Financial Risk Manager Part 1
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A portfolio manager is trying to determine the correlation between the return of two assets. Given the following data about the yearly returns of the stocks, he decides to calculate the Kendall's τ correlation coefficient for the returns of these assets.
| Year | Return of Asset X | Return of Asset Y |
|---|---|---|
| 1 | 3% | 8% |
| 2 | 1% | 5% |
| 3 | –4% | 6% |
| 4 | 5% | 2% |
| 5 | 2% | 9% |
What is Kendall's τ correlation coefficient for the returns of the two assets?
Exam-Like
Community
TTanishq
A
-0.1
B
-0.2
C
0
D
0.1
Explanation:
Calculation of Kendall's τ Correlation Coefficient
Kendall's τ measures the ordinal association between two variables. It is calculated as:
τ = (number of concordant pairs - number of discordant pairs) / (total number of pairs)
Step 1: List the data pairs
- Year 1: (3, 8)
- Year 2: (1, 5)
- Year 3: (-4, 6)
- Year 4: (5, 2)
- Year 5: (2, 9)
Step 2: Compare all pairs
Total number of pairs = C(5,2) = 10
Concordant pairs (both variables move in same direction):
-
(3,8) vs (1,5): X decreases (3→1), Y decreases (8→5) → Concordant
-
(3,8) vs (-4,6): X decreases (3→-4), Y decreases (8→6) → Concordant
-
(3,8) vs (5,2): X increases (3→5), Y decreases (8→2) → Discordant
-
(3,8) vs (2,9): X decreases (3→2), Y increases (8→9) → Discordant
-
(1,5) vs (-4,6): X decreases (1→-4), Y increases (5→6) → Discordant
-
(1,5) vs (5,2): X increases (1→5), Y decreases (5→2) → Discordant
-
(1,5) vs (2,9): X increases (1→2), Y increases (5→9) → Concordant
-
(-4,6) vs (5,2): X increases (-4→5), Y decreases (6→2) → Discordant
-
(-4,6) vs (2,9): X increases (-4→2), Y increases (6→9) → Concordant
-
(5,2) vs (2,9): X decreases (5→2), Y increases (2→9) → Discordant
Step 3: Count concordant and discordant pairs
- Concordant pairs: 4
- Discordant pairs: 6
Step 4: Calculate τ
τ = (4 - 6) / 10 = -2 / 10 = -0.2
Therefore, Kendall's τ correlation coefficient is -0.2, indicating a weak negative correlation between the returns of Asset X and Asset Y.
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