Actuarial science is both an application and combination mathematics and stats that assesses financial risks usually in insurance companies and fields.

Some key things that actuaries answer and deal with are:

• What is the probability a policy holder files a claim?
• What is the expected cost of that claim?
• How much premium should be charged to remain stable?

Actuaries use statistical models to turn uncertainty and unknown variables into known and quantifiable risk.

The foundation of actuarial science is arguably Expected Value:

\[E[X] = \sum_{i} x_i \cdot P(X = x_i)\]

And for a continuous loss distribution (if needed or necessary):

\[E[X] = \int_{0}^{\infty} x \cdot f(x) \, dx\]

The pure premium (the fair price before expenses/profit) is:

\[\text{Pure Premium} = E[\text{Loss}] = E[\text{Frequency}] \times E[\text{Severity}]\] Both Expected Loss and Pure premium are essential for acturaries in order for them to develop and report to companies whether the risks they are taking are good or bad.

Actuaries model loss by using:

• Exponential — simple claims
• Gamma — right-skewed loss
• Log — loss severity modeling
• Pareto — heavy-tailed, and loss potential

For example, if losses \(X \sim \text{Exponential}(\lambda)\):

\[f(x) = \lambda e^{-\lambda x}, \quad x \geq 0\]

\[E[X] = \frac{1}{\lambda}, \quad \text{Var}(X) = \frac{1}{\lambda^2}\]

The aggregate loss \(S\) is the total claims paid by an insurer:

\[S = X_1 + X_2 + \cdots + X_N\]

where \(N\) is the # of claims and \(X_i\) are the actual loss amounts.

From this we get that:

\[E[S] = E[N] \cdot E[X]\]

\[\text{Var}(S) = E[N] \cdot \text{Var}(X) + \text{Var}(N) \cdot (E[X])^2\]

This model separates the how often from the how much (frequency vs severity).

set.seed(42)
policy <- 10000
freq_lambda <- 0.1
smean <- 500
claims <- rpois(policy, lambda = freq_lambda)

#total loss/policy
alosses <- sapply(claims, function(n) {
  if (n == 0) return(0)
  sum(rexp(n, rate = 1 / smean))
})

mean(alosses)
var(alosses)

(Expected loss)/policy: \(E[S] = 0.1 \times 500 = \$50\)

Charging exactly \(E[S]\) is not actually safe, as the insurer would most likely expect to become bankrupt at least 50% of the time.

So, actuaries add a safety loading \(\theta\) in order to avoid this:

\[\text{Gross Premium} = (1 + \theta) \cdot E[S]\]

Using the Value at Risk (VaR) at confidence level \(\alpha\):

\[\text{VaR}_\alpha(S) = F_S^{-1}(\alpha)\]

For example, at \(\alpha = 0.95\), the premium covers losses 95% of the time.

With both premium pricing and safety loading actuaries are able to further inform companies about the risks they are taking including if they are worthwhile or not.