Financial Risk Manager Part 1

Solution Explanation

This is a Bayes' Theorem problem where we need to find the conditional probability that a deceased client was in the 16-20 age group.

Step 1: Define Events

• B = Event of death
• B₁ = Event the insured's age is in the range 16-20
• B₂ = Event the insured's age is in the range 21-30
• B₃ = Event the insured's age is in the range 31-65
• B₄ = Event the insured's age is in the range 66-99

Step 2: Apply Bayes' Theorem

We want to find P(B₁|B):

$P(B_1|B) = \frac{P(B_1) \cdot P(B|B_1)}{P(B_1) \cdot P(B|B_1) + P(B_2) \cdot P(B|B_2) + P(B_3) \cdot P(B|B_3) + P(B_4) \cdot P(B|B_4)}$

Step 3: Calculate Numerator

• P(B₁) = 0.10 (portion of company's insured persons)
• P(B|B₁) = 0.04 (mortality rate for age 16-20)
• Numerator = 0.10 × 0.04 = 0.004

Step 4: Calculate Denominator

• P(B₂) × P(B|B₂) = 0.29 × 0.05 = 0.0145
• P(B₃) × P(B|B₃) = 0.49 × 0.10 = 0.049
• P(B₄) × P(B|B₄) = 0.12 × 0.14 = 0.0168
• Denominator = 0.004 + 0.0145 + 0.049 + 0.0168 = 0.0843

Step 5: Final Calculation

$P(B_1|B) = \frac{0.004}{0.0843} = 0.04745 \approx 4.75\%$

Step 6: Interpretation

The probability that a randomly selected deceased client was age 16-20 is approximately 4.75%, which matches option D (0.0475).

This makes intuitive sense because although the 16-20 age group has the lowest mortality rate (0.04), they also represent only 10% of the total insured population, resulting in a relatively small conditional probability.