Interpreting Negative Zero Rates and Discount Factors in Curve Bootstrapping

Summary

The document addresses whether negative zero rates obtained while bootstrapping a euro swap curve indicate a calculation error. Its response is that negative interest rates can be economically possible, so a negative zero rate alone does not prove the bootstrap is wrong. It recommends checking the algorithm with inputs that yield positive rates and locating where negative results begin to assess the calculation.

It distinguishes negative rates from negative discount factors. Under the zero-rate convention described, negative rates correspond to discount factors above one; a negative discount factor is not a valid outcome. The exchange does not work through the bootstrap step by step or inspect the market quotes, day-count conventions, or implementation, so it cannot diagnose the specific program. Its main lesson is to validate the implementation and interpret rates and discount factors consistently.

Key ideas

• Negative zero rates can occur and do not by themselves establish a bootstrap error.
• With negative rates, discount factors can exceed one.
• A negative discount factor indicates a problem rather than an ordinary consequence of negative rates.
• Testing the bootstrap with inputs producing positive rates can help isolate implementation issues.

Tags

# negative discount and zero rate on swap bootstraping


# negative discount and zero rate on swap bootstraping












Hi I am writing a program to Bootstrap a EURO zero swap-curve for tenor 3M and 6M with given bid and ask. When I run the program , I get a negative zero rate and discount factor from 5Y till 30Y for both Swap 3m and 6m. Is it normal or am I doing something wrong in calculation?

For swap part, I use this formula to calculate the discount: $D_T=\frac{D_{T/N}-R_T^{par} \sum_{i=1}^{T-1}D_t}{1+R_T^{par}}$

and to calculate zero rate I use the following formula: $Z_T=((\frac{1}{D_T})^{\frac{365}{days}}-1) $

I really appreciate any tips or help. Bests

## Answer by dg_risk (score 1)

https://quant.stackexchange.com/a/15723

The formula seems to be correct. Negative interest rates are not impossible in these days.

http://www.bloombergview.com/quicktake/negative-interest-rates

Have you checked the algorithm with values that produce positive rates? And in what area lie the negative ones?

In the case of negative interest rates the discount factors should be greater than one, of course. Negative discount factors are (even in these days) not possible.

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