Why Risk is so Hard to Measure

Overlapping comparisons

The tables include three cases

  • HOverl, H day overlapping estimation
  • sqrtH1day, square root of H time scaling
  • NoOverHday, use non-overlapping H day data

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.

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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]

© All rights reserved, Jon Danielsson, 2018