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Stefanie🍅 · 2021年02月06日

问贝叶斯公式的一道题No.PZ2020010301000006 (问答题)

  1. FRM I
  2. Quantitative

NO.PZ2020010301000006

问题如下：

Suppose that 10% of fund managers are superstars. Superstars have a 20% chance of beating their benchmark by more than 5% each year(high return), whereas normal fund managers have only a 5% chance of beating their benchmark by more than 5%.

Continue the application of Bayes’ rule to compute the probability that a manager is a superstar after observing two years of “high” returns.

选项：

解释：

Consider the three scenarios: (High, High), (High, Low) and (Low, Low). We are interested in Pr (Star|High, High) using Bayes’ rule, this is equal to

Pr(High, High|Star)Pr(Star) /Pr(High, High).

Stars produce high returns in 20% of years, and so Pr(High, High|Star) = 20% * 20% Pr (Star) is still 10%.

Finally, we need to compute Pr (High, High), which is Pr(High, High|Star) Pr(Star) + Pr(High, High|Normal)Pr(Normal).

This value is 20% * 20% * 10% + 5% * 5% * 90% = 0.625%. Combing these values,

20% * 20% * 10%/0.625%=64%

This is a large increase from the 30% chance after one year.

Suppose that 10% of fund managers are superstars. Superstars have a 20% chance of beating their benchmark by more than 5% each year(high return), whereas normal fund managers have only a 5% chance of beating their benchmark by more than 5%.

Continue the application of Bayes’ rule to compute the probability that a manager is a superstar after observing two years of “high” returns.


解析

Consider the three scenarios: (High, High), (High, Low) and (Low, Low). We are interested in Pr (Star|High, High) using Bayes’ rule, this is equal to

Pr(High, High|Star)Pr(Star) /Pr(High, High).

Stars produce high returns in 20% of years, and so Pr(High, High|Star) = 20% * 20% Pr (Star) is still 10%.

Finally, we need to compute Pr (High, High), which is Pr(High, High|Star) Pr(Star) + Pr(High, High|Normal)Pr(Normal).

This value is 20% * 20% * 10% + 5% * 5% * 90% = 0.625%. Combing these values,

20% * 20% * 10%/0.625%=64%

This is a large increase from the 30% chance after one year.


我是用画图最后得出这题的答案是64%。但是看答案底下还有个30%不太明白那个是答案吗？

我是(10%*20*20%)/(5%*5%*90+10%*20%*20%)=0.004/0.00625=64%


就是想问问解析最后一句30%是干啥的。。

谢谢

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3 个答案

品职答疑小助手雍 · 2023年04月19日

是有什么问题么？还是在验算

大山 · 2023年04月18日

30.76%，2/6.5

品职答疑小助手雍 · 2021年02月07日

嗨，爱思考的PZer你好：


它这个只是追加了一句对比性的描述，就是如果你只看到1年high的话，这个人（约）30%是一个superstar。

不过如果连续2年它都high的话，这个人按这题的答案有64%是superstar。


-------------------------------
努力的时光都是限量版，加油！


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