  The financial losses from extreme weather events can have a disastrous
effect, often costing billions of pounds. While changes in the disposition of
individual events is of importance to both the insurance and re-insurance
industries, these companies are often concerned with the aggregate risk posed
in a season. This project explores how the statistical properties of aggregate
risk measures may change when, to reflect the earth's changing climate, models
are made time dependent. Historical random sum equations by Wald (1945) and
Blackwell and Girshick (1947) are used to develop a relationship between the
frequency of events and the aggregate risk.
  The covariance between the occurrence of events and aggregate risk is found
to be the product of the expected value of the aggregate risk and the
dispersion statistic. Furthermore, a new equation (the "J-equation") relates
the correlation between the frequency of events and aggregate risk to the shape
of the distribution of storm intensities and the dispersion statistic. This
equation highlights that the correlation between the two variables is invariant
to a change in the scale of the severity distribution. The theory is applied to
a simulated future dataset from the 2020 Norwegian climate model NorESM2-LM. As
the data observes the theory presented, this opens the door for these results
to be applied to wider geographical regions in future studies.
