VaR estimation using skewed mixture models and various mixtures of factor analyzers<sup>†</sup>
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(2) 770. Kwangyee Ko · Jangsun Baek. ᄌᄃ ᆼ ᅩ ᅮᅥ ᄁᄋ ᆫᄁ ᅮ ᅩᄅ ᅵᄅ ᆯᄀ ᅳ ᆽᄀ ᅡ ᅩᄋ ᆻᄀ ᅵ ᅥᄂ ᅡᄋ ᅫᄃ ᅩᅪ ᄋᄎ ᆷᄃ ᅥ ᅩᄀ ᅡᄌ ᆫᄌ ᅩ ᅢᄒ ᆫᄃ ᅡ ᅡ. ᄄ ᅡᄅ ᅡᄉ ᅥᄃ ᅡᄇ ᆫᄅ ᅧ ᆼᄌ ᅣ ᆼᄀ ᅥ ᅲᄇ ᆫᄑ ᅮ ᅩᄅ ᆯᄀ ᅳ ᅡᄌ ᆼᄒ ᅥ ᆫᄀ ᅡ ᅵᄌ ᆫᄋ ᅩ ᅴᄇ ᆼᄇ ᅡ ᆸ ᅥ ᆯᄉ ᅳ ᄋ ᆯᄌ ᅵ ᅦᄌ ᅡᄅ ᅭᄋ ᅦᄌ ᆨᄋ ᅥ ᆼᄒ ᅭ ᆻᄋ ᅢ ᆯᄄ ᅳ ᅢᄆ ᆫᄌ ᅡ ᆨᄉ ᅩ ᅳᄅ ᆸᄌ ᅥ ᅵᄆ ᆺᄒ ᅩ ᆫᄀ ᅡ ᆯᄀ ᅧ ᅪᄅ ᆯᄂ ᅳ ᅡᄐ ᅡᄂ ᆫᄃ ᅢ ᅡᄀ ᅩᄆ ᆭᄋ ᅡ ᆫᄆ ᅳ ᆫᄒ ᅮ ᆫᄋ ᅥ ᅦᄉ ᅥᄇ ᅩᄀ ᅩᄒ ᅡᄀ ᅩᄋ ᆻᄃ ᅵ ᅡ. ᄇ ᆫᄑ ᅮ ᅩᄋ ᅦ ᅢᄒ ᄃ ᅡᅧ ᄋᄇ ᅩᄃ ᅡᄋ ᆼᄐ ᅲ ᆼᄉ ᅩ ᆼᄋ ᅥ ᆻᄂ ᅵ ᆫᄀ ᅳ ᅡᄌ ᆼᄋ ᅥ ᆯᄒ ᅳ ᅡᄀ ᅵᄋ ᅱᄒ ᅢᄉ ᅥᄌ ᆼᄀ ᅥ ᅲᄒ ᆫᄒ ᅩ ᆸᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᆯᄉ ᅳ ᅡᄋ ᆼᄒ ᅭ ᆫᅧ ᅡ ᆼ ᄀᄋ ᅮᄋ ᅱᄒ ᆷᄎ ᅥ ᆨᄃ ᅳ ᅩᄋ ᅦᄃ ᅢᄒ ᆫᅥ ᅡ ᆼ ᄌᄒ ᆨᄉ ᅪ ᆼᄋ ᅥ ᅵ ᅩᄀ ᄌ ᆷᄀ ᅳ ᅢᄉ ᆫᄃ ᅥ ᅬᄋ ᆻᄃ ᅥ ᅡ (Venkataraman, 1997). Koᄋ ᅪ Son (2015)ᄋ ᆫᄌ ᅳ ᅮᄉ ᆨᄉ ᅵ ᅮᄋ ᆨᄅ ᅵ ᆯᄋ ᅲ ᅴᄇ ᆫᄃ ᅧ ᆼᄋ ᅩ ᅭᄋ ᆫᄋ ᅵ ᆯᄀ ᅳ ᅵᄋ ᆸᄀ ᅥ ᅩᄋ ᅲᄋ ᅭ ᆫᄀ ᅵ ᄋ ᅪᄀ ᆼᄀ ᅧ ᅵᄇ ᆫᄃ ᅧ ᆼᄀ ᅩ ᆼᄐ ᅩ ᆼᄋ ᅩ ᅭᄋ ᆫᄋ ᅵ ᅳᄅ ᅩᄀ ᅮᄇ ᆫᄒ ᅮ ᆫᄑ ᅡ ᅩᄐ ᅳᄑ ᆯᄅ ᅩ ᅵᄋ ᅩᄋ ᅴ VaR ᄋ ᅨᄎ ᆨᅳ ᅳ ᆯ ᄋᄌ ᅦᄋ ᆫᄒ ᅡ ᅡᄋ ᆻᄋ ᅧ ᅳᄆ ᅧ, Hong ᄃ ᆼ (2016)ᄋ ᅳ ᆫᄃ ᅳ ᅡᄇ ᆫ ᅧ ᆼ Vector at Riskᄅ ᅣ ᄅ ᆯᄌ ᅳ ᅦᄋ ᆫᄒ ᅡ ᅡᄀ ᅩᄐ ᆨᄌ ᅳ ᆼᄑ ᅥ ᅩᄐ ᅳᄑ ᆯᄅ ᅩ ᅵᄋ ᅩᄀ ᅡᄉ ᆯᅥ ᅥ ᆼ ᄌᄃ ᆫᄀ ᅬ ᆼᄋ ᅧ ᅮᄃ ᅢᄋ ᆫᄌ ᅡ ᆨᄋ ᅥ ᆫ VaRᄋ ᅵ ᆯᄎ ᅳ ᅮᄌ ᆼᄒ ᅥ ᅡᄋ ᆻᄃ ᅧ ᅡ. Soltykᄋ ᅪ Gupta (2011)ᄋ ᆫ ᄋ ᅳ ᅫᄃ ᅩᄆ ᅩᄉ ᅮᄅ ᆯ ᄃ ᅳ ᅩᄋ ᆸᄒ ᅵ ᆫ ᄉ ᅡ ᆼᄇ ᅥ ᆫᄇ ᅮ ᆫᄑ ᅮ ᅩᄅ ᆯ ᄀ ᅳ ᅡᄌ ᆫ ᄃ ᅵ ᅡᄇ ᆫᄅ ᅧ ᆼ ᄀ ᅣ ᅵᄋ ᆯᄋ ᅮ ᅥᄌ ᆫ ᄌ ᅵ ᆼᄀ ᅥ ᅲᄒ ᆫᄒ ᅩ ᆸᄆ ᅡ ᅩᄒ ᆼ (skew normal ᅧ mixture model)ᄋ ᆯ ᄌ ᅳ ᆨᄋ ᅥ ᆼᄒ ᅭ ᆫ ᅧ ᅡ ᆼ ᄀᄋ ᅮ ᄉ ᆼᄂ ᅥ ᆼᄋ ᅳ ᅵ ᄒ ᆼᅡ ᅣ ᆼ ᄉᄃ ᅬᄋ ᆻᄃ ᅥ ᅡᄀ ᅩ ᄇ ᅩᄀ ᅩᄒ ᅡᄋ ᆻᄃ ᅧ ᅡ. ᄀ ᅳᄅ ᅥᄂ ᅡ ᄐ ᅮᄌ ᅡᄉ ᅮᄋ ᆨᄋ ᅵ ᅴ ᄇ ᆫᄑ ᅮ ᅩᄂ ᆫ ᅮ ᅳ ᆫ ᄇᄑ ᅩᄋ ᅴ ᅵᄃ ᄇ ᅢᅵ ᄎᄉ ᆼ ᆼᄋ ᅥ ᅵᅬ ᄋᄋ ᅦᄃ ᅩᄃ ᅮᄁ ᅥᄋ ᆫᄁ ᅮ ᅩᄅ ᅵᄂ ᅡᄎ ᆷᄃ ᅥ ᅩᄃ ᆼᄋ ᅳ ᅴᄐ ᆨᄌ ᅳ ᆼᄋ ᅵ ᆯᄀ ᅳ ᅡᄌ ᅵᄀ ᅩᄋ ᆻᄂ ᅵ ᆫᄀ ᅳ ᆼᄋ ᅧ ᅮᄀ ᅡᄉ ᆯᄌ ᅵ ᅦᄌ ᅡᄅ ᅭᄋ ᅦᄉ ᅥᄌ ᅡᄌ ᅮᄂ ᅡᄐ ᅡᄂ ᆫᄃ ᅡ ᅡ. ᅩᅡ ᄀ ᄎᄋ ᆫᄌ ᅯ ᅡᄅ ᅭᄇ ᆫᄑ ᅮ ᅩᄋ ᅴᄀ ᅪᄃ ᅡᄒ ᆫᄆ ᅡ ᅩᄉ ᅮᄅ ᆯᅮ ᅳ ᆯ ᄌᄋ ᅵᄀ ᅵᄋ ᅱᅡ ᆫ 허 ᆯ ᄌᄋ ᆨᄌ ᅣ ᆨᄋ ᅥ ᆫᄒ ᅵ ᆫᄒ ᅩ ᆸᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᅳᄅ ᅩᄉ ᅥᄃ ᅡᄇ ᆫᄅ ᅧ ᆼᄌ ᅣ ᆼᄀ ᅥ ᅲᄇ ᆫᄑ ᅮ ᅩᄅ ᆯᄀ ᅳ ᅡᄌ ᆼᄒ ᅥ ᆫ ᅡ ᅭᄋ ᄋ ᆫᄇ ᅵ ᆫᄉ ᅮ ᆨᄌ ᅥ ᅡᄒ ᆫᄒ ᅩ ᆸᄆ ᅡ ᅩᄒ ᆼ (mixtures of factor analyzers; MFA)ᄋ ᅧ ᅵ Ghahramaniᄋ ᅪ Hinton (1997)ᄋ ᅦᄋ ᅴᄒ ᅡ ᅧᄌ ᄋ ᅦᄋ ᆫᄃ ᅡ ᅬᄋ ᆻᄋ ᅥ ᅳᄆ ᅧ, ᄀ ᅳᄋ ᅵᄒ ᅮᄅ ᅩᄇ ᅵᄌ ᆼᄀ ᅥ ᅲᄇ ᆫᄑ ᅮ ᅩᄋ ᅦᄃ ᅢᄒ ᆫ MFA ᄆ ᅡ ᅩᄒ ᆼᄃ ᅧ ᆯᄋ ᅳ ᅵᄋ ᆫᄀ ᅧ ᅮᅬ ᄃᄋ ᅥᄋ ᆻᄃ ᅪ ᅡ (McLachlan ᄃ ᆼ, 2007; ᅳ Murray ᄃ ᆼ, 2014a). MFA ᄇ ᅳ ᅩᄃ ᅡᄃ ᅩᄃ ᅥᄌ ᆯᄋ ᅥ ᆨᄌ ᅣ ᆨᄋ ᅥ ᆫᄀ ᅵ ᆼᄐ ᅩ ᆼᄋ ᅩ ᅭᄋ ᆫᄇ ᅵ ᆫᄉ ᅮ ᆨᄌ ᅥ ᅡᄒ ᆫᄒ ᅩ ᆸᄆ ᅡ ᅩᄒ ᆼ (mixtures of common factor ᅧ analyzers; MCFA)ᄋ ᅵᄉ ᆼᄇ ᅥ ᆫᅮ ᅮ ᆫ ᄇᄑ ᅩᄀ ᅡᄃ ᅡᄇ ᆫᄅ ᅧ ᆼᄌ ᅣ ᆼᄀ ᅥ ᅲᄇ ᆫᄑ ᅮ ᅩᄋ ᆫᅧ ᅵ ᆼ ᄀᄋ ᅮ (Baek ᄃ ᆼ, 2010)ᄋ ᅳ ᅪᄃ ᅡᄇ ᆫᄅ ᅧ ᆼ t−ᄇ ᅣ ᆫᄑ ᅮ ᅩᄋ ᆫᅧ ᅵ ᆼ ᄀᄋ ᅮ (Baekᄋ ᅪ McLachlan, 2011) ᄀ ᆨᄀ ᅡ ᆨᄌ ᅡ ᅦᄋ ᆫᄃ ᅡ ᅬᄋ ᅥᄀ ᅩᄎ ᅡᄋ ᆫᄋ ᅯ ᅲᄌ ᆫᄌ ᅥ ᅡᄀ ᆫᄌ ᅮ ᆸᄇ ᅵ ᆫᄉ ᅮ ᆨᄋ ᅥ ᅦᄉ ᆼᄀ ᅥ ᆼᄌ ᅩ ᆨᄋ ᅥ ᅳᄅ ᅩᄌ ᆨᄋ ᅥ ᆼᄃ ᅭ ᅬᄋ ᆻᄃ ᅥ ᅡ. ᄎ ᅬᄀ ᆫᄋ ᅳ ᅦ ᆫ ᄀ ᅳ ᄂ ᅵᄋ ᆯᄋ ᅮ ᅥᄌ ᆫ t−ᄇ ᅵ ᆫᄑ ᅮ ᅩᄅ ᆯ ᄋ ᅳ ᅵᄋ ᆼᄒ ᅭ ᆫ ᄀ ᅡ ᆼᄐ ᅩ ᆼ ᄀ ᅩ ᅵᄋ ᆯᄋ ᅮ ᅥᄌ ᆫ−t ᄋ ᅵ ᅭᄋ ᆫᄇ ᅵ ᆫᄉ ᅮ ᆨᄌ ᅥ ᅡ ᄒ ᆫᄒ ᅩ ᆸᄆ ᅡ ᅩᄒ ᆼ (mixtures of common skew-t ᅧ factor analyzers; MCStFA)ᄀ ᅡ Murray ᄃ ᆼ (2014b)ᄋ ᅳ ᅦᄋ ᅴᄒ ᅡᄋ ᅧᄌ ᅦᄋ ᆫᄃ ᅡ ᅬᄋ ᆻᄃ ᅥ ᅡ. ᆫᄋ ᅩ ᄇ ᆫᄀ ᅧ ᅮᄋ ᅦᄉ ᅥᄂ ᆫᄑ ᅳ ᅩᄐ ᅳᄑ ᆯᄅ ᅩ ᅵᄋ ᅩᄅ ᆯᄀ ᅳ ᅮᄉ ᆼᄒ ᅥ ᅡᄂ ᆫᄐ ᅳ ᅮᄌ ᅡᄉ ᅮᄋ ᆨᄋ ᅵ ᅴᄀ ᅢᄉ ᅮᄀ ᅡᄆ ᅢᄋ ᅮᄆ ᆭᄋ ᅡ ᆫᄀ ᅳ ᆼᄋ ᅧ ᅮ, ᄃ ᅡᄇ ᆫᄅ ᅧ ᆼᄀ ᅣ ᅵᄋ ᆯᄋ ᅮ ᅥᄌ ᆫᅥ ᅵ ᆼ ᄌᄀ ᅲᄇ ᆫᄑ ᅮ ᅩ ᆫᄒ ᅩ ᄒ ᆸᅩ ᅡ ᄆᄒ ᆼ (MSN), ᄃ ᅧ ᅡᄇ ᆫᄅ ᅧ ᆼ−tᄇ ᅣ ᆫᄑ ᅮ ᅩᄒ ᆫᄒ ᅩ ᆸᄆ ᅡ ᅩᄒ ᆼ (MSt), MFA, MCFA, MCtFAᄅ ᅧ ᆯᄌ ᅳ ᆨᄋ ᅥ ᆼᄒ ᅭ ᅡᄋ ᅧᄑ ᅩᄐ ᅳᄑ ᆯᄅ ᅩ ᅵᄋ ᅩᄇ ᆫᄑ ᅮ ᅩ ᆯᄌ ᅳ ᄅ ᆼᄒ ᅥ ᆨᄒ ᅪ ᅡᄀ ᅦᄎ ᅮᄌ ᆼᄒ ᅥ ᆫᄒ ᅡ ᅮᄋ ᅥᄂ ᅳᄆ ᅩᄒ ᆼᄋ ᅧ ᅵᄉ ᆫᄅ ᅵ ᅬᄉ ᆼᄋ ᅥ ᆻᄂ ᅵ ᆫ VaRᄋ ᅳ ᆯᄋ ᅳ ᅨᄎ ᆨᄒ ᅳ ᅡᄂ ᆫᄌ ᅳ ᅵᄇ ᆫᄉ ᅮ ᆨᄒ ᅥ ᅡᄀ ᅩᄌ ᅡᄒ ᆫᄃ ᅡ ᅡ.. 2. 고차원 수익률분포 적합 ᄎᄀ ᅬ ᆫᄀ ᅳ ᅵᄋ ᆯᄋ ᅮ ᅥᄌ ᆫᄌ ᅵ ᆼᄀ ᅥ ᅲᄇ ᆫᄑ ᅮ ᅩ (skew normal distribution)ᄂ ᅡᄀ ᅵᄋ ᆯᄋ ᅮ ᅥᄌ ᆫ t−ᄇ ᅵ ᆫᄑ ᅮ ᅩ (skew-t distribution)ᄋ ᅪᄀ ᇀ ᅡ ᅵᄀ ᄋ ᅵᄋ ᆯᄋ ᅮ ᅥᄌ ᆫᄇ ᅵ ᆫᄑ ᅮ ᅩᄌ ᆨᄋ ᅩ ᅵᄀ ᆫᄉ ᅪ ᆷᄋ ᅵ ᆯᄆ ᅳ ᆭᄋ ᅡ ᅵᄇ ᆮᄀ ᅡ ᅩᄋ ᆻᄃ ᅵ ᅡ. ᄋ ᅵᄇ ᆫᄑ ᅮ ᅩᄌ ᆨᄋ ᅩ ᆫᄀ ᅳ ᅵᄋ ᆯᄋ ᅮ ᅥᄌ ᆫᄌ ᅵ ᆼᄀ ᅥ ᅲᄇ ᆫᄑ ᅮ ᅩ (Azzalini, 1985)ᄋ ᅦᄋ ᅫ ᅩᄃ ᄃ ᆼᄃ ᅳ ᅡᄅ ᆫᄐ ᅳ ᆨᄌ ᅳ ᆼᄃ ᅵ ᆯᅳ ᅳ ᆯ ᄋᄀ ᅲᄌ ᆼᄒ ᅥ ᅡᄂ ᆫᄎ ᅳ ᅮᄀ ᅡᄌ ᆨᆫ ᅥ ᄋ ᅵᄆ ᅩᄉ ᅮᄃ ᆯᅳ ᅳ ᆯ ᄋᄃ ᅩᄋ ᆸᄒ ᅵ ᅡᄋ ᅧᄒ ᆨᄌ ᅪ ᆼᄒ ᅡ ᆫᄀ ᅡ ᆺᄋ ᅥ ᅵᄃ ᅡ (Genton, 2004; ArellanoValleᄋ ᅪ Azzalini, 2006; Genton, 2006). ᆨᄅ ᅪ ᄒ ᆯᄇ ᅲ ᆨᄐ ᅦ ᅥ Yᄀ ᅡᄋ ᅱᄎ ᅵᄇ ᆨᄐ ᅦ ᅥ µ, ᄎ ᆨᄃ ᅥ ᅩᄒ ᆼᅧ ᅢ ᆯ ᄅ Σ, ᄀ ᅳᄅ ᅵᄀ ᅩᄋ ᅫᄃ ᅩᄆ ᅩᄉ ᅮ αᄅ ᆯᄀ ᅳ ᆽᄂ ᅡ ᆫᄃ ᅳ ᅡᄇ ᆫᄅ ᅧ ᆼᄀ ᅣ ᅵᄋ ᆯᄋ ᅮ ᅥᄌ ᆫᅥ ᅵ ᆼ ᄌᄀ ᅲᄇ ᆫᄑ ᅮ ᅩᄅ ᆯᄄ ᅳ ᅡ ᆫᄃ ᅳ ᄅ ᅡᄆ ᆫ, ᄀ ᅧ ᅳᄀ ᆺᄋ ᅥ ᅴᄒ ᆨᄅ ᅪ ᆯᄆ ᅲ ᆯᄃ ᅵ ᅩᄒ ᆷᄉ ᅡ ᅮᄂ ᆫᄃ ᅳ ᅡᄋ ᆷᄀ ᅳ ᅪᄀ ᇀᄃ ᅡ ᅡ. f (y; µ, Σ, α) = 2ϕp (y; µ, Σ)Φp (αT Σ−1 (y − µ); 0, 1 − αT Σ−1 α).. (2.1). ᄋᄄ ᅵ ᅢ, ϕp (·; µ, Σ)ᄂ ᆫᄑ ᅳ ᆼᄀ ᅧ ᆫᄇ ᅲ ᆨᄐ ᅦ ᅥ µᄋ ᅪᄀ ᆼᄇ ᅩ ᆫᄉ ᅮ ᆫᄒ ᅡ ᆼᅧ ᅢ ᆯ ᄅ Σᄅ ᆯᄀ ᅳ ᆽᄂ ᅡ ᆫ pᄎ ᅳ ᅡᄋ ᆫᄌ ᅯ ᆼᄀ ᅥ ᅲᄇ ᆫᄑ ᅮ ᅩᄒ ᆨᄅ ᅪ ᆯᄆ ᅲ ᆯᄃ ᅵ ᅩᄒ ᆷᄉ ᅡ ᅮᄋ ᅵᄆ ᅧ, Φp (·; µ, Σ)ᄂ ᆫ ᅳ ᅳᄋ ᄀ ᅦᄉ ᆼᄋ ᅡ ᆼᄒ ᅳ ᅡᄂ ᆫᅮ ᅳ ᆫ ᄇᄑ ᅩᄒ ᆷᄉ ᅡ ᅮᄋ ᅵᄃ ᅡ. ᆫᆯ ᅡ ᄆ ᅵ Yᄀ ᄋ ᅡᄃ ᅡᄇ ᆫᄅ ᅧ ᆼᄀ ᅣ ᅵᄋ ᆯᄋ ᅮ ᅥᄌ ᆫ−t ᄇ ᅵ ᆫᄑ ᅮ ᅩᄅ ᆯᄄ ᅳ ᅡᄅ ᆫᄃ ᅳ ᅡᄆ ᆫ, ᄀ ᅧ ᅳᄀ ᆺᄋ ᅥ ᅴᄒ ᆨᄅ ᅪ ᆯᄆ ᅲ ᆯᄃ ᅵ ᅩᄒ ᆷᄉ ᅡ ᅮᄂ ᆫᄃ ᅳ ᅡᄋ ᆷᄀ ᅳ ᅪᄀ ᇀᄃ ᅡ ᅡ. p (−ν−p)/4 ν ν/2 K(−ν−p)/2 ( [αT Σ−1 α][ν + δ(y, µ, Σ)]) ν + δ(y, µ, Σ) . (2.2) f (y; µ, Σ, α, ν) = T −1 α Σ α (2π)p/2 |Σ|1/2 Γ(ν/2)2ν/2−1 exp{(µ − y)T Σ−1 α} ᄋᄄ ᅵ ᅢ, µ, Σ, α, νᄂ ᆫᄀ ᅳ ᆨᄀ ᅡ ᆨᄋ ᅡ ᅱᄎ ᅵ, ᄎ ᆨᄃ ᅥ ᅩᄒ ᆼᄅ ᅢ ᆯ, ᄋ ᅧ ᅫᄃ ᅩᄆ ᅩᄉ ᅮ, ᄀ ᅳᄅ ᅵᄀ ᅩᄌ ᅡᄋ ᅲᄃ ᅩᄋ ᅵᄆ ᅧ, δ(y, µ, Σ) = (y − µ)T Σ−1 (y − µ)ᄋ ᅵᅡ ᄃ. ᄄ ᅩᄒ ᆫ, Kλ ᄂ ᅡ ᆫᄌ ᅳ ᅵᄉ ᅮ λᄋ ᆯᄀ ᅳ ᆽᄂ ᅡ ᆫᄉ ᅳ ᅮᄌ ᆼ Bessel ᄒ ᅥ ᆷᄉ ᅡ ᅮᄋ ᅵᄃ ᅡ. ᆫᄋ ᅩ ᄇ ᆫᄀ ᅧ ᅮᄋ ᅦᄉ ᅥᄂ ᆫᅩ ᅳ ᆫ ᄒᄒ ᆸᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᅦᄋ ᅴᄒ ᅡᄋ ᅧᄐ ᅮᄌ ᅡᄉ ᅮᄋ ᆨᄇ ᅵ ᆫᄑ ᅮ ᅩᄅ ᆯᄎ ᅳ ᅮᄌ ᆼᄒ ᅥ ᅡᄅ ᅧᄀ ᅩᄒ ᆫᄃ ᅡ ᅡ. gᄀ ᅢᄋ ᅴᄉ ᆼᄇ ᅥ ᆫᅳ ᅮ ᆯ ᄋᄀ ᆽᄂ ᅡ ᆫᅩ ᅳ ᆫ ᄒᄒ ᆸᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᅴ ᆫᄑ ᅮ ᄇ ᅩᄂ ᆫᄃ ᅳ ᅡᄋ ᆷᄀ ᅳ ᅪᄀ ᇀᄋ ᅡ ᅵ gᄀ ᅢᄋ ᅴᄉ ᆼᄇ ᅥ ᆫᅮ ᅮ ᆫ ᄇᄑ ᅩᄋ ᅴᄉ ᆫᅧ ᅥ ᆼ 혀 ᆯ ᄀᄒ ᆸᄋ ᅡ ᅳᄅ ᅩᄌ ᆼᄋ ᅥ ᅴᄃ ᆫᄃ ᅬ ᅡ. f (y; Ψ) =. g X i=1. πi f (y; θi )..
(3) VaR estimation using skewed mixture models and various mixtures of factor analyzers. 771. ᄋᄄ ᅵ ᅢ , θi ᄂ ᆫ iᄇ ᅳ ᆫᄍ ᅥ ᅢᄉ ᆼᄇ ᅥ ᆫᅮ ᅮ ᆫ ᄇᄑ ᅩᄋ ᅴᄆ ᅵᄌ ᅵᄋ ᅴᄆ ᅩᄃ ᆫᄆ ᅳ ᅩᄉ ᅮᄃ ᆯᅳ ᅳ ᆯ ᄋᄀ ᅳᄋ ᆫᄋ ᅡ ᅦᄑ ᅩᄒ ᆷᄒ ᅡ ᆫᄇ ᅡ ᆨᄐ ᅦ ᅥᄋ ᅵᄆ ᅧ, ᄒ ᆫᄒ ᅩ ᆸᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᅴᄌ ᆫᄎ ᅥ ᅦᄆ ᅩᄉ ᅮᄇ ᆨ ᅦ T T T ᅥᄂ ᄐ ᆫ Ψ = (π1 , · · · , πg−1 , θ1 , · · · , θg ) ᄋ ᅳ ᅵᄃ ᅡ. ᄒ ᆫᄒ ᅩ ᆸᄇ ᅡ ᅵᄋ ᆯ πi ᄃ ᅲ ᆯᄋ ᅳ ᆫᄇ ᅳ ᅵᄋ ᆷᄉ ᅳ ᅮᄋ ᅵᄀ ᅩ, ᄆ ᅩᄃ ᅮᄒ ᆸᄒ ᅡ ᅡᄆ ᆫ 1ᄋ ᅧ ᅵᄃ ᅡ. 2.1. 적합모형 ᄇᄋ ᆫ ᅩ ᆫᄀ ᅧ ᅮᄋ ᅦᄉ ᅥᄂ ᆫᄐ ᅳ ᅮᄌ ᅡᄉ ᅮᄋ ᆨᄅ ᅵ ᆯᄇ ᅲ ᆨᄐ ᅦ ᅥ Y = (Y1 , Y2 , · · · , Yp )T ᄅ ᆯᄇ ᅳ ᅵᄃ ᅢᄎ ᆼᄌ ᅵ ᆨᄇ ᅥ ᆫᄑ ᅮ ᅩᄅ ᆯᄉ ᅳ ᆼᄇ ᅥ ᆫᅮ ᅮ ᆫ ᄇᄑ ᅩᄅ ᅩᄀ ᅡᄌ ᆫᄃ ᅵ ᅡᄉ ᆺᄀ ᅥ ᅡ P ᅴ VaRᄋ ᆯᄋ ᅳ ᅨᄎ ᆨᄒ ᅳ ᅡᄋ ᅧ ᅵᄒ ᄌ ᆫᄒ ᅩ ᆸᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᅦᄌ ᆨᄒ ᅥ ᆸᄉ ᅡ ᅵᄏ ᅵᄀ ᅩ, ᄀ ᆨᄀ ᅡ ᆨᄋ ᅡ ᅴᄎ ᅮᄌ ᆼᄆ ᅥ ᅩᄒ ᆼᄋ ᅧ ᅦᄄ ᅡᄅ ᆫᄎ ᅳ ᆼᄐ ᅩ ᅮᄌ ᅡᄉ ᅮᄋ ᆨᄅ ᅵ ᆯ YR = pi=1 Yi ᄋ ᅲ ᅡᄌ ᄀ ᆼᄌ ᅡ ᆼᄒ ᅥ ᆨᄃ ᅪ ᅩᄀ ᅡᄂ ᇁᄋ ᅩ ᆫᄆ ᅳ ᅩᄒ ᆼᄋ ᅧ ᆯᄉ ᅳ ᆫᅢ ᅥ ᆨ ᄐᄒ ᆫᄃ ᅡ ᅡ. Y ᄀ ᅡ gᄀ ᅢᄉ ᆼᄇ ᅥ ᆫᄋ ᅮ ᆯᄀ ᅳ ᅡᄌ ᆫᄃ ᅵ ᅡᄇ ᆫᄅ ᅧ ᆼᄀ ᅣ ᅵᄋ ᆯᄋ ᅮ ᅥᄌ ᆫᅥ ᅵ ᆼ ᄌᄀ ᅲᄇ ᆫᄑ ᅮ ᅩᄒ ᆫᄒ ᅩ ᆸᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᆯᄄ ᅳ ᅡ ᆫᄃ ᅳ ᄅ ᅡᄆ ᆫ, ᄉ ᅧ ᆨ (2.1)ᄅ ᅵ ᅩᄑ ᅭᄒ ᆫᄃ ᅧ ᆫᄒ ᅬ ᆨᄅ ᅪ ᆯᄆ ᅲ ᆯᄃ ᅵ ᅩᄒ ᆷᄉ ᅡ ᅮᄅ ᆯᄀ ᅳ ᆽᄂ ᅡ ᆫ iᄇ ᅳ ᆫᄍ ᅥ ᅢᄉ ᆼᄇ ᅥ ᆫᅮ ᅮ ᆫ ᄇᄑ ᅩᄅ ᆯ f (y; µi , Σi , αi )ᄅ ᅳ ᅡᄀ ᅩᄒ ᆯᄄ ᅡ ᅢ, ᄀ ᅵᄋ ᆯᄋ ᅮ ᅥᄌ ᆫ ᅵ ᆼᄀ ᅥ ᄌ ᅲᄇ ᆫᄑ ᅮ ᅩᄒ ᆫᄒ ᅩ ᆸᄆ ᅡ ᅩᄒ ᆼ (MSN: mixtures of skew-normal distribution)ᄋ ᅧ ᅴᄆ ᆯᄃ ᅵ ᅩᄒ ᆷᄉ ᅡ ᅮᄂ ᆫᄃ ᅳ ᅡᄋ ᆷᄀ ᅳ ᅪᄀ ᇀᄃ ᅡ ᅡ. f (y; Ψ) =. g X. πi f (y; µi , Σi , αi ).. (2.3). i=1. ᄃᄇ ᅮ ᆫᄍ ᅥ ᅢᄌ ᆨᄋ ᅥ ᆼᄒ ᅭ ᆯᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᆫᄀ ᅳ ᅵᄋ ᆯᄋ ᅮ ᅥᄌ ᆫtᄒ ᅵ ᆫᄒ ᅩ ᆸᄆ ᅡ ᅩᄒ ᆼ (MSt: mixtures of skew-t distribution)ᄋ ᅧ ᅵᄃ ᅡ. Yᄀ ᅡ gᄀ ᅢ ᆼᄇ ᅥ ᄉ ᆫᅳ ᅮ ᆯ ᄋᄀ ᅡᄌ ᆫᄃ ᅵ ᅡᄇ ᆫᄅ ᅧ ᆼᄀ ᅣ ᅵᄋ ᆯᄋ ᅮ ᅥᄌ ᆫtᄒ ᅵ ᆫᄒ ᅩ ᆸᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᆯᄄ ᅳ ᅡᄅ ᆫᄃ ᅳ ᅡᄆ ᆫ, ᄉ ᅧ ᆨ (2.2)ᄅ ᅵ ᅩᄑ ᅭᄒ ᆫᄃ ᅧ ᆫᄒ ᅬ ᆨᄅ ᅪ ᆯᄆ ᅲ ᆯᄃ ᅵ ᅩᄒ ᆷᄉ ᅡ ᅮᄅ ᆯᄀ ᅳ ᆽᄂ ᅡ ᆫ iᄇ ᅳ ᆫᄍ ᅥ ᅢᄉ ᆼ ᅥ ᆫᅮ ᅮ ᄇ ᆫ 보 ᄑᄅ ᆯ f (y; µi , Σi , αi , νi )ᄅ ᅳ ᅡᄀ ᅩᄒ ᆯᄄ ᅡ ᅢ, ᄀ ᅵᄋ ᆯᄋ ᅮ ᅥᄌ ᆫtᄒ ᅵ ᆫᄒ ᅩ ᆸᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᅴᄆ ᆯᄃ ᅵ ᅩᄒ ᆷᄉ ᅡ ᅮᄂ ᆫᄃ ᅳ ᅡᄋ ᆷᄀ ᅳ ᅪᄀ ᇀᄃ ᅡ ᅡ: f (y; Ψ) =. g X. πi f (y; µi , Σi , αi , νi ).. (2.4). i=1. ᄃᄋ ᅡ ᆷᄋ ᅳ ᅳᄅ ᅩᄀ ᅩᄎ ᅡᄋ ᆫᄌ ᅯ ᅡᄅ ᅭᄇ ᆫᄑ ᅮ ᅩᄅ ᆯᄌ ᅳ ᆨᄒ ᅥ ᆸᄉ ᅡ ᅵᄏ ᅵᄀ ᅵᄋ ᅱᅡ ᆫ ᄒᄌ ᆯᄋ ᅥ ᆨᄆ ᅣ ᅩᄒ ᆼᄋ ᅧ ᆫ MFA, MCFA, MCtFA ᄆ ᅵ ᅩᄒ ᆼᄋ ᅧ ᅴᄀ ᆫᄀ ᅪ ᅨᄉ ᆨᄀ ᅵ ᅪ ᆫᄒ ᅩ ᄒ ᆸᄇ ᅡ ᆫᄑ ᅮ ᅩᄂ ᆫᄀ ᅳ ᆨᄀ ᅡ ᆨᄃ ᅡ ᅡᄋ ᆷᄀ ᅳ ᅪᄀ ᇀᄃ ᅡ ᅡ. M F A : y = µi + Λi Ui + ϵi , Ui ∼ N (0, Iq ), ϵi ∼ N (0, Di ), f (y; Ψ) =. g X. πi fi (y; µi , Λi ΛTi + Di ).. (2.5). i=1. M CF A : y = ΛUi + ϵi , Ui ∼ N (ξi , Ωi ), ϵi ∼ N (0, D), f (y; Ψ) =. g X. πi fi (y; Λξi , ΛΩi ΛT + D).. (2.6). i=1. M CtF A : y = ΛUi + ϵi , Ui ∼ tq (ξi , Ωi , νi ), ϵi ∼ tp (0, D, νi ), f (y; Ψ) =. g X. πi tp (y; Λξi , ΛΩi ΛT + D, νi ).. (2.7). i=1. ᆫᄋ ᅩ ᄇ ᆫᄀ ᅧ ᅮᄋ ᅦᄉ ᅥᄂ ᆫ KOSPI 200 ᄌ ᅳ ᅵᄉ ᅮᄅ ᆯᄋ ᅳ ᅵᄅ ᅮᄂ ᆫᄉ ᅳ ᅵᄀ ᅡᄎ ᆼᄋ ᅩ ᆨᄉ ᅢ ᆼᄋ ᅡ ᅱ 200ᄀ ᅢᄒ ᅬᄉ ᅡᄋ ᅴᄌ ᅮᄉ ᆨᄋ ᅵ ᆯᄃ ᅳ ᅢᄉ ᆼᄋ ᅡ ᅳᄅ ᅩ 28ᄀ ᅢᄌ ᅮᄉ ᆨᄋ ᅵ ᆯ ᅳ 새 ᆫ ᅥ ᆨ 태 ᄒᄉ ᅥᄐ ᅮᄌ ᅡᄑ ᅩᄐ ᅳᄑ ᆯᄅ ᅩ ᅵᄋ ᅩᄅ ᆯᄀ ᅳ ᅮᄉ ᆼᄒ ᅥ ᅡᄀ ᅩ, ᄀ ᅳᄀ ᆺᄋ ᅥ ᅴᄇ ᆫᄑ ᅮ ᅩᄅ ᆯᄋ ᅳ ᅱᄋ ᅴ MFA, MCFA, MCtFA, MSN, ᄀ ᅵᄋ ᆯᄋ ᅮ ᅥᄌ ᆫ t ᅵ ᆫᄒ ᅩ ᄒ ᆸᅩ ᅡ ᄆᄒ ᆼ (MSt)ᄋ ᅧ ᆯᄌ ᅳ ᆨᄋ ᅥ ᆼᄒ ᅭ ᅡᄋ ᅧᄎ ᅮᄌ ᆼᄒ ᅥ ᆫᄃ ᅡ ᅡ. 2.2. EM 알고리즘을 이용한 모수추정 배 ᅵ ᄃᄎ ᆼᄒ ᅵ ᆫᄒ ᅩ ᆸᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᅴᄆ ᅩᄉ ᅮᄃ ᆯᄋ ᅳ ᆫ EM (expectation-maximization) ᄋ ᅳ ᆯᄀ ᅡ ᅩᄅ ᅵᄌ ᆷ (Dempster ᄃ ᅳ ᆼ, 1997)ᄋ ᅳ ᅦᄋ ᅴ ᅡᄋ ᄒ ᅧᄒ ᅭᅪ ᄀᄌ ᆨᄋ ᅥ ᅳᄅ ᅩᄎ ᅮᄌ ᆼᄃ ᅥ ᆯᄉ ᅬ ᅮᄋ ᆻᄃ ᅵ ᅡ. EM ᄋ ᆯᄀ ᅡ ᅩᄅ ᅵᄌ ᆷᄋ ᅳ ᆫᅩ ᅳ ᆫ ᄒᄒ ᆸᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᅴᄆ ᅩᄉ ᅮᄋ ᅦᄃ ᅢᄒ ᆫᄎ ᅡ ᅬᄋ ᅮᄎ ᅮᄌ ᆼᄎ ᅥ ᅵᄅ ᆯᄀ ᅳ ᅨᄉ ᆫᅡ ᅡ ᆯ ᄒᄄ ᅢᄉ ᅡ ᆼᄃ ᅭ ᄋ ᅬᄂ ᆫᄑ ᅳ ᅭᄌ ᆫᄌ ᅮ ᆨᄋ ᅥ ᆫᄃ ᅵ ᅩᄀ ᅮᄋ ᅵᄃ ᅡ. ᄋ ᅵᄇ ᆼᄇ ᅡ ᆸᄋ ᅥ ᆫᄆ ᅳ ᅩᄉ ᅮᄎ ᅮᄌ ᆼᄎ ᅥ ᅵᄃ ᆯᄋ ᅳ ᅵᄉ ᅮᄅ ᆷᄃ ᅧ ᆯᄄ ᅬ ᅢᄁ ᅡᄌ ᅵ E-ᄃ ᆫᄀ ᅡ ᅨᅪ ᄋ M-ᄃ ᆫᄀ ᅡ ᅨᄅ ᆯᄀ ᅳ ᅭᄃ ᅢᄅ ᅩᄌ ᆨᄋ ᅥ ᆼ ᅭ ᅡᄋ ᄒ ᅧᄃ ᆫᄀ ᅡ ᅨᄌ ᆨᄋ ᅥ ᅳᄅ ᅩᄎ ᅮᄌ ᆼᄒ ᅥ ᅢᄀ ᆫᄃ ᅡ ᅡ. Y = (Y1 , Y2 , · · · , Yp )T ᄀ ᅡᄐ ᅮᄌ ᅡᄑ ᅩᄐ ᅳᄑ ᆯᄅ ᅩ ᅵᄋ ᅩᄅ ᆯᄀ ᅳ ᅮᄉ ᆼᄒ ᅥ ᅡᄂ ᆫᄌ ᅳ ᅮᄉ ᆨᄉ ᅵ ᅮᄋ ᆨᄅ ᅵ ᆯᄋ ᅲ ᅵᄅ ᅡᄀ ᅩ.
(4) 772. Kwangyee Ko · Jangsun Baek. ᄒᄀ ᅡ ᅩ, ᄋ ᅵᄌ ᅡᄅ ᅭᄃ ᆯᄋ ᅳ ᆯ MFA, MCFA, MCtFA, MSN, MSt ᄆ ᅳ ᅩᄒ ᆼᄋ ᅧ ᅦᄌ ᆨᄒ ᅥ ᆸᄉ ᅡ ᅵᄏ ᆫᄒ ᅵ ᅮᄀ ᆨᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᅴᄆ ᅩᄉ ᅮᄎ ᅮᄌ ᆼᄎ ᅥ ᅵᄃ ᆯᅳ ᅳ ᆯ ᄋ Pp ᆯᅳ ᅳ ᆯ ᄋ ᅡᄌ ᄀ ᆫᅡ ᅵ ᆨ ᄀᄆ ᅩᄒ ᆼᄋ ᅧ ᅴᄇ ᆫᄑ ᅮ ᅩᄅ ᅩᄇ ᅮᄐ ᅥᄆ ᅩᄋ ᅴᄌ ᅡᄅ ᅭᄅ ᆯᄆ ᅳ ᆭᄋ ᅡ ᅵᄉ ᆼᅥ ᅢ ᆼ ᄉᄒ ᆫᄃ ᅡ ᅡᄋ ᆷᄋ ᅳ ᅦᄀ ᆨᄀ ᅡ ᆨᄋ ᅡ ᅴᄉ ᆼᅥ ᅢ ᆼ ᄉᄌ ᅡᄅ ᅭᄋ ᅦᄃ ᅢᄒ ᆫ Y = i=1 Yi ᄃ ᅡ ᅨᄉ ᄀ ᆫᅡ ᅡ ᄒᄀ ᅩ α = 0.01ᄋ ᅦᄉ ᆼᄋ ᅡ ᆼᄒ ᅳ ᅡᄂ ᆫᄇ ᅳ ᆨᄇ ᅢ ᆫᄋ ᅮ ᅱᄉ ᅮ FR−1 (α)ᄅ ᆯᄎ ᅳ ᆽᄋ ᅡ ᅡᄉ ᅥ VaRᄋ ᆯᄀ ᅳ ᅨᄉ ᆫᄒ ᅡ ᆫᄃ ᅡ ᅡ. ᄉ ᆯᄌ ᅵ ᅦᄋ ᆫᄀ ᅧ ᅮᄋ ᅦᄉ ᅥᄉ ᅡᄋ ᆼᄃ ᅭ ᆯᄉ ᅬ ᅮ ᆨᄅ ᅵ ᄋ ᆯᄌ ᅲ ᅡᄅ ᅭᄋ ᅴᄌ ᆼᄋ ᅥ ᅴᄋ ᅪᄇ ᆷᄋ ᅥ ᅱᄂ ᆫᄃ ᅳ ᅡᄋ ᆷᄀ ᅳ ᅪᄀ ᇀᄃ ᅡ ᅡ. Yjt ᄅ ᆯᄌ ᅳ ᅮᄉ ᆨᄑ ᅵ ᅩᄐ ᅳᄑ ᆯᄅ ᅩ ᅵᄋ ᅩᄅ ᆯᄀ ᅳ ᅮᄉ ᆼᄒ ᅥ ᅡᄂ ᆫ jᄇ ᅳ ᆫᄍ ᅥ ᅢᄌ ᅮᄉ ᆨᄋ ᅵ ᅴ tᄉ ᅵᄌ ᆷᄋ ᅥ ᅦᄉ ᅥᄋ ᅴ ᅮᄋ ᄉ ᆨᄅ ᅵ ᆯᄋ ᅲ ᅵᄅ ᅡᄀ ᅩᄒ ᅡᄌ ᅡ. Pjt ᄅ ᆯ tᄉ ᅳ ᅵᄌ ᆷᄋ ᅥ ᅦᄉ ᅥᄋ ᅴᄌ ᅮᄀ ᅡᄅ ᅡᄀ ᅩᄒ ᆯᄄ ᅡ ᅢ, Yjt ᄂ ᆫᄋ ᅳ ᅡᄅ ᅢᅪ ᄋᄀ ᇀᄋ ᅡ ᅵᄀ ᅨᄉ ᆫᄃ ᅡ ᆫᄃ ᅬ ᅡ. Pjt yjt = log . Pj(t−1) ᄇ ᄋ ᆫ ᅩ ᆫᄀ ᅧ ᅮᄋ ᅦᄉ ᅥ ᄉ ᅡᄋ ᆼᄃ ᅭ ᆫ ᄌ ᅬ ᅮᄀ ᅡᄌ ᅡᄅ ᅭᄂ ᆫ 01/01/2000 ∼ 12/31/2014 ᄉ ᅳ ᅡᄋ ᅵᄋ ᅴ ᄋ ᆯᄇ ᅯ ᆯ ᄉ ᅧ ᅮᄋ ᆨᄅ ᅵ ᆯᄌ ᅲ ᅡᄅ ᅭᄅ ᆯ ᄀ ᅳ ᅨᄉ ᆫᄒ ᅡ ᅡᄋ ᅧ VaRᄋ ᆯ ᄎ ᅳ ᅮᄌ ᆼᄒ ᅥ ᅡᄋ ᆻᄃ ᅧ ᅡ. ᄀ ᆨᄀ ᅡ ᆨᄋ ᅡ ᅴ ᄒ ᆫᄒ ᅩ ᆸᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᅦ ᄃ ᅢᄒ ᅡᄋ ᅧ EM ᄋ ᆯᄀ ᅡ ᅩᄅ ᅵᄌ ᆷᄋ ᅳ ᅦ ᄋ ᅴᅡ ᆫ ᄒ ᄆ ᅩᄉ ᅮᄎ ᅮᄌ ᆼᄋ ᅥ ᆫ ᄋ ᅳ ᅧᄅ ᅥ ᄋ ᆫᄀ ᅧ ᅮᄌ ᅡᄋ ᅦ ᄋ ᅴ ᅢᄉ ᄒ ᅥRᄋ ᆫᄋ ᅥ ᅥᄅ ᅩᄀ ᅢᄇ ᆯᄃ ᅡ ᆫᄑ ᅬ ᅢᄏ ᅵᄌ ᅵᄃ ᆯᄋ ᅳ ᆯᄉ ᅳ ᅡᄋ ᆼᄒ ᅭ ᅡᄋ ᆻᄃ ᅧ ᅡ. MFA ᄆ ᅩᄒ ᆼᄋ ᅧ ᅦᄃ ᅢᄒ ᅢᄉ ᅥᄂ ᆫ ‘EMMIXmfa’, MCFA, MCtFA ᅳ ᅩᄒ ᄆ ᆼᅦ ᅧ ᄋ ᄃ ᅢᄒ ᅡᄋ ᅧᄂ ᆫ ‘EMMIXmcfa’, ᄀ ᅳ ᅳᄅ ᅵᄀ ᅩ MSN, MSt ᄆ ᅩᄒ ᆼᄋ ᅧ ᅦ ᄃ ᅢᄒ ᅡᄋ ᅧᄂ ᆫ ‘EMMIXskew’ᄀ ᅳ ᅡ ᄉ ᅡᄋ ᆼᄃ ᅭ ᅬᄋ ᆻᄃ ᅥ ᅡ. VaR ᄀ ᅨᄉ ᆫᄌ ᅡ ᆯᄎ ᅥ ᅡᄂ ᆫᄃ ᅳ ᆨᄌ ᅩ ᅡᄌ ᆨᄋ ᅥ ᅳᄅ ᅩRᄋ ᆫᄋ ᅥ ᅥᄅ ᆯᄋ ᅳ ᅵᄋ ᆼᄒ ᅭ ᅡᄋ ᅧᄑ ᅳᄅ ᅩᄀ ᅳᄅ ᆷᄋ ᅢ ᆯᄌ ᅳ ᆨᄉ ᅡ ᆼᄒ ᅥ ᅡᄋ ᆻᄃ ᅧ ᅡ. 2.3. VaR의 정확도 평가 VaRᄋ ᅴᄌ ᆼᄒ ᅥ ᆨᄃ ᅪ ᅩᄅ ᆯᄑ ᅳ ᆼᄀ ᅧ ᅡᄒ ᆯᄉ ᅡ ᅮᄋ ᆻᄂ ᅵ ᆫᄋ ᅳ ᅧᄅ ᅥᄀ ᅡᄌ ᅵᄇ ᆼᄇ ᅡ ᆸᄋ ᅥ ᅵᄌ ᅦᄋ ᆫᄃ ᅡ ᅬᄋ ᅥᄋ ᆻᄋ ᅪ ᅳᄂ ᅡᄇ ᆫᄋ ᅩ ᆫᄀ ᅧ ᅮᄋ ᅦᄉ ᅥᄂ ᆫᄀ ᅳ ᅡᄌ ᆼᄂ ᅡ ᆯᄅ ᅥ ᅵᄉ ᅡᄋ ᆼᄃ ᅭ ᅬ ᄂᄉ ᆫ ᅳ ᅡᅮ ᄒᄀ ᆷᄌ ᅥ ᆼ (backtesting)ᄀ ᅥ ᅪᄃ ᆨᄅ ᅩ ᆸᄉ ᅵ ᆼᅥ ᅥ ᆷ 거 ᆼ ᄌ (independence test)ᄅ ᆯᄌ ᅳ ᆨᄋ ᅥ ᆼᄒ ᅭ ᅡᄀ ᅵᄅ ᅩᄒ ᆫᄃ ᅡ ᅡ. ᄉ ᅡᄒ ᅮᄀ ᆷᅥ ᅥ ᆼ ᄌ (Kupiec, 1995)ᄂ ᆫᄉ ᅳ ᆯᄌ ᅵ ᅦᄅ ᅩ VaR ᄋ ᅵᄉ ᆼᄃ ᅡ ᅥᄏ ᆫᄀ ᅳ ᅪᄃ ᅡᄉ ᆫᄉ ᅩ ᆯᄋ ᅵ ᆯᄋ ᅳ ᆸᄋ ᅵ ᆫᄇ ᅳ ᅵᄋ ᆯᄋ ᅲ ᅵᄀ ᅵᄃ ᅢᅬ ᄃᄂ ᆫᄇ ᅳ ᅵᄋ ᆯ αᄋ ᅲ ᅪᄃ ᆼᄋ ᅩ ᆯᄒ ᅵ ᆫᄌ ᅡ ᅵᄅ ᆯᄀ ᅳ ᆷᄌ ᅥ ᆼᄒ ᅥ ᅡᄀ ᅵᄋ ᅱ ᆫᅥ ᅡ ᄒ ᆺ 기 ᄋᄃ ᅡ. nᄋ ᅵᄎ ᆼᄌ ᅩ ᅡᄅ ᅭᄋ ᅴᄉ ᅮᄋ ᅵᄀ ᅩ vᄀ ᅡᄉ ᆯᄌ ᅵ ᅦᄅ ᅩᅪ ᄀᄀ ᅥᄌ ᅡᄅ ᅭᄌ ᆼᄋ ᅮ ᅦᄉ ᅥᄀ ᅪᄃ ᅡᄉ ᆫᄉ ᅩ ᆯᄋ ᅵ ᅵᄇ ᆯᄉ ᅡ ᆼᄒ ᅢ ᆫᄒ ᅡ ᆺᄉ ᅬ ᅮᄅ ᅡᄀ ᅩᄒ ᆯᄄ ᅡ ᅢ, ᄀ ᅪ ᅡᄉ ᄃ ᆫᆯ ᅩ ᅵᄇ ᄉ ᅵᄋ ᆯᄋ ᅲ ᆫ r = v/nᄋ ᅳ ᅵᄃ ᅡ. ᄉ ᅡᄒ ᅮᄀ ᆷᅥ ᅥ ᆼ ᄌᄋ ᆫᄋ ᅳ ᅵᄇ ᅵᄋ ᆯᄋ ᅲ ᅵᄀ ᅵᄃ ᅢᄇ ᅵᄋ ᆯ αᄋ ᅲ ᅪᄃ ᆼᄋ ᅩ ᆯᄒ ᅵ ᆫᄌ ᅡ ᅵᄀ ᅱᄆ ᅮᄀ ᅡᄉ ᆯ H0 : r = αᄋ ᅥ ᅦᄃ ᅢ ᆫᄃ ᅡ ᄒ ᅢᄅ ᆸᄀ ᅵ ᅡᄉ ᆯ H1 : r ̸= αᄅ ᅥ ᆯᄀ ᅳ ᆷᅥ ᅥ ᆼ ᄌᄒ ᆫᄃ ᅡ ᅡ. ᄆ ᅩᄒ ᆼᄋ ᅧ ᅵᄌ ᆨᄌ ᅥ ᆯᄒ ᅥ ᅡᄃ ᅡᄂ ᆫᄀ ᅳ ᅱᄆ ᅮᄀ ᅡᄉ ᆯᄒ ᅥ ᅡᄋ ᅦᄉ ᅥ vᄂ ᆫᄋ ᅳ ᅵᄒ ᆼᄇ ᅡ ᆫᄑ ᅮ ᅩᄅ ᆯᄄ ᅳ ᅡᄅ ᅳᄆ ᅳᄅ ᅩᄃ ᅡ ᆷᄀ ᅳ ᄋ ᅪᄀ ᇀᄋ ᅡ ᆫᄋ ᅳ ᅮᄃ ᅩᄇ ᅵᄀ ᆷᅥ ᅥ ᆼ ᄌᄐ ᆼᄀ ᅩ ᅨᄅ ᆼᄋ ᅣ ᆫᄌ ᅳ ᅡᄋ ᅲᄃ ᅩ 1ᄋ ᆫᄏ ᅵ ᅡᄋ ᅵᄌ ᅦᄀ ᆸᅮ ᅩ ᆫ ᄇᄑ ᅩᄅ ᆯᄄ ᅳ ᅡᄅ ᆫᄃ ᅳ ᅡ. " n−v # 1−r r LRbt = 2log . 1−α α ᄀᄅ ᅳ ᅥᅳ ᄆᄅ ᅩᄉ ᅡᄒ ᅮᄀ ᆷᄌ ᅥ ᆼᄋ ᅥ ᅦᄋ ᅴᄒ ᅡᄋ ᅧᅪ ᄀᄃ ᅡᄉ ᆫᄉ ᅩ ᆯᄒ ᅵ ᆺᄉ ᅬ ᅮᄀ ᅡᄂ ᅥᄆ ᅮᄆ ᆭᄀ ᅡ ᅥᄂ ᅡᄂ ᅥᄆ ᅮᄌ ᆨᄋ ᅥ ᆫᄀ ᅳ ᆼᄋ ᅧ ᅮᄒ ᅢᄃ ᆼ VaR ᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᆯᄀ ᅳ ᅵᄀ ᆨᄒ ᅡ ᅡᄀ ᅦ ᆫᄃ ᅬ ᄃ ᅡ. ᆨᆸ ᅩ ᄃ ᅵᅥ ᄅ ᆼ 서 ᆷ ᄀᄌ ᆼ (Christoffersen, 1998)ᄋ ᅥ ᆫ VaR ᄀ ᅳ ᅪᄃ ᅡᄉ ᆫᄉ ᅩ ᆯ ᄀ ᅵ ᅪᄌ ᆼᄋ ᅥ ᅵ ᄉ ᅵᄀ ᅨᄋ ᆯᄌ ᅧ ᆨᄋ ᅥ ᅳᄅ ᅩ ᄌ ᆼᄉ ᅩ ᆨᄌ ᅩ ᆨᄋ ᅥ ᆫᄌ ᅵ ᅵᄅ ᆯ ᄀ ᅳ ᆷᅥ ᅥ ᆼ ᄌᄒ ᆫᄃ ᅡ ᅡ. VaR ᅩ ᄆᄒ ᆼᄋ ᅧ ᅵᄋ ᆯᄇ ᅵ ᆫᄌ ᅡ ᆨᄋ ᅥ ᅳᄅ ᅩᄌ ᆼᄒ ᅥ ᆨᄒ ᅪ ᅡᄃ ᅡᄆ ᆫ VaR ᄀ ᅧ ᅪᄃ ᅡᄉ ᆫᄉ ᅩ ᆯᄋ ᅵ ᅵᄇ ᆯᄉ ᅡ ᆼᄒ ᅢ ᆯᄒ ᅡ ᆨᄅ ᅪ ᆯᄋ ᅲ ᅵᅪ ᄀᄀ ᅥᄋ ᅦᅪ ᄀᄃ ᅡᄉ ᆫᄉ ᅩ ᆯᄋ ᅵ ᅵᄇ ᆯᄉ ᅡ ᆼᅢ ᅢ ᆻ ᄒᄂ ᆫᄌ ᅳ ᅵᄋ ᅧ ᅮᄋ ᄇ ᅦᅡ ᄄᄅ ᅡᄌ ᆼᄉ ᅩ ᆨᄃ ᅩ ᅬᄌ ᅵᄋ ᆭᄋ ᅡ ᅡᄋ ᅣᄒ ᆫᄃ ᅡ ᅡ. ᄌ ᆨᄀ ᅳ ᅪᄃ ᅡᄉ ᆫᄉ ᅩ ᆯᄃ ᅵ ᆯᄋ ᅳ ᅵᄃ ᆨᄅ ᅩ ᆸᄌ ᅵ ᆨᄋ ᅥ ᅵᄋ ᅥᄋ ᅣᄒ ᆫᄃ ᅡ ᅡ. ᄃ ᆨᄅ ᅩ ᆸᅥ ᅵ ᆼ ᄉᄋ ᅦᄃ ᅢᄒ ᆫᄋ ᅡ ᅮᄃ ᅩᄇ ᅵᄀ ᆷᅥ ᅥ ᆼ ᄌᄐ ᆼᄀ ᅩ ᅨ ᆼᄋ ᅣ ᄅ ᆫᅡ ᅳ ᄃᄋ ᆷᄀ ᅳ ᅪᄀ ᇀᄃ ᅡ ᅡ. (1 − q)N1 +N2 q N3 +N4 LRind = −2log . (1 − q1 )N1 q1N3 (1 − q2 )N2 q2N4 ᄋᄄ ᅵ ᅢ, N1 ᄋ ᆫᄀ ᅳ ᅪᄃ ᅡᄉ ᆫᄉ ᅩ ᆯᄋ ᅵ ᅵᄋ ᆹᄂ ᅥ ᆫᄀ ᅳ ᆼᄋ ᅧ ᅮᄀ ᅳᄃ ᅡᄋ ᆷᄋ ᅳ ᆨᄉ ᅧ ᅵᄀ ᅪᄃ ᅡᄉ ᆫᄉ ᅩ ᆯᄋ ᅵ ᅵᄇ ᆯᄉ ᅡ ᆼᄒ ᅢ ᅡᄌ ᅵᄋ ᆭᄋ ᅡ ᆫᄒ ᅳ ᆺᄉ ᅬ ᅮ , N2 ᄂ ᆫᄀ ᅳ ᅪᄃ ᅡᄉ ᆫᄉ ᅩ ᆯᄋ ᅵ ᅵᄋ ᆹ ᅥ ᆻᄋ ᅥ ᄋ ᅳᄂ ᅡᄀ ᅳᄃ ᅡᄋ ᆷᄋ ᅳ ᅦᄂ ᆫᄀ ᅳ ᅪᄃ ᅡᄉ ᆫᄉ ᅩ ᆯᄋ ᅵ ᅵᄇ ᆯᄉ ᅡ ᆼᄒ ᅢ ᆫᄒ ᅡ ᆺᄉ ᅬ ᅮ, N3 ᄂ ᆫᄀ ᅳ ᅪᄃ ᅡᄉ ᆫᄉ ᅩ ᆯᄇ ᅵ ᆯᄉ ᅡ ᆼᄒ ᅢ ᅮᄀ ᅳᄃ ᅡᄋ ᆷᄋ ᅳ ᅦᅪ ᄀᄃ ᅡᄉ ᆫᄉ ᅩ ᆯᄋ ᅵ ᅵᄇ ᆯᄉ ᅡ ᆼᄒ ᅢ ᅡᄌ ᅵ ᆭᄋ ᅡ ᄋ ᆫᄒ ᅳ ᆺᄉ ᅬ ᅮ, N4 ᄂ ᆫᄀ ᅳ ᅪᄃ ᅡᄉ ᆫᄉ ᅩ ᆯᄇ ᅵ ᆯᄉ ᅡ ᆼᄒ ᅢ ᅮᄀ ᅳᄃ ᅡᄋ ᆷᄀ ᅳ ᅪᄃ ᅡᄉ ᆫᄉ ᅩ ᆯᄋ ᅵ ᅵᄄ ᅩᄇ ᆯᄉ ᅡ ᆼᄒ ᅢ ᆫᄒ ᅡ ᆺᄉ ᅬ ᅮᄅ ᆯᄂ ᅳ ᅡᄐ ᅡᄂ ᆫᄃ ᅢ ᅡ. q, q1 , q2 ᄂ ᆫᄀ ᅳ ᆨᄀ ᅡ ᆨᄀ ᅡ ᅪ ᅡᄉ ᄃ ᆫᆯ ᅩ ᅵᄋ ᄉ ᅵᄇ ᆯᄉ ᅡ ᆼᄒ ᅢ ᆫᄌ ᅡ ᅡᄅ ᅭᄋ ᅴᄇ ᅵᄋ ᆯ, ᄋ ᅲ ᅵᄌ ᆫᄋ ᅥ ᅦᅪ ᄀᄃ ᅡᄉ ᆫᄉ ᅩ ᆯᄋ ᅵ ᅵᄇ ᆯᄉ ᅡ ᆼᄒ ᅢ ᅡᄌ ᅵᄋ ᆭᄋ ᅡ ᆻᄋ ᅡ ᅳᄂ ᅡᄀ ᅳᄃ ᅡᄋ ᆷᄋ ᅳ ᅦᄂ ᆫᄇ ᅳ ᆯᄉ ᅡ ᆼᄒ ᅢ ᆫᄌ ᅡ ᅡᄅ ᅭᄋ ᅴᄇ ᅵᄋ ᆯ, ᅲ ᅵᄌ ᄋ ᆫᅦ ᅥ 와 ᄀᄃ ᅡᄉ ᆫᄉ ᅩ ᆯᄋ ᅵ ᅵᄇ ᆯᄉ ᅡ ᆼᄒ ᅢ ᅡᄀ ᅩᄀ ᅳᄃ ᅡᄋ ᆷᄋ ᅳ ᅦᄃ ᅩᅪ ᄀᄃ ᅡᄉ ᆫᄉ ᅩ ᆯᄋ ᅵ ᅵᄇ ᆯᄉ ᅡ ᆼᄒ ᅢ ᆫᄌ ᅡ ᅡᄅ ᅭᄋ ᅴᄇ ᅵᄋ ᆯᅳ ᅲ ᆯ ᄋᄂ ᅡᄐ ᅡᄂ ᆫᄃ ᅢ ᅡ. ᄌ ᆨ ᅳ q=. N3 N4 N3 + N4 , q1 = , q2 = N1 + N2 + N3 + N4 N1 + N3 N2 + N4. ᄋᄃ ᅵ ᅡ. ᄋ ᅱᄋ ᅴᄀ ᆷᅥ ᅥ ᆼ ᄌᄐ ᆼᄀ ᅩ ᅨᄅ ᆼ LRind ᄂ ᅣ ᆫᄀ ᅳ ᅱᄆ ᅮᄀ ᅡᄉ ᆯ H0 : q1 = q2 ᄋ ᅥ ᅦᄃ ᅢᄒ ᅡᄋ ᅧᄃ ᅢᄅ ᆸᄀ ᅵ ᅡᄉ ᆯ H1 : 1ᄎ ᅥ ᅡ Markovᄌ ᆼᄉ ᅩ ᆨᅳ ᅩ ᆯ ᄋᄀ ᆷ ᅥ ᆼᄒ ᅥ ᄌ ᅡᄆ ᅧ, ᄀ ᅱᄆ ᅮᄀ ᅡᄉ ᆯᄒ ᅥ ᅡᄋ ᅦᄉ ᅥᄌ ᅡᄋ ᅲᄃ ᅩ 1ᄋ ᆫᄏ ᅵ ᅡᄋ ᅵᄌ ᅦᄀ ᆸᅮ ᅩ ᆫ ᄇᄑ ᅩᄅ ᆯᄄ ᅳ ᅡᄅ ᆫᄃ ᅳ ᅡ..
(5) VaR estimation using skewed mixture models and various mixtures of factor analyzers. 773. 미 ᅡ ᄌᄆ ᆨᄋ ᅡ ᅳᄅ ᅩᄌ ᆼᄒ ᅥ ᆨᄃ ᅪ ᅩᄑ ᆼᄀ ᅧ ᅡᄎ ᆨᄃ ᅳ ᅩᄅ ᅩᄉ ᅥᄃ ᅡᄋ ᆷᄀ ᅳ ᅪᅡ ᄀᄋ ᇀ ᅵᄌ ᆼᄋ ᅥ ᅴᄃ ᅬᄂ ᆫᄎ ᅳ ᅩᅪ ᄀᄇ ᅵᄋ ᆯ ER (exceeding ratio)ᄋ ᅲ ᆯᄀ ᅳ ᅩᄅ ᅧᄒ ᆫᄃ ᅡ ᅡ (Choiᄋ ᅪ Min, 2011). v ER = . αn ERᄀ ᆹᄋ ᅡ ᅵ 1ᄇ ᅩᄃ ᅡᄏ ᅳᄆ ᆫᄒ ᅧ ᅢᄃ ᆼᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᅵ VaRᄋ ᆯᄀ ᅳ ᅪᄉ ᅩᄎ ᅮᄌ ᆼᄒ ᅥ ᆷᄋ ᅡ ᆯᄋ ᅳ ᅴᄆ ᅵᄒ ᅡᄀ ᅩ, ᄇ ᆫᄆ ᅡ ᆫᄋ ᅧ ᅦ 1ᄇ ᅩᄃ ᅡᄌ ᆨᄋ ᅡ ᅳᄆ ᆫ VaRᄋ ᅧ ᆯᄀ ᅳ ᅪᄃ ᅢᄎ ᅮ ᆼᄒ ᅥ ᄌ ᆷᄋ ᅡ ᆯᄋ ᅳ ᅴᄆ ᅵᄒ ᆫᄃ ᅡ ᅡ. VaR ᄀ ᆯᅥ ᅧ ᆼ ᄌᄋ ᆯᄋ ᅳ ᅱᅡ ᆫ ᄒᄎ ᅬᄉ ᆫᄋ ᅥ ᅴᄆ ᅩᄒ ᆼᄋ ᅧ ᆯᄉ ᅳ ᆫᅢ ᅥ ᆨ ᄐᄒ ᅡᄀ ᅵᄋ ᅱᅡ ᆫ 허 ᆯ ᄌᄎ ᅡᄅ ᆯᄋ ᅳ ᅭᄋ ᆨᄒ ᅣ ᅡᄆ ᆫᄃ ᅧ ᅡᄋ ᆷᄀ ᅳ ᅪᄀ ᇀᄃ ᅡ ᅡ. 1) ᄉ ᆨ (2.3) - (2.7)ᄅ ᅵ ᅩᄀ ᆨᄀ ᅡ ᆨᄆ ᅡ ᆯᄃ ᅵ ᅩᄒ ᆷᄉ ᅡ ᅮᄀ ᅡᄌ ᅮᄋ ᅥᄌ ᆫᄒ ᅵ ᆫᄒ ᅩ ᆸᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᅦᄃ ᅢᄒ ᅡᄋ ᅧ EM ᄋ ᆯᄀ ᅡ ᅩᄅ ᅵᄌ ᆷᅳ ᅳ ᆯ ᄋᄋ ᅵᄋ ᆼᄒ ᅭ ᅡᄋ ᅧᄆ ᅩᄉ ᅮᄅ ᆯ ᅳ ᅮᄌ ᄎ ᆼᅡ ᅥ ᄒᄀ ᅩᄀ ᆨᄀ ᅡ ᆨ VaRᄋ ᅡ ᆯᄀ ᅳ ᅨᄉ ᆫᅡ ᅡ ᆫ ᄒᄃ ᅡ. 2) ᅡ ᆨ ᄀᄀ ᆨᄋ ᅡ ᅴᄆ ᅩᄒ ᆼᄋ ᅧ ᅦᄄ ᅡᄅ ᅡᄎ ᅮᄌ ᆼᄃ ᅥ ᆫ VaRᄋ ᅬ ᅵᄌ ᆼᄒ ᅥ ᆨᄒ ᅪ ᆫᄌ ᅡ ᅵᄉ ᅡᄒ ᅮᄀ ᆷᅥ ᅥ ᆼ ᄌᄀ ᅪᄃ ᆨᄅ ᅩ ᆸᅥ ᅵ ᆼ 서 ᆷ 거 ᆼ ᄌᄋ ᆯᄉ ᅳ ᆯᄉ ᅵ ᅵᄒ ᅡᄋ ᅧᄆ ᅩᄒ ᆼᄋ ᅧ ᅴᄌ ᆨᄒ ᅥ ᆸᅥ ᅡ ᆼ ᄉᄋ ᆯ ᅳ ᆫᅡ ᅡ ᄑ ᆫ 다 ᆫ ᄒᄃ ᅡ. 3) ᅡ ᄆᄌ ᅵᄆ ᆨᄋ ᅡ ᅳᄅ ᅩ α = 0.01 ∼ 0.05ᄋ ᅦᄃ ᅢᄒ ᅡᄋ ᅧᄀ ᆨᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᅴ VaRᄃ ᆯᅳ ᅳ ᆯ ᄋᄎ ᅮᄌ ᆼᄒ ᅥ ᅡᄋ ᅧᄉ ᆯᄌ ᅵ ᅦᄌ ᅡᄅ ᅭᄅ ᅩᄇ ᅮᄐ ᅥᄀ ᅮᄒ ᆫᅧ ᅡ ᆼ ᄀᄒ ᆷᄌ ᅥ ᆨ ᅥ VaRᄃ ᆯᄀ ᅳ ᅪᄀ ᅡᄌ ᆼᄀ ᅡ ᅡᄁ ᅡᄋ ᆫᄆ ᅮ ᅩᄒ ᆼᄋ ᅧ ᅵᄋ ᅥᄄ ᆫᅥ ᅥ ᆺ ᄀᆫ ᄋ ᅵᄌ ᅵᄇ ᆰᄒ ᅡ ᅵᄀ ᅩ, ᄌ ᅡᄅ ᅭᄋ ᅴᄀ ᆼᄒ ᅧ ᆷᄌ ᅥ ᆨ VaRᄃ ᅥ ᆯᄀ ᅳ ᅪᄀ ᅡᄌ ᆼᄀ ᅡ ᅡᄁ ᆸᄀ ᅡ ᅦᄎ ᅮᄌ ᆼᄒ ᅥ ᅡᄂ ᆫᄆ ᅳ ᅩᄒ ᆼ ᅧ ᆯᄎ ᅳ ᄋ ᅮᄒ ᅮᄐ ᅮᄌ ᅡᄑ ᅩᄐ ᅳᄑ ᆯᄅ ᅩ ᅵᄋ ᅩ VaR ᄀ ᆯᅥ ᅧ ᆼ ᄌᄋ ᆯᄋ ᅳ ᅱᅡ ᆫ ᄒᄎ ᅬᄉ ᆫᄋ ᅥ ᅴᄆ ᅩᄒ ᆼᄋ ᅧ ᅳᄅ ᅩᄌ ᅦᄋ ᆫᄒ ᅡ ᆫᄃ ᅡ ᅡ.. 3. 실증분석 3.1. 자료 ᆫᄌ ᅩ ᄇ ᆯᄋ ᅥ ᅦᄉ ᅥᄂ ᆫᄑ ᅳ ᅩᄐ ᅳᄑ ᆯᄅ ᅩ ᅵᄋ ᅩᄅ ᆯᄀ ᅳ ᅮᄉ ᆼᄒ ᅥ ᅡᄀ ᅩᄐ ᅮᄌ ᅡᄌ ᅮᄉ ᆨᄋ ᅵ ᅴᄉ ᅮᄀ ᅡᄆ ᆭᄋ ᅡ ᆫᄀ ᅳ ᆼᄋ ᅧ ᅮ, ᄋ ᇁᄋ ᅡ ᅦᄉ ᅥᄀ ᅩᄅ ᅧᄃ ᆫᄋ ᅬ ᅧᄅ ᅥᄀ ᅡᄌ ᅵᄆ ᅩᄒ ᆼᄃ ᅧ ᆯᅮ ᅳ ᆼ ᄌ ᄋᄉ ᅦ ᅥᄋ ᅥᄄ ᆫᄆ ᅥ ᅩᄒ ᆼᄋ ᅧ ᅵᄑ ᅩᄐ ᅳᄑ ᆯᄅ ᅩ ᅵᄋ ᅩ VaR ᄎ ᅮᄌ ᆼᄋ ᅥ ᅦᄋ ᆻᄋ ᅵ ᅥᄉ ᅥᄀ ᅡᄌ ᆼᄋ ᅡ ᅮᄉ ᅮᄒ ᆫᅥ ᅡ ᆼ ᄉᄂ ᆼᅳ ᅳ ᆯ ᄋᄇ ᅩᄋ ᅵᄂ ᆫᄌ ᅳ ᅵᄇ ᆫᄉ ᅮ ᆨᄒ ᅥ ᆫᄃ ᅡ ᅡ. ᄒ ᆫᄀ ᅡ ᆨᄋ ᅮ ᅴᄏ ᅩᄉ ᅳ ᅵᄉ ᄑ ᅵᄌ ᆼᄋ ᅡ ᅦᄉ ᅥ 2015ᄂ ᆫᄒ ᅧ ᆫᄒ ᅡ ᅢᄃ ᆼᄋ ᅩ ᆫᄋ ᅡ ᅬᄀ ᆨᄋ ᅮ ᆫ, ᄀ ᅵ ᅵᄀ ᆫ, ᄀ ᅪ ᅢᄋ ᆫᄌ ᅵ ᆸᄃ ᅵ ᆫᄋ ᅡ ᅵᄀ ᆨᄀ ᅡ ᆨᄀ ᅡ ᅡᄌ ᆼᄆ ᅡ ᆭᄋ ᅡ ᅵᄐ ᅮᄌ ᅡᄒ ᆫᄉ ᅡ ᆼᄋ ᅡ ᅱ 28ᄀ ᅢᄒ ᅬᄉ ᅡᄌ ᅮᄉ ᆨ ᅵ ᆯᄉ ᅳ ᄋ ᆫᅥ ᅥ ᆼ ᄌᄒ ᅡᄋ ᅧ 2010ᄂ ᆫ 1ᄋ ᅧ ᆯᄇ ᅯ ᅮᄐ ᅥ 2014ᄂ ᆫ 12ᄋ ᅧ ᆯᄁ ᅯ ᅡᄌ ᅵᄆ ᅢᄋ ᆯᄆ ᅯ ᆯᄌ ᅡ ᆼᄀ ᅩ ᅡᄅ ᆯᄅ ᅳ ᅩᄀ ᅳᄎ ᅡᄇ ᆫᄒ ᅮ ᆫᄋ ᅡ ᆯᄉ ᅯ ᅮᄋ ᆨᄅ ᅵ ᆯᄌ ᅲ ᅡᄅ ᅭ (ᄎ ᆼ 179 ᄀ ᅩ ᆫ ᅪ ᆨᄀ ᅳ ᄎ ᆹ)ᄅ ᅡ ᆯᄑ ᅳ ᅭᄇ ᆫᄌ ᅩ ᅡᄅ ᅭᄅ ᅩᄒ ᆯᄋ ᅪ ᆼᄒ ᅭ ᅡᄋ ᆻᄃ ᅧ ᅡ. Figure 3.1ᄋ ᆫᄑ ᅳ ᅩᄐ ᅳᄑ ᆯᄅ ᅩ ᅵᄋ ᅩᄅ ᆯᄀ ᅳ ᅮᄉ ᆼᄒ ᅥ ᅡᄀ ᅩᄋ ᆻᄂ ᅵ ᆫ 28ᄀ ᅳ ᅢᄌ ᅮᄉ ᆨᄐ ᅵ ᅮᄌ ᅡᄉ ᅮᄋ ᆨᄅ ᅵ ᆯᄃ ᅲ ᆯᅳ ᅳ ᆯ ᄋ ᅩᄃ ᄆ ᅮᄒ ᆸᄒ ᅡ ᆫᄎ ᅡ ᆼᄐ ᅩ ᅮᄌ ᅡᄉ ᅮᄋ ᆨᄅ ᅵ ᆯᄃ ᅲ ᆯᄋ ᅳ ᅦᄃ ᅢᄒ ᆫᄉ ᅡ ᅵᄀ ᅨᄋ ᆯᄀ ᅧ ᅳᄅ ᅢᄑ ᅳᄋ ᅵᄃ ᅡ.. Figure 3.1 Time series plot of the total rate of return. Table 3.1ᄋ ᆫᄋ ᅳ ᅵᄃ ᆯᄎ ᅳ ᆼᄐ ᅩ ᅮᄌ ᅡᄉ ᅮᄋ ᆨᄅ ᅵ ᆯᄌ ᅲ ᅡᄅ ᅭᄋ ᅦᄃ ᅢᄒ ᆫᄀ ᅡ ᅵᄎ ᅩᄐ ᆼᄀ ᅩ ᅨᄅ ᆼᄀ ᅣ ᆹᄃ ᅡ ᆯᅳ ᅳ ᆯ ᄋᄂ ᅡᄐ ᅡᄂ ᅢᄀ ᅩᄋ ᆻᄃ ᅵ ᅡ. ᄎ ᆼᄉ ᅩ ᅮᄋ ᆨᄅ ᅵ ᆯᄋ ᅲ ᅴᄑ ᆼᄀ ᅧ ᆫᄋ ᅲ ᆫ ᅳ 0.28ᄋ ᅦᄀ ᆫᄉ ᅳ ᅡᄒ ᅡᄀ ᅩᄋ ᅫᄃ ᅩᄀ ᅡᄋ ᆷᄉ ᅳ ᅮᄅ ᅩᄉ ᅥᄎ ᆼᄉ ᅩ ᅮᄋ ᆨᄅ ᅵ ᆯᄋ ᅲ ᆫᄋ ᅳ ᆫᄍ ᅬ ᆨᄋ ᅩ ᅳᄅ ᅩᄀ ᅵᄋ ᆯᄋ ᅮ ᅥᄌ ᆫᄇ ᅵ ᅵᄃ ᅢᄎ ᆼᄇ ᅵ ᆫᄑ ᅮ ᅩᄅ ᆯᄄ ᅳ ᅡᄅ ᆫᄃ ᅳ ᅡᄀ ᅩᄒ ᆯᄉ ᅡ ᅮᄋ ᆻᄃ ᅵ ᅡ..
(6) 774. Kwangyee Ko · Jangsun Baek. Table 3.1 Simple statistics of (log return) statistic mean sd skewness kurtosis Jarque Bera test (p-value). Y (log return) 0.279 2.085 -0.619 1.978 1.16E-05. Jarque-Bera ᄀ ᆷᄌ ᅥ ᆼᅧ ᅥ ᆯ ᄀᄀ ᅪ p-ᄀ ᆹᄋ ᅡ ᅵ 0ᄋ ᅦᄀ ᅡᄁ ᅡᄋ ᅮᄆ ᅳᄅ ᅩᄎ ᆼᄉ ᅩ ᅮᄋ ᆨᄅ ᅵ ᆯᄋ ᅲ ᅴᄇ ᆫᄑ ᅮ ᅩᄂ ᆫᄌ ᅳ ᆼᄀ ᅥ ᅲᄇ ᆫᄑ ᅮ ᅩᄅ ᆯᄄ ᅳ ᅡᄅ ᆫᄃ ᅳ ᅡᄀ ᅩᄇ ᆯᄉ ᅩ ᅮᄋ ᆹᄃ ᅥ ᅡ. ᄌ ᆨ ᅳ ᅧᄅ ᄋ ᅥᅢ ᄀᄋ ᅴᄐ ᅮᄌ ᅡᄉ ᅮᄋ ᆨᄅ ᅵ ᆯᄅ ᅲ ᅩᄋ ᅵᄅ ᅮᄋ ᅥᄌ ᆫᄎ ᅵ ᆼᄉ ᅩ ᅮᄋ ᆨᄅ ᅵ ᆯᄋ ᅲ ᆫᄌ ᅳ ᆼᄀ ᅥ ᅲᄇ ᆫᄑ ᅮ ᅩᄀ ᅡᄋ ᅡᄂ ᆫᄋ ᅵ ᆫᄍ ᅬ ᆨᄋ ᅩ ᅳᄅ ᅩᄀ ᅵᄋ ᆯᄋ ᅮ ᅥᄌ ᆫᄒ ᅵ ᆼᄐ ᅧ ᅢᄋ ᅴᄇ ᆫᄑ ᅮ ᅩᄅ ᆯᄄ ᅳ ᅡᄅ ᆫ ᅳ ᅡᄀ ᄃ ᅩᄒ ᆯᄉ ᅡ ᅮᄋ ᆻᄃ ᅵ ᅡ. ᄋ ᅵᄂ ᆫ Figure 3.2ᄋ ᅳ ᅴᄌ ᅪᄎ ᆨᄎ ᅳ ᆺᅥ ᅥ ᆫ ᄇᄍ ᅢᄀ ᆼᄒ ᅧ ᆷᄇ ᅥ ᆫᄑ ᅮ ᅩᄀ ᅳᄅ ᆷᄋ ᅵ ᅦᄉ ᅥᄃ ᅩᄒ ᆨᄋ ᅪ ᆫᄒ ᅵ ᆯᄉ ᅡ ᅮᄋ ᆻᄃ ᅵ ᅡ. ᄃ ᅢᄎ ᅦᄅ ᅩᄆ ᅩᄃ ᆫ ᅳ ᅩᄒ ᄆ ᆼᅳ ᅧ ᆯ ᄃᄋ ᅵᄌ ᅮᄉ ᆨᄐ ᅵ ᅮᄌ ᅡᄉ ᅮᄋ ᆨᄅ ᅵ ᆯᄌ ᅲ ᅡᄅ ᅭᄋ ᅦᄇ ᅵᄉ ᆺᄒ ᅳ ᅡᄀ ᅦᄌ ᆯᄌ ᅡ ᆨᄒ ᅥ ᆸᄃ ᅡ ᆷᄋ ᅬ ᆯᄋ ᅳ ᆯᄉ ᅡ ᅮᄋ ᆻᄃ ᅵ ᅡ. 3.2. VaR 추정 및 모형 성능 평가 ᅩᄐ ᄑ ᅳᄑ ᆯᄅ ᅩ ᅵᄋ ᅩᄅ ᆯ ᄀ ᅳ ᅮᄉ ᆼᄒ ᅥ ᅡᄂ ᆫ ᄐ ᅳ ᅮᄌ ᅡᄉ ᅮᄋ ᆨᄅ ᅵ ᆯᄋ ᅲ ᅵ ᄋ ᅧᄅ ᅥ ᄀ ᅢᄋ ᅵᄀ ᅩ ᄎ ᆼᄐ ᅩ ᅮᄌ ᅡᄉ ᅮᄋ ᆨᄅ ᅵ ᆯᄋ ᅲ ᅵ ᄀ ᅵᄋ ᆯᄋ ᅮ ᅥᄌ ᆫ ᄇ ᅵ ᆫᄑ ᅮ ᅩᄅ ᆯ ᄂ ᅳ ᅡᄐ ᅡᄂ ᅢᄀ ᅩ ᄋ ᆻᄂ ᅵ ᆫ ᅳ ᄉᄌ ᆯ ᅵ ᆼᅡ ᅳ ᄌᄅ ᅭᄋ ᅦ MSN, MSt, MFA, MCFA, MCtFA ᄆ ᅩᄒ ᆼᄋ ᅧ ᆯ ᄀ ᅳ ᆨᄀ ᅡ ᆨ ᄌ ᅡ ᆨᄒ ᅥ ᆸᄉ ᅡ ᅵᄏ ᅵᄀ ᅩ ᄀ ᅳᄃ ᆯᄋ ᅳ ᅴ ᄉ ᆼᄂ ᅥ ᆼᅳ ᅳ ᆯ ᄋ ᄇ ᅵᄀ ᅭᄒ ᅡᄋ ᆻᄃ ᅧ ᅡ. ᆨᄀ ᅡ ᄀ ᆨᅴ ᅡ ᄋ ᄒ ᆫᄒ ᅩ ᆸᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᅴ ᄎ ᅬᄌ ᆨ ᅥ ᅥ ᆼ ᄉᄇ ᆫᄀ ᅮ ᅢᄉ ᅮ (g)ᄅ ᆯ ᄀ ᅳ ᆯᄌ ᅧ ᆼᄒ ᅥ ᅡᄀ ᅵ ᄋ ᅱᄒ ᅢ BIC ᄀ ᅵᄌ ᆫᅳ ᅮ ᆯ ᄋ ᄋ ᅵᄋ ᆼᄒ ᅭ ᅡᄋ ᆻᄃ ᅧ ᅡ. MSNᄀ ᅪ MStᄋ ᅴ ᄀ ᆼ ᅧ ᅮ gᄀ ᄋ ᆹᄋ ᅡ ᆯ 1ᄋ ᅳ ᅦᄉ ᅥᄇ ᅮᄐ ᅥ 4ᄁ ᅡᄌ ᅵ ᄇ ᆫᄒ ᅧ ᅪᄉ ᅵᄏ ᅵᄆ ᆫᄉ ᅧ ᅥ ᄀ ᆨ ᄆ ᅡ ᅩᄒ ᆼᄃ ᅧ ᆯᅳ ᅳ ᆯ ᄋ ᄌ ᆨᄒ ᅥ ᆸᄉ ᅡ ᅵᄏ ᅧᄉ ᅥ ᅬ ᄎᄉ ᅩ BIC ᄀ ᆹᄋ ᅡ ᆯ ᄀ ᅳ ᆽᄂ ᅡ ᆫ gᄀ ᅳ ᆹᄋ ᅡ ᆯ ᄉ ᅳ ᆫᅢ ᅥ ᆨ ᄐᄒ ᅡ ᅧᄒ ᄋ ᅢᄃ ᆼᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᅴᄎ ᅬᄌ ᆨᄆ ᅥ ᅩᄒ ᆼᄋ ᅧ ᅳᄅ ᅩᄉ ᆫᅥ ᅥ ᆼ ᄌᄒ ᅡᄋ ᆻᄃ ᅧ ᅡ. ᄄ ᅩᄒ ᆫ MFA, MCFA, MCtFA ᄃ ᅡ ᆼᄋ ᅳ ᅭᄋ ᆫᄇ ᅵ ᆫᄉ ᅮ ᆨᄌ ᅥ ᅡᄆ ᅩᄒ ᆼᄋ ᅧ ᅴᄀ ᆼᄋ ᅧ ᅮᄋ ᅭ ᆫᄋ ᅵ ᄋ ᅴᅡ ᄎᄋ ᆫ qᄃ ᅯ ᅩ 1ᄋ ᅦᄉ ᅥᄇ ᅮᄐ ᅥ 4ᄁ ᅡᄌ ᅵᄇ ᆫᄒ ᅧ ᅪᄉ ᅵᄏ ᅧᄀ ᅡᄆ ᆫᄉ ᅧ ᅥ g = 1, 2, 3, 4ᄋ ᅪᄌ ᅩᄒ ᆸᄒ ᅡ ᅡᄋ ᅧᄀ ᅡᄌ ᆼᄎ ᅡ ᅬᄉ ᅩᄋ ᅴ BIC ᄀ ᆹᄋ ᅡ ᆯᄀ ᅳ ᆽᄂ ᅡ ᆫ ᅳ (g, q) ᅩ ᄆᄒ ᆼᄋ ᅧ ᆯᄎ ᅳ ᅬᄌ ᆨᄆ ᅥ ᅩᄒ ᆼᄋ ᅧ ᅳᄅ ᅩᄉ ᆫᄌ ᅥ ᆼᄒ ᅥ ᅡᄋ ᆻᄃ ᅧ ᅡ. MSNᄋ ᅪ MSt ᄆ ᅩᄒ ᆼᄋ ᅧ ᅴᄀ ᆼᄋ ᅧ ᅮᅬ ᄎᄌ ᆨᄉ ᅥ ᆼᄇ ᅥ ᆫᄉ ᅮ ᅮᄂ ᆫᄆ ᅳ ᅩᄃ ᅮ g =1 ᄋ ᅵᄋ ᆻᄀ ᅥ ᅩ, MFAᄂ ᆫ (g, q) = (1, 1), MCFAᄂ ᅳ ᆫ (g, q) = (2, 2), MCtFAᄂ ᅳ ᆫ (g, q) = (5, 2)ᄋ ᅳ ᅵᄋ ᆻᄃ ᅥ ᅡ. ᄑ ᅭᄇ ᆫᄌ ᅩ ᅡᄅ ᅭᄋ ᅦᄃ ᅢᄒ ᆫᅧ ᅡ ᆼ ᄀ ᆷᄇ ᅥ ᄒ ᆫᅩ ᅮ ᄑᄋ ᅪᄀ ᆨᄎ ᅡ ᅬᄌ ᆨᄆ ᅥ ᅩᄒ ᆼᄋ ᅧ ᅴᄎ ᅮᄌ ᆼᄃ ᅥ ᆫᄒ ᅬ ᆫᄒ ᅩ ᆸᄇ ᅡ ᆫᄑ ᅮ ᅩᄃ ᆯᄋ ᅳ ᅵ Figure 3.2ᄋ ᅦᄀ ᅳᄅ ᅧᄌ ᅧᄋ ᆻᄃ ᅵ ᅡ. VaRᄋ ᅦᄃ ᅢᄒ ᆫᄎ ᅡ ᅮᄌ ᆼᄀ ᅥ ᆹᄋ ᅡ ᆫᄀ ᅳ ᆨᄎ ᅡ ᅬᄌ ᆨᅥ ᅥ ᆼ ᄉᄇ ᆫᅳ ᅮ ᆯ ᄋᄀ ᆽᄂ ᅡ ᆫᄆ ᅳ ᅩᄒ ᆼᄋ ᅧ ᆯᄑ ᅳ ᅭᄇ ᆫᄌ ᅩ ᅡᄅ ᅭᄋ ᅦᄌ ᆨᄒ ᅥ ᆸᄉ ᅡ ᅵᄏ ᅧᄒ ᅢᄃ ᆼᄆ ᅡ ᅩᄉ ᅮᄃ ᆯᄋ ᅳ ᆯᄎ ᅳ ᅮᄌ ᆼᄒ ᅥ ᅡᄀ ᅩ, ᄎ ᅮ ᆼᄃ ᅥ ᄌ ᆫᅩ ᅬ ᄆᄒ ᆼᄇ ᅧ ᆫᄑ ᅮ ᅩᄅ ᅩᄇ ᅮᄐ ᅥᄆ ᅩᄋ ᅴᄉ ᆯᄒ ᅵ ᆷᄌ ᅥ ᅡᄅ ᅭᄃ ᆯᅳ ᅳ ᆯ ᄋᄉ ᆼᄉ ᅢ ᆼᄒ ᅥ ᅡᄋ ᅧᄋ ᆮᄀ ᅥ ᅦᄃ ᆫᄃ ᅬ ᅡ. (Soltykᄋ ᅪ Gupta, 2011). Table 3.2 Performance of various mixture models on estimating 1% VaR of the total 28 Korean stocks rate of returns. The backtesting and independence values refer to the p-values of the corresponding test, respectively. VaR(0.01) ER Backtesting Independent test. MFA -4.788 1.117 1 0.831. MCFA -5.901 1.117 1 0.831. MCtFA -4.670 1.117 1 0.831. MSN -4.851 1.117 1 0.831. MSt -4.169 1.117 1 0.831. Table 3.3 Performance of various mixture models on estimating 5% VaR of the total 28 Korean stocks rate of returns. The backtesting and independence values refer to the p-values of the corresponding test, respectively. VaR(0.05) ER Backtesting Independent test. MFA -3.179 1.117 1 0.572. MCFA -3.740 0.782 0.0640 0.449. MCtFA -3.008 1.117 1 0.572. MSN -3.468 0.894 0.195 0.385. MSt -2.680 1.230 1 0.696. α = 0.01ᄀ ᅪ α = 0.05ᄋ ᅦᄃ ᅢᄋ ᆼᄒ ᅳ ᅡᄂ ᆫᄀ ᅳ ᆨᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᆯᄋ ᅳ ᅵᄋ ᆼᄒ ᅭ ᆫ VaR ᄎ ᅡ ᅮᄌ ᆼᄀ ᅥ ᆹᄃ ᅡ ᆯᄋ ᅳ ᅵ Table 3.2ᄋ ᅪ Table 3.3ᄋ ᅴᄎ ᆺᅥ ᅥ ᆫ ᄇ ᅢᅢ ᄍ 헤 ᆼ ᄋᄀ ᆨᄀ ᅡ ᆨᄌ ᅡ ᆼᄅ ᅥ ᅵᅬ ᄃᄋ ᅥᄋ ᆻᄃ ᅵ ᅡ. α = 0.01ᄋ ᅴᄀ ᆼᄋ ᅧ ᅮᄉ ᆯᄌ ᅵ ᅦᄌ ᅡᄅ ᅭᄅ ᅩᄇ ᅮᄐ ᅥᄀ ᅨᄉ ᆫᄃ ᅡ ᆫᄀ ᅬ ᆼᄒ ᅧ ᆷᄌ ᅥ ᆨ 1% VaRᄋ ᅥ ᅵ -4.567ᄋ ᅵᄆ ᅳ.
(7) VaR estimation using skewed mixture models and various mixtures of factor analyzers. 775. Figure 3.2 The empirical distribution of YR and the estimated distributions of MFA, MCFA, MCtFA, MSN and MSt. ᄅ MCtFA ᄆ ᅩ ᅩᄒ ᆼᄋ ᅧ ᅵᄀ ᅡᄌ ᆼᄀ ᅡ ᆫᄌ ᅳ ᆸᄒ ᅥ ᆫᄎ ᅡ ᅮᄌ ᆼᄀ ᅥ ᆹ (-4.670)ᄋ ᅡ ᆯᄉ ᅳ ᆼᄉ ᅢ ᆼᄒ ᅥ ᅡᄋ ᆻᄃ ᅧ ᅡ. ᄐ ᆼᄉ ᅩ ᆼᄌ ᅡ ᅡᄅ ᅭᄋ ᅴᄎ ᅬᄉ ᅩᄀ ᆹᄋ ᅡ ᆯᄀ ᅳ ᆼᄒ ᅧ ᆷᄌ ᅥ ᆨ 0% ᄇ ᅥ ᆫᄋ ᅮ ᅱ ᅮ, ᄀ ᄉ ᅳᅵ ᄅᄀ ᅩᄎ ᅬᄃ ᅢᄀ ᆹᄋ ᅡ ᆯᄀ ᅳ ᆼᄒ ᅧ ᆷᄌ ᅥ ᆨ 100% ᄇ ᅥ ᆫᄋ ᅮ ᅱᄉ ᅮᄅ ᅡᄀ ᅩᄒ ᅡᄆ ᅧ, ᄌ ᅡᄅ ᅭᄋ ᅴᄎ ᆼᄀ ᅩ ᅢᄉ ᅮᄀ ᅡ 179ᄀ ᅢᄋ ᅵᄆ ᅳᄅ ᅩ 1%ᄇ ᆫᄋ ᅮ ᅱᄉ ᅮᄂ ᆫᄎ ᅳ ᅬᄉ ᅩᄀ ᆹ ᅡ ᆯᄌ ᅳ ᄋ ᅦᅬ ᄋᄒ ᅡᄀ ᅩ 1.79ᄇ ᆫᄍ ᅥ ᅢᄅ ᅩᄌ ᆨᄋ ᅡ ᆫᄀ ᅳ ᆹᄋ ᅡ ᅳᄅ ᅩᄉ ᅥᄃ ᅮᄇ ᆫᄍ ᅥ ᅢᅪ ᄋᄉ ᅦᄇ ᆫᄍ ᅥ ᅢᄌ ᆨᄋ ᅡ ᆫᄀ ᅳ ᆹᄋ ᅡ ᅴᄀ ᅡᄌ ᆼᄑ ᅮ ᆼᄀ ᅧ ᆫᄒ ᅲ ᆫᄀ ᅡ ᆹᄋ ᅡ ᅳᄅ ᅩᄀ ᅨᄉ ᆫᄃ ᅡ ᆫᄃ ᅬ ᅡ. ᄄ ᅡ ᅡᄉ ᄅ ᅥᄀ ᆼᄒ ᅧ ᆷᄌ ᅥ ᆨ 1% VaR -4.567ᄋ ᅥ ᆫᄃ ᅳ ᅮᄇ ᆫᄍ ᅥ ᅢᄅ ᅩᄌ ᆨᄋ ᅡ ᆫᄀ ᅳ ᆹ -6.356ᄀ ᅡ ᅪᄉ ᅦᄇ ᆫᄍ ᅥ ᅢᄅ ᅩᄌ ᆨᄋ ᅡ ᆫᄀ ᅳ ᆹ -4.061ᄋ ᅡ ᅴᄀ ᅡᄌ ᆼᄑ ᅮ ᆼᄀ ᅧ ᆫᄀ ᅲ ᆹ ᅡ ᅵᄃ ᄋ ᅡ. 5ᄀ ᅢᄆ ᅩᄒ ᆼᄋ ᅧ ᅦᄃ ᅢᄒ ᆫᄉ ᅡ ᅡᄒ ᅮᄀ ᆷᅥ ᅥ ᆼ ᄌᄀ ᅪᄃ ᆨᄅ ᅩ ᅵ ᆸᅥ ᆼ ᄉᄀ ᆷᅥ ᅥ ᆼ ᄌᄀ ᆯᄀ ᅧ ᅪ (p-ᄀ ᆹ)ᄀ ᅡ ᅡ Table 3.2ᄋ ᅴ 3ᄇ ᆫᄍ ᅥ ᅢᅪ ᄋ 4ᄇ ᆫᄍ ᅥ ᅢᄒ ᆼᄋ ᅢ ᅦᄌ ᆼᄅ ᅥ ᅵᄃ ᅬ ᅥᄋ ᄋ ᆻᄃ ᅵ ᅡ. ᄋ ᅲᄋ ᅴᄉ ᅮᄌ ᆫ 0.05 ᄒ ᅮ ᅡᄋ ᅦᄉ ᅥᄆ ᅩᄃ ᆫᄆ ᅳ ᅩᄒ ᆼᄋ ᅧ ᅴ p-ᄀ ᆹᄋ ᅡ ᅵᄋ ᅲᄋ ᅴᄉ ᅮᄌ ᆫᄇ ᅮ ᅩᄃ ᅡᄏ ᅥᄉ ᅥᄀ ᅱᄆ ᅮᄀ ᅡᄉ ᆯᄋ ᅥ ᆯᄀ ᅳ ᅵᄀ ᆨᄉ ᅡ ᅵᄏ ᅵᄌ ᅵᄆ ᆺᄒ ᅩ ᅡᄆ ᅳ ᅩᄆ ᄅ ᅩᄃ ᆫᄆ ᅳ ᅩᄒ ᆼᄃ ᅧ ᆯᄋ ᅳ ᅵᄌ ᆨᄒ ᅥ ᆸᄒ ᅡ ᅡᄃ ᅡᄀ ᅩᄒ ᆯᄉ ᅡ ᅮᄋ ᆻᄃ ᅵ ᅡ. ᄎ ᅩᅪ ᄀᄇ ᅵᄋ ᆯ (ER)ᄋ ᅲ ᆫ 1.117ᄅ ᅳ ᅩᄉ ᅥᄆ ᅩᄃ ᅮᄃ ᆼᄋ ᅩ ᆯᄒ ᅵ ᅡᄆ ᅧᄋ ᆨᄀ ᅣ ᆫᄀ ᅡ ᅪᄉ ᅩᄎ ᅮᄌ ᆼᄒ ᅥ ᅡ ᅩᄋ ᄀ ᆻᅳ ᅵ ᆷ 으 ᆯ ᄋᄇ ᅩᄋ ᅧᄌ ᆫᄃ ᅮ ᅡ. ᄋ ᅵᄂ ᆫᄀ ᅳ ᆨᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᅴ 1% VaR ᄎ ᅮᄌ ᆼᄀ ᅥ ᆹᄇ ᅡ ᅩᄃ ᅡᄌ ᆨᄋ ᅡ ᆫᄉ ᅳ ᆯᄌ ᅵ ᅦᄉ ᅮᄋ ᆨᄅ ᅵ ᆯᄌ ᅲ ᅡᄅ ᅭᄋ ᅴᄀ ᅢᄉ ᅮᄀ ᅡᄃ ᅮᄀ ᅢᄅ ᅩᄉ ᅥ ᅩᄃ ᄆ ᅮᄃ ᆼᄋ ᅩ ᆯᄒ ᅵ ᅡᄀ ᅵᄄ ᅢᄆ ᆫᄋ ᅮ ᅵᄃ ᅡ. ᄉ ᅡᄒ ᅮᄀ ᆷᅥ ᅥ ᆼ ᄌᄋ ᅴ p-ᄀ ᆹᄋ ᅡ ᅵᄆ ᅩᄃ ᅮ 1ᄋ ᅵᄆ ᅧᄋ ᅵᄂ ᆫᄆ ᅳ ᅩᄃ ᆫᄆ ᅳ ᅩᄒ ᆼᄋ ᅧ ᅦᄉ ᅥᄉ ᆯᄌ ᅵ ᅦᄅ ᅩ VaR ᄋ ᅵᄉ ᆼᄃ ᅡ ᅥᄏ ᆫᄀ ᅳ ᅪ ᅡᄉ ᄃ ᆫᄉ ᅩ ᆯᄋ ᅵ ᆯᄋ ᅳ ᆸᄋ ᅵ ᆫᄇ ᅳ ᅵᄋ ᆯᄋ ᅲ ᅵᄀ ᅵᄃ ᅢᄃ ᅬᄂ ᆫᄇ ᅳ ᅵᄋ ᆯ αᄋ ᅲ ᅪᄃ ᆼᄋ ᅩ ᆯᄒ ᅵ ᆷᄋ ᅡ ᆯᄂ ᅳ ᅡᄐ ᅡᄂ ᆫᄃ ᅢ ᅡ. ᄄ ᅩᄒ ᆫᄃ ᅡ ᆨᄅ ᅩ ᆸᄉ ᅵ ᆼᅥ ᅥ ᆷ 거 ᆼ ᄌᄋ ᅴ p-ᄀ ᆹᄋ ᅡ ᅵᄆ ᅩᄃ ᅮ 0.831ᄅ ᅩ ᆼᄋ ᅩ ᄃ ᆯᅡ ᅵ ᄒᄆ ᅧᄆ ᅩᄃ ᆫᄆ ᅳ ᅩᄒ ᆼᄋ ᅧ ᅦᄉ ᅥᅪ ᄀᄃ ᅡᄉ ᆫᄉ ᅩ ᆯᄀ ᅵ ᅪᄌ ᆼᄋ ᅥ ᅵᄉ ᅵᄀ ᅨᄋ ᆯᄌ ᅧ ᆨᄋ ᅥ ᅳᄅ ᅩᄃ ᆨᄅ ᅩ ᆸᄌ ᅵ ᆨᄋ ᅥ ᅵᄃ ᅡ. α = 0.05ᄋ ᅴᄀ ᆼᄋ ᅧ ᅮᄉ ᆯᄌ ᅵ ᅦᄌ ᅡᄅ ᅭᄅ ᅩᄇ ᅮᄐ ᅥᄀ ᅨᄉ ᆫᄃ ᅡ ᆫᄀ ᅬ ᆼᄒ ᅧ ᆷᄌ ᅥ ᆨ 5% VaRᄋ ᅥ ᆫ -3.340ᄋ ᅳ ᅳᄅ ᅩᄉ ᅥ MSN ᄆ ᅩᄒ ᆼᄋ ᅧ ᅵᄀ ᅡᄌ ᆼᄀ ᅡ ᅡᄁ ᅡ ᆫᄎ ᅮ ᄋ ᅮᄌ ᆼᄀ ᅥ ᆹ (-3.468)ᄋ ᅡ ᆯᄉ ᅳ ᆼᄉ ᅢ ᆼᄒ ᅥ ᅡᄋ ᆻᄃ ᅧ ᅡ. 5ᄀ ᅢᄆ ᅩᄒ ᆼᄋ ᅧ ᅦᄃ ᅢᄒ ᆫᄉ ᅡ ᅡᄒ ᅮᄀ ᆷᅥ ᅥ ᆼ ᄌᄀ ᅪᄃ ᆨᄅ ᅩ ᆸᄉ ᅵ ᆼᅥ ᅥ ᆷ 거 ᆼ ᄌᄀ ᆯᄀ ᅧ ᅪ (p-ᄀ ᆹ)ᄀ ᅡ ᅡ Table 3.3ᄋ ᅴ 3ᄇ ᆫᄍ ᅥ ᅢᅪ ᄋ 4ᄇ ᆫᄍ ᅥ ᅢᄒ ᆼᄋ ᅢ ᅦᄌ ᆼᄅ ᅥ ᅵᄃ ᅬᄋ ᅥᄋ ᆻᄃ ᅵ ᅡ. ᄋ ᅲᄋ ᅴᄉ ᅮᄌ ᆫ 0.05ᄒ ᅮ ᅡᄋ ᅦᄉ ᅥᄆ ᅩᄃ ᆫᄆ ᅳ ᅩᄒ ᆼᄋ ᅧ ᅴ p-ᄀ ᆹᄋ ᅡ ᅵᄋ ᅲᄋ ᅴᄉ ᅮᄌ ᆫᄇ ᅮ ᅩᄃ ᅡᄏ ᅥᄉ ᅥᄀ ᅱᄆ ᅮ ᅡᄉ ᄀ ᆯᄋ ᅥ ᆯᄀ ᅳ ᅵᄀ ᆨᄉ ᅡ ᅵᄏ ᅵᄌ ᅵᄆ ᆺᄒ ᅩ ᅡᄆ ᅳᄅ ᅩᄆ ᅩᄃ ᆫᄆ ᅳ ᅩᄒ ᆼᄃ ᅧ ᆯᄋ ᅳ ᅵᄌ ᆨᄒ ᅥ ᆸᄒ ᅡ ᅡᄃ ᅡᄀ ᅩᄒ ᆯᄉ ᅡ ᅮᄋ ᆻᄃ ᅵ ᅡ. ᄀ ᅳᄅ ᅥᄂ ᅡᄉ ᅡᄒ ᅮᄀ ᆷᅥ ᅥ ᆼ ᄌᄋ ᅴᄀ ᆼᄋ ᅧ ᅮ MCFAᄋ ᅪ MSN ᄆ ᅩᄒ ᆼᄋ ᅧ ᅴ p-ᄀ ᆹᄋ ᅡ ᅵᄀ ᆨᄀ ᅡ ᆨ 0.06ᄀ ᅡ ᅪ 0.19ᄅ ᅩᄉ ᅥᄃ ᅡᄅ ᆫᄆ ᅳ ᅩᄒ ᆼᄃ ᅧ ᆯᄋ ᅳ ᅴ p-ᄀ ᆹ (1)ᄋ ᅡ ᅦᄇ ᅵᄒ ᅡᄋ ᅧᄉ ᆼᄃ ᅡ ᅢᄌ ᆨᄋ ᅥ ᅳᄅ ᅩᄌ ᆨᄋ ᅡ ᆫᄀ ᅳ ᆹᄋ ᅡ ᆯᄀ ᅳ ᅡ ᅵᄀ ᄌ ᅩᄋ ᆻᄋ ᅵ ᅳᄆ ᅳᄅ ᅩᄃ ᅮᄆ ᅩᄒ ᆼᄋ ᅧ ᆫᄉ ᅳ ᆯᄌ ᅵ ᅦᄅ ᅩ VaR ᄋ ᅵᄉ ᆼᄃ ᅡ ᅥᄏ ᆫᄀ ᅳ ᅪᄃ ᅡᄉ ᆫᄉ ᅩ ᆯᄋ ᅵ ᆯᄋ ᅳ ᆸᄋ ᅵ ᆫᄇ ᅳ ᅵᄋ ᆯᄋ ᅲ ᅵᄀ ᅵᄃ ᅢᄃ ᅬᄂ ᆫᄇ ᅳ ᅵᄋ ᆯ αᄋ ᅲ ᅪᄃ ᆼᄋ ᅩ ᆯᄒ ᅵ ᆯ ᅡ ᆺᄋ ᅥ ᄀ ᅵᄅ ᅡᄂ ᆫᄉ ᅳ ᆫᄅ ᅵ ᅬᄀ ᆷᄋ ᅡ ᅵᄃ ᅡᄉ ᅩᄄ ᆯᄋ ᅥ ᅥᄌ ᆫᄃ ᅵ ᅡᄀ ᅩᄒ ᆯᄉ ᅡ ᅮᄋ ᆻᄃ ᅵ ᅡ. MFA, MCtFA, MSt ᄆ ᅩᄒ ᆼᄋ ᅧ ᆫᄎ ᅳ ᅩᅪ ᄀᄇ ᅵᄋ ᆯ (ER)ᄋ ᅲ ᅵ 1ᄇ ᅩᄃ ᅡ ᅥᄉ ᄏ ᅥᅡ ᄃᄉ ᅩᅪ ᄀᄉ ᅩᄎ ᅮᄌ ᆼᄃ ᅥ ᅬᄋ ᆻᄋ ᅥ ᅳᄆ ᅧ, ᄇ ᆫᄆ ᅡ ᆫᄋ ᅧ ᅦ MCFAᄋ ᅪ MSNᄋ ᆫ 1ᄇ ᅳ ᅩᄃ ᅡᄌ ᆨᄋ ᅡ ᅡᄉ ᅥᄃ ᅡᄉ ᅩᅪ ᄀᄃ ᅢᄎ ᅮᄌ ᆼᄃ ᅥ ᅬᄋ ᆻᄃ ᅥ ᅡᄀ ᅩᄒ ᆯᄉ ᅡ ᅮᄋ ᆻ ᅵ ᅡ. ᅩ ᄃ ᆨ ᄃᄅ ᆸᅥ ᅵ ᆼ 서 ᆷ 거 ᆼ ᄌᄋ ᅴ p-ᄀ ᆹᄋ ᅡ ᅵᄆ ᅩᄃ ᅮᄋ ᅲᄋ ᅴᄉ ᅮᄌ ᆫ 0.05ᄇ ᅮ ᅩᄃ ᅡᄏ ᅳᄆ ᅳᄅ ᅩᄆ ᅩᄃ ᆫᄆ ᅳ ᅩᄒ ᆼᄋ ᅧ ᅦᄉ ᅥᄀ ᅪᄃ ᅡᄉ ᆫᄉ ᅩ ᆯᄀ ᅵ ᅪᄌ ᆼᄋ ᅥ ᅵᄉ ᅵᄀ ᅨᄋ ᆯᄌ ᅧ ᆨᄋ ᅥ ᅳᄅ ᅩ.
(8) 776. Kwangyee Ko · Jangsun Baek. ᆨᄅ ᅩ ᄃ ᆸᄌ ᅵ ᆨᄋ ᅥ ᅵᄃ ᅡ. Figure 3.3ᄋ ᅦᄂ ᆫ α = 0.01ᄀ ᅳ ᅪ α = 0.05 ᄉ ᅡᄋ ᅵᄋ ᅦᄉ ᅥ 100ᄀ ᅢᄋ ᅴ αᄀ ᆹᄋ ᅡ ᅵᄌ ᆼᄀ ᅳ ᅡᄒ ᆷᄋ ᅡ ᅦᄄ ᅡᄅ ᅡᄀ ᅳᄋ ᅦᄃ ᅢᄋ ᆼᄒ ᅳ ᅡᄂ ᆫᄀ ᅳ ᆼᄒ ᅧ ᆷᄌ ᅥ ᆨ ᅥ VaRᄀ ᅪᄀ ᆨᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᅴ VaR ᄎ ᅮᄌ ᆼᄀ ᅥ ᆹᄃ ᅡ ᆯᄋ ᅳ ᅵᄀ ᅳᄅ ᅧᄌ ᅧᄋ ᆻᄃ ᅵ ᅡ. ᄃ ᅢᄎ ᅦᄅ ᅩ MFA, MCtFA, MSN ᄆ ᅩᄒ ᆼᄋ ᅧ ᅵ MStᄂ ᅡ MCFA ᄆ ᅩ ᆼᄋ ᅧ ᄒ ᅦᄇ ᅵᄒ ᅡᄋ ᅧᄀ ᆼᄒ ᅧ ᆷᄌ ᅥ ᆨ VaRᄋ ᅥ ᅦᄇ ᅩᄃ ᅡᄃ ᅥᄀ ᅡᄁ ᆸᄀ ᅡ ᅦᄎ ᅮᄌ ᆼᄒ ᅥ ᅡᄀ ᅩᄋ ᆻᄋ ᅵ ᆷᅳ ᅳ ᆯ ᄋᄋ ᆯᄉ ᅡ ᅮᄋ ᆻᄃ ᅵ ᅡ. αᄀ ᆹᄋ ᅡ ᅴᄇ ᆫᄒ ᅧ ᅪᄋ ᅦᄄ ᅡᄅ ᅡᄀ ᆨᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᅴ VaR ᄎ ᅮᄌ ᆼᄀ ᅥ ᆹ (V aRi,α , i = MFA, MCFA, MCtFA, MSN, MSt)ᄃ ᅡ ᆯᄋ ᅳ ᅵᄀ ᆼᄒ ᅧ ᆷᄌ ᅥ ᆨ VaR ᄀ ᅥ ᆹ (V aRe,α )ᄀ ᅡ ᅪᅡ ᆼ ᄉᄃ ᅢᄌ ᆨ ᅥ P.05 2 (V aR − V aR ) /|V aR |ᄅ ᆯ ᅳ ᅳᄅ ᄋ ᅩᄋ ᆯᄆ ᅥ ᅡᄂ ᅡᄎ ᅡᄋ ᅵᄀ ᅡᄂ ᅡᄂ ᆫᄌ ᅳ ᅵᄋ ᆯᄋ ᅡ ᅡᄇ ᅩᄀ ᅵᄋ ᅱᄒ ᅢᄉ ᆼᄃ ᅡ ᅢᄌ ᆨᄌ ᅥ ᅦᄀ ᆸᄎ ᅩ ᅡᄋ ᅵ i,α e,α e,α α=.01 ᅨᄉ ᄀ ᆫᅡ ᅡ ᄒᄆ ᆫ MFA (0.756), MCtFA (1.592), MSN (2.259), MSt (4.159), MCFA (20.849) ᄉ ᅧ ᆫᄋ ᅮ ᅵᄃ ᅡ. ᄄ ᅡᄅ ᅡᄉ ᅥ α = 0.01ᄀ ᅪ α = 0.05 ᄉ ᅡᄋ ᅵᄋ ᅴ VaR ᄎ ᅮᄌ ᆼᄋ ᅥ ᅦᄋ ᆻᄋ ᅵ ᅥᄉ ᅥ MFAᄋ ᅪ MCtFA ᄆ ᅩᄒ ᆼᄋ ᅧ ᅵᄃ ᅡᄅ ᆫᄆ ᅳ ᅩᄒ ᆼᄃ ᅧ ᆯᄋ ᅳ ᅦᄇ ᅵᄒ ᅢᄉ ᅥᄉ ᆼᄃ ᅡ ᅢᄌ ᆨ ᅥ ᅳᄅ ᄋ ᅩᅡ ᄌᄅ ᅭᄋ ᅴᄀ ᆼᄒ ᅧ ᆷᄌ ᅥ ᆨ VaRᄋ ᅥ ᅦᄌ ᅩᄀ ᆷᄃ ᅳ ᅥᄀ ᅡᄁ ᅡᄋ ᆫᄎ ᅮ ᅮᄌ ᆼᄀ ᅥ ᆹᄋ ᅡ ᆯᄃ ᅳ ᅩᄎ ᆯᄒ ᅮ ᅡᄋ ᆻᄃ ᅧ ᅡ.. Figure 3.3 Comparison of VaRs. 4. 결론 ᅮᅡ ᄐ ᄌᄋ ᅱᄒ ᆷᄋ ᅥ ᆯᄀ ᅳ ᆫᄅ ᅪ ᅵᄒ ᅢᄋ ᅣᄒ ᅡᄂ ᆫᄀ ᅳ ᆷᅲ ᅳ ᆼ ᄋᄀ ᅵᄀ ᆫᄋ ᅪ ᅦᄉ ᅥᄃ ᅡᄉ ᅮᄋ ᅴᄐ ᅮᄌ ᅡᄉ ᅮᄋ ᆨᄋ ᅵ ᅳᄅ ᅩᄀ ᅮᄉ ᆼᄃ ᅥ ᆫᄑ ᅬ ᅩᄐ ᅳᄑ ᆯᄅ ᅩ ᅵᄋ ᅩ VaRᄅ ᆯᄌ ᅳ ᆼᄒ ᅥ ᆨᄒ ᅪ ᅵᄎ ᅮᄌ ᆼ ᅥ ᄒᄂ ᅡ ᆫᄀ ᅳ ᆺᄋ ᅥ ᆫᄆ ᅳ ᅢᄋ ᅮᄌ ᆼᄋ ᅮ ᅭᄒ ᅡᄃ ᅡ. ᄇ ᆫᄋ ᅩ ᆫᄀ ᅧ ᅮᄋ ᅦᄉ ᅥᄂ ᆫᄀ ᅳ ᅩᄎ ᅡᄋ ᆫᄌ ᅯ ᅡᄅ ᅭᄇ ᆫᄑ ᅮ ᅩᄌ ᆨᄒ ᅥ ᆸᄋ ᅡ ᅦᄋ ᅲᄋ ᆼᄒ ᅭ ᆫᄋ ᅡ ᅧᄅ ᅥᄀ ᅡᄌ ᅵᄒ ᆫᄒ ᅩ ᆸᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᆯᄋ ᅳ ᅵᄋ ᆼ ᅭ ᅡᄋ ᄒ ᅧᅩ ᆼ ᄎᄐ ᅮᄌ ᅡᄉ ᅮᄋ ᆨᄅ ᅵ ᆯᄋ ᅲ ᅴ VaRᄅ ᆯᄎ ᅳ ᅮᄌ ᆼᄒ ᅥ ᅡᄋ ᆻᄃ ᅧ ᅡ. ᄃ ᅡᄉ ᅮᄋ ᅴᄐ ᅮᄌ ᅡᄉ ᅮᄋ ᆨᄅ ᅵ ᆯᅮ ᅲ ᆫ ᄇᄑ ᅩᄅ ᅩᄉ ᅥᄇ ᅵᄃ ᅢᄎ ᆼᄌ ᅵ ᆨᄒ ᅥ ᆫᄒ ᅩ ᆸᄆ ᅡ ᅩᄒ ᆼᄈ ᅧ ᆫᄆ ᅮ ᆫᄋ ᅡ ᅡᄂ ᅵ ᅡᄀ ᄅ ᅩᅡ ᄎᄋ ᆫᄌ ᅯ ᅡᄅ ᅭᄇ ᆫᄑ ᅮ ᅩᄋ ᅴᄆ ᅩᄉ ᅮᄎ ᅮᄌ ᆼᄋ ᅥ ᅦᄇ ᅩᄃ ᅡᄌ ᆯᄋ ᅥ ᆨᄌ ᅣ ᆨᄋ ᅥ ᆫᄃ ᅵ ᅡᄋ ᆼᄒ ᅣ ᆫᄋ ᅡ ᅭᄋ ᆫᄇ ᅵ ᆫᄉ ᅮ ᆨᄌ ᅥ ᅡᄆ ᅩᄒ ᆼᄃ ᅧ ᆯᄋ ᅳ ᅵᄀ ᅩᄅ ᅧᅬ ᄃᄋ ᆻᄃ ᅥ ᅡ. ᄉ ᆯᄌ ᅵ ᆼᅮ ᅳ ᆫ ᄇᄉ ᆨ ᅥ ᆯᄀ ᅧ ᄀ ᅪᄀ ᅩᄅ ᅧᄃ ᆫᄒ ᅬ ᆫᄒ ᅩ ᆸᄆ ᅡ ᅩᄒ ᆼᄃ ᅧ ᆯᄋ ᅳ ᅵ 1%ᄋ ᅦᄉ ᅥ 5%ᄉ ᅡᄋ ᅵ VaR ᄎ ᅮᄌ ᆼᄋ ᅥ ᅦᄋ ᆻᄋ ᅵ ᅥᄉ ᅥᄆ ᅩᄃ ᅮᄇ ᅵᄉ ᆺᄒ ᅳ ᆫᅥ ᅡ ᆼ ᄉᄂ ᆼᅳ ᅳ ᆯ ᄋᄇ ᅩᄋ ᅧᄌ ᅮᄀ ᅩᄋ ᆻᄋ ᅵ ᅳᄂ ᅡ, MFAᄋ ᅪ MCtFA ᄆ ᅩᄒ ᆼᄋ ᅧ ᅵᄃ ᅡᄅ ᆫᄆ ᅳ ᅩᄒ ᆼᄃ ᅧ ᆯᄋ ᅳ ᅦᄇ ᅵᄒ ᅢᄉ ᅥᄉ ᆼᄃ ᅡ ᅢᄌ ᆨᄋ ᅥ ᅳᄅ ᅩᄌ ᅡᄅ ᅭᄋ ᅴᄀ ᆼᄒ ᅧ ᆷᄌ ᅥ ᆨ VaRᄋ ᅥ ᅦᄌ ᅩᄀ ᆷᄃ ᅳ ᅥᄀ ᅡᄁ ᅡᄋ ᆫᄎ ᅮ ᅮᄌ ᆼ ᅥ ᆹᄋ ᅡ ᄀ ᆯᄃ ᅳ ᅩᄎ ᆯᄒ ᅮ ᅡᄋ ᆻᄃ ᅧ ᅡ. ᄄ ᅩᄒ ᆫᄉ ᅡ ᆼᄃ ᅡ ᅢᄌ ᆨᄋ ᅥ ᅳᄅ ᅩᄋ ᅲᄋ ᆫᄒ ᅧ ᆫᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᆫ MSN, MSt, MFAᄋ ᅵ ᅴᄀ ᆼᄋ ᅧ ᅮᄒ ᆫᄒ ᅩ ᆸᅥ ᅡ ᆼ ᄉᄇ ᆫᄋ ᅮ ᅴᄀ ᅢᄉ ᅮᄀ ᅡᄒ ᆫᄀ ᅡ ᅢ ᆫᄆ ᅵ ᄋ ᅩᄒ ᆼᄋ ᅧ ᅵᅬ ᄎᄌ ᆨᄋ ᅥ ᅳᄅ ᅩᄌ ᆨᄒ ᅥ ᆸᄃ ᅡ ᆫᄇ ᅬ ᆫᄆ ᅡ ᆫ, ᄇ ᅧ ᅩᄃ ᅡᄌ ᅦᄒ ᆫᄌ ᅡ ᆨᄋ ᅥ ᆫᄆ ᅵ ᅩᄒ ᆼᄋ ᅧ ᆫ MCFA, MCtFAᄂ ᅵ ᆫᄌ ᅳ ᅡᄅ ᅭᄋ ᅴᄋ ᅫᄃ ᅩᄋ ᅪᄃ ᅮᄁ ᅥᄋ ᆫᄁ ᅮ ᅩᄅ ᅵ ᆨᄉ ᅳ ᄐ ᆼᄋ ᅥ ᆯᄇ ᅳ ᆫᄋ ᅡ ᆼᄒ ᅧ ᅡᄀ ᅵᄋ ᅱᄒ ᅢᄉ ᆼᄇ ᅥ ᆫᄋ ᅮ ᅴᄀ ᅢᄉ ᅮᄀ ᅡᄀ ᆨᄀ ᅡ ᆨ 2ᄀ ᅡ ᅢ, 5ᄀ ᅢᄋ ᆫᄒ ᅵ ᆫᄒ ᅩ ᆸᄆ ᅡ ᅩᄒ ᆼᄋ ᅧ ᅵᅬ ᄎᄌ ᆨᄆ ᅥ ᅩᄒ ᆼᄋ ᅧ ᅳᄅ ᅩᄉ ᆫᄌ ᅥ ᆼᄃ ᅥ ᅬᄋ ᅥᄌ ᆨᄒ ᅥ ᆸᄃ ᅡ ᅬᄋ ᆻᄃ ᅥ ᅡ..
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(10) Journal of the Korean Data & Information Science Society 2018, 29(3), 769–778. http://dx.doi.org/10.7465/jkdi.2018.29.3.769 ᆫᄀ ᅡ ᄒ ᆨᄃ ᅮ ᅦᄋ ᅵᄐ ᅥᄌ ᆼᄇ ᅥ ᅩᅪ ᄀᄒ ᆨᄒ ᅡ ᅬᄌ ᅵ. VaR estimation using skewed mixture models and various mixtures of factor analyzers. †. Kwangyee Ko1 · Jangsun Baek2 12. Department of Statistics, Chonnam National University. Received 15 May 2018, revised 24 May 2018, accepted 25 May 2018. Abstract It is very important to estimate VaR (Value-at-Risk) of the portfolio with many returns on investment for risk management. The distribution of returns has often thick tails, skewness or kurtosis. In order to estimate VaR of the total investment return, we consider not only mixture of skewed distributions models such as MSN (Mixtures of skew-normal distribution) and MSt (Mixtures of skew-t distribution), but also various parsimonious mixtures of factor analyzers models of MFA (Mixtures of factor analyzers), MCFA (Mixtures of common factor analyzers) and MCtFA (Mixtures of common skew-t factor analyzers). Application of the models to the KOSPI returns data showed that all models have the similar performance for estimating 1% - 5% VaR, but the estimates of MFA and MCtFA are more close to the empirical VaRs of the data. Keywords: EM algorithm, mixtures of factor analyzers, portfolio, skewed mixture model, VaR.. †. This study was financially supported by Chonnam National University, 2015. Lecturer, Department of Statistics, Chonnam National University, Gwangju 61186, Korea. 2 Corresponding author: Professor, Department of Statistics, Chonnam National University, Gwangju 61186, Korea. E-mail: [email protected] 1.
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