IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Mortality projection
Lee-Carter k(t) is modelled as a random walk with drift: k(t+1) = k(t) + d + e(t), where e(t) is independent N(0, s²). Estimated d = -1.5 and s = 2.0, and the latest value is k(2024) = -30. What is the 95% prediction interval for k(2029), ignoring parameter uncertainty in d and s (use 1.96, with sqrt5 = 2.236)?
The five-year-ahead mean is -30 + 5×(-1.5) = -37.5, and the variance is 5×4 = 20, so the standard deviation is 4.472. The 95% interval is therefore -37.5 ± 1.96×4.472, which is about -37.5 ± 8.77.
- A-37.5 ± 8.77Correct
- B-37.5 ± 3.92
- C-37.5 ± 19.6
- D-30 ± 8.77
- -37.5 ± 17.5
Explanation
Five steps ahead: mean = -30 + 5(-1.5) = -37.5. Variance = 5 × 4 = 20, sd = 4.472. Half-width = 1.96 × 4.472 = 8.77. Option 1 uses a one-step sd of 2, wrongly ignoring accumulation; option 3 uses variance 20 as sd... not square-rooted.
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