IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Loss distributions, with and without risk sharing
Claims follow a lognormal distribution with mu = 8 and sigma = 1.2. Claims inflate uniformly by 25%. What are the parameters of the inflated claim distribution?
The inflated claims are lognormal with mu = 8 + ln 1.25 and sigma = 1.2. Taking logarithms turns multiplication by 1.25 into adding ln 1.25 to the normal mean, so mu shifts by that amount while sigma is unchanged.
- Amu = 8, sigma = 1.5
- Bmu = 8.25, sigma = 1.2
- Cmu = 8 + ln 1.25, sigma = 1.2Correct
- Dmu = 10, sigma = 1.2
- mu = 8 + ln 1.25, sigma = 1.2 x 1.25
Explanation
If Y = 1.25X, ln Y = ln X + ln 1.25, which shifts the normal mean by ln 1.25 and leaves the variance unchanged. So mu becomes 8 + ln 1.25 and sigma stays 1.2. Multiplying mu or sigma directly by 1.25 confuses a multiplicative change with an additive shift on the log scale.
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