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Modeling Historical VaR Returns for a Zero-Coupon German Bond

Article Quant Q&A · Author: equanimity
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Summary

The document raises a risk-modeling question about constructing historical returns for a long-dated German government bond with a zero coupon. The author is building a historical Value at Risk model and is concerned that very low yields, including yields at zero or below, may make a return series based directly on yield changes misleading or mathematically problematic. A short sequence of yield observations illustrates the issue, with values moving around zero.

No answer or return-construction method is supplied. In particular, the document does not resolve whether to model bond price changes, yield changes, or another risk factor, nor does it give a VaR result or discuss duration and convexity. It is therefore best read as a prompt identifying an input-modeling challenge: yield is not itself a bond return, and the stated concern about a zero denominator remains unanswered here.

Key ideas

  • The author needs historical returns for a long-dated zero-coupon government bond to estimate VaR.
  • Yield observations close to, at, or below zero raise concerns about using yield-based returns.
  • The document does not provide a return calculation or VaR modeling recommendation.
  • It leaves open how price sensitivity and other bond risk factors should enter the historical series.

Tags

Fixed incomeRisk managementStatistics
Full text
# How to compute return series for a German government bond with a 0% coupon?


# How to compute return series for a German government bond with a 0% coupon?












Recently, the German government issued a long-dated bond with a 0% coupon.

I'm trying to implement a historical VaR model and would like to know the best way to model the historical returns of this bond.

What's tripping me up is the possibility that the return series could be "corrupted" due to:

a) the low level of rates

b) the possibility of having 0% in the denominator

Let's assume the returns are:

t_0 = 0% YTM

t_1 = 0.01% YTM

t_2 = 0.025% YTM

t_3 = 0% YTM

t_4 = -0.01% YTM

t_5 = 0% YTM

t_6 = 0.03% YTM

What's the best way to model these returns?

Thanks!

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.