Question

Based on your calculations of the spot rates below, calculate:

a. The forward rate for a one-year zero issued one year from today, f(1,1)

b. The forward rate for a one-year zero issued 2 years from today, f(2,1)

c. The forward rate for a two-year zero issued two years from today, f(2,2)

Maturity	Par Rate	Spot Rate
1	2.00%	2.00%
2	2.57%	2.5774%
3	3.11%	3.1331%
4	3.88%	3.9490%

Please show work!

Answer 1

a)

(1 + 2 year spot rate)² = (1 + 1 year spot rate) * (1 + forward rate for a one-year zero issued one year from today)

(1 + 2.5774%)² = (1 + 2%) * (1 + forward rate for a one-year zero issued one year from today)

(1 + forward rate for a one-year zero issued one year from today) = (1 + 2.5774%)² / (1 + 2%)

(1 + forward rate for a one-year zero issued one year from today) = 1.031581

forward rate for a one-year zero issued one year from today = 3.1581%

b)

(1 + 3 year spot rate)³ = (1 + 2 year spot rate)² * (1 + forward rate for a one-year zero issued 2 years from today)

(1 + 3.1331%)³ = (1 + 2.5774%)² * (1 + forward rate for a one-year zero issued 2 years from today)

(1 + forward rate for a one-year zero issued 2 years from today) = (1 + 3.1331%)³ / (1 + 2.5774%)²

(1 + forward rate for a one-year zero issued 2 years from today) = 1.042535

forward rate for a one-year zero issued 2 years from today = 4.2535%

c)

(1 + 4 year spot rate)⁴ = (1 + 2 year spot rate)² * (1 + forward rate for a two-year zero issued two years from today)²

(1 + 3.9490%)⁴ = (1 + 2.5774%)² * (1 + forward rate for a two-year zero issued two years from today)²

(1 + forward rate for a two-year zero issued two years from today)² = (1 + 3.9490%)⁴ / (1 + 2.5774%)²

(1 + forward rate for a two-year zero issued two years from today)² = 1.109629

(1 + forward rate for a two-year zero issued two years from today) = (1.109629)^{(1 / 2)}

(1 + forward rate for a two-year zero issued two years from today) = 1.053389

forward rate for a two-year zero issued two years from today​​​​​​​​​​​​​​ = 5.3389%