Curtate Life Expectancy
Expected remaining whole years of life from survival probabilities.
Formula
Variables
Example
Summed from a life table's survival probabilities.
How it works
Curtate life expectancy counts how many complete years a person is expected to live from age x, ignoring any fraction of the final year. The formula works because the expected value of a nonnegative integer-valued lifetime can be written as the sum of tail probabilities: the chance of surviving at least 1 more year, plus at least 2 more years, and so on. Each term {}_k p_x adds the probability that the person is still alive at the start of year k, so the total gives the average number of whole years lived beyond age x.
When to use it
Use this when you need an actuarial expectation for integer-valued future lifetime, especially in life tables, pension calculations, and insurance models based on annual survival data. It is most useful when benefits, premiums, or decrements are measured in complete years rather than exact ages.
Worked example, step by step
Suppose a 60-year-old has the following survival probabilities from a life table: {}_1 p_{60} = 0.985 {}_2 p_{60} = 0.970 {}_3 p_{60} = 0.952 {}_4 p_{60} = 0.930 {}_5 p_{60} = 0.905 and probabilities beyond 5 years are negligible for this example. Use the formula: e_{60} = \sum_{k=1}^{\infty} {}_k p_{60} Approximate with the given terms: e_{60} \approx 0.985 + 0.970 + 0.952 + 0.930 + 0.905 Compute the sum: e_{60} \approx 4.742 So the curtate life expectancy at age 60 is about 4.742 years, meaning the expected number of complete years lived after age 60 is about 4.74 years.
Common mistakes to avoid
- Using exact lifetime expectancy instead of curtate life expectancy and including fractional years by accident
- Starting the sum at k = 0, which incorrectly adds 1 for everyone
- Confusing {}_k p_x with q_x or with the probability of dying within k years
Did You Know?
Life tables, first built by Edmond Halley in 1693, remain the backbone of pricing life insurance.
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