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Sagot :
The "break-even" interest rate for year n that equates the return on an n-period zero-coupon bond to that of an n - 1 - period zero-coupon bond rolled over into a one-year bond in year n is defined as the forward rate.
A forward rate is a specified price agreed by all parties involved for the delivery of a good at a specific date in the future. The use of forward rates can be speculative if a buyer believes the future price of a good will be greater than the current forward rate. Alternatively, sellers use forward rates to mitigate the risk that the future price of a good materially decreases.
Regardless of the prevailing spot rate at the time the forward rate meets maturity, the agreed-upon contract is executed at the forward rate. For example, on January 1st, the spot rate of a case of iceberg lettuce is $50. The restaurant and the farmer agree to the delivery of 100 cases of iceberg lettuce on July 1st at a forward rate of $55 per case. On July 1st, even if the price per case has decreased to $45/case or increased to $65/case, the contract will proceed at $55/case.
To extract the forward rate, we need the zero-coupon yield curve.
We are trying to find the future interest rate [tex]{\displaystyle r_{1,2}}{\displaystyle r_{1,2}}[/tex] for time period
[tex]{\displaystyle (t_{1},t_{2})}(t_1, t_2), {\displaystyle t_{1}}t_{1}[/tex] and [tex]{\displaystyle t_{2}}t_{2}[/tex] expressed in years, given the rate [tex]{\displaystyle r_{1}}r_{1}[/tex] for
time period [tex]{\displaystyle (0,t_{1})}(0, t_1)[/tex] and rate [tex]{\displaystyle r_{2}}r_{2}[/tex] for time period [tex]{\displaystyle (0,t_{2})}(0, t_2)[/tex]. To do this, we use the property that the proceeds from investing at rate [tex]{\displaystyle r_{1}}r_{1}[/tex] for
time period [tex]{\displaystyle (0,t_{1})}(0, t_1)[/tex] and then reinvesting those proceeds at rate
[tex]{\displaystyle r_{1,2}}{\displaystyle r_{1,2}}[/tex] for time period [tex]{\displaystyle (t_{1},t_{2})}(t_1, t_2)[/tex] is equal to the proceeds from
investing at rate [tex]{\displaystyle r_{2}}r_{2}[/tex] for time period [tex]{\displaystyle (0,t_{2})}(0, t_2)[/tex].
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