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# Zero coupon bond

A zero coupon bond (ZCB) is a binding contract who pays one unit of currency at some pre-specified future time $T$. $T$ is called maturity time. The ZCB is also called a pure discount bond. If the interest rate is positive, (which it is in realistic cases), the value of ZCB is always less then one before time $T$. If the
interest rate (short rate) is constant, r say, the value of the ZCB at a time $t$ than maturity time $T$, $p(t,T)$ equals $e^{-r(T-t)}$. If we model the short rate with an affine term structure model the value of the ZCB is given by
\[
p(t,T)=e^{A(t,T)-r(t)B(t,T)},\]
where A and B are deterministic functions satisfying a system of ordinary differential equations.
Since $P(T,T)\equiv 1$ for all $r(T)$ we must have that $A(T,T)=0$ and $B(T,T)=0$.

The central banks issue contracts of ZCB-type on a regular basis. These contracts are then called treasury bills or T-Bills for short. The maturity times are usually less or equal to one year. For longer maturities usually coupon bonds are issued. These contracts are called treasury bonds.

The central banks issue contracts of ZCB-type on a regular basis. These contracts are then called treasury bills or T-Bills for short. The maturity times are usually less or equal to one year. For longer maturities usually coupon bonds are issued. These contracts are called treasury bonds.

Questions: Magnus Wiktorsson

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