死亡の確率 qx アクチュアリー科学

Understanding the Probability of Death (qx) in Actuarial Science

In actuarial science, one of the fundamental concepts used in risk assessment and life insurance calculations is the probability of death, denoted as qx. This metric provides insights into the likelihood of an individual of a certain age dying before reaching the next age. By understanding qx, actuaries can estimate premiums, calculate reserves, and design pension plans. Let's break down the components of this crucial formula.

Defining the Formula

qx = (dx / lx)

In this equation, qx represents the probability of death within a specified interval, typically one year. Here are the key inputs:

• dx: The number of deaths within the specified age interval. This value is crucial for calculating the probability and should be expressed as a whole number.
• lx: The number of lives at the beginning of the age interval. Like dx, this should also be expressed as a whole number.

Inputs and Outputs

The inputs dx and lx must both be numeric values, typically drawn from life tables or actuarial tables. The output qx is a probability and is expressed as a decimal value between 0 and 1:

• Inputs:
• dx: Number of deaths (numeric, whole number)
• lx: Number of lives at the beginning (numeric, whole number)
• Output:
• qx: Probability of death (decimal)

Example Values:

• dx = 50
• lx = 1,000

Using these values, qx is calculated as:

Therefore, the probability of death within the specified age interval is 0.05, or 5%.

Real-Life Application

Imagine an insurance company that wants to calculate the premium for a life insurance policy for individuals aged 40. By understanding the probability of death for this age group, the company can estimate how many policyholders might die before reaching age 41. Suppose they determine that, historically, 20 out of every 1,000 individuals aged 40 die before turning 41. This would mean:

qx = 20 / 1000 = 0.02

This 2% probability of death can then be used to set premiums that balance risk and profitability.

Conclusion

Understanding the probability of death (qx) is essential for actuaries, particularly those involved in life insurance and pensions. By accurately calculating qx, these professionals ensure that financial models are both viable and fair. Whether you're setting premiums, planning for retirement, or assessing risk, the formula for qx provides a foundational tool for making informed decisions.