### 数学代考|金融数学代考Financial Mathematics代写|UPPER AND LOWER BOUNDS

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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

## 数学代考|金融数学代考Financial Mathematics代写|ANNUITIES AND LIFE INSURANCE

In this section, the idea proposed by Cohen (2011) is extended to the case of continuous annuities, as set out in Proposition 1. In addition, the discrete case is also presented in Proposition 2. However, it is first necessary to define the present value of a given annuity-certain, immediate, temporary per $n$ years and paying one monetary unit per year, for both the continuous and the discrete (in advance) cases, as expressed in equations (11) and (12), respectively:
$$\begin{gathered} \bar{a}{n \mid}=\int{0}^{n} e^{-\delta t} d t=\left{\begin{array}{c} n, \delta=0 \ \frac{1-e^{-t}}{\delta}, \delta>0 . \end{array}\right. \ \ddot{a}{n \mid}=\sum{t=0}^{n-1} v^{t}=\left{\begin{array}{c} n, i=0 \ \frac{1-v^{n}}{1-v}, i>0 . \end{array}\right. \end{gathered}$$
Proposition 1 (continuous case): being $x$ and $x+n$ two ages such that $0 \leq$ $x \leq x+n<\omega$, it is possible to state that
$$\left(\bar{a}{n \mid}+\bar{a}{x+n} \cdot e^{-\delta n}\right) \cdot{ }{n} p{x} \leq \bar{a}{x} \leq \bar{a}{n \mid}+{ }{n} p{x} \cdot e^{-\delta n} \cdot \bar{a}_{x+n}$$

## 数学代考|金融数学代考Financial Mathematics代写|APPLICATION AND DISCUSSIONS

This section suggests applications and provides numerical examples about the results demonstrated in section 3. In addition, the effects of variation in the interest rate and the data gap (i.e. the value of $n$ ) are discussed, as well as the retangularization of the survival curve.Initially, for illustrative purposes, intervals for the values of $\bar{a}_{20}$ and $A 20$ are calculated by considering the complete availability of data from the ages of $20,30,40,50,60$, and 70 years, i.e. the intervals were calculated by means of different values for the data gap. In addition, still for illustrative purposes, the central point of the interval is considered an estimate of the actuarial fair value of the respective financial products and, based on this estimate, the error (estimated value less actual value, which uses the complete data since 20 years) is also computed. For the calculations, the ‘IBGE 2015 mortality table’ was considered for both sexes – extrapolated for ages over 80 years, and an effective interest rate of $3 \%$ per year. The results for are summarized in Table 1, while the results for $A 20$ are set out in Table $2 .$

At this point, two aspects deserve to be highlighted: first, it would only be possible to calculate the error of having the actual values of financial products; but, surely, such values would not be available in situations involving incomplete mortality data. Second, the error would depend on the value used as an estimate. If, for instance, a researcher decided to conservatively use values greater than the central value of the interval as a way of analyzing the expected present value of annuities and life insurance policies, then the results on error might be different. Therefore, it is useful to evaluate how the length of the interval produced behaves in face of variations in the parameters, as the next subsection does.

## 数学代考|金融数学代考Financial Mathematics代写|The Length of Intervals

Once some illustrative results have been shown, it is worth formally discussing the impact of certain parameters on the length of the interval (upper bound minus lower bound) produced. Being $T a$ the length of the interval formed by the bounds in (15), so $T a$ is defined by the expression:
$$T_{a}=\ddot{a}{n \mid}\left(1-{ }{n} p_{x}\right) .$$
Using Equation (17), it is noticed that, everything else being constant, $T a$ increases with increases in $n$, since $\mathrm{a}^{-} 20 \mathrm{a}^{-} 20$ is an increasing function of $n$ and $n p x$ is a non-increasing function of $n$ (see example in Table 1); just as we observe that $T a$ decreases with increases in $i$, since $\mathrm{a}^{-} 20 \mathrm{a}^{-} 20$ is a decreasing function of $i$ (Faro, 2006). This result is shown in Table $3 .$ Finally, $T a$ decreases with increases in $n p x$, since, as stated, $n p x$ is a nonincreasing function of $n$.

This last result refers to the phenomenon of retangularization of the survival curve. The rectangularization process, as well shown by Wilmoth and Horiuchi (1999), is characterized by high survival rates in childhood and adulthood, and rapid mortality in advanced ages. Thus, in a hypothetical case where $n p x=1$, then the lower and upper bounds in (15) are equal and, indeed, $\ddot{a}{x}=\ddot{a}{n \mid}+\ddot{a}_{x+n} \cdot v^{n}$. This fact reinforces the result that the more rectangular the survival curve, the smaller distance between the lower and upper bounds of ax, i.e. the smaller interval is produced.

Just as in the discussion of the bounds of an annuity, it is also relevant to analyze the bounds for a life insurance. For this product, being TA the length of the interval formed by the bounds in (16), then TA is defined by:
$$T_{A}=\left(1-v^{n}\right) .\left(1-{ }{n} p{x}\right) .$$
For (18), it is observed that TA increases with increases in $\mathrm{n}$ (see Table 2) and decreases with increases in ${ }{n} p{x}$, similarly to what happens with Ta. However, TA increases with increases in $i$, since $(1-v n)$ is an increasing function of $i$ (see Table 4).

## 数学代考|金融数学代考Financial Mathematics代写|ANNUITIES AND LIFE INSURANCE

$$\begin{gathered} \bar{a}{n \mid}=\int{0}^{n} e^{-\delta t} dt=\left{ n,d=0 1−和−吨d,d>0.\正确的。\ \ddot{a}{n \mid}=\sum{t=0}^{n-1} v^{t}=\left{ n,一世=0 1−在n1−在,一世>0.\正确的。 \结束{聚集} 磷r○p○s一世吨一世○n1(C○n吨一世n在○在sC一个s和):b和一世nGX一个ndX+n吨在○一个G和ss在CH吨H一个吨0≤$$X≤X+n<ω$,一世吨一世sp○ss一世bl和吨○s吨一个吨和吨H一个吨 \left(\bar{a}{n \mid}+\bar{a}{x+n} \cdot e^{-\delta n}\right) \cdot{ }{n} p{x} \leq \bar{a}{x} \leq \bar{a}{n \mid}+{ }{n} p{x} \cdot e^{-\delta n} \cdot \bar{a}_{x +n}$\$

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