### 数学代考|金融数学代考Financial Mathematics代写|Upper and Lower Bounds

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

This study aimed to set upper and lower bounds for the expected present value of whole life annuities and whole life insurance policies from incomplete mortality data, generalizing previous results on life expectancy. Since its inception, in the $17^{\text {th }}$ century, actuarial science has been devoted to the study of annuities and insurance plans. Thus, setting intervals that provide an initial idea about the cost of these products using incomplete mortality data represents a theoretical contribution to the area and this may have major

applications in markets lacking historical records or those having little reliability of mortality data, as well as in new markets still poorly explored. For both the continuous and discrete cases, upper and lower bounds were constructed for the expected present value of whole life annuities and whole life insurance policies, contracted by a person currently aged $x$, based on information about the expected present value of these respective financial products subscribed to by a person of age $x+n$ and the probability that an individual of age $x$ survives to at least age $x+n$. Through the bounds of a continuous annuity, in an environment where the instantaneous interest rate is equal to zero, the results shown also set bounds for the complete life expectancy, which implies that the contribution of this research generalizes previous results in the literature. It was also found that, for both annuities and insurance plans, the length of constructed intervals increases as the data gap size increases and it decreases as the survival curve becomes more rectangular. Illustratively, bounds for life expectancy at 40 and 60 years of age, for the 10 municipalities showing the highest life expectancy at birth in Brazil in 2010, were constructed by using data available in the Atlas of Human Development in Brazil.

Keywords:-Actuarial mathematics; actuarial science; annuities; life insurance; mortality table.

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

According to Pitacco, Denuit, Haberman and Olivieri (2009), actuarial science flourished in the mid-17th century, based on compound interest rate theory and probability theory, as well as observations on mortality. Also according to these authors, one of the first themes to be addressed by this new science was calculating the expected present value of annuities. Such interest arose because governments used to sell whole life annuities as a way to finance public enterprises.

Pitacco et al. (2009) also indicate that Jan de Witt, in 1671, was the precursor in the calculation of annuities using a hypothetical mortality table and a constant interest rate. However, as well emphasized by Haberman and Sibbet (1995), de Witt’s contribution had little repercussion at the time. The same, in turn, cannot be said of Edmund Halley’s work, in 1693, which, in addition to developing a mortality table by means of actual observations, also introduced a method for calculating the cost of annuities that reverberates up to the present day. The reader interested in historical aspects of actuarial science will benefit from the reading of Hald (1990) and Haberman and Sibbet (1995).

Thus, it can be noticed that actuarial science, since its inception, has taken as one of its central foci the study of annuities. To calculate the expected present value in an annuity for any given individual of age $x$, it is necessary to have access to a complete mortality table (having information from at least age $x$ ) or a survival function representative of the population to which the person belongs. In some instances, it is also possible to generate a complete table by means of abridged life tables and use the resulting complete table to calculate the cost of desired annuity or insurance (Baili, Micheli, Montanari, \& Capocaccia, 2005; Ibrahim, 2008).

However, when there is no data on the mortality probabilities (age after age) since age $x$, calculating the expected present value of desired annuity is compromised. The impossibility to obtain detailed mortality data for a sequence of ages may occur due to lack of historical records or little reliability of existing data or also because this is a new market that is still poorly explored, for instance.

The absence of complete mortality tables also hinders the calculation of longevity measures, as in the case of the complete life expectancy. In this way, Cohen (2011) established upper and lower bounds on life expectancy at a given age $x$, knowing only detailed mortality data from age $x+n$ (and, indeed, life expectancy at age $x+n$ ), as well as the probability that a person aged $x$ survives to at least age $x+n$.

## 数学代考|金融数学代考Financial Mathematics代写|THEORETICAL FRAMEWORK

Suppose a complete mortality table that has $\omega$ as its maximum age, i.e. no individual is supposed to be alive at age $\omega$. Being $q x$ the probability that a person of age $x$ dies along that age, this implies that $q \omega-1=1$. In addition, $t p x$, for $0 \leq x \leq \omega$ an $t \geq 0$, indicates the probability that a person of age $x$ survives to at least age $x+t$. In the continuous case, $t p x$ is named as survival function. Of course, as a survival function, $t p x$ is a non-increasing function of $t$, i.e. as $t$ increases, $t p x$ decreases or remains at the same value. In addition, $t p x=1$ and $t p x=0$ whenever $\mathrm{t} \geq \omega-x$. In addition, being $\mu_{2}(t)=\lim _{d t \rightarrow 0} \frac{\Delta d x}{d t}$ the force of mortality (or instantaneous mortality rate) at age $x+$ $t$, for $\mathrm{t}>0$, then, the probability that a person of age $x$ survives to at least age $x$ $+t$ and dies instantaneously thereafter is defined by $t p x \mu x(t) d t$ (Dickson, Hardy, \& Waters, 2013). Finally, $\mathrm{i} \geq 0$ is the annual effective interest rate in a compound capitalization regime and $\delta \geq 0$ is the instantaneous interest rate in the continuous capitalization regime, so that $\ln (1+i)=\delta$. In this way, the financial decapitalization factor is defined as $v=1 /(1+i)=e-\delta$.

So, as well taught by Dickson, Hardy and Waters (2013), the expected present value of whole life continuous annuity subscribed to by a person aged $x$ is given by:
$$\bar{a}{z}=\int{0}^{\omega-x} p_{x} \cdot e^{-t d} d t .$$
The discrete case, i.e. the net single premium for a whole life annuitydue subscribed to by a person aged $x$ is defined as:
$$\ddot{a}{x}=\sum{t=0}^{\omega-x-1}{ }{t} p{x} \cdot v^{t} \text {. }$$
(2)

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

Pitacco 等人。(2009) 还指出，1671 年的 Jan de Witt 是使用假设死亡率表和恒定利率计算年金的先驱。然而，正如 Haberman 和 Sibbet (1995) 所强调的那样，de Witt 的贡献在当时没有什么反响。反过来，埃德蒙·哈雷 (Edmund Halley) 在 1693 年的工作也不能说同样的话，他除了通过实际观察制定死亡率表外，还引入了一种计算年金成本的方法，这种方法一直流传至今。对精算科学的历史方面感兴趣的读者将从阅读 Hald (1990) 以及 Haberman 和 Sibbet (1995) 中受益。

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