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·K· · 2023年05月22日

为什么c选项是highest variance？

  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.

MAD=∑i=1n∣Xi−Xˉ∣nMAD = \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.

知道了range variance 还有mad，但我不知道怎么比较得出选项的？

选项的意思能不能翻译下，谢谢老师。

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星星_品职助教 · 2023年05月23日

同学你好，

C选项Fund ABC if the measure of dispersion is the mean absolute deviation 意为如果以MAD为基准来衡量离散程度，则fund ABC的（绝对）离散程度最高（highest absolute dispersion）。

代入MAD的公式，可分别计算出三只fund的MAD为：

Fund ABC：14.4%；

Fund XYZ：9.8%；

Fund PQR：8.8%

由于MAD越大代表离散程度越高，所以最高的Fund ABC的离散程度最大，C选项的描述正确。

A、B选项的翻译和判断和上述逻辑是一样的。

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