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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!!

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  • $\begingroup$ What is $c_i$? Make sure that $c_i$ is equal to the i-th coupon, except for the last one $c_i,i=n$ which equals 100+coupon. $\endgroup$
    – Alex C
    May 1, 2016 at 13:50
  • $\begingroup$ Yes, sure! $c_i$ is exactly what you said :) $\endgroup$
    – Fred G.
    May 1, 2016 at 15:45

1 Answer 1

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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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  • $\begingroup$ Ok, I see your point. The only problem is that this exercise is "academic" and I really do not think that negative interest rates (default risk,...) are included. $\endgroup$
    – Fred G.
    May 1, 2016 at 15:47
  • $\begingroup$ Anyway, did I use the dirty price correctly? What if I considered $P_{clean}(t)=\sum_i c_i P(t,T_i)$? $\endgroup$
    – Fred G.
    May 1, 2016 at 15:48
  • $\begingroup$ It should be Dirty Price = ∑iciP(t,Ti) + 100 P(t, Tn) - dont forget to include the return of principal at the end! $\endgroup$
    – dm63
    May 3, 2016 at 3:03

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