Actuarial Quiz Questions

A Jupyter notebook demonstrating actuarial concepts through two quiz questions covering Year of Account Exposure and Stop Loss pricing.

Setup

python3 -m venv .venv
source .venv/bin/activate
pip install jupyter numpy scipy matplotlib
jupyter notebook actuarial_quiz.ipynb

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Q1: Integral of Year of Account Exposure

This models a year of account with:

• Even attachment of policies throughout the year
• One-year policy terms

At time x=0 (start of year), no exposure has been earned. Exposure builds linearly as policies attach throughout the first year, peaking at x=1 (end of year 1) when all policies are on-risk. During year 2, policies begin to expire, and exposure decreases linearly until x=2 when all policies have run off.

The Year of Account Exposure is defined as:

• For 0 ≤ x ≤ 1: Exposure(x) = x
• For 1 < x ≤ 2: Exposure(x) = 1 - (x - 1) = 2 - x

Step 1: Plot the Year of Account Exposure

A triangular exposure peaking at x = 1.

Step 2: Integrate Exposure(x) to get Incurred(x)

This assumes that claims are proportional to exposure, that all claims are reported instantly and case reserved perfectly.

Incurred(x) = ∫₀ˣ Exposure(t) dt

x	Incurred(x)
0	0.0000
1	0.5000
2	1.0000

Step 3: Plot Year of Account Exposure and Incurred Together

Combined line chart showing both the YOA Exposure and its cumulative integral.

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Q2: Stop Loss at Sigma Excess of Mean Plus Sigma

Normal distribution with:

• Mean (μ) = 60%
• Standard deviation (σ) = 6%

Layer: 6% xs 66% (attachment at μ + σ, width of σ)

PDF Visualization

The probability density function is plotted with:

• Vertical lines at 66% (attachment) and 72% (exhaustion)
• Shaded region indicating the stop loss layer

Step 1: Approximate Loss Cost (Trapezoidal Rule)

Average of 1 - CDF(x) at 66% and 72%, multiplied by layer width (6%).

1 - CDF(66%) = 15.9%
1 - CDF(72%) =  2.3%
Average survival probability = 9.1%

Approximate loss cost (trapezoidal) = 0.544%

Step 2: Integrate 1 - CDF(x) between 66% and 72%

The expected loss to the layer computed as:

E[Layer Loss] = ∫ (1 - F(x)) dx, from 66% to 72%

Integrated loss cost = 0.449%

Step 3: Monte Carlo Simulation

Simulate x 10,000 times and calculate the average of min(max(0, x - 66%), 6%).

Simulated loss cost (n=10,000) = 0.452%
Standard error = 0.014%

Step 4: Compare All Answers

======================================================
COMPARISON OF METHODS
======================================================
Method                             Loss Cost   % Diff
------------------------------------------------------
1. Trapezoidal Approximation          0.544%        -
2. Numerical Integration              0.449%   -17.5%
3. Monte Carlo Simulation             0.452%   -16.9%
======================================================

• The integration method is the most accurate.
• Trapezoidal rule provides a quick approximation.
• Simulation converges to the true value with more iterations.

Step 5: Simulation Convergence

Demonstrates how the Monte Carlo simulation converges to the numerical integration result as the number of simulations increases (up to 100,000).

Final simulation result (n=100,000): 0.448%
Numerical integration result:        0.449%
Difference:                          0.001%

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License

MIT