# 寿险精算中的 $m_x$ 代表什么？

• 题主在国外某精算高校就读，在做寿险精算的往年真题时，看到题目中要求在UDD假设下计算 $m_{55}$, 但想不起来这是哪个符号了。求问！

• $m_x$ 是中心死亡率（central rates of mortality），定义为：

$$m_{x}=\frac{q_{x}}{\int_{0}^{1} t p_{x} d t}$$

在英精小黄书（Tables）的 121 页可以查到 $m_{x}$ 的另外一个式子:

$$m_{x}=\frac{d_{x}}{\int_{0}^{1} l_{x+t} d t}$$

在 UDD 假设下， $m_{x}$ 可以进一步近似为：

$$m_{x} \approx \frac{q_{x}}{1-\frac{1}{2}q_x}$$

这个知识点在部分英国学校里会放在寿险精算科目。而在新体系的英国精算师考试中，该知识点在 CS2 的 Chapter 06: Survival models 中。

$q_x$ is the initial rate of mortality. This measures the number of deaths divided by the number of lives alive at age x. The problem with this is that it assumes that there are "$l_x$" person years lived between age x->x+1 (Obviously lives will die durring the year of age, and not be exposed to risk for the whole year). $m_x$ is the same thing, but it is dx divided by the expected number of person years lived between ages x->x+1. This is Integral from 0-1 $l_{x+t} dt$. The meaning of the two is very different. $q_x$ is the probability a life now aged x dies within the next year given alive at time x. $m_x$ is the probability a life aged anywhere between x,x+1 dies before attaining age x+1. I dont think it's used very much in actuarial calculations but it is handy in dermograpic/population studies because of the nice property that $m_x * ExC$ for any age equals the number of deaths, this is not true for qx (Try this for yourself).
• 翻译过来，按照我的理解：$q_x$的死亡率是$x$到$x+1$之间死亡人数除以在$x$岁活着的人数，而$m_x$的死亡率是$x$到$x+1$之间死亡人数除以平均在$x$到$x+1$岁之间活着的人数