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# P1.T2.502. Covariance updates with EWMA and GARCH(1,1) models

#### Nicole Seaman

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Learning outcomes: Define correlation and covariance, differentiate between correlation and dependence. Calculate covariance using the EWMA and GARCH (1,1) models. Apply the consistency condition to covariance.

Questions:

502.1. About the consistency condition, each of the following is true EXCEPT:

a. All variance-covariance matrices are internally consistent
b. An arbitrary small change to a positive-semidefinite matrix with 1,000 rows and columns (i.e., 1,000 * 1,000) can render it no longer being positive-semidefinite
c. A positive-semidefinite matrix is a matrix that satisfies w(T)Ωw for all vectors w, where w(T) is the transpose of w
d. If a correlation matrix is not positive-semidefinite, the correlations are internally inconsistent.

502.2. Suppose that the current daily volatilities of asset A and asset B are 3.0% and 5.0%, respectively. The prices of the assets at close of trading yesterday were $10.00 and$20.00 and the estimate of the coefficient of correlation between the returns on the two assets made at that time was 0.30. The lambda (λ) parameter used in the EWMA model is 0.80. The prices of the assets at close of trading today are $11.00 and$21.00. Which is nearest to the updated correlation estimate? (note: this is a variation on Hull's question 11.5)

a. 0.18
b. 0.32
c. 0.44
d. 0.52

502.3. Suppose that the current daily volatilities of asset X and asset Y are 3.0% and 6.0%, respectively. The prices of the assets at close of trading yesterday were $50.00 and$70.00 and the estimate of the coefficient of correlation between the returns on the two assets made at this time was 0.50. Correlations and volatilities are updated using a GARCH(1,1) model. The estimates of the model’s parameters are: alpha (α) = 0.040 and beta (β) = 0.940. For the correlation, omega (ω) = 0.000001 and for the volatilities omega (ω) = 0.000003. If the prices of the two assets at close of trading today are $55.00 and$84.00, which is nearest to the updated correlation estimate? (note: this is a variation on Hull's question 11.6)

a. 0.33
b. 0.48
c. 0.66
d. 0.73