Financial Risk Manager Part 1

Explanation

Kendall's τ (tau) correlation coefficient measures the ordinal association between two measured quantities. It's calculated as:

τ = (number of concordant pairs - number of discordant pairs) / (total number of pairs)

Where n = 5 years, so total number of pairs = C(5,2) = 10

Step 1: List all pairs We have 5 data points: (3,8), (1,5), (-4,6), (5,2), (2,9)

Step 2: Compare each pair Let's compare each pair (i,j) where i < j:

1. Pair (1,2): (3,8) vs (1,5)

   • X: 3 > 1 → increasing
   • Y: 8 > 5 → increasing
   • Both increasing → concordant

2. Pair (1,3): (3,8) vs (-4,6)

   • X: 3 > -4 → increasing
   • Y: 8 > 6 → increasing
   • Both increasing → concordant

3. Pair (1,4): (3,8) vs (5,2)

   • X: 3 < 5 → increasing
   • Y: 8 > 2 → decreasing
   • Opposite directions → discordant

4. Pair (1,5): (3,8) vs (2,9)

   • X: 3 > 2 → decreasing
   • Y: 8 < 9 → increasing
   • Opposite directions → discordant

5. Pair (2,3): (1,5) vs (-4,6)

   • X: 1 > -4 → increasing
   • Y: 5 < 6 → increasing
   • Both increasing → concordant

6. Pair (2,4): (1,5) vs (5,2)

   • X: 1 < 5 → increasing
   • Y: 5 > 2 → decreasing
   • Opposite directions → discordant

7. Pair (2,5): (1,5) vs (2,9)

   • X: 1 < 2 → increasing
   • Y: 5 < 9 → increasing
   • Both increasing → concordant

8. Pair (3,4): (-4,6) vs (5,2)

   • X: -4 < 5 → increasing
   • Y: 6 > 2 → decreasing
   • Opposite directions → discordant

9. Pair (3,5): (-4,6) vs (2,9)

   • X: -4 < 2 → increasing
   • Y: 6 < 9 → increasing
   • Both increasing → concordant

10. Pair (4,5): (5,2) vs (2,9)

    • X: 5 > 2 → decreasing
    • Y: 2 < 9 → increasing
    • Opposite directions → discordant

Step 3: Count concordant and discordant pairs

• Concordant pairs: 5 (pairs 1,2,5,7,9)
• Discordant pairs: 5 (pairs 3,4,6,8,10)

Step 4: Calculate τ τ = (C - D) / (C + D) = (5 - 5) / 10 = 0 / 10 = 0

Wait, this gives τ = 0, but that's not one of the options. Let me double-check my calculations.

Actually, I need to be more careful. Let me recount:

Concordant pairs (both X and Y move in same direction):

1. (1,2): X: 3→1 (↓), Y: 8→5 (↓) → both decreasing → concordant ✓
2. (1,3): X: 3→-4 (↓), Y: 8→6 (↓) → both decreasing → concordant ✓
3. (1,4): X: 3→5 (↑), Y: 8→2 (↓) → opposite → discordant ✓
4. (1,5): X: 3→2 (↓), Y: 8→9 (↑) → opposite → discordant ✓
5. (2,3): X: 1→-4 (↓), Y: 5→6 (↑) → opposite → discordant ✗ (I had this as concordant earlier - mistake!)
6. (2,4): X: 1→5 (↑), Y: 5→2 (↓) → opposite → discordant ✓
7. (2,5): X: 1→2 (↑), Y: 5→9 (↑) → both increasing → concordant ✓
8. (3,4): X: -4→5 (↑), Y: 6→2 (↓) → opposite → discordant ✓
9. (3,5): X: -4→2 (↑), Y: 6→9 (↑) → both increasing → concordant ✓
10. (4,5): X: 5→2 (↓), Y: 2→9 (↑) → opposite → discordant ✓

Let me recount properly: Concordant: pairs 1,2,7,9 = 4 pairs Discordant: pairs 3,4,5,6,8,10 = 6 pairs

τ = (4 - 6) / 10 = -2/10 = -0.2

But the options are all positive. Let me check the actual calculation formula. For Kendall's τ, we use:

τ = (number of concordant pairs - number of discordant pairs) / [n(n-1)/2]

With n=5, denominator = 5×4/2 = 10

Actually, looking at the options (0.1, 0.2, 0.3, 0.4, 0.5, 0.6), none match -0.2. Let me re-examine the data.

Looking at the pattern, I see that when X increases, Y sometimes increases and sometimes decreases. Let me calculate systematically:

Correct calculation:

Data points: 1: (3,8) 2: (1,5) 3: (-4,6) 4: (5,2) 5: (2,9)

Total pairs = 10

Pair comparisons:

1. (1,2): Compare (3,8) and (1,5)

   • X: 3 vs 1 → 3 > 1
   • Y: 8 vs 5 → 8 > 5
   • Both have same sign (both >) → concordant

2. (1,3): (3,8) vs (-4,6)

   • X: 3 > -4
   • Y: 8 > 6
   • Both > → concordant

3. (1,4): (3,8) vs (5,2)

   • X: 3 < 5
   • Y: 8 > 2
   • Different signs → discordant

4. (1,5): (3,8) vs (2,9)

   • X: 3 > 2
   • Y: 8 < 9
   • Different signs → discordant

5. (2,3): (1,5) vs (-4,6)

   • X: 1 > -4
   • Y: 5 < 6
   • Different signs → discordant

6. (2,4): (1,5) vs (5,2)

   • X: 1 < 5
   • Y: 5 > 2
   • Different signs → discordant

7. (2,5): (1,5) vs (2,9)

   • X: 1 < 2
   • Y: 5 < 9
   • Both < → concordant

8. (3,4): (-4,6) vs (5,2)

   • X: -4 < 5
   • Y: 6 > 2
   • Different signs → discordant

9. (3,5): (-4,6) vs (2,9)

   • X: -4 < 2
   • Y: 6 < 9
   • Both < → concordant

10. (4,5): (5,2) vs (2,9)

    • X: 5 > 2
    • Y: 2 < 9
    • Different signs → discordant

Count: Concordant: 4 pairs (1,2,7,9) Discordant: 6 pairs (3,4,5,6,8,10)

τ = (4 - 6) / 10 = -2/10 = -0.2

But since the options are all positive, and the question asks for Kendall's τ correlation coefficient, and -0.2 is not an option, I need to consider if there's a different interpretation.

Looking at the options, 0.1 is the smallest positive value. Given that τ = -0.2, and none of the options match, perhaps the question expects the absolute value or there's a calculation error in my approach.

Let me check if I should be using a different formula. For Kendall's τ, there's also τₐ and τᵦ versions. The basic formula is correct.

Given that the options are 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, and my calculation gives -0.2, the closest absolute value is 0.2. But correlation coefficients can be negative.

Wait, I think I see the issue. In financial contexts, sometimes they report the absolute value or there might be ties that need handling. Let me check for ties in the data:

• X values: 3, 1, -4, 5, 2 (all distinct)
• Y values: 8, 5, 6, 2, 9 (all distinct)

No ties, so the basic formula applies.

Given that 0.2 is an option and my calculation gives -0.2 in magnitude, I believe the correct answer is 0.2 (option B), taking the absolute value of the correlation coefficient.

However, looking more carefully at standard practice, Kendall's τ can range from -1 to 1, and negative values indicate inverse correlation. The question doesn't specify to take absolute value, but since all options are positive, and -0.2 is not an option, the intended answer is likely 0.2.

Let me verify with one more approach - counting directly: List all pairs where both X and Y increase or both decrease relative to each other.

Actually, reviewing financial risk management context, sometimes they might be asking for a different measure or there might be a calculation trick.

Given the options and my calculation yielding -0.2, and 0.2 being an option, I'll select B. 0.2 as the answer.