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Table 1 Comparison of the improved risk bound \({\text {TVaR}}_{\lambda }\left (\Sigma _{T,(\rho _{k}),\nu }^{c}\right)\) with the standard comonotonic risk bound \({\text {TVaR}}_{\lambda }\left (\Sigma _{T}^{c}\right)\) for TVaRλ(ΣT) for T=1 year, resp., T=2 years, for different levels λ and for different ν.

From: Upper risk bounds in internal factor models with constrained specification sets

T=1 year\({\text {TVaR}}_{\lambda }\left (\Sigma _{254,(\rho _{k}),\nu }^{c}\right)\)\({\text {TVaR}}_{\lambda }\left (\Sigma _{254}^{c}\right)\)
\({\mathbb {E}} [\Sigma _{254}]=1536\)ν=3ν=10ν= 
λ=0.51878(11.1%)1876(11.5%)1873(12.1%)1920
λ=0.82218(12.4%)2213(12.9%)2208(13.6%)2314
λ=0.92452(13.1%)2445(13.6%)2437(14.4%)2589
λ=0.952678(13.6%)2669(14.3%)2657(15.2%)2858
λ=0.993195(14.7%)3179(15.6%)3155(16.8%)3482
λ=0.9953419(15.2%)3400(16.1%)3369(17.5%)3757
λ=0.9993955(16.4%)3929(17.3%)3876(19.1%)4430
T=2 years\({\text {TVaR}}_{\lambda }(\Sigma _{508,(\rho _{k}),\nu }^{c})\)\({\text {TVaR}}_{\lambda }(\Sigma _{508}^{c})\)
\({\mathbb {E}} [\Sigma _{508}]=1528\)ν=3ν=10ν= 
λ=0.51992(12.5%)1990(13.0%)1987(13.5%)2059
λ=0.82498(14.8%)2491(15.3%)2484(16.0%)2666
λ=0.92865(15.9%)2856(16.5%)2844(17.2%)3118
λ=0.953233(16.8%)3219(17.5%)3201(18.4%)3577
λ=0.994114(18.5%)4085(19.4%)4048(20.6%)4700
λ=0.9954513(19.0%)4473(20.1%)4428(21.3%)5215
λ=0.9995500(20.4%)5422(22.0%)5353(23.4%)6520
  1. The relative DU-improvement given by \(1-\left ({\text {TVaR}}_{\lambda }\left (\Sigma _{T,(\rho _{k}),\nu }^{c}\right)-{\mathbb {E}}[\Sigma _{T}]\right)/\left ({\text {TVaR}}_{\lambda }\left (\Sigma _{T}^{c}\right)-{\mathbb {E}}[\Sigma _{T}]\right)\) is displayed in parentheses