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西红柿面 · 2024年04月10日

计算PV of expected loss的DF的时候用的DF为啥是 无风险收益

  1. CFA II
  2. Fixed Income

* 问题详情，请 查看题干

NO.PZ201812310200000101

问题如下：

The market price of bond B1 is€875. The bond is:

选项：

A.

fairly valued.

B.

overvalued.

C.

undervalued.

解释：

B is correct.

The following table shows that the credit valuation adjustment (CVA) for the bond is

€36.49, the sum of the present values of expected loss. The steps taken to complete the table are as follows.

Step 1: Exposure at Date T is 1000 (1+r) 4−T , where r is 3%. That is, exposure is computed by discounting the face value of the bond using the risk-free rate and the number of years until maturity.

Step 2: Recovery = Exposure × Recovery rate

Step 3: Loss given default (LGD) = Exposure – Recovery

Step 4: Probability of default (POD) on Date 1 is 1.50%, the assumed hazard rate. The probability of survival (POS) on Date 1 is 98.50%.

For subsequent dates, POD is calculated as the hazard rate multiplied by the previous date’s POS.

For example, to determine the Date 2 POD (1.4775%), the hazard rate of (1.50%) is multiplied by the Date 1 POS (98.50%).

Step 5: POS in Dates 2–4 = POS in the previous year – POD

(That is, POS in Year T= POS in year [ T– 1] – POD in Year T.)

POS can also be determined by subtracting the hazard rate from 100% and raising it to the power of the number of years:

(100% – 1.5000%)1 = 98.5000%

(100% – 1.5000%)2 = 97.0225%

(100% – 1.5000%)3 = 95.5672%

(100% – 1.5000%)4 = 94.1337%

Step 6: Expected loss = LGD × POD

Step 7: Discount factor (DF) for Date T is 1 (1+r) T , where r is 3%.

Step 8: PV of expected loss = Expected loss × DF

Value of the bond if the bond were default free would be 1,000 × DF for Date 4 = €888.49.

Fair value of the bond considering CVA = €888.49 – CVA = €888.49 – €36.49 = €852.00.

Because the market price of the bond (€875) is greater than the fair value of €852, B is correct.

A is incorrect because the market price of the bond differs from its fair value. C is incorrect because although the bond’s value if the bond were default free is greater than the market price, the bond has a risk of default, and CVA lowers its fair value to below the market price.

是因为利率没有波动才用的无风险收益吗？如果利率有波动就要用表格里的DF？

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pzqa31 · 2024年04月13日

嗨，爱思考的PZer你好：


嗯嗯，何老师课上说CVA是用sport rate折现，这道题是基于这个假设，所以统一用3%来折现。


没波动用YTM，有波动用spot rate，DF是spot rate的倒数，同学理解的没错哈。

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加油吧，让我们一起遇见更好的自己！

pzqa31 · 2024年04月10日

嗨，努力学习的PZer你好：


因为这里是假设风险中性的，讲义上也有写到：

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虽然现在很辛苦，但努力过的感觉真的很好，加油！

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