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果儿 · 2019年12月14日

问一道题:NO.PZ2017092702000088

问题如下:

A portfolio has an expected return of 7% with a standard deviation of 13%. For an investor with a minimum annual return target of 4%, the probability that the portfolio return will fail to meet the target is closest to:

选项:

A.

33%.

B.

41%.

C.

59%

解释:

B is correct.

There are three steps, which involve standardizing the portfolio return: First, subtract the portfolio mean return from each side of the inequality: P(Portfolio return – 7%) ≤ 4% – 7%). Second, divide each side of the inequality by the standard deviation of portfolio return: P[(Portfolio return – 7%)/13% ≤ (4% – 7%)/13%] = P(Z ≤ –0.2308) = N(–0.2308). Third, recognize that on the left-hand side we have a standard normal variable, denoted by Z and N(–x) = 1 – N(x). Rounding –0.2308 to –0.23 for use with the cumulative distribution function (cdf) table, we have N(–0.23) = 1 – N(0.23) = 1 – 0.5910 = 0.409, approximately 41 percent. The probability that the portfolio will underperform the target is about 41 percent.

为何不是 (7-4)/13得0.23。即F(0.23)为概率落在高于0.23外的概率。请解答,谢谢!

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已采纳答案

星星_品职助教 · 2019年12月16日

同学你好,

这道题的考点是正态分布标准化的过程,标准化的公式为Z=(X-μ)/σ,这道题里μ是7%,直接从公式角度出发解题即可。

另外标准正态分布的Z表是个累计概率分布函数的概念,例如你提到的F(0.23),是0.23左侧的所有面积之和(所有低于0.23的概率之和)。而这道题实际上求的是-0.23左侧的面积之和。也就是F(-0.23),或者1-F(0.23),注意1-F(0.23)才是概率落在高于0.23外的概率,加油~

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