Question

Business Finance Question

5-16.

Suppose today is January 2, 2019, and investors expect the annual risk-free interest rates in 2023 and 2024 to be:

Year    One-Year Rate

2023    4.5%

2024    2.3

Currently, a four-year Treasury bond that matures on December 31, 2022, has an interest rate equal to 2.5 percent. What is the yield to maturity for Treasury bonds that mature at the end of (a) 2023 (a five-year bond) and (b) 2024 (a six-year bond)? Assume the bonds have no risks.

Answer 1

a) Interest rate on four year Treasury bond or for bond maturing in Dec 31, 2022 = R₄ = 2.5%,

One year risk free interest rate for 2023 = F₁ = 4.5%

Yield to maturity for Treasury bond that matures at end of 2023 or five year Treasury bond = R₅

Since all bonds are risk free and We know that

(1+R₅)⁵ = (1+R₄)⁴ (1+F₁)

(1+R₅)⁵ = (1+2.5%)⁴ (1+4.5%)

(1+R₅)⁵ = 1.10381289 x 1.045 = 1.1534844

R₅ = (1.1534844)^1/5 - 1 = 1.028969 - 1 = 0.028969 = 2.8969% = 2.90% (rounded to two decimal places)

Yield to maturity of Treasury bond that matures in five years or at end of 2023 = 2.90%

b Interest rate on four year Treasury bond or for bond maturing in Dec 31, 2022 = R₄ = 2.5%,

One year risk free interest rate for 2023 = F₁ = 4.5%, One year risk free interest rate for 2024 = F₂ = 2.3%

Yield to maturity for Treasury bond that matures at end of 2024 or six year Treasury bond = R₆

Since the all bonds are risk free and We know that

(1+R₆)⁶ = (1+R₄)⁴ (1+F₁)(1+F₂)

(1+R₆)⁶ = (1+2.5%)⁴ (1+4.5%)(1+2.3%)

(1+R₆)⁶ = 1.10381289 x 1.045 x 1.023 = 1.18001461

R₆ = (1.18001461)^1/6 - 1 = 1.027971 - 1 = 0.027971 = 2.7971% = 2.80% (rounded to two decimal places)

Hence Yield to maturity of Treasury bond that matures in six years or at end of 2024 = 2.80%