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徐威廉 · 2022年05月25日

ΔYTM 为什么是var的分位点？

  1. CFA III
  2. Fixed Income

* 问题详情，请 查看题干

NO.PZ202112010200002202

问题如下：

What is the approximate VaR for the bond position at a 99% confidence interval (equal to 2.33 standard deviations) for one month (with 21 trading days) if daily yield volatility is 0.015% and returns are normally distributed?

选项：

A.

$1,234,105

B.

$2,468,210

C.

$5,413,133

解释：

A is correct. The expected change in yield based on a 99% confidence interval for the bond and a 0.015% yield volatility over 21 trading days equals 16 bps = (0.015% × 2.33 standard deviations × √21).

We can quantify the bond’s market value change by multiplying the familiar (–ModDur × ∆Yield) expression by bond price to get $1,234,105 = ($75 million × 1.040175 ⨯ (–9.887 × .0016)).

为什么收益率改变即 Δ YTM=u-z*standard deviation， 这不是VaR的分位点吗？

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lynn_品职助教 · 2022年05月26日

嗨，从没放弃的小努力你好：


Δ YTM=u-z*standard deviation， 这不是VaR的分位点吗？


这道题是这样，他让我们计算在一个月里，99% confidence interval下，债券价格的VaR。


也就是说，我们需要计算：在一个月里，在99%的概率下，债券的最大损失；或者按照VAR的另外一个角度理解就是：在一个月里，在1%概率下，债券的最小损失。


下跌Price多少金额 = - duration × Yield%变动幅度 × market value。


在上面公式里面，债券的Duration与market value是题干已知信息，所以要计算Price最少下跌多少金额，现在就转化成了计算Yield%最少上升多少幅度，求出了Yield%最少上升多少幅度，我们就可以计算出Price最少下跌多少金额。所以我们现在要计算在一个月里面，在1%的概率下，Yield%最少会上升多少幅度。题干刚好给的是Yield%的相关信息，所以现在我们可以套用题干yield%的信息来计算了。


根据Var的公式，|μmonthly-2.33σmonthly|，这个公式得到是以μ为原点，向左、向右的最大值，向左得到的是最大亏损，向右得到的是最大收益。


如果已知y的μ和σ，-2.33σmonthly得到就是△y取负的最大值，2.33σmonthly得到的就是△y取正的最大值


所以，要用2.33*0.015%*21^(1/2)得到△y取正的最大值，进而根据△P=-P*md*△y来计算出bond position的最大损失。


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