Mathematical and Statistical Methods for Actuarial Sciences and Finance

6 L. Ballester, R. Ferrer, and C. Gonz ́alez

Ta b le 3 .Maximum likelihood estimates of the GARCH-M extended model

3-month interest rate changes

ωi λi θi γi α 0 α 1 βδi

Portfolio L

− 0. 01 ∗∗∗ 0.96∗∗∗ − 1. 12 0.004∗∗∗ 0.0003∗∗∗ 0.09∗∗∗ 0.82∗∗∗ − 15. 04 ∗∗∗

(4.99) (17.60) (− 0 .09) (9.47) (10.22) (5.67) (54.94) (− 8 .56)

Portfolio M

0.02∗∗∗ 0.50∗∗∗ − 1. 31 0.002∗∗∗ 0.0004∗∗∗ 0.15∗∗∗ 0.66∗∗∗ − 13. 88 ∗∗∗

(11.04) (10.16) (− 1 .17) (8.87) (18.63) (4.74) (27.83) (− 12 .98)

Portfolio S

− 0. 15 ∗∗∗ 0.27∗∗∗ − 1. 31 − 0. 02 ∗∗∗ 0.00009∗∗∗0.03∗∗∗ 0.89∗∗∗ —

(− 53 .73) (5.12) (− 1 .17) (− 58 .56) (12.89) (6.33) (148.20) —

10-year interest rate changes

ωi λi θi γi α 0 α 1 βδi

Portfolio L

0.03∗∗∗ 0.89∗∗∗− 6. 80 ∗∗∗ 0.003∗∗∗ 0.0004∗∗∗ 0.14∗∗∗ 0.79∗∗∗ − 45. 34 ∗∗∗

(9.08) (15.38) (− 7 .25) (6.41) (11.02) (6.65) (43.33) (− 8 .97)

Portfolio M

0.04∗∗∗ 0.48∗∗∗− 3. 19 ∗∗∗ 0.005∗∗∗ 0.0003∗∗∗ 0.11∗∗∗ 0.78∗∗∗ − 30. 49 ∗∗∗

(14.29) (8.82) (− 3 .04) (12.18) (19.48) (5.19) (48.76) (− 10 .36)

Portfolio S

− 0. 11 ∗∗∗ 0.25∗∗∗− 3. 28 ∗∗∗− 0. 01 ∗∗∗ 0.00009∗∗∗0.04∗∗∗ 0.87∗∗∗ —

(− 42 .01) (4.26) (− 3 .37) (− 46 .50) (13.17) (6.80) (138.17) —

Interest rate spread

ωi λi θi γi α 0 α 1 βδi

Portfolio L

0.13∗∗∗ 0.95∗∗∗ − 0. 32 0.018∗∗∗ 0.0003∗∗∗ 0.10∗∗ 0.80∗∗∗ − 10. 00 ∗∗∗

(3.24) (15.82) (− 0 .90) (3.10) (2.17) (2.08) (11.16) (− 3 .79)

Portfolio M

0.05∗∗∗ 0.51∗∗∗ 0.03 0.007∗∗∗ 0.0001∗∗∗ 0.08∗∗∗ 0.83∗∗∗ − 9. 10 ∗∗∗

(18.84) (9.71) (0.18) (16.78) (12.45) (5.39) (66.60) (− 5 .40)

Portfolio S

− 0. 18 ∗∗∗ 0.26∗∗∗− 0. 61 ∗∗∗− 0. 03 ∗∗∗ 0.00008∗∗∗0.02∗∗∗ 0.89∗∗∗ —

(− 66 .93) (5.20) (− 3 .23) (− 75 .23) (12.67) (5.90) (155.75) —

This table shows the maximum likelihood estimates of the GARCH(1,1)-M extended model
for the different interest rate proxies based on equations (1)–(3). Values oft-statistics are in
parentheses and∗∗∗,∗∗and∗denote statistical significance at the 1%, 5% and 10% levels,
respectively.