A Nonparametric Operational Risk Modeling Approach Based on Cornish-Fisher Expansion

Table 4

VaR of operational risk by using Cornish-Fisher expansion

Orders	Cornish-Fisher expansion	VaR (99.9%)
1 order; (𝜅1, 𝜅2)	𝑥𝑝=𝑣𝑝	3380
2 order; (𝜅1, 𝜅2, 𝜅3)	𝑥𝑝=𝑣𝑝+16(𝑣^2𝑝−1)𝜅3	13290
3 order; (𝜅1, 𝜅2, 𝜅3, 𝜅4)	𝑥𝑝=𝑣𝑝+16(𝑣^2𝑝−1)𝜅3+124(𝑣^3𝑝−3𝑣𝑝)𝜅4	84
4 order; (𝜅1, 𝜅2, 𝜅3, 𝜅4)	𝑥𝑝=𝑣𝑝+16(𝑣^2𝑝−1)𝜅3+124(𝑣^3𝑝−3𝑣𝑝)𝜅4−136(2𝑣^3𝑝−5𝑣𝑝)𝜅^23	82
5 order; (𝜅1, 𝜅2, 𝜅3, 𝜅4, 𝜅5)	𝑥𝑝=𝑣𝑝+16(𝑣^2𝑝−1)𝜅3+124(𝑣^3𝑝−3𝑣𝑝)𝜅4−136(2𝑣^3𝑝−5𝑣𝑝)𝜅^23+1120(𝑣^4𝑝−6𝑣^2𝑝+3)𝜅5	67
6 order; (𝜅1, 𝜅2, 𝜅3, 𝜅4, 𝜅5)	𝑥𝑝=𝑣𝑝+16𝑣^2𝑝−1𝜅3+1𝑣24^3𝑝−3𝑣𝑝𝜅4−1362𝑣^3𝑝−5𝑣𝑝𝜅^23; +1120(𝑣^4𝑝−6𝑣^2𝑝+3)𝜅5−124(𝑣^4𝑝−5𝑣^2𝑝+2)𝜅3𝜅4	82
7 order; (𝜅1, 𝜅2, 𝜅3, 𝜅4, 𝜅5)	𝑥𝑝=𝑣𝑝+16𝑣^2𝑝−1𝜅3+1𝑣24^3𝑝−3𝑣𝑝𝜅4−1362𝑣^3𝑝−5𝑣𝑝𝜅^23; +1120(𝑣^4𝑝−6𝑣^2𝑝+3)𝜅5−124(𝑣^4𝑝−5𝑣^2𝑝+2)𝜅3𝜅4+1324(12𝑣^4𝑝−53𝑣^2𝑝+17)𝜅^33	82