Spot Rates and Forward Rates

The spot yield curve includes a one-year spot rate of 3.4%, three-year spot rate of 4.2%, and a four-year spot rate of 4.5%. What is the value of the forward price $$F_{3,1}$$ ?

Not quite. This is the forward contract price $$F_{1,3}$$, using the one-year spot rate and the four-year spot rate to find the forward contract price starting in one year and lasting three more years. But the question was looking for the $$F_{3,1}$$ forward price.

That's not it. This value can be calculated if you use the correct pair of spot rates, but then set up the calculation from that point as looking for $$F_{1,3}$$ rather than $$F_{3,1}$$.

You got it! This can be calculated using the forward pricing model, $$\displaystyle DF_B = DF_A \times F_{A,B-A} $$ The forward price $$F_{3,1}$$ is for a loan which starts in three years and lasts for one year. So only the three-year spot rate and four-year spot rates are needed. These can be used to calculate the prices or discount factors as $$\displaystyle \frac{1}{(1 + 0.045)^4} = \frac{1}{(1 + 0.042)^3} \times F_{3,1} $$. The forward price is then found after some simplification as $$\displaystyle F_{3,1} = \frac{0.838561}{0.883887} \approx 0.9487 $$.

0.8671

0.8738

0.9487