Deriving Zero-Coupon Discount Factors from Coupon Bond Prices

Summary

The document asks how to infer zero-coupon discount factors from clean and dirty prices of coupon bonds with annual coupons and staggered maturities. It proposes expressing a bond's dirty price as the sum of its coupon and principal cash flows, each discounted to the present, then solving recursively for discount factors and fitting a parametric term structure by least squares. The questioner is puzzled because the inferred factors exceed one.

The answer clarifies that a discount factor above one is possible: it corresponds to a negative interest rate, as observed in some European government bond markets. This resolves the apparent impossibility but does not work through the numerical example or explain practical details such as accrued interest, cash-flow conventions, or curve fitting. The note therefore offers a useful interpretation of the result, while leaving the full bootstrapping procedure unspecified.

Key ideas

• A coupon bond's dirty price reflects the present value of its coupon and principal cash flows.
• The proposed approach solves for discount factors recursively from bonds with progressively longer maturities.
• Discount factors above one are possible when the corresponding interest rates are negative.
• The answer interprets the result but does not provide a numerical calculation or detailed curve-building conventions.

Tags

# How can I compute zero coupon bond prices from dirty/clean prices of coupon bonds?


# How can I compute zero coupon bond prices from dirty/clean prices of coupon bonds?












I am having problems with computing zero-coupon bond prices. The question is the following:

Today is $t$=14.4.2016 and I know dirty and clean prices of coupon bonds expiring at maturities: 4.7.2016, 4.7.2017, 4.7.2018,4.7.2019, 4.7.2020,4.7.2021. Coupons are paid annually on the date of maturity.

How can we determine the term structure of zero-coupon bond prices?

My idea is simply to apply the formula:

$ P_{dirty}(t) = \sum_i^n c_i P(t,T_i)$

Starting from the bond A expiring on the 4th July 2016, we should have

$P_{dirty}^A (t)=c^A P(t,4.7.2016)$

from which we can compute the first discount factor. Then, considering the other bonds, recursively, we should be able to compute them all.

Finally, employing these results and the method of least squares (assuming a parametric form of the term structure) we should be able to estimate the term structure.

The issue is that the discount factors $P(t,T_i)$ turn out to be bigger than $1$, which impossible. Can you help me?

Is the formula above correct?

I can provide you with all the numerical values, if you need them.

Thank you very much for any help you can provide!!

## Answer by dm63 (score 2)

https://quant.stackexchange.com/a/25747

Discount factors>1 is not impossible. It just means that rates are negative, which is indeed the case in several Government bond markets in Europe.

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