Given a discount curve, \(D(T)\), one can extract continuously compounded zero-coupon rates for any date via

\[D(T)=e^{-T \times Z(T)} \Leftrightarrow Z(T)=-\frac{1}{T} \ln D(T)\]

when interest rates are constant, \(Z(T)=r\) for a constant \(r\). The graph of \(Z(T)\) versus \(T\) is known as the zero-coupon curve.

(a) Given two dates \(0

(b) Calculate the " 1 -year forward rate, 2 years forward," \(f_{c}([2,3])\), for three cases:

i. \(Z(2)=4 \%, Z(3)=5 \%\)

ii. \(Z(2)=4 \%, Z(3)=4 \%\)

iii. \(Z(2)=4 \%, Z(3)=3 \%\)

Explain the resulting forward rates in each case.