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

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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|Introduction

In recent years there has been a significant growth of inyestment products aimed at attracting investors who are worried about the downside potential of the financial markets for pension investments. The main feature of these products is a minimum guaranteed return together with exposure to the upside movements of the market.
There are several different guarantees available in the market. The one most commonly used is the nominal guarantee which guarantees a fixed percentage of the initial investment. However there also exist funds with a guarantee in real terms which is linked to an inflation index. Another distinction can be made between fixed and flexible guarantees, with the fixed guarantee linked to a particular rate and the flexible to for instance a capital market index. Real guarantees are a special case of flexible guarantees. Sometimes the guarantee of a minimum rate of return is even set relative to the performance of other pension funds.

Return guarantees typically involve hedging or insuring. Hedging involves eliminating the risk by sacrificing some or all of the potential for gain, whereas insuring involves paying an insurance premium to eliminate the risk of losing a large amount.
Many government and private pension schemes consist of defined benefit plans. The task of the pension fund is to guarantee benefit payments to retiring clients by investing part of their current wealth in the financial markets. The responsibility of the pension fund is to hedge the client’s risk, while meeting the solvency requirements in such a way that all benefit payments are met. However at present there are significant gaps between fund values, contributions made by employees, and pension obligations to retirees.

One way in which the guarantee can be achieved is by investing in zero-coupon Treasury bonds with a maturity equal to the time horizon of the investment product in question. However using this option foregoes all upside potential. Even though the aim is protect the investor from the downside, a reasonable expectation of returns higher than guaranteed needs to remain.

In this paper we will consider long-term nominal minimum guaranteed return plans with a fixed time horizon. They will be closed end guarantee funds; after the initial contribution there is no possibility of making any contributions during the lifetime of the product. The main focus will be on how to optimally hedge the risks involved in order to avoid having to buy costly insurance.

However this task is not straightforward, as it requires long-term forecasting for all investment classes and dealing with a stochastic liability. Dynamic stochastic programming is the technique of choice to solve this kind of problem as such a model will automatically hedge current portfolio allocations against the future uncertainties in asset returns and liabilities over a long horizon (see e.g. Dempster et $a l, 2003$ ). This will lead to more robust decisions and previews of possible future benefits and problems contrary to, for instance, static portfolio optimization models such as the Markowitz (1959) mean-variance allocation model.

Consiglio et al. (2007) have studied fund guarantees over single investment periods and Hertzog et al. $(2007)$ treat dynamic problems with a deterministic risk barnier. However a practical method should have the flexibility to take into account multiple time periods, portfolio constraints such as prohibition of short selling and varying degrees of risk aversion. In addition, it should be based on a realistic representation of the dynamics of the relevant factors such as asset prices or returns and should model the changing market dynamics of risk management. All these factors have been carefully addressed here and are explained further in the sequel.

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

This chapter looks at several methods to optimally allocate assets for a minimum guaranteed return fund using expected average and expected maximum shortfall risk measures relative to the current value of the guarantee. The models will be applied to eight different assets: coupon bonds with maturity equal to $1,2,3,4,5$, 10 and 30 years and an equity index, and the home currency is the euro. Extensions incorporated into these models are the presence of coupon rates directly dependent on the term structure of bond returns and the annual rolling of the coupon-bearing bonds.

We consider a discrete time and space setting. The time interval considered is given by $\left{0, \frac{1}{12}, \frac{2}{12}, \ldots, T\right}$, where the times indexed by $t=0,1, \ldots, T-1$ correspond to decision times at which the fund will trade and $T$ to the planning horizon at which no decision is made, see Figure 1. We will be looking at a five-year horizon.
Uncertainty $\Omega$ is represented by a scenario tree, in which each path through the tree corresponds to a scenario $\omega$ in $\Omega$ and each node in the tree corresponds to a time along one or more scenarios. An example scenario tree is given in Figure 2 . The probability $p(\omega)$ of scenario $\omega$ in $\Omega$ is the reciprocal of the total number of scenarios as the scenarios are generated by Monte Carlo simulation and are hence equiprobable.

The stock price process $\mathbf{S}$ is (initially) assumed to follow a geometric Brownian motion, i.e.
$$\frac{d \mathbf{S}{t}}{\mathbf{S}{t}}=\mu_{S} d t+\sigma_{S} d \mathbf{W}{t}^{S}$$ where $d \mathbf{W}{t}^{S}$ is correlated with the $d \mathbf{W}_{t}$ terms driving the three term structure factors discussed in Section $3 .$

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

Let (see Table 1)

• $h_{t}(\omega)$ denote the shortfall at time tand scenario $\omega$, i.e.
$$h_{t}(\omega):=\max \left(0, L_{t}(\omega)-W_{t}(\omega)\right) \quad \forall \omega \in \Omega \quad t \in T^{\text {total }}$$
• $H(\omega):=\max {t \in T^{\text {total }}} h{t}(\omega)$ denote the maximum shortfall over time for scenario $\omega$.
The constraints considered for the minimum guaranteed return problem are:
• cash balance constraints. These constraints ensure that the net cash flow at each time and at each scenario is equal to zero
$$\sum_{a \in A} f P_{0, a}^{b u y}(\omega) x_{0, a}^{+}(\omega)=W_{0} \quad \omega \in \Omega$$
Designing Minimum Guaranteed Return Funds
25
$$\sum_{a \in A \backslash{S}} \frac{1}{2} \delta_{t-1}^{a}(\omega) F^{a} x_{t, a}^{-}(\omega)+\sum_{a \in A} g P_{t, a}^{s e l l}(\omega) x_{t, a}^{-}(\omega)=\sum_{a \in A} f P_{t, a}^{b u y}(\omega) x_{t, a}^{+}(\omega)$$
$\omega \in \Omega \quad 1 \in T^{d} \backslash{0} .$
In (4) the left-hand side represents the cash freed up to be reinvested at time $t \in T^{d} \backslash{0}$ and consists of two distinct components. The first term represents the semi-annual coupons received on the coupon-bearing Treasury bonds held between time $t-1$ and $t$, the second term represents the cash obtained from selling part of the portfolio. This must equal the value of the new assets bought given by the right hand side of (4).

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

Consiglio 等人。(2007) 研究了单一投资期间的基金担保，Hertzog 等人。(2007)用确定性风险边界处理动态问题。然而，一种实用的方法应该能够灵活地考虑多个时间段、投资组合限制，例如禁止卖空和不同程度的风险规避。此外，它应基于对资产价格或收益等相关因素动态的真实表示，并应模拟风险管理的不断变化的市场动态。所有这些因素都在这里得到了仔细的解决，并在续集中进行了进一步的解释。

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

d小号吨小号吨=μ小号d吨+σ小号d在吨小号在哪里d在吨小号与d在吨驱动本节讨论的三个期限结构因素的术语3.

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

• H吨(ω)表示时间和情景的短缺ω， IE
H吨(ω):=最大限度(0,大号吨(ω)−在吨(ω))∀ω∈Ω吨∈吨全部的
• $H(\omega):=\max {t \in T^{\text {total }}} h {t}(\omega)d和n这吨和吨H和米一种X一世米你米sH这r吨F一种一世一世这v和r吨一世米和F这rsC和n一种r一世这\欧米茄$。
考虑最小保证回报问题的约束是：
• 现金余额限制。这些约束确保在每个时间和每个场景下的净现金流等于零
∑一种∈一种F磷0,一种b你是(ω)X0,一种+(ω)=在0ω∈Ω
设计最低保证回报基金
25
∑一种∈一种∖小号12d吨−1一种(ω)F一种X吨,一种−(ω)+∑一种∈一种G磷吨,一种s和一世一世(ω)X吨,一种−(ω)=∑一种∈一种F磷吨,一种b你是(ω)X吨,一种+(ω)
ω∈Ω1∈吨d∖0.
在（4）中，左侧代表当时释放出来可用于再投资的现金吨∈吨d∖0并由两个不同的组件组成。第一项代表在不同时期持有的附息国债收到的半年度息票吨−1和吨，第二项代表出售部分投资组合获得的现金。这必须等于 (4) 右侧给出的购买的新资产的价值。

## 广义线性模型代考

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