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

BlackTom AI · Accepted Answer

The problem asks for the forward rate for the third year, given continuous compounding.
First, translate the zero-coupon rates into discount factors: P(0,2) = e^{-0.06 * 2} = e^{-0.12} and P(0,3) = e^{-0.065 * 3} = e^{-0.195}.
Next, relate the three-year discount factor to the two-year discount factor and the one-year forward rate starting in year 2: P(0,3) = P(0,2) * e^{-f_{2,3} * 1}, where f_{2,3} is the forward rate for year 3.
Substitute the discount factors: e^{-0.195} = e^{-0.12} * e^{-f_{2,3}}.
Divide both sides by e^{-0.12} (or subtract exponents): e^{-(0.195 - 0.12)} = e^{-f_{2,3}} => e^{-0.075} = e^{-f_{2,3}}.
Take natural logs: f_{2,3} = 0.075, i.e., 7.5%.
Therefore, the forward rate for the third year is 7.5% when using continuous compounding.

Note on the provided options: the visible option corresponding to the computed value is 'd. 7.5%'. Since only that option is presented in the provided data, there are no additional distractors shown to critique here. In a typical multiple-choice setup, other plausible-looking forward rates (e.g., 6%, 6.5%, or a value derived from different compounding assumptions) would be evaluated against the same discount-factor relationship, but with the current information we can confirm the forward rate as 7.5% using the continuous-compounding formula.

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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