Why Risk is so Hard to Measure

Overlapping comparisons

The tables include three cases

The first two cases are reported in the paper.

We show the mean, standard deviation and empirical 99% confidence bound, for both ES and VaR, for two levels of H, 10 and 50 days. Both the Student-t and t-GARCH results are reported.

Student-t

VaR H 10

N alpha HOverlVaR HOverl HOverl NoOverHdayVaR NoOverHday NoOverHday sqrtH1dayVaR sqrtH1day sqrtH1day
300 days 2.50 1.01 (0.81) [0.43,4.48] 1.13 (0.33) [0.62,2.49]
2 years 2.50 1.12 (0.88) [0.50,5.31] 1.08 (0.23) [0.68,1.91]
4 years 2.50 1.24 (1.19) [0.60,6.66] 1.39 (1.26) [0.57,6.97] 1.04 (0.15) [0.75,1.53]
10 years 2.50 1.03 (0.21) [0.70,1.92] 1.23 (0.40) [0.70,3.05] 1.03 (0.089) [0.83,1.29]
50 years 2.50 1.00 (0.080) [0.84,1.26] 1.03 (0.099) [0.82,1.35] 1.02 (0.039) [0.92,1.12]
1000 years 2.50 1.00 (0.017) [0.96,1.05] 1.00 (0.020) [0.95,1.05] 1.01 (0.009) [0.99,1.04]
300 days 3.00 1.00 (0.49) [0.48,3.25] 1.16 (0.29) [0.69,2.30]
2 years 3.00 1.06 (0.54) [0.56,3.65] 1.12 (0.20) [0.75,1.84]
4 years 3.00 1.11 (0.62) [0.65,4.21] 1.25 (0.68) [0.62,4.55] 1.09 (0.13) [0.82,1.52]
10 years 3.00 1.01 (0.15) [0.75,1.61] 1.16 (0.27) [0.74,2.34] 1.08 (0.081) [0.90,1.32]
50 years 3.00 1.00 (0.060) [0.87,1.19] 1.02 (0.078) [0.85,1.26] 1.07 (0.035) [0.98,1.17]
1000 years 3.00 1.00 (0.013) [0.97,1.03] 1.00 (0.016) [0.96,1.04] 1.07 (0.008) [1.05,1.09]
300 days 4.00 0.98 (0.29) [0.53,2.23] 1.17 (0.23) [0.75,2.02]
2 years 4.00 1.02 (0.28) [0.61,2.32] 1.13 (0.17) [0.81,1.70]
4 years 4.00 1.03 (0.26) [0.70,2.33] 1.14 (0.35) [0.66,2.72] 1.11 (0.11) [0.88,1.46]
10 years 4.00 1.00 (0.10) [0.80,1.35] 1.11 (0.18) [0.78,1.78] 1.10 (0.068) [0.94,1.30]
50 years 4.00 1.00 (0.043) [0.90,1.13] 1.02 (0.061) [0.88,1.19] 1.09 (0.030) [1.02,1.17]
1000 years 4.00 1.00 (0.010) [0.98,1.02] 1.00 (0.013) [0.97,1.03] 1.09 (0.007) [1.07,1.11]

ES H 10

N alpha HOverlES HOverl HOverl NoOverHdayES NoOverHday NoOverHday sqrtH1dayES sqrtH1day sqrtH1day
300 days 2.50 0.88 (0.71) [0.38,3.93] 1.06 (0.41) [0.57,2.78]
2 years 2.50 0.95 (0.67) [0.45,4.10] 1.06 (0.33) [0.64,2.32]
4 years 2.50 0.97 (0.50) [0.53,3.19] 1.02 (0.62) [0.49,3.76] 1.05 (0.21) [0.72,1.87]
10 years 2.50 0.99 (0.32) [0.64,2.38] 1.00 (0.34) [0.61,2.43] 1.05 (0.14) [0.82,1.55]
50 years 2.50 1.00 (0.14) [0.78,1.57] 1.00 (0.15) [0.77,1.57] 1.05 (0.060) [0.93,1.25]
1000 years 2.50 1.00 (0.032) [0.93,1.11] 1.00 (0.033) [0.93,1.11] 1.05 (0.013) [1.02,1.09]
300 days 3.00 0.91 (0.45) [0.44,2.97] 1.12 (0.30) [0.66,2.38]
2 years 3.00 0.95 (0.43) [0.52,3.01] 1.11 (0.23) [0.74,2.03]
4 years 3.00 0.97 (0.31) [0.61,2.38] 1.01 (0.37) [0.56,2.75] 1.10 (0.16) [0.82,1.69]
10 years 3.00 0.99 (0.20) [0.71,1.85] 0.99 (0.21) [0.67,1.88] 1.11 (0.10) [0.91,1.46]
50 years 3.00 1.00 (0.088) [0.84,1.33] 1.00 (0.093) [0.82,1.34] 1.11 (0.045) [1.01,1.25]
1000 years 3.00 1.00 (0.020) [0.96,1.06] 1.00 (0.021) [0.95,1.06] 1.11 (0.010) [1.08,1.14]
300 days 4.00 0.93 (0.27) [0.51,2.09] 1.14 (0.22) [0.75,1.96]
2 years 4.00 0.96 (0.24) [0.59,2.04] 1.14 (0.17) [0.82,1.73]
4 years 4.00 0.98 (0.17) [0.68,1.71] 1.00 (0.22) [0.62,1.92] 1.13 (0.11) [0.89,1.51]
10 years 4.00 0.99 (0.11) [0.78,1.42] 0.99 (0.13) [0.74,1.46] 1.13 (0.073) [0.97,1.36]
50 years 4.00 1.00 (0.051) [0.89,1.17] 1.00 (0.058) [0.87,1.18] 1.13 (0.033) [1.06,1.22]
1000 years 4.00 1.00 (0.011) [0.97,1.03] 1.00 (0.013) [0.97,1.04] 1.13 (0.007) [1.11,1.15]

VaR H 50

N alpha HOverlVaR HOverl HOverl NoOverHdayVaR NoOverHday NoOverHday sqrtH1dayVaR sqrtH1day sqrtH1day
300 days 2.50 1.01 (0.81) [0.43,4.48] 1.13 (0.33) [0.62,2.49]
2 years 2.50 1.12 (0.88) [0.50,5.31] 1.08 (0.23) [0.68,1.91]
4 years 2.50 1.24 (1.19) [0.60,6.66] 1.39 (1.26) [0.57,6.97] 1.04 (0.15) [0.75,1.53]
10 years 2.50 1.03 (0.21) [0.70,1.92] 1.23 (0.40) [0.70,3.05] 1.03 (0.089) [0.83,1.29]
50 years 2.50 1.00 (0.080) [0.84,1.26] 1.03 (0.099) [0.82,1.35] 1.02 (0.039) [0.92,1.12]
1000 years 2.50 1.00 (0.017) [0.96,1.05] 1.00 (0.020) [0.95,1.05] 1.01 (0.009) [0.99,1.04]
300 days 3.00 1.00 (0.49) [0.48,3.25] 1.16 (0.29) [0.69,2.30]
2 years 3.00 1.06 (0.54) [0.56,3.65] 1.12 (0.20) [0.75,1.84]
4 years 3.00 1.11 (0.62) [0.65,4.21] 1.25 (0.68) [0.62,4.55] 1.09 (0.13) [0.82,1.52]
10 years 3.00 1.01 (0.15) [0.75,1.61] 1.16 (0.27) [0.74,2.34] 1.08 (0.081) [0.90,1.32]
50 years 3.00 1.00 (0.060) [0.87,1.19] 1.02 (0.078) [0.85,1.26] 1.07 (0.035) [0.98,1.17]
1000 years 3.00 1.00 (0.013) [0.97,1.03] 1.00 (0.016) [0.96,1.04] 1.07 (0.008) [1.05,1.09]
300 days 4.00 0.98 (0.29) [0.53,2.23] 1.17 (0.23) [0.75,2.02]
2 years 4.00 1.02 (0.28) [0.61,2.32] 1.13 (0.17) [0.81,1.70]
4 years 4.00 1.03 (0.26) [0.70,2.33] 1.14 (0.35) [0.66,2.72] 1.11 (0.11) [0.88,1.46]
10 years 4.00 1.00 (0.10) [0.80,1.35] 1.11 (0.18) [0.78,1.78] 1.10 (0.068) [0.94,1.30]
50 years 4.00 1.00 (0.043) [0.90,1.13] 1.02 (0.061) [0.88,1.19] 1.09 (0.030) [1.02,1.17]
1000 years 4.00 1.00 (0.010) [0.98,1.02] 1.00 (0.013) [0.97,1.03] 1.09 (0.007) [1.07,1.11]

ES H 50

N alpha HOverlES HOverl HOverl NoOverHdayES NoOverHday NoOverHday sqrtH1dayES sqrtH1day sqrtH1day
300 days 2.50 0.88 (0.71) [0.38,3.93] 1.06 (0.41) [0.57,2.78]
2 years 2.50 0.95 (0.67) [0.45,4.10] 1.06 (0.33) [0.64,2.32]
4 years 2.50 0.97 (0.50) [0.53,3.19] 1.02 (0.62) [0.49,3.76] 1.05 (0.21) [0.72,1.87]
10 years 2.50 0.99 (0.32) [0.64,2.38] 1.00 (0.34) [0.61,2.43] 1.05 (0.14) [0.82,1.55]
50 years 2.50 1.00 (0.14) [0.78,1.57] 1.00 (0.15) [0.77,1.57] 1.05 (0.060) [0.93,1.25]
1000 years 2.50 1.00 (0.032) [0.93,1.11] 1.00 (0.033) [0.93,1.11] 1.05 (0.013) [1.02,1.09]
300 days 3.00 0.91 (0.45) [0.44,2.97] 1.12 (0.30) [0.66,2.38]
2 years 3.00 0.95 (0.43) [0.52,3.01] 1.11 (0.23) [0.74,2.03]
4 years 3.00 0.97 (0.31) [0.61,2.38] 1.01 (0.37) [0.56,2.75] 1.10 (0.16) [0.82,1.69]
10 years 3.00 0.99 (0.20) [0.71,1.85] 0.99 (0.21) [0.67,1.88] 1.11 (0.10) [0.91,1.46]
50 years 3.00 1.00 (0.088) [0.84,1.33] 1.00 (0.093) [0.82,1.34] 1.11 (0.045) [1.01,1.25]
1000 years 3.00 1.00 (0.020) [0.96,1.06] 1.00 (0.021) [0.95,1.06] 1.11 (0.010) [1.08,1.14]
300 days 4.00 0.93 (0.27) [0.51,2.09] 1.14 (0.22) [0.75,1.96]
2 years 4.00 0.96 (0.24) [0.59,2.04] 1.14 (0.17) [0.82,1.73]
4 years 4.00 0.98 (0.17) [0.68,1.71] 1.00 (0.22) [0.62,1.92] 1.13 (0.11) [0.89,1.51]
10 years 4.00 0.99 (0.11) [0.78,1.42] 0.99 (0.13) [0.74,1.46] 1.13 (0.073) [0.97,1.36]
50 years 4.00 1.00 (0.051) [0.89,1.17] 1.00 (0.058) [0.87,1.18] 1.13 (0.033) [1.06,1.22]
1000 years 4.00 1.00 (0.011) [0.97,1.03] 1.00 (0.013) [0.97,1.04] 1.13 (0.007) [1.11,1.15]

t-GARCH

VaR H 10

N HOverlVaR HOverl HOverl NoOverHdayVaR NoOverHday NoOverHday sqrtH1dayVaR sqrtH1day sqrtH1day
300 days 1.01 (0.33) [0.49,2.38] 1.14 (0.29) [0.68,2.33]
2 years 1.02 (0.29) [0.57,2.28] 1.11 (0.22) [0.72,2.00]
4 years 1.02 (0.22) [0.65,1.95] 1.15 (0.35) [0.62,2.62] 1.08 (0.16) [0.79,1.68]
10 years 1.01 (0.14) [0.75,1.52] 1.12 (0.22) [0.74,2.00] 1.06 (0.100) [0.86,1.40]
50 years 1.00 (0.059) [0.87,1.18] 1.02 (0.076) [0.85,1.25] 1.05 (0.044) [0.95,1.18]
1000 years 1.00 (0.013) [0.97,1.03] 1.00 (0.016) [0.96,1.04] 1.05 (0.010) [1.02,1.07]

ES H 10

N HOverlES HOverl HOverl NoOverHdayES NoOverHday NoOverHday sqrtH1dayES sqrtH1day sqrtH1day
300 days 0.95 (0.30) [0.47,2.20] 1.09 (0.27) [0.67,2.18]
2 years 0.97 (0.26) [0.55,2.06] 1.08 (0.22) [0.72,1.95]
4 years 0.98 (0.20) [0.64,1.80] 1.01 (0.25) [0.58,2.02] 1.07 (0.16) [0.78,1.68]
10 years 0.99 (0.14) [0.74,1.52] 0.99 (0.16) [0.70,1.57] 1.07 (0.11) [0.86,1.45]
50 years 1.00 (0.064) [0.86,1.21] 1.00 (0.072) [0.84,1.23] 1.06 (0.048) [0.96,1.22]
1000 years 1.00 (0.015) [0.96,1.04] 1.00 (0.016) [0.96,1.04] 1.06 (0.011) [1.04,1.09]

VaR H 50

N HOverlVaR HOverl HOverl NoOverHdayVaR NoOverHday NoOverHday sqrtH1dayVaR sqrtH1day sqrtH1day
300 days 1.01 (0.33) [0.49,2.38] 1.14 (0.29) [0.68,2.33]
2 years 1.02 (0.29) [0.57,2.28] 1.11 (0.22) [0.72,2.00]
4 years 1.02 (0.22) [0.65,1.95] 1.15 (0.35) [0.62,2.62] 1.08 (0.16) [0.79,1.68]
10 years 1.01 (0.14) [0.75,1.52] 1.12 (0.22) [0.74,2.00] 1.06 (0.100) [0.86,1.40]
50 years 1.00 (0.059) [0.87,1.18] 1.02 (0.076) [0.85,1.25] 1.05 (0.044) [0.95,1.18]
1000 years 1.00 (0.013) [0.97,1.03] 1.00 (0.016) [0.96,1.04] 1.05 (0.010) [1.02,1.07]

ES H 50

N HOverlES HOverl HOverl NoOverHdayES NoOverHday NoOverHday sqrtH1dayES sqrtH1day sqrtH1day
300 days 0.95 (0.30) [0.47,2.20] 1.09 (0.27) [0.67,2.18]
2 years 0.97 (0.26) [0.55,2.06] 1.08 (0.22) [0.72,1.95]
4 years 0.98 (0.20) [0.64,1.80] 1.01 (0.25) [0.58,2.02] 1.07 (0.16) [0.78,1.68]
10 years 0.99 (0.14) [0.74,1.52] 0.99 (0.16) [0.70,1.57] 1.07 (0.11) [0.86,1.45]
50 years 1.00 (0.064) [0.86,1.21] 1.00 (0.072) [0.84,1.23] 1.06 (0.048) [0.96,1.22]
1000 years 1.00 (0.015) [0.96,1.04] 1.00 (0.016) [0.96,1.04] 1.06 (0.011) [1.04,1.09]