Financial Risk Manager Part 1

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Explanation:

The correct answer is B: 23.43

Explanation of the Calculation

Covariance between two random variables $X_1$ (Company A profits) and $X_2$ (Company B profits) is given by:

\text{Cov}(X_1, X_2) = E(X_1 X_2) - E(X_1) \cdot E(X_2)

You are given $E(X_1 X_2) = 43.23$ .

Step 1: Compute $E(X_1)$

$X_1$ takes values: -1, 0, 2, 4 (in millions).

Using the joint probability table:

E(X_1) = \sum \sum x_1 \cdot P(X_1 = x_1, X_2 = x_2)

This yields:
$E(X_1) \approx 1.2567$ million

Step 2: Compute $E(X_2)$

$X_2$ takes values: -50, 0, 10, 100 (in millions).

E(X_2) = \sum \sum x_2 \cdot P(X_1 = x_1, X_2 = x_2)

This yields:
$E(X_2) \approx 15.752$ million

Step 3: Compute the Covariance

\text{Cov}(X_1, X_2) = 43.23 - (1.2567 \times 15.752) \approx 43.23 - 19.7955 \approx \mathbf{23.4345}

Rounded to two decimal places, this is 23.43.

(Note: The joint probabilities sum to approximately 1.0005 due to minor rounding in the table, but this has negligible impact on the final result.)

This matches option B exactly.

Key Takeaway for FRM Part 1

Always use the definition of covariance when a joint probability distribution is provided.
You must first calculate the individual expectations $E(X_1)$ and $E(X_2)$ from the margins before subtracting from the given $E(X_1X_2)$ .
This concept frequently appears in the Quantitative Analysis section of the exam.

Explanation:

The correct answer is B: 23.43

Explanation of the Calculation

Covariance between two random variables $X_1$ (Company A profits) and $X_2$ (Company B profits) is given by:

\text{Cov}(X_1, X_2) = E(X_1 X_2) - E(X_1) \cdot E(X_2)

You are given $E(X_1 X_2) = 43.23$ .

Step 1: Compute $E(X_1)$

$X_1$ takes values: -1, 0, 2, 4 (in millions).

Using the joint probability table:

E(X_1) = \sum \sum x_1 \cdot P(X_1 = x_1, X_2 = x_2)

This yields:
$E(X_1) \approx 1.2567$ million

Step 2: Compute $E(X_2)$

$X_2$ takes values: -50, 0, 10, 100 (in millions).

E(X_2) = \sum \sum x_2 \cdot P(X_1 = x_1, X_2 = x_2)

This yields:
$E(X_2) \approx 15.752$ million

Step 3: Compute the Covariance

\text{Cov}(X_1, X_2) = 43.23 - (1.2567 \times 15.752) \approx 43.23 - 19.7955 \approx \mathbf{23.4345}

Rounded to two decimal places, this is 23.43.

(Note: The joint probabilities sum to approximately 1.0005 due to minor rounding in the table, but this has negligible impact on the final result.)

This matches option B exactly.

Key Takeaway for FRM Part 1

Always use the definition of covariance when a joint probability distribution is provided.
You must first calculate the individual expectations $E(X_1)$ and $E(X_2)$ from the margins before subtracting from the given $E(X_1X_2)$ .
This concept frequently appears in the Quantitative Analysis section of the exam.

Comments (1)

Yash ParwaniApr 13, 2026 6:03 AM

There are no tables provided in the question

Q.3747 The yearly profits of the two firms A and B can be summarized in the following probability matrix.

Company A (X₁) Profits
-1 Million 0 Million 2 Million 4 Million
Company B (X₂) Profits -50 Million 0.0197 0.0395 0.010 0.002
0 Million 0.0390 0.230 0.124 0.0298
10 Million 0.011 0.127 0.144 0.0662
100 Million 0 0.0309 0.0656 0.0618

What is the covariance of company A and B given that E(X₁X₂) = 43.23?

		Company A (X₁) Profits
		-1 Million	0 Million	2 Million	4 Million
Company B (X₂) Profits	-50 Million	0.0197	0.0395	0.010	0.002
	0 Million	0.0390	0.230	0.124	0.0298
	10 Million	0.011	0.127	0.144	0.0662
	100 Million	0	0.0309	0.0656	0.0618

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Last updated: April 13, 2026 at 19:12

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Financial Risk Manager Part 1

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Explanation of the Calculation

Step 1: Compute E(X1)E(X_1)E(X1​)

Step 2: Compute E(X2)E(X_2)E(X2​)

Step 3: Compute the Covariance

Key Takeaway for FRM Part 1

Explanation of the Calculation

Step 1: Compute E(X1)E(X_1)E(X1​)

Step 2: Compute E(X2)E(X_2)E(X2​)

Step 3: Compute the Covariance

Key Takeaway for FRM Part 1

Comments (1)

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Comments (1)

Step 1: Compute $E(X_1)$

Step 2: Compute $E(X_2)$

Step 1: Compute $E(X_1)$

Step 2: Compute $E(X_2)$