Question

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:_________

Answer 1

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 ${\displaystyle r_{1,2}}{\displaystyle r_{1,2}}$  for time period  

${\displaystyle (t_{1},t_{2})}(t_1, t_2), {\displaystyle t_{1}}t_{1}$  and  ${\displaystyle t_{2}}t_{2}$  expressed in years, given the rate  ${\displaystyle r_{1}}r_{1}$  for

time period  ${\displaystyle (0,t_{1})}(0, t_1)$  and rate ${\displaystyle r_{2}}r_{2}$  for time period ${\displaystyle (0,t_{2})}(0, t_2)$.  To do this, we use the property that the proceeds from investing at rate  ${\displaystyle r_{1}}r_{1}$ for

time period ${\displaystyle (0,t_{1})}(0, t_1)$  and then reinvesting those proceeds at rate

${\displaystyle r_{1,2}}{\displaystyle r_{1,2}}$  for time period  ${\displaystyle (t_{1},t_{2})}(t_1, t_2)$  is equal to the proceeds from

investing at rate ${\displaystyle r_{2}}r_{2}$ for time period ${\displaystyle (0,t_{2})}(0, t_2)$.

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