Actuarial Science applies mathematics, statistics, and financial theory to evaluate and manage risk in:

• Insurance
• Pensions
• Health Care
• Finance
• Catastrophe Modeling

Actuaries use statistical models to predict the likelihood of future events.

Actuaries often estimate expected future costs.

The expected loss is calculated as:

\[ E(L) = \sum_{i=1}^{n} p_i \cdot c_i \]

Where:

• \(p_i\) = probability of event \(i\)
  
• \(c_i\) = cost of event \(i\)

This formula is fundamental for pricing insurance policies and managing risk.

Two key random variables used by actuaries:

- Frequency: number of claims \[ N \sim \text{Poisson}(\lambda) \] - Severity: size of each claim \[ X_i \sim \text{Gamma}(\alpha, \theta) \]

Total loss: \[ S = \sum_{i=1}^{n} X_i \]

This is called a compound distribution, central in actuarial modeling.

We simulate claim frequency and severity data for 200 observations:

\(Frequency \sim \text{Poisson}(\lambda = 5)\)
\(Severity \sim \text{Gamma}(\alpha = 3, \theta = 1000)\)

  frequency severity
1         4 1502.918
2         7 2559.937
3         4 4705.641
4         8 1694.868
5         9 2429.312
6         2 3265.805

Code Used: ggplot(act_data, aes(frequency, severity)) + geom_point(alpha = 0.7) + theme_minimal() + labs(title = “Claim Frequency vs Severity”, x = “Number of Claims”, y = “Claim Size(dollars)”)

An insurer must charge a premium P that covers:

• Expected Claim
• Variability Risk
• Profit and Expenses

A simple premium model:

\[ P = E(S) + K \cdot \sqrt(Var(S)) \]

where k adjusts for risk tolerance.

Fit a simple linear model relating severity to frequency

Call:
lm(formula = severity ~ frequency, data = act_data)

Residuals:
    Min      1Q  Median      3Q     Max 
-2665.0 -1281.1  -287.7   865.4  5585.0 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  3138.50     313.73  10.004   <2e-16 ***
frequency     -36.00      57.46  -0.627    0.532    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1667 on 198 degrees of freedom
Multiple R-squared:  0.001979,  Adjusted R-squared:  -0.003061 
F-statistic: 0.3926 on 1 and 198 DF,  p-value: 0.5316

Actuarial workflows often include:

• Modeling claim frequency and claim severity
• Simulating compound distributions for total loss
• Determining premiums with risk loading
• Validating models on historical data

Actuarial Science is about quantifying uncertainty and managing risk effectively.