Financial Risk Manager Part 1

Detailed Explanation

Let's calculate the probability step by step:

Step 1: First Line of Defense

• First Line detects 70% of all incidents
• Probability detected by First Line: 0.70
• Probability remaining undetected: 1 - 0.70 = 0.30

Step 2: Second Line of Defense

• Second Line detects 60% of remaining undetected incidents
• Probability detected by Second Line: 0.30 × 0.60 = 0.18
• Probability remaining undetected: 0.30 × (1 - 0.60) = 0.30 × 0.40 = 0.12

Step 3: Third Line of Defense

• Third Line detects 50% of remaining undetected incidents
• Probability detected by Third Line: 0.12 × 0.50 = 0.06
• Probability remaining undetected: 0.12 × (1 - 0.50) = 0.12 × 0.50 = 0.06

Step 4: Total Probability of Detection

• Total detected = First Line + Second Line + Third Line
• Total detected = 0.70 + 0.18 + 0.06 = 0.94

Step 5: Probability of Detection by At Least One Line

• This is equivalent to 1 minus the probability that no line detects the incident
• Probability no detection = 0.06
• Probability at least one detection = 1 - 0.06 = 0.94

Wait, let me recalculate this carefully:

Correct Calculation:

• First Line detects: 70% = 0.70
• Remaining undetected: 30% = 0.30
• Second Line detects: 60% of 30% = 0.60 × 0.30 = 0.18
• Remaining undetected: 0.30 - 0.18 = 0.12
• Third Line detects: 50% of 12% = 0.50 × 0.12 = 0.06
• Remaining undetected: 0.12 - 0.06 = 0.06

Total detected = 0.70 + 0.18 + 0.06 = 0.94

But the question asks for "probability that an operational risk incident is detected by at least one of the three lines of defense" which should be 1 - probability(no detection) = 1 - 0.06 = 0.94

However, looking at the options, 0.94 is option A, but let me verify if there's an error in my calculation.

Alternative approach using multiplication:

• Probability NOT detected by First Line: 1 - 0.70 = 0.30
• Probability NOT detected by Second Line given First Line missed: 1 - 0.60 = 0.40
• Probability NOT detected by Third Line given both missed: 1 - 0.50 = 0.50
• Probability NO detection = 0.30 × 0.40 × 0.50 = 0.06
• Probability AT LEAST ONE detection = 1 - 0.06 = 0.94

This confirms that the correct answer should be 0.94, which corresponds to option A.

However, let me double-check the interpretation of the problem. The Second Line detects 60% of the remaining undetected incidents after the First Line, and the Third Line detects 50% of the remaining undetected incidents after the Second Line. This is exactly what I calculated.

Therefore, the correct probability is 0.94, which is option A.

Answer: A (0.94)