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13995173783 · 2023年06月08日

请老师画图并用汉语讲解一下这个题目谢谢！

  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.

这个题目的提问内容意思我没有完全看懂，另外请老师用中文，并用图示法讲解一下，谢谢！


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品职答疑小助手雍 · 2023年06月09日

同学你好，我觉得画图反而不如把情况列清楚更看得清晰。

因为题目给的前提是发现了一个人连续两年高收益，star连续两年高收益的概率是20%*20%=4%，普通人连续两年高收益的概率是5%*5%=0.25%，然后10%的人是star，90%的是普通，所以随意一个人连续高收益的概率就是4%*10%+0.25%*90%=0.625%。

其中这人是star的概率就是4%*10%=0.4%，

现在已知连续高收益的话，star的概率就是0.4/0.625

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