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天才出于勤奋 · 2021年12月25日

求连续两年的概率，为什么不能先求一年的，然后再连续两年？

  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.

即先按照一年20%*10%/20%*10%+90%*5%，算出来一年的概率，连续两年就平方

这样结果算出来是不一样的，为什么？

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2 个答案

品职答疑小助手雍 · 2022年01月16日

我先回答第二个问题：对star来说，1年期发生事件（5%超额收益）的概率是20%，连续两年发生这个事情的概率不就是20%*20%么，普通基金经理同理，想不出这里会有其他的歧义。

第一个问题：根据第二个问题的结果：Pr(H,H|S)=4%，Pr（H，H|N）=0.25%，而S的数量占总体的10%，N的数量占90%，所以Pr(H,H)=4%*10%+0.25%*90%=0.625%。 而Pr (high,high| star)*Pr(star)则等于4%*10%=0.4%。最终结果就是0.4%/0.625%=64%


这题我觉得可以简化处理的点就是把上述的第二个问题连续两年超额收益的概率单独算出来（4%和0.25%那两个结果），重新作为题目的条件，那结构就简单了。

品职答疑小助手雍 · 2021年12月26日

同学你好，题意要求的是，如果连续两年beat market，他是superstar的概率是多少。

你是想算一年beat market的话，superstar的概率是多少，然后再平方。这方法你写成平方后写出来就会发现你都不知道分子分母代表的意义了。所以是不对的。

起码就分母来说，star和非star 打败市场的概率应该是相互独立的，但是平方之后还带着相关性就更无法理解了。列完式子最好还是看一下能不能解释出它的含义，解释不出显然就是有问题的了。

Clare · 2022年01月16日

老师好，请问这道题 可以画图吗，关于公式有2点不明 1.Pr(High, High|Star) 为什么= 20% * 20% Pr (Star) is still 10%.(计算上和公式表达不太明白），我觉得按照bayes公式 Pr(star|high,high)= Pr (high,high| star)*Pr(star)/Pr(high,high)= 20%*10%/ Pr(high,high)。2.Pr（H，H)=Pr(H,H|S)*Pr(S)+Pr(H,H|N)*Pr(N)其中为什么算Pr（H，H|S)是20%*20%？Pr（H，H|N）=5%*5%？ 谢谢老师

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