Financial Risk Manager Part 1

This is a Bayes' theorem problem where we need to find the conditional probability that a client was in age range 31-65 given that they died.

Given data:

• Age 16-20: Mortality = 0.04, Portion = 0.1
• Age 21-30: Mortality = 0.05, Portion = 0.29
• Age 31-65: Mortality = 0.10, Portion = 0.49
• Age 66-99: Mortality = 0.14, Portion = 0.12

Let:

• B = event that a client dies
• B₁ = client is in age range 16-20
• B₂ = client is in age range 21-30
• B₃ = client is in age range 31-65
• B₄ = client is in age range 66-99

We need: P(B₃ | B)

Using Bayes' theorem: P(B₃ | B) = [P(B₃) × P(B | B₃)] / [P(B₁) × P(B | B₁) + P(B₂) × P(B | B₂) + P(B₃) × P(B | B₃) + P(B₄) × P(B | B₄)]

Calculations:

• P(B₁) × P(B | B₁) = 0.1 × 0.04 = 0.004
• 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

Total probability of death: 0.004 + 0.0145 + 0.049 + 0.0168 = 0.0843

P(B₃ | B) = 0.049 / 0.0843 = 0.5813 ≈ 0.58 or 58%

Why other options are incorrect:

• B (0.172): This is approximately P(B₂ | B) = 0.0145/0.0843 ≈ 0.172
• C (0.168): This is approximately P(B₄ | B) = 0.0168/0.0843 ≈ 0.199, but 0.168 is actually the numerator for B₄
• D (0.047): This is P(B₃) × P(B | B₃) = 0.049 (rounded to 0.047), which is just the numerator, not the conditional probability