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oceanskytang · 2024年05月04日

第二问的理解

  1. FRM I
  2. Quantitative

NO.PZ2020010303000012

问题如下：

The monthly return on a hedge fund portfolio with USD 1 billion in assets is N(.02, .0003). What is the distribution of the gain in a month?

a. The fund has access to a USD 10 million line of credit that does not count as part of its portfolio. What is the chance that the firm’s loss in a month exceeds this line of credit?

b. What would the line of credit need to be to ensure that the firm’s loss was less than the line of credit in 99.9% of months (or equivalently, larger than the LOC in 0.1% of months)?

解释：

a. The monthly return is 2%, and the monthly standard deviation is 1.73%. In USD, the monthly change in portfolio value has a mean of 2% * USD 1 billion = USD 20 million and a standard deviation of 1.73% * USD 1 billion = USD 17.3 million. The probability that the portfolio loses more than USD 10 million is than (working in millions)

Pr(V<−10)=Pr(V−2017.3<−10−2017.3)=Pr(Z<−1.73)Pr(V<-10)=Pr(\frac{V-20}{17.3}<\frac{-10-20}{17.3})=Pr(Z<-1.73)

Using the normal table, Pr(Z<-1.73)=4.18%

b. Here we work in the other direction. First, we find the quantile where Pr(Z < z) = 99.9%, which gives z = -3.09. This is then scaled to the distribution of the change in the value of the portfolio by multiply-ing by the standard deviation and adding the mean, 17.3 * -3.09 + 20 = -33.46. The fund would need a line of credit of USD 33.46 million to have a 99.9% change of having a change above this level.

第一问中，the portfolio loses more than USD 10 million，求的是Pr(V<−10)；（我理解是亏损大于10m，即return<-10)

对比着第二问中，or equivalently, larger than the LOC in 0.1% of months，解答中为什么不是Pr(Z < z) = 0.1%，而是Pr(Z < z) = 99.9%，

我理解只有Pr(Z < z) = 0.1%，z = -3.09

如果Pr(Z < z) = 99.9%，z=3.09


此处理解错误在哪里，麻烦老师指正

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pzqa39 · 2024年05月05日

嗨，努力学习的PZer你好：


a题问的是这个portfolio的亏损大于this line of credit，也就是portfolio的亏损大于10million的概率，而题目给的概率分布，是它的收益分布，所以亏损大于10million对应着收益小于-10million。


第二问这里解析写得有点不严谨，99.9%对应的分位点是3.09。只是这里根据实际的题目含义，题目是收益率服从这个分布，然后题目这里是问0.1%分位点的情况，所以要对3.09取个反（由对称看左边的分位点）


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