Financial Risk Manager Part 1

Financial Risk Manager Part 1

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The following sample autocorrelation estimates are obtained using 250 data points:

Lag123
Coefficient0.3-0.15-0.10

Compute the value of the Ljung Box Q statistic.

TTanishq

Explanation:

Explanation

The Ljung Box statistic, also known as the modified Box Pierce statistic, is a function of the accumulated autocorrelations, ρi\rho_i, up to time lag mm. It's calculated as:

Q(m)=n(n+2)βˆ‘i=1m(ρi2nβˆ’i)Q(m) = n(n + 2)\sum_{i=1}^{m} \left( \frac{\rho_i^2}{n - i} \right)

Given:

  • Sample size (n) = 250
  • Time lag (m) = 3
  • Autocorrelations: ρ₁ = 0.3, ρ₂ = -0.15, ρ₃ = -0.10

Calculation:

Q(3)=250Γ—252(0.32249)+250Γ—252((βˆ’0.15)2248)+250Γ—252((βˆ’0.1)2247)Q(3) = 250 \times 252 \left( \frac{0.3^2}{249} \right) + 250 \times 252 \left( \frac{(-0.15)^2}{248} \right) + 250 \times 252 \left( \frac{(-0.1)^2}{247} \right)

Breaking it down:

  1. First term: 250Γ—252Γ—0.09249=63,000Γ—0.0003614=22.77250 \times 252 \times \frac{0.09}{249} = 63,000 \times 0.0003614 = 22.77
  2. Second term: 250Γ—252Γ—0.0225248=63,000Γ—0.0000907=5.71250 \times 252 \times \frac{0.0225}{248} = 63,000 \times 0.0000907 = 5.71
  3. Third term: 250Γ—252Γ—0.01247=63,000Γ—0.0000405=2.55250 \times 252 \times \frac{0.01}{247} = 63,000 \times 0.0000405 = 2.55

Total: 22.77+5.71+2.55=31.0322.77 + 5.71 + 2.55 = 31.03

The Ljung Box Q statistic is approximately 31, which matches option B._

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