的特定债券因素。

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Market and Credit Risk Model

v吨,吨一世一世=∑吨<米≤吨一世C米一世和−是吨,吨一世(米−吨)

dr吨(ω)=μrd吨+σr(吨,r)d在吨r(ω) d圆周率吨到(ω)=μ到d吨+σ到(吨,圆周率到)∑一世∈到q到一世d在吨一世(ω)∀到 d这吨一世(ω)−μ这d吨+σ一世d在吨一世(ω)+b一世(ω)dΨ吨一世(λ一世)∀一世

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考| Forward-Looking ALM Tests

statistics-lab™ 为您的留学生涯保驾护航 在代写金融中的随机方法Stochastic Methods in Finance方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融中的随机方法Stochastic Methods in Finance方面经验极为丰富，各种代写金融中的随机方法Stochastic Methods in Finance相关的作业也就用不着说。

• 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|Forward-Looking ALM Tests

From a strategic vantage point for an investor or pension sponsor, rule-based simulators are the instruments of choice for ongoing strategic risk control and management. These approaches allow for covering with very high degree of accuracy all aspects of the complex, individual situation in which strategic financial decisions are made. Realistic simulations of an investments vehicle’s stochastic behavior is a rich and reliable source for the information needed in ongoing strategic risk control activities as well for regularly revising decisions on risk optimal strategies allocation.

Most strategic decisions occur across a multi-periodic context, often with complex, path-dependent rules for rebalancing, contribution, and withdrawals. In most cases such conditional rebalancing rules can be found for each single asset; it is necessary to include them in order to represent each asset’s marginal contribution. Such rules are not only an essential aspect of asset allocation, but also the rules offer the opportunity to design and optimize them as an integral part of the strategy. In that sense one has to define strategic asset allocation not only as the composition of a portfolio, which would be sufficient if the world were a static one, but rather as an asset allocation strategy, which includes portfolio compositions and management rules. On the level of the strategy, these rules should not depend on exogenous conditions of single markets, but rather on the overall goal achievement compared to the individual preference structure, e.g., at a certain high funding level, a de-risking of the strategic portfolio happens, since no more risk taking is necessary to achieve the overall goals.

Such management rules always depend on additional sets of evaluation rules, internally or externally given, to evaluate the development of the results of the strategies, for example, under commercial law balance sheet and profit/loss calculations, taxation, or various other aspects. These rules produce incentives to favor one allocation strategy over another. Thus it is a basic requirement in order to find individually optimal strategies, to work with rule simulators, which represent relevant management and evaluation rules adequately and with the least level of simplification. This requirement is matched by modern rule simulators such as the PROTINUS Strategy Cockpit $^{\text {TM }}$ – described below and employed for the forward-looking tests.

When the multi-period environment is represented accurately, it becomes possible to design individually specific sets of objectives, constraints, and bounds. This is a major advantage of rule simulators, for which it is a condition qua sine non to have optimization and simulation in a single model setting. The necessity of optimizing strategies based on individual objective functions comes from the fact that any strategic investment is done to serve not a general common goal, but always to fulfill goals depending on individual conditions, rule sets, and preference structure. Systems like the PROTINUS Strategy Cockpit ” allow for setting up any objective functions derived from the variable of the rule set and perform desired calculation on them.

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

Protinus Strategy Cockpit is an integrated development environment for modeling multi-period stochastic planning problems in finance. It has been employed successfully as a strategic risk controlling and risk management tool over the past decade. It is a policy-rule simulator and optimizer, working in all MS Windows and MS Server ${ }^{\text {TM }}$ environments developed by the consulting firm PROTINUS in Munich, Germany. Ziemba and Mulvey (1998) described such systems in general, which they called decision rule approaches in comparison to alternative approaches, at a time when only very small number of implementations of such approaches existed. Predecessors of the PROTINUS Strategy Cockpit developed by the former Princeton-based organization Lattice Financial LLC were implemented, for example, as the first and, for several years, primary ALM system employed at the large German corporate Siemens.

The PROTINUS Strategy Cockpit ” generates user-specific rules by assembling a library of building block rules, ranging from simple arithmetic operations to complex accounting standards. The building blocks are put into the desired order via an execution schedule and where necessary a quantitative condition for their execution. PROTINUS Strategy Cockpit iM includes a powerful meta-heuristic, non-convex optimization algorithm based on early concepts of Glover’s Tabu search (Glover and Laguna 1997 ). The analyses are updated from past runs by saving solution information as templates and by grouping into reports. PROTINUS Strategy Cockpit ${ }^{\text {TM }}$ includes project management tools typically needed when working with large scenario spaces and complex rule sets and performing risk measurement and allocation studies. Figures $3.6$ and $3.7$ are screenshots from the financial engineers’ version. There is also a configuration available, which provides a fully automatic version, called PROTINUS Strategy Factory ${ }^{\text {TM }}$. Over the past several years, this system has been implemented to individually optimize on a quarterly basis, in a fully automatic fashion, several tens of thousands of portfolios for unit-linked insurance contracts provided by the Insurance Company Skandia in several countries in central Europe.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Evaluating Several Versions of the Global DEO Strategy

This section has two purposes. First, it provides brief examples of how a rule simulator can help determine the effects of addition-specific investment products to a given asset universe, a question investors as well as providers of such products have

to answer regularly. Second, we will analyze specifically how the DEO strategies perform in a prospective context.

The overlay strategies are evaluated in PROTINUS Strategy Cockpit ${ }^{\text {TM }}$ with three case studies: an asset-only context, a pension plan with IFRS balance sheet calculations, and the same pension plan rule with an added conditional contribution rule. All three cases are based on scenarios for the DEO overlay strategies and a group of broad, standard markets, i.e., US and Euro-equity as well as Eurogovernment and Euro-corporate bonds. For the pension plan cases, IFRS liability scenarios are produced from the scenarios for inflation and discount rates of the economic model. The basic liability data representing the plan’s population with all its biometric characteristics come from a modified real data set. We choose to generate a 10 -year, 1000 -scenario space with a quarterly scaling, since all shorter time steps are not of the typical interest for the strategic investor. In addition, we employ a commodity overlay strategy, called the Princeton Index (Mulvey and Vural 2010), to provide additional diversification benefits.

For these analyses, we apply a cascade structure economic model which allows for the generation of consistent scenario spaces for market returns and fundamentals and liabilities. The economic model is again made out of a group of simple building blocks. Early versions of such models are described by Mulvey (1996). The model includes basic equations with mean reverting assumptions and derives most returns processes implicitly from the cascade of fundamental processes. These return processes are either produced by functional dependencies or via advanced random number drawing techniques. For all processes the first four moments, including skewness and kurtosis, and the interactions among the variables can be defined explicitly. The cascade represents one core macroeconomic environment, typically representing the investor’s home region in order to include the local inflation and the rate, especially discount rate environment. These variables, along with scenarios for the liability variables such as IFRS defined benefit obligation (DBO), service cost, and pension payments are produced, so that a complete and fully consistent set of scenario spaces results, which represents accurately relevant exogenous risk factors.

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

Protinus Strategy Cockpit 是一个集成开发环境，用于对金融领域的多期随机规划问题进行建模。在过去的十年中，它已成功地用作战略风险控制和风险管理工具。它是一个策略规则模拟器和优化器，适用于所有 MS Windows 和 MS ServerTM值 由德国慕尼黑的咨询公司 PROTINUS 开发的环境。Ziemba 和 Mulvey (1998) 概括地描述了这样的系统，他们将其称为与替代方法相比的决策规则方法，当时这种方法的实现数量很少。例如，由前普林斯顿组织 Lattice Financial LLC 开发的 PROTINUS Strategy Cockpit 的前身已被实施，作为德国大型企业西门子多年来使用的第一个 ALM 系统，也是多年来的主要 ALM 系统。

PROTINUS Strategy Cockpit”通过组装一个构建块规则库来生成用户特定的规则，范围从简单的算术运算到复杂的会计标准。构建块通过执行计划以及在必要时执行的定量条件被放入所需的顺序。PROTINUS Strategy Cockpit iM 包括一个强大的元启发式非凸优化算法，该算法基于 Glover 的禁忌搜索的早期概念（Glover 和 Laguna 1997）。通过将解决方案信息保存为模板并分组到报告中，可以从过去的运行中更新分析。PROTINUS 策略驾驶舱TM值 包括处理大型场景空间和复杂规则集以及执行风险测量和分配研究时通常需要的项目管理工具。数据3.6和3.7是金融工程师版本的截图。还有一个配置可用，它提供了一个全自动版本，称为 PROTINUS Strategy FactoryTM值 . 在过去的几年中，该系统已实施，以全自动方式按季度单独优化斯堪迪亚保险公司在中欧几个国家提供的数以万计的单位连结保险合同组合。

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考| Multi-objective Functions and Solution Strategies

statistics-lab™ 为您的留学生涯保驾护航 在代写金融中的随机方法Stochastic Methods in Finance方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融中的随机方法Stochastic Methods in Finance方面经验极为丰富，各种代写金融中的随机方法Stochastic Methods in Finance相关的作业也就用不着说。

• 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|Multi-objective Functions and Solution Strategies

Setting an investment strategy for a DB pension plan is complicated by conflicting requirements and the diverse goals of the stakeholders. Each of the interested groups is served by several of the defined $Z$-objective functions. Especially relevant is the relationship between the pension plan and the sponsoring organization. In the USA, DB pension plans fall under the auspices of the Departments of Labor and Tax, and the requirement of the 1974 Employee Retirement and Security Act ERISA (with ongoing modifications by changing regulations and Congressional action). Thus, a US-based DB pension plan must undergo annual valuations by certified actuaries, who compute the various ratios including the accumulated benefit obligations $(\mathrm{ABO})$, the projected benefit obligations $(\mathrm{PBO})$, and funding ratios. These valuation exercises help determine the requirements for contributions by the sponsoring organization and the fees to be paid to the quasi-governmental organization PBGC (whose job is to take over pensions from bankrupt companies).

We will employ the following five objective functions (Mulvey et al. 2005a, $2008) .$

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

The first function, called economic value, is a combination of the expected riskadjusted discounted value of future contributions (Black 1995) and the discounted value of the surplus/deficit of the pension plan at the horizon, time $=T$. The first part of this objective provides a measure for the long-run cost of the pension trust:
$$Z_{1_{-} A}=\sum_{s \in S} \pi_{s} \sum_{t \in T} y_{t, s}^{\mathrm{CONT}} /\left(1+r_{t, s}\right)$$
where the risk-adjusted discount rate equals $r_{t, s}$ and is based on actuarial and economic judgment. The second part involves the discounted value of the pension’s surplus wealth at the end of the planning horizon:
$$Z_{1} B=\sum_{s \in S} \pi_{s} S w_{\mathrm{r}+1, s}$$
This part focuses on the investment strategy and contribution policy of the pension trust so that the highest average surplus value is achieved. Thus, the first objective function is to maximize economic value:
$$Z_{1}=Z_{1_{-} B}-Z_{1_{-} A}$$

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Advantages of Futures Market Strategies

At the strategic level, the futures markets does not require any direct capital investment and is thereby distinguished from traditional asset variables ${A}$. A prominent example involves commitments made in the futures/forward/swap markets (Mulvey et al. 2007). Here, for example, the investor may engage in a contract to purchase or sell a designated amount of a commodity such as corn at a designated date in the future. The investors (buyer and seller) must, of course, conform to exchange requirements for initial and margin capital and must participate in the mark-to-themarket mechanisms at the end of each trading day. We do not model these tactical issues on our strategic ALM model. Rather, we treat securities in the futures market as adjuncts to the core assets and assume that all margin calls are managed in the standard way to avoid margin calls. These investments are defined via “overlay variables.” Implicitly we assume that the size of the overlay variables is relatively

modest in scale and diverse enough to treat them at the strategic level without the need for tactical issues.

There are several advantages to futures market investments. First, it is straightforward to “go” short or long on a particular contract without the burden and costs of borrowing the security (traditional shorting). Second, long/short investments in commodities, currencies, and fixed income combinations can assist the investor in achieving the goal of achieving wide diversification. For a DB pension plan, there are additional advantages. In particular, a pension plan must maintain a health funding ratio in order to minimize contributions from the sponsoring organization to the pension plan. As mentioned, the funding ratio and pension surplus depend upon not only upon the market value of assets but also on the discounted value of estimated future cash flows (liabilities to pay retirees). The discount rate has a large impact on the funding ratio. During major economic downturns, for instance, the risk-free rate can drop by substantial amounts – with a possible commensurate decrease in the funding ratio. Accordingly, the duration of assets and the duration of liabilities will contribute to the management of a DP pension plan. A duration mismatch can be addressed by engaging in swaps or other futures/forward market operations. Mulvey et al. $(2007,2010)$ provide further discussions of overlay strategies.

We add futures strategies via a new set ${A-O}$ and associated decision variables $x_{j, t, s}$ for $j \varepsilon{A-O]$ – to the ALM model:
$$\sum_{j \in A-O} x_{j, t, s}^{*}\left(r_{j, t, s}\right)=x_{t, s}^{\text {Overlay }} \quad \forall s \in S, t=1, \ldots, T+1$$
The total return of the futures variables in time $t$ and under scenario $s$ is defined by $r_{j, t, s}$. We include the return from these variables in the cash flow constraint (3.6) as follows:
$$x_{1, r, s}=x_{1, t-1, s}^{\mapsto}+\sum_{i \neq 1} x_{i, t-1, s}^{\mathrm{SHLL}}\left(1-\sigma_{i, t-1}\right)-\sum_{i \neq 1} x_{i, t-1, s}^{\mathrm{BUY}}-b_{i-1, s}+y_{t-1, s}^{\mathrm{CONT}}+x_{i-1, s}^{\mathrm{BORR}}+x_{t-1, s}^{\text {averlay }}$$
$$\forall s \in \mathbf{S}, t=1, \ldots, T+1 .$$

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考| Performance Enhancements for Defined Benefit Pension Plans

statistics-lab™ 为您的留学生涯保驾护航 在代写金融中的随机方法Stochastic Methods in Finance方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融中的随机方法Stochastic Methods in Finance方面经验极为丰富，各种代写金融中的随机方法Stochastic Methods in Finance相关的作业也就用不着说。

• 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|Defined Benefit Pension Plans

Traditional pension trusts, called defined benefit (DB) plans herein, are threatened by a number of forces. The factors include (1) the loss of funding surpluses occurring over the past 10 years and current underfunded ratios, (2) a demographic nightmare – long-lived retirees and a shrinking workforce, (3) changing regulations, (4) greater emphasis on individual responsibilities for managing personal affairs such as retirement, and $(\supset)$ inefficient financial planning. 1he days of someone working for a single organization-IBM for example-for their entire career and then retiring with comfort supported by the company’s contributions are largely gone (except for public sector employees in certain cases).

This chapter takes up the lack of effective risk management and financial planning by DB pension plans. The $2001 / 2002$ economic contraction showed that the ample pension plan surpluses that existed in 1999 could be lost during an equity market downturn and a commensurate drop in interest rates which raises

the market value of liabilities (Mulvey et al. 2005b; Ryan and Fabozzi 2003). The loss of surplus could have been largely avoided by applying modern asset and liability management models to the problem of DB pension plans. Boender et al. (1998), Bogentoft et al. (2001), Cariño et al. (1994), Dempster et al. (2003, 2006), Dert (1995), Hilli et al. (2003), Kouwenberg and Zenios (2001), Mulvey et al. (2000, 2008), Zenios and Ziemba (2006), and Ziemba and Mulvey (1998) describe the methodology and show examples of successful applications. The Kodak pension plan (Olson 2005), for example, implemented an established ALM system for pensions in 1999, protecting its surplus over the subsequent recession. The situation repeated itself during the 2008 crash when most pension plan funding ratios dropped further. Again, systematic risk management via ALM models would have largely protected the pension plans.

Over the past decade, there has been considerable debate regarding the appropriate level of risk for a DB pension plan. On one side, advocates of conservative investments, called liability-driven investing or LDI in this chapter, have proposed a portfolio tilted to fixed income securities, similar to the portfolio of an insurance company. These proponents argue that a pension plan must fulfill its obligations to the retirees over long-time horizons and accordingly should reduce risks to the maximum degree possible.

To minimize risks, pension liabilities are “immunized” by the purchase of assets with known (or predictable) cash flows which are “adequate” to pay future liabilities. The goal is to maintain a surplus for the pension plan: Surplus/deficit = value(assets) – PV(liabilities), where the liability discount rate is prescribed by regulations such as promulgated by the Department of Labor in the United States. To protect the pension surplus ${ }^{1}$ requires an analysis of the future liabilities for the pension plan, i.e., expected payments to the plan retirees throughout a long time period – 40 or 50 or even 60 years in the future. Clearly with an ongoing organization, these liabilities are uncertain due to longevity risks, to future inflation, to possible modifications of payments for changing policy, and to other contingencies. Importantly, interest rate uncertainty plays a major role since the value of the liabilities (and many assets) will depend directly on interest rate movements. For these reasons, the asset mix will need to be modified at frequent intervals under LDI, for example, as part of the annual valuation exercises conducted by qualified actuaries. Similar to an insurance company, the duration of assets and the duration of liabilities should be matched (approximately) at least if the pension plan is to ensure that the surplus does not disappear due to interest rate movements.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|An Asset–Liability Management Model for DB Pension Plans

This section defines our asset and liability management (ALM) model for a defined benefit pension plan. We follow the framework established in Mulvey et al. (2005a, 2008) via multi-stage stochastic program. This framework allows for realistic conditions to be modeled such as the requirement for funding contributions when the pension deficit exceeds a specified limit and addressing transaction costs. However, as with any multi-stage stochastic optimization model, the number of decision variables grows exponentially with the number of stages and state variables. To compensate and to reduce the execution time to a manageable amount, we will apply a set of policy rules within a Monte Carlo simulation. The quality of the policy rules can be evaluated by means of an “equivalent” stochastic program. See Mulvey et al. (2008) and Section $3.5$ for further details.

To start, we establish a sequence of time stages for the model: $t=[1,2, \ldots, T]$. Typically, since a pension plan must maintain solvency and be able to pay its liabilities over long time periods, we generate a long-horizon model – over $10-40$ years with annual or quarterly time steps. To defend the pension plan over short time periods, we employ the DEO overlay strategies – which are dynamically adjusted over days or weeks. However, the target level of DEO is set by the strategic ALM model. In effect, the DEO provides a tactical rule for protecting the pension plan during turbulent conditions.

We define a set of generic asset categories ${A}$ for the pension plan. The categories must be well posed so that either a passive index can be invested in, or so that a benchmark can be established for an active manager. In the ALM model, the investment allocation is revised at the end of each time period with possible transaction costs. For convenience, dividends and interest payments are reinvested in the originating asset classes. Also, we assume that the variables depicting asset categories are non-negative. Accordingly, we include “investment strategies” in ${A}$, such as long-short equity or buy-write strategies in the definition of “asset categories.” The need for investment strategies in ${A}$ has become evident as standard long-only securities in 2008 became almost completely correlated (massive contagion). The investment strategies themselves may take action (revise their own investment allocations) more frequently and dynamically than as indicated by the strategy ALM model.

Next, a set of scenarios ${S}$ is generated as the basis of the forward-looking financial planning system. The scenarios should be built along several guiding principles. First, importantly, the time paths of economic variables should be plausible and should to the degree possible depict a comprehensive range of plausible outcomes. Second, the historical data should provide evidence of reasonable statistical properties, for example, the historical volatilities of the stock returns over the scenarios should be consistent with historical volatilities. Third, current market conditions should be considered when calibrating the model’s parameters. As an example, interest rate models ought to begin (time $=0$ ) with the current spot rate or forward rate curves. Third, as appropriate, expert judgment should be taken into account. The expected returns for each asset category should be evaluated by the

institution’s economists. There should be consistency among the various parties in a financial organization or at least the differences should be explainable. A number of scenario generators have been successfully applied over the past 25 years for asset and liability management models (Dempster et al. 2003; Heyland and Wallace 2001; Mulvey 1996).

For each $i \in{A}, t=[1,2, \ldots, T], s \in{S}$, we define the following parameters and decision variables in the basic ALM model:

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Multi-objective Functions and Solution Strategies

Setting an investment strategy for a DB pension plan is complicated by conflicting requirements and the diverse goals of the stakeholders. Each of the interested groups is served by several of the defined $Z$-objective functions. Especially relevant is the relationship between the pension plan and the sponsoring organization. In the USA, DB pension plans fall under the auspices of the Departments of Labor and Tax, and the requirement of the 1974 Employee Retirement and Security Act ERISA (with ongoing modifications by changing regulations and Congressional action). Thus, a US-based DB pension plan must undergo annual valuations by certified actuaries, who compute the various ratios including the accumulated benefit obligations (ABO), the projected benefit obligations (PBO), and funding ratios. These valuation exercises help determine the requirements for contributions by the sponsoring organization and the fees to be paid to the quasi-governmental organization PBGC (whose job is to take over pensions from bankrupt companies).

We will employ the following five objective functions (Mulvey et al. 2005a, 2008).

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。