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=3 | ν=10 | ν=∞ |  | |||
λ=0.5 | 1878 | (11.1%) | 1876 | (11.5%) | 1873 | (12.1%) | 1920 |
λ=0.8 | 2218 | (12.4%) | 2213 | (12.9%) | 2208 | (13.6%) | 2314 |
λ=0.9 | 2452 | (13.1%) | 2445 | (13.6%) | 2437 | (14.4%) | 2589 |
λ=0.95 | 2678 | (13.6%) | 2669 | (14.3%) | 2657 | (15.2%) | 2858 |
λ=0.99 | 3195 | (14.7%) | 3179 | (15.6%) | 3155 | (16.8%) | 3482 |
λ=0.995 | 3419 | (15.2%) | 3400 | (16.1%) | 3369 | (17.5%) | 3757 |
λ=0.999 | 3955 | (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=3 | ν=10 | ν=∞ |  | |||
λ=0.5 | 1992 | (12.5%) | 1990 | (13.0%) | 1987 | (13.5%) | 2059 |
λ=0.8 | 2498 | (14.8%) | 2491 | (15.3%) | 2484 | (16.0%) | 2666 |
λ=0.9 | 2865 | (15.9%) | 2856 | (16.5%) | 2844 | (17.2%) | 3118 |
λ=0.95 | 3233 | (16.8%) | 3219 | (17.5%) | 3201 | (18.4%) | 3577 |
λ=0.99 | 4114 | (18.5%) | 4085 | (19.4%) | 4048 | (20.6%) | 4700 |
λ=0.995 | 4513 | (19.0%) | 4473 | (20.1%) | 4428 | (21.3%) | 5215 |
λ=0.999 | 5500 | (20.4%) | 5422 | (22.0%) | 5353 | (23.4%) | 6520 |