Question

The yield on 1-year Treasury securities is 6%, 2-year securities yield 6.2%, 3-year securities yield 6.3%, and 4-year securities yield 6.5%. There is no maturity risk premium. Using expectations theory and geometric averages, forecast the yields on the following securities:

A 1-year security, 1 year from now?

A 1-year security, 2 years from now?

A 2-year security, 1 year from now?

A 3-year security, 1 year from now?

Answer 1

As per expectation theory, long term interest rate is simple a weighted average of current and expected future short-term interest rate provided MRP is zero.

1)

(1 + two-year yield)² = (1 + one-year yield) x (1+ future one-year yield)

(1+ 0.062) ² = (1 + 0.06) x (1+ future one-year yield)

(1+ 0.062) ² / (1 + 0.06) = 1+ future one-year yield

1+ future one-year yield = (1.062) ²/ (1.06)

                                           = 1.127844/1.06

                                          = 1.06400377358491

Future one-year yield = 1.06400377358491 – 1 = 0.06400377358491 or 6.4 %

2)

(1 + three-year yield)³ = (1 + two-year yield)² x (1+ future one-year yield)

(1 + 0.063)³ = (1 + 0.062) x (1+ future one-year yield)

(1.063)³ = (1.062) ² x (1+ future one-year yield)

(1.063)³ / (1.062) ² = (1+ future one-year yield)

1+ future one-year yield = 1.201157047/1.127844

                                               = 1.0650028257454

Future one-year yield = 1.0650028257454 – 1 = 0.0650028257454 or 6.5 %

3)

(1 + three-year yield)³ = (1 + one-year yield) x (1+ future two-year yield)²

(1 +0.063)³ = (1 + 0.06) x (1+ future two-year yield)²

(1.063)³ = (1.06) x (1+ future two-year yield)²

(1.063)³/ (1.06) = (1+ future two-year yield)²

(1+ future two-year yield)² = 1.201157047/1.06

                                               = 1.1331670254717

Future two-year yield = √ (1.1331670254717) – 1

                                        = 1.06450318246199 – 1

                                        = 0.06450318246199 or 6.45 %

4)

(1 + four-year yield)⁴ = (1+ one-year yield) x (1 + future three-year yield)³

(1 +0.065)⁴ = (1 + 0.06) x (1 + future three-year yield)³

(1.065)⁴ = (1.06) x (1 + future three-year yield)³

(1.065)⁴ / (1.06) = (1 + future three-year yield)³

(1 + future three-year yield)³ = 1.286466350625/1.06

                                               = 1.21364750058962

1 + future three-year yield = ³ √ 1.21364750058962 = 1.06667190227378

Future three-year yield =1.06667190227378 -1 = 0.06667190227378 or 6.67 %