Question

Consider the following spot interest rates for maturities of one, two, three, and four years.

           r₁ = 3.9%    r₂ = 4.5%     r₃ = 5.2%     r₄ = 6.0%

What are the following forward rates, where f_{1, k} refers to a forward rate for the period beginning in one year and extending for k years? (Do not round intermediate calculations. Enter your answers as a percent rounded to 2 decimal places. Omit the "%" sign in your response.)

f1,1	%
f1,2	%
f1,3	%

Answer 1

This question requires application of pure expectations hypothesis theory

f1,1 is 1 year rate, 1 year from now.

(1 + 2 Yr Rate)² = (1 + 1 Yr Rate) * (1 + 1 Yr Rate 1 Yr from now)

(1 + 4.5%)² = (1 + 3.9%) * (1 + 1 Yr Rate 1 Yr from now)

1.092025 = 1.039 * (1 + 1 Yr Rate 1 Yr from now)

(1 + 1 Yr Rate 1 Yr from now) = 1.0510

1 Yr Rate 1 Yr from now = 5.10%

f1,2 is 2 year rate, 1 year from now.

(1 + 3 Yr Rate)³ = (1 + 1 Yr Rate) * (1 + 2 Yr Rate 1 Yr from now)²

(1 + 5.2%)² = (1 + 3.9%) * (1 + 2 Yr Rate 1 Yr from now)²

1.1643 = 1.039 * (1 + 2 Yr Rate 1 Yr from now)²

(1 + 2 Yr Rate 1 Yr from now)² = 1.120551

(1 + 2 Yr Rate 1 Yr from now) = 1.0586

2 Yr Rate 1 Yr from now = 5.86%

f1,3 is 3 year rate, 1 year from now.

(1 + 4 Yr Rate)⁴ = (1 + 1 Yr Rate) * (1 + 3 Yr Rate 1 Yr from now)³

(1 + 6.0%)⁴ = (1 + 3.9%) * (1 + 3 Yr Rate 1 Yr from now)³

1.262477 = 1.039 * (1 + 3 Yr Rate 1 Yr from now)³

(1 + 3 Yr Rate 1 Yr from now)³ = 1.215089

(1 + 3 Yr Rate 1 Yr from now) = 1.0671

3 Yr Rate 1 Yr from now = 6.71%