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blade8932 · 2022年09月12日

这道题用排除法做的，思路应该没问题吧，A和B的选项都比较好算或者直接辨认,C的话还需要老老实实的计算（应该没有直接得出MAD的捷径吧）老师对吗？

  1. CFA I
  2. Quantitative

NO.PZ2021061603000025

问题如下：

Annual returns and summary statistics for three funds are listed in the following exhibit:

The fund with the highest absolute dispersion is:

选项：

A.Fund PQR if the measure of dispersion is the range B.Fund XYZ if the measure of dispersion is the variance C.Fund ABC if the measure of dispersion is the mean absolute deviation

解释：

C is correct. The mean absolute deviation (MAD) of Fund ABC's returns is greater than the MAD of both of the other funds.

MDA=∑i=1n∣Xi−Xˉ∣nMDA = \frac{{\sum\limits_{i = 1}^n {\left| {{X_i} - \bar X} \right|} }}{n}, where Xˉ{\bar X} is the arithmetic mean of the series.

MAD for Fund ABC =

∣−20−(−4)∣+∣23−(−4)∣+∣−14−(−4)∣+∣5−(−4)∣+∣−14−(−4)∣5=14.4%\frac{{\left| { - 20 - ( - 4)} \right| + \left| {23 - ( - 4)} \right| + \left| { - 14 - ( - 4)} \right| + \left| {5 - ( - 4)} \right| + \left| { - 14 - ( - 4)} \right|}}{5} = 14.4\%

MAD for Fund XYZ=

∣−33−(−10.8)∣+∣−12−(−10.8)∣+∣−12−(−10.8)∣+∣−8−(−10.8)∣+∣11−(−10.8)∣5=9.8%\frac{{\left| { - 33 - ( - 10.8)} \right| + \left| { - 12 - ( - 10.8)} \right| + \left| { - 12 - ( - 10.8)} \right| + \left| { - 8 - ( - 10.8)} \right| + \left| {11 - ( - 10.8)} \right|}}{5} = 9.8\%

MAD for Fund PQR=

∣−14−(−5)∣+∣−18−(−5)∣+∣6−(−5)∣+∣−2−(−5)∣+∣3−(−5)∣5=8.8%\frac{{\left| { - 14 - ( - 5)} \right| + \left| { - 18 - ( - 5)} \right| + \left| {6 - ( - 5)} \right| + \left| { - 2 - ( - 5)} \right| + \left| {3 - ( - 5)} \right|}}{5} =8.8\%

A and B are incorrect because the range and variance of the three funds are as follows:


The numbers shown for variance are understood to be in "percent squared" terms so that when taking the square root, the result is standard deviation in percentage terms. Alternatively, by expressing standard deviation and variance in decimal form, one can avoid the issue of units. In decimal form, the variances for Fund ABC, Fund XYZ, and Fund PQR are 0.0317, 0.0243, and 0.0110, respectively.

RT

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星星_品职助教 · 2022年09月14日

MAD的公式其实很好记。MAD全称是Mean Absolute Deviation，这里的每个单词都有代指。

其中deviation就是每个xi的取值和均值x bar的偏离程度（所以MAD这个指标衡量的是离散程度）。

Deviation写成数学符号就是xi - x bar。

现在希望衡量的是总的离散程度，所以就要把各个xi- x bar的值加起来。但是，由于xi可能高于x bar，也可能低于x bar，所以正负值会互相抵消。这是我们不希望看到的，因为无论是xi高于均值还是xi低于均值，都属于对于均值的偏离，抵消后没法合理反应出真实的偏离水平。

所以就要在xi - x bar的外面套一层绝对值，这样就可以去衡量数值上的偏离程度，这就是MAD中的absolute的含义。

最后再除以n，得到平均的偏离水平，这就是MAD中“mean”的含义。

所以通过MAD的名字就可以记住公式。

-----

方差的公式也很类似，同样还是因为deviation，即xi - x bar 有正有负，不希望抵消。所以除了绝对值之外的另一种处理就是给每一项都平方，然后再求这个平方的均值。所以方差也被称为 mean squared deviation。


星星_品职助教 · 2022年09月12日

同学你好，

对的，本题可以使用排除法，MAD只能手算。

但建议在练习的时候多练练MAD的公式，考试的时候不一定每道题都能用排除法。

blade8932 · 2022年09月14日

好的，谢谢老师。

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2024-08-06 10:23 1 · 回答

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2023-05-22 22:38 1 · 回答

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2023-04-22 23:57 1 · 回答

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2023-01-11 12:00 1 · 回答