题错了吧

问题如下：

A risk manager is estimating the market risk of a portfolio using both the normal distribution and the lognormal distribution assumptions. The manager gathers the following data on the portfolio:  Annual mean: 15%  Annual volatility: 35%  Current portfolio value: EUR 4,800,000  Trading days in a year: 252 Which of the following statements is correct?

选项：

A.Lognormal 95% VaR is less than normal 95% VaR at the 1-day holding period by 0.13% B.Lognormal 95% VaR is less than normal 95% VaR at the 1-year holding period by 7.91% C.Lognormal 99% VaR is less than normal 99% VaR at the 1-day holding period by 1.43% D.Lognormal 99% VaR is less than normal 99% VaR at the 1-year holding period by 13.86%

解释：

Annual return = 0.15; Daily return = 0.15/252 = 0.000595

Annual volatility = 0.35; Daily volatility = 0.35/sqrt(252) = 0.022048

The normal VaR and lognormal VaR, in percentage terms, are calculated as follows (ignoring negative signs):

1-day normal 95% VaR = (R_p – zs) = (0.000595 – 1.645 * 0.022048) = 3.57%

1-day lognormal 95% VaR = (1 – e^{[Rp – zs]}) = (1 – exp[0.000595 – 1.645*0.022048])m = 3.51%

1-year normal 95% VaR = (R_p – zs) = (0.15 – 1.645 * 0.35) = 42.58%

1-year lognormal 95% VaR = (1 – exp[0.15 – (1.645*0.35)]) = 34.67%

Also,

1-day normal 99% VaR = (R_p – zs) = (0.000595 – 2.326 * 0.022048) = 5.07%

1-day lognormal 99% VaR = (1 – e^{[Rp – zs]}) = (1 – exp[0.000595 – 2.326*0.022048]) = 4.94%

1-year normal 99% VaR = (R_p – zs) = (0.15 – 2.326 * 0.35) = 66.41%

1-year lognormal 99% VaR = (1 – exp[0.15 – (2.326*0.35)]) = 48.53%

Hence,

1-day holding period: Lognormal 95% VaR is smaller than Normal 95% VaR by: 3.57 – 3.51 = 0.06%. So, A is incorrect. (See explanation for C below).

1-year holding period: Lognormal 95% VaR is smaller than Normal 95% VaR by: 42.58 – 34.67 = 7.91%. So, B is correct.

1-day holding period: Lognormal 99% VaR is smaller than Normal 99% VaR by: 5.07 – 4.94 = 0.13%. So, C is incorrect (4.94 – 3.51 = 1.43%).

1-year holding period: Lognormal 99% VaR is smaller than Normal 99% VaR by: 66.41 – 48.53 = 17.88%. So, D is incorrect (48.53 – 34.67 = 13.86%).

The normal VaR and lognormal VaR, in value terms, are calculated as follows (ignoring negative signs):

1-day normal 95% VaR = (R_p – zs)*V = (0.000595 – 1.645 * 0.022048)*4.8m = EUR 171,235

1-day lognormal 95% VaR = (1 – e^{[Rp – zs]})*V = (1 – exp[0.000595 – 1.645*0.022048])*4.8m = EUR 168,217

1-year normal 95% VaR = (R_p – zs)*V = (0.15 – 1.645 * 0.35)*4.8m = EUR 2,043,600

1-year lognormal 95% VaR = (1 – exp[0.15 – (1.645*0.35)])*4.8m = EUR 1,664,258
Also,1-day normal 99% VaR = (R_p – zs)*V = (0.000595 – 2.326 * 0.022048)*4.8m = EUR 243,306

1-day lognormal 99% VaR = (1 – e^{[Rp – zs]})*V = (1 – exp[0.000595 – 2.326*0.022048])*4.8m = EUR 237,242

1-year normal 99% VaR = (R_p – zs)*V = (0.15 – 2.326 * 0.35)*4.8m = EUR 3,187,680

1-year lognormal 99% VaR = (1 – exp[0.15 – (2.326*0.35)])*4.8m = EUR 2,329,264
Hence,1-day holding period: Lognormal 95% VaR is smaller than Normal 95% VaR by: 171,235 – 168,217 = EUR 3,018.

1-year holding period: Lognormal 95% VaR is smaller than Normal 95% VaR by: 2,043,600 – 1,664,258 = EUR 379,342.

1-day holding period: Lognormal 99% VaR is smaller than Normal 99% VaR by: 243,306 – 237,242 = EUR 6,064.

1-year holding period: Lognormal 99% VaR is smaller than Normal 99% VaR by: 3,187,680 – 2,329,264 = EUR 858,416.