The Automated Bias-Corrected and Accelerated Bootstrap Confidence Intervals for Risk Measures

Figures & data

Figure 1. Two-Sided Confidence Limits for VaR for Security Level: (a) p = 0.95 and (b) p = 0.99. Note: Different line styles indicate the method used: BCA-VaR (solid line), par-comp (long dashed line), and non-par (dot-dashed line).

Figure 2. Two-Sided Confidence Limits for CTE for Security Level: (a) p = 0.95 and (b) p = 0.99. Note: Different line styles indicate the method used: BCA-CTE (solid line), par-comp (long dashed line), and non-par (dot-dashed line).

Table 1 Summary of the Interval Estimates for VaR and CTE Based on the Secura Re Dataset

Method	$1-\alpha$	VaR	CTE
p = 0.95
non-par	0.95	(3.660, 5.093)	(4.597, 6.173)
par-comp	0.95	(3.745, 4.407)	(4.529, 5.608)
BCA	0.95	(3.759, 4.528)	(4.550, 5.759)
non-par	0.99	(3.586, 5.127)	(4.349, 6.420)
par-comp	0.99	(3.642, 4.510)	(4.360, 5.777)
BCA	0.99	(3.674, 4.703)	(4.366, 6.046)
p = 0.99
non-par	0.95	(5.343, 7.899)	(6.612, 7.942)
par-comp	0.95	(4.992, 6.308)	(5.804, 7.688)
BCA	0.95	(4.988, 6.525)	(5.903, 8.296)
non-par	0.99	(5.100, 7.899)	(6.403, 8.151)
par-comp	0.99	(4.785, 6.514)	(5.508, 7.984)
BCA	0.99	(4.766, 6.840)	(5.654, 8.742)

Table 2 Summary of the Computational Parametric and BCA Estimates for VaR and CTE Based on the Secura Re Dataset

Measure	p	${\widehat{\sigma }}_{\mathit{par}}$	${\widehat{\sigma }}_{\mathit{boot}}$	${\widehat{z}}_0$	$\widehat{a}$	Skew
VaR	0.95	0.1687	0.1928	0.0790	0.0364	0.2518
	0.99	0.3356	0.3747	0.0301	0.0371	0.1622
CTE	0.95	0.2751	0.3205	0.0238	0.0372	0.2043
	0.99	0.4807	0.5627	0.0916	0.0363	0.2879

Figure 3. Coverage Probability for the Left-Truncated Lognormal Distribution of the Three Estimation Methods: BCA (solid), non-par (dotted), par-comp (dashed). Note: Dashed gray lines denote the expected coverage probability.

Figure 4. Average Width of the Confidence Interval for the Left-Truncated Lognormal Distribution of the Three Estimation Methods: BCA (solid), non-par (dotted), par-comp (dashed).

Table 3 Comparison of the Implementation Methods (%): BCA (Top) and Parametric Asymptotic (Bottom).

Measure	Δwidth	Δcov	${\Delta }_{\text{cov},1}$	${\Delta }_{\text{cov},2}$
BCA approach
VaR	(–0.08, 0.15)	(–0.37, 0.25)	(–4.26, 0.08)	(–4.18, 0.18)
CTE	(–0.63, 0.05 )	(–0.44, 0.34)	(–4.29, 0.34)	(–4.51, 0.23)
Parametric asymptotic approach
VaR	(–8.44, –5.34)	(–6.29, –0.51)	(–6.29, –0.58)	(–3.09, 0.50)
CTE	(–9.27, –6.89)	(–7.00, –0.84)	(–6.66, –1.06)	(–3.51, 0.89)

Note: Δ_width denotes the range of relative differences in percent between method 1 and method 2 for the width of the confidence intervals. Δ_cov denotes the range of relative differences in percent between method 1 and method 2 for the coverage of the confidence intervals. ${\Delta }_{\text{cov},1}$ denotes the range of relative differences in percent between method 1 and the true coverage of the confidence intervals. ${\Delta }_{\text{cov},2}$ denotes the range of relative differences in percent between method 2 and the true coverage of the confidence intervals.