What is the price of a forward zero coupon bond.

Posted on November 9, 2015 by qiprimer

A zero coupon bond is effectively a loan; when you buy a zero coupon bond at time \bf{ t=0} you give the seller certain amount of money (bond price \bf{ P(T)}) and get back bond par amount when the bond matures at time \bf{ T}. In other words, you are getting an interest rate \bf{ r_T} on your loan such that

\bf{ P(T)(1 + r_T T) = 1 }

A forward zero coupon bond is, by the same token, a forward loan. If we consider a forward zero coupon bond that starts at time \bf{ T_1} and matures at \bf{ T_2}, then such a bond is equivalent to two future cash flows – a negative one at time \bf{ T_1} of the amount \bf{ X}, which is a price of the forward bond at time \bf{ T_1}, and positive cash flow of \bf{ 1} at \bf{ T_2}. If we can replicate those cash flows with a portfolio of spot zero coupon bonds (the bonds that start at \bf{ t=0}) we can compute \bf{ X}.

To replicate a negative \bf{ X} cash flow at time \bf{ T_1} we need to sell \bf{ X} of zero coupon bond that matures at \bf{ T_1}. To get \bf{ 1} at \bf{ T_2} we just need to buy a zero coupon bond that matures at \bf{ T_2}. Thus, the replicating portfolio price at time \bf{ t=0} is

\bf{ X P(T_1) - P(T_2) }

The forward contract on a zero coupon bond is always constructed in such way that its price at time \bf{ t=0} is zero. Thus we have

\bf{ X P(T_1) - P(T_2) = 0 }

or

\bf{ X = \frac{P(T_2)}{P(T_1)} }

This \bf{ X} is the price of the forward zero coupon bond at time \bf{ T_1}. Thus, the discount rate of that bond \bf{ r_F} can be computed from

\bf{ X(1 + r_F (T_2 - T_1)) = 1 }

leading to the expression for the forward discount rate:

\bf{ r_F = \frac{1}{T_2 - T_1}\frac{P(T_1) - P(T_2)}{P(T_2)} }

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