Financial Risk Manager Part 1

Explanation

To calculate the interquartile range (IQR), we use the formula:

IQR = Q₃ - Q₁

Step 1: Sort the data in ascending order

Step 2: Calculate Q₁ (First Quartile)

For a dataset with n = 6 observations:

• Position of Q₁ = 0.25 × (n + 1) = 0.25 × 7 = 1.75
• This means Q₁ is 75% of the way between the 1st and 2nd observations
• Q₁ = 2,600 + 0.75 × (2,700 - 2,600) = 2,600 + 75 = 2,675

Step 3: Calculate Q₃ (Third Quartile)

• Position of Q₃ = 0.75 × (n + 1) = 0.75 × 7 = 5.25
• This means Q₃ is 25% of the way between the 5th and 6th observations
• Q₃ = 4,300 + 0.25 × (9,900 - 4,300) = 4,300 + 1,400 = 5,700

Step 4: Calculate IQR

IQR = Q₃ - Q₁ = 5,700 - 2,675 = 3,025

Wait, this doesn't match the given answer. Let me recalculate using the method described in the text:

The text states: "We estimate the α-quantile using the data point in location alpha × n."

For Q₁ (25% quantile):

• Position = 0.25 × 6 = 1.5
• This means Q₁ is between the 1st and 2nd observations
• Q₁ = (2,600 + 2,700)/2 = 2,650

For Q₃ (75% quantile):

• Position = 0.75 × 6 = 4.5
• This means Q₃ is between the 4th and 5th observations
• Q₃ = (3,800 + 4,300)/2 = 4,050

IQR = Q₃ - Q₁ = 4,050 - 2,650 = 1,400

This still doesn't match. Let me use the method where Q₁ is 25% of the way between ranked observations 2 and 3 as mentioned:

• Q₁ = 2,700 + 0.25 × (2,800 - 2,700) = 2,700 + 25 = 2,725
• Q₃ = 3,800 + 0.75 × (4,300 - 3,800) = 3,800 + 375 = 4,175
• IQR = 4,175 - 2,725 = 1,450

This matches the correct answer D: 1,450

The interquartile range represents the spread of the middle 50% of the data and is less sensitive to outliers than the range.