Explanation

Kendall's τ correlation coefficient measures the ordinal association between two variables. It's calculated as:

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

Step 1: Rank the data First, let's rank the returns for X and Y:

Step 2: Count concordant and discordant pairs For n=5 observations, total pairs = C(5,2) = 10

Let's examine each pair:

1. (2010,2011): Rank X: 3→5 (increasing), Rank Y: 1→2 (increasing) → Concordant
2. (2010,2012): Rank X: 3→2 (decreasing), Rank Y: 1→4 (increasing) → Discordant
3. (2010,2013): Rank X: 3→1 (decreasing), Rank Y: 1→5 (increasing) → Discordant
4. (2010,2014): Rank X: 3→4 (increasing), Rank Y: 1→3 (increasing) → Concordant
5. (2011,2012): Rank X: 5→2 (decreasing), Rank Y: 2→4 (increasing) → Discordant
6. (2011,2013): Rank X: 5→1 (decreasing), Rank Y: 2→5 (increasing) → Discordant
7. (2011,2014): Rank X: 5→4 (decreasing), Rank Y: 2→3 (increasing) → Discordant
8. (2012,2013): Rank X: 2→1 (decreasing), Rank Y: 4→5 (increasing) → Discordant
9. (2012,2014): Rank X: 2→4 (increasing), Rank Y: 4→3 (decreasing) → Discordant
10. (2013,2014): Rank X: 1→4 (increasing), Rank Y: 5→3 (decreasing) → Discordant

Counts:

• Concordant pairs: 2
• Discordant pairs: 8

Step 3: Calculate Kendall's τ τ = (2 - 8) / 10 = -6 / 10 = -0.6

Wait, let me double-check the calculation...

Actually, τ = (number of concordant - number of discordant) / [n(n-1)/2] = (2 - 8) / 10 = -6/10 = -0.6

But the answer choices show -0.6 as option C, and the correct answer is marked as B (-0.2). Let me recalculate more carefully.

Alternative calculation method: Using the formula: τ = (C - D) / √[(n(n-1)/2 - T_x)(n(n-1)/2 - T_y)]

Where T_x and T_y account for ties (no ties in this data).

Actually, for Kendall's τ with no ties: τ = (C - D) / [n(n-1)/2] = (2 - 8) / 10 = -0.6

However, let me verify the ranking and pairs again:

Looking at the actual return values:

• When X increases from 2010 to 2011 (5%→50%), Y increases from -10% to -5% → Concordant
• When X decreases from 2010 to 2012 (5%→-10%), Y increases from -10% to 20% → Discordant
• When X decreases from 2010 to 2013 (5%→-20%), Y increases from -10% to 40% → Discordant
• When X increases from 2010 to 2014 (5%→30%), Y increases from -10% to 15% → Concordant
• When X decreases from 2011 to 2012 (50%→-10%), Y increases from -5% to 20% → Discordant
• When X decreases from 2011 to 2013 (50%→-20%), Y increases from -5% to 40% → Discordant
• When X decreases from 2011 to 2014 (50%→30%), Y increases from -5% to 15% → Discordant
• When X decreases from 2012 to 2013 (-10%→-20%), Y increases from 20% to 40% → Discordant
• When X increases from 2012 to 2014 (-10%→30%), Y decreases from 20% to 15% → Discordant
• When X increases from 2013 to 2014 (-20%→30%), Y decreases from 40% to 15% → Discordant

Yes, concordant = 2, discordant = 8, τ = -0.6

Given the answer choices and the marked correct answer being B (-0.2), there might be an error in either the question or the provided answer. Based on my calculation, the correct Kendall's τ should be -0.6.