### 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

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

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

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## MATLAB代写

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