### 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
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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).

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