The two-year zero rate is 6% and the three-year zero rate is 6.5%. What is the forward rate for the third year? All rates are continuously compounded.

Question

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We start by identifying the given inputs and the goal: the two-year zero rate is 6% (z2 = 0.06) and the three-year zero rate is 6.5% (z3 = 0.065). All rates are continuously compounded. We want the forward rate for year 3, denoted f(2,3), i.e., the annualized rate applicable during the third year such that investing for 2 years at z2 and then one more year at f(2,3) yields the same amount as investing for 3 years at z3.

Use the continuous compounding pricing relation: e^{z3 * 3} = e^{z2 * 2} * e^{f(2,3)}. This expresses that the value of a 3-year zero at rate z3 equals the value of first locking in a 2-year zero at z2 and then rolling into a forward rate for year 3.

Solve for the forward rate: e^{f(2,3)} = e^{z3 * 3} / e^{z2 * 2} = e^{(z3 * 3 - z2 * 2)}. Therefore f(2,3) = z3 * 3 - z2 * 2.

Plug in the numbers: z3 * 3 = 0.065 * 3 = 0.195, and z2 * 2 = 0.06 * 2 = 0.12. Subtracting gives f(2,3) = 0.195 - 0.12 = 0.075, or 7.5%.

Option assessments:
- Option a: 6.75%. This would correspond to a forward rate of 6.75% if the difference z3*3 - z2*2 equaled 0.0675. But the actual difference is 0.075, so 6.75% is too low and does not satisfy the no-arbitrage relationship above.
- Option b: 7.0%. This would imply f(2,3) = 7.0%, which would require z3*3 - z2*2 to equal 0.07. Since the computed difference is 0.075, this value underestimates the required forward rate.
- Option c: 7.25%. Similarly, this value would require z3*3 - z2*2 = 0.0725, which is not equal to the actual 0.075. It is closer but still not correct relative to the given z2 and z3.
- Option d: 7.5%. This matches the calculation f(2,3) = z3 * 3 - z2 * 2 = 0.195 - 0.12 = 0.075, i.e., 7.5%. It satisfies the no-arbitrage condition linking the 2-year and 3-year zero rates to the 1-year forward rate starting in year 3.

Additional notes:
- The concept relies on the continuity-compounding framework, where exponential discount factors translate linear differences in (time × rate) into forward rates.
- If you recompute using the given z2 and z3, you should consistently arrive at 7.5% for the forward rate for the third year; any other option would violate the fundamental relation between spot rates and forward rates in this setup.

FIN 413 (LEC B01 B02 B03 Winter 2025) Quiz 2

The two-year zero rate is 6% and the three-year zero rate is 6.5%. What is the forward rate for the third year? All rates are continuously compounded.

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