What is the continuously compounded forward rate, Mathematics

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At time t an investor shorts a $1 face value zero coupon bond that matures at time T = t

and uses the entire proceeds to purchase a zero coupon bond that matures at time S = T.

(a) In what quantity is the zero coupon bond that matures at time S purchased? Your answer should be expressed in terms of the time t prices P(t;T) and P (t;S).

(b) Explain why these transactions are equivalent to agreeing to lend over the future period [T, S] at a rate that is determined at time t.

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