### 数学代考|金融数学代考Financial Mathematics代写|Defining Bounds for e40 and e60 Using the Atlas

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## 数学代考|金融数学代考Financial Mathematics代写|Human Development in Brazil

This subsection presents an actual application of the results of Cohen (2011) and, consequently, of this research. This is the calculation of bounds for e40 and e60 using information from the Atlas of Human Development in Brazil (Programa das Nações Unidas para o Desenvolvimento Humano, 2013a).Life expectancy at birth is the main longevity measure and it is commonly used as an indicator of human development (Mayhew \& Smith, 2015). Due to their importance, data on life expectancy at birth are typically set out in research on demography, public health, etc. However, when there is no detailed data on mortality, such availability does not occur as easily for life expectancy at other ages. Mathers, Stevens, Boerma, White and Tobias (2015) point out that life expectancy at the age of 60 years, for instance, is a relevant indicator of longevity for older people and knowing it for a given population is key to public planning in social security and health, among other areas.Originally using the result of Cohen (2011), it has been found that we can estimate upper and lower bounds for complete life expectancy at age $x$, only knowing the probability that a person aged $x$ is alive at age $x+n$ and the complete life expectancy at age $x+n$. Of course, using the same argument, bounds for $e_{x+n}$ are constructed by means of prior knowledge about $n p x$ and ex. That is, it follows directly from (8) that:

$$\frac{\stackrel{\AA}{\dot{\varepsilon}}{x}-n}{{ }^{n} p{x}} \leq \dot{e}{x+n} \leq \frac{\dot{e}{x}}{n p_{x}}-n .$$
The Atlas of Human Development in Brazil (Programa das Nações Unidas para o Desenvolvimento Humano, 2013b) is a tool for accessing the Municipal Human Development Index, for the years 1991, 2000, and 2010 and it is available at Programa das Nações Unidas para o Desenvolvimento Humano (2013a). Using this tool, it is possible, for instance, for the mentioned years, to obtain information about life expectancy at birth, the probability that a newborn survives until the age of 40 years, and the probability that a newborn survives until the age of 60 years, for all Brazilian municipalities. It is worth noticing that, in 2010 , Brazil had 5,565 municipalities. So, it is possible to construct intervals for life expectancy values at ages 40 and 60 years for Brazilian municipalities in the respective years,Illustratively, Table 5 displays the bounds for $e_{40}$ and $e_{60}$ for the 10 municipalities with the highest life expectancy at birth in Brazil, in the year 2010. Curiously, all these municipalities belong to the state of Santa Catarina. Data were collected in the Atlas of Human Development in Brazil Programa das Nações Unidas para o Desenvolvimento Humano, 2013a) on December 22, 2017 .

## 数学代考|金融数学代考Financial Mathematics代写|INTRODUCTION

Over the past few decades, defined benefits pension schemes have largely been converted into defined contributions pension schemes without or with lower guarantees. Especially the recent financial crisis and increasing life expectancies affect the sustainability of pension systems that include guarantees. Therefore, there is a rising1 number of products available in the market that explicitly let these risks be borne by the individual rather than

the employer or insurer. If the pension payments in the decumulation phase include risk, we refer to these designs as variable annuities. Fixed annuities are those for which the payments are not uncertain.

There is a wide literature on variable annuities including investigating different embedded guarantees (Mahayni and Schneider, 2012; Chen et al., 2015), pricing variable annuities (Bauer et al., 2008; Bacinello et al., 2011; Nirmalendran et al., 2014), hedging variable annuities (Coleman et al., 2006; Trottier et al., 2018) or combinations of these (Kling et al., 2011; Bernard et al., 2014). Moreover, optimal demand for different annuity products is also investigated (Horneff et al., 2009; Blake et al., 2014; Peijnenburg et al., 2016). Designs in which equity exposure is incorporated in the annuity product is shown to increase welfare by, for example, Koijen et al. (2011).
We investigate variable annuities, where the variability arises due to risky investment returns. We study the relationship between the so-called assumed interest rate (AIR) and the (expected) annuity payments. The AIR effectively determines the decumulation speed of financial wealth over the payout phase: a larger AIR leads to higher early payments and lower later payments. In case the AIR equals the expected return on the underlying portfolio, the income during the payout phase is, in expectation, constant. See Dellinger (2006) and Horneff et al. (2010) for more details on the usage of the AIR concept in insurance pricing. We provide a novel way to solve the optimal consumption problem and, in doing so, derive the AIR that optimizes lifetime utility of consumption. We also analyze the utility loss of investors with constant relative risk aversion (CRRA) preferences who allocate their wealth suboptimally over their life cycle and/or have a suboptimal risk exposure. Under the assumption of an optimally chosen risk exposure, we find that a restriction to a (suboptimal) constant expected pension income does not lead to large utility losses. We also show that pension payments with a horizon of “only” 10 years are fairly insensitive to the choice of the AIR. As a result, unlike common practice, communication about the effect of choosing an AIR is preferably based on results for horizons closer to 20 years. We also investigate how financial shocks can be smoothed and what the effect is on the variable annuity. In that case, we find that a horizon-dependent AIR ensures a constant expected pension income.

## 数学代考|金融数学代考Financial Mathematics代写|VARIABLE ANNUITIES

The financial market that we consider is described by the seminal work of Merton (1971). This implies that in the standard Black-Scholes/Merton setting there is a risk-free asset with a constant interest rate $r$, there is a risky asset with price $\mathrm{St}$ at time t that evolves by the diffusion process.
\begin{aligned} \mathrm{d} S_{t} &=\mu S_{t} \mathrm{~d} t+\sigma S_{t} \mathrm{~d} Z_{t} \ &=(r+\lambda \sigma) S_{l} \mathrm{~d} t+\sigma S_{l} \mathrm{~d} Z_{t} . \end{aligned}
Thus, we assume that the stock price $S_{t}$ follows a geometric Brownian motion, where $\mu$ stands for the expected return, $\sigma$ is the stock volatility, $\lambda$ is the Sharpe ratio
$$\lambda=(\mu-r) / \sigma,$$
and $Z$ is a standard Brownian motion on the probability space $(\Omega, F, P)$
Moreover, we assume the isoelastic (power) function for utility that exhibits a CRRA and is given by
$$u(x)= \begin{cases}\frac{x^{1-\gamma}}{1-\gamma} & \text { if } \gamma>0, \gamma \neq 1 \ \ln (x) & \text { if } \gamma=1\end{cases}$$
where $\gamma$ is the relative risk aversion level. The more risk averse the investor is, the higher $\gamma$. We exclude negative risk aversion levels which would imply risk loving preferences. Since additive constant terms in objective functions do not affect optimal decisions, a term of minus one is omitted from the numerator which would be needed to show that the limiting case of $\gamma$ to one converges to logarithmic utility,
In general, the investor is endowed with initial wealth $\mathrm{W}{0}$ which can be used for consumption and the remainder is invested in the financial market. The wealth process is given by $$\mathrm{d} W{t}=\left(\left(r+w_{t}(\mu-r)\right) W_{t}-c_{t}\right) \mathrm{d} t+\sigma w_{t} W_{t} \mathrm{~d} Z_{t},$$
where $w_{t}$ is the fraction invested in the risky asset and $c_{t}$ is the withdrawal

(consumption) rate. For the CRRA utility function, the optimal time-varying risk exposure $w_{t}$ is known to be
$$w^{*}=\frac{\lambda}{\gamma \sigma} ;$$
see, for example Theorem 3.8.8. in Karatzas and Shreve (1998). That is, the optimal exposure is state- and time-independent. Concerning the optimal consumption choice, we represent the withdrawal via the AIR which determines the allocation of initial wealth to the (optimal) consumption at various horizons. We formulate this problem using a discrete number of consumption dates which effectively means that we solve H separate terminal wealth problems. The novelty in this setup allows us to directly cast optimal consumption questions into AIRs in variable annuities.

As an example, consider a retiree who enters retirement with total wealth $\mathrm{W}_{0}$ at time 0 and who needs to finance $\mathrm{H}$ annual pension payments at times. For ease of exposition, we assume $\mathrm{H}$ to be given; that is, we consider fixed term instead of lifelong variable annuities. Think of $\mathrm{H}$ as the remaining life expectancy at retirement age. ${ }^{2}$

The pension payments at each horizon $\mathrm{h}=0, \ldots, \mathrm{H}-1$ have to be financed from the initial total pension wealth $\mathrm{W}_{0}$. This simple idea is formalized by the notation in the next definition.

## 数学代考|金融数学代考Financial Mathematics代写|Human Development in Brazil

e˙\AAX−nnpX≤和˙X+n≤和˙XnpX−n.

## 数学代考|金融数学代考Financial Mathematics代写|VARIABLE ANNUITIES

Merton (1971) 的开创性著作描述了我们所考虑的金融市场。这意味着在标准 Black-Scholes/Merton 设置中，存在利率不变的无风险资产r, 存在有价格的风险资产小号吨在时间 t 由扩散过程演变。

d小号吨=μ小号吨 d吨+σ小号吨 d从吨 =(r+λσ)小号l d吨+σ小号l d从吨.

λ=(μ−r)/σ,

d在吨=((r+在吨(μ−r))在吨−C吨)d吨+σ在吨在吨 d从吨,

（消费）率。对于 CRRA 效用函数，最佳时变风险敞口在吨已知是

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## MATLAB代写

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