已知员工各年龄段1年退休概率，如何计算其未来5年退休概率？

Hey there! Let's walk through how to figure out the probability that an employee retires at any point in the next 5 years using your existing age-to-next-year-retirement-probability dataset. This is a classic sequential probability problem, and here's a straightforward way to tackle it:

Core Logic #

The key idea is that retiring in the next 5 years means one of these mutually exclusive events happens:

• The employee retires in the first year
• The employee doesn't retire in the first year, but retires in the second year
• The employee doesn't retire in the first two years, but retires in the third year
• And so on, up to the fifth year

Since these events can't occur simultaneously, we calculate each event's probability and sum them all together.

Step-by-Step Calculation #

Let's define some terms first to make it concrete:

• current_age: The employee's current age
• P_i: The retirement probability for the employee's age at the start of year i (so P_1 is the probability of retiring in the next year from current_age, P_2 is the probability from current_age + 1, etc.)
• S_i: The probability the employee hasn't retired before year i (this is the product of (1 - P_1) * (1 - P_2) * ... * (1 - P_{i-1}))

The total 5-year retirement probability formula is:

Total_Prob = P_1 + (S_1 * P_2) + (S_2 * P_3) + (S_3 * P_4) + (S_4 * P_5)

Example Walkthrough #

Suppose we have an employee aged 45, with these annual retirement probabilities:

• P_1 (45 → 46): 5% (0.05)
• P_2 (46 → 47): 8% (0.08)
• P_3 (47 → 48): 12% (0.12)
• P_4 (48 → 49): 18% (0.18)
• P_5 (49 → 50): 25% (0.25)

Calculating each term:

1. Year 1 retirement probability: 0.05
2. Year 2 retirement probability: (1 - 0.05) * 0.08 = 0.95 * 0.08 = 0.076
3. Year 3 retirement probability: (1 - 0.05) * (1 - 0.08) * 0.12 = 0.95 * 0.92 * 0.12 ≈ 0.10488
4. Year 4 retirement probability: 0.95 * 0.92 * 0.88 * 0.18 ≈ 0.1371
5. Year 5 retirement probability: 0.95 * 0.92 * 0.88 * 0.82 * 0.25 ≈ 0.1593

Adding these up: 0.05 + 0.076 + 0.10488 + 0.1371 + 0.1593 ≈ 0.5273 (or ~52.73%)

That's the total probability the employee retires at any point in the next 5 years.

Key Notes #

• Double-check that you're pulling the correct P_i value each year: the employee's age increases by 1 for every year they haven't retired.
• This approach assumes annual retirement probabilities are independent, which is a standard assumption unless your internal data suggests otherwise.

内容的提问来源于stack exchange，提问作者Christian