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梦梦 · 2024年07月30日

关于EL和UL的概念

  1. FRM I
  2. Valuation And Risk Models

NO.PZ2020011303000127

问题如下：

A USD 1million loan has a probability of 0.5% of defaulting in a year. The recovery rate is estimated to be 40%.

The expected loss in USD is 0.003. This is USD 3,000.

The variance of the loss is 0.001791. The standard deviation is the square root of this, or USD 0.04232 million. This is USD 42,320.

Suppose that there are three USD 1 million loans. The correlation between any pair of the loans is 0.2. What is the mean and standard deviation of the portfolio credit loss?

解释：

The expected loss on the three loans is three times the expected loss on one loan or USD 9,000. The variance of the portfolio loss is

3 × 0.001791+ 6 × 0.001791 × 0.2 = 0.007522

The standard deviation of the portfolio loss is the square root of this or USD 0.086731. This is USD 86,731. (Note that the standard deviation is less than three times the standard deviation calculated for one loan because there are some diversification benefits.)

题目问：假设有三个1m的贷款，每个贷款的一年的违约概率是0.5%，recover rate是40%，任意两个贷款之间的相关系数是0.2，求这个贷款组合的EL和损失的标准差。

EL=n*PD*EAD*LGD=3*0.5%*1m*(1-40%)=0.009

$形式9000

组合方差=3*单个方差+6*ρ*单个方差=3 × 0.001791+ 6 × 0.001791 × 0.2 = 0.007522

标准差=0.007522^0.5=0.086731

dollar形式USD 86,731

1、老师，只要题目问到“standard deviation of the portfolio credit loss”究竟是不是等于资产组合的“UL”？

2、EL是credit loss的均值？

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4 个答案
已采纳答案

pzqa39 · 2024年08月08日

嗨，努力学习的PZer你好：


你这个公式变形一下就可以得到我的公式了，在这个题目里面X1、X2、X3你可以看成是同质的single loan。V（X1）+V（X2）+V（X3）其实就是我公式的前半部分，2COV（X1，X2）+2COV（X1，X3）+2COV（X1，X3）可以合成我公式的后半部分。

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梦梦 · 2024年08月22日

好的，谢谢

pzqa39 · 2024年08月07日

嗨，从没放弃的小努力你好：


我写的公式就是你的这个公式合并得到的，对于同质的贷款可以直接用我发的这个公式

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努力的时光都是限量版，加油！

梦梦 · 2024年08月07日

那也就是说我的公式也是对的？那为啥我用我写的这个公式计算不等于答案呀😭😭😭

pzqa39 · 2024年08月01日

嗨，爱思考的PZer你好：


这道题是三个独立且同质的贷款，我们可以用这个公式去计算。

你提到的公式中 0.4232是单个贷款的标准差吧？不应该直接用于方差计算。组合方差的计算基于贷款的方差，而不是贷款的标准差。

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梦梦 · 2024年08月06日

为啥？假设贷款X1，X2和X3，组合的方差为什么不是V（X1+X2+X3）=V（X1）+V（X2）+V（X3）+2COV（X1，X2）+2COV（X1，X3）+2COV（X1，X3）？

pzqa39 · 2024年07月31日

嗨，爱思考的PZer你好：


1、是的，UL 可以理解为 loss 的标准差, UL = 根号下 Var (loss)

2、对的，expected loss是均值

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就算太阳没有迎着我们而来，我们正在朝着它而去，加油！

梦梦 · 2024年07月31日

明白了，谢谢，但是为什么是3 × 0.001791+ 6 × 0.001791 × 0.2 = 0.007522而不是3 × 0.001791+ 6 × 0.4232 × 0.2 = 0.007522？以资产组合由A和B两项资产组成为例，V（A+B）不是应该等于V（A）+V（B）+2*p*西格玛A*西格玛B？

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