Question

Suppose the term structure of interest rates has these spot interest rates: r1 = 6.9%. r2 = 6.7%, r3 = 6.5%, and r4 = 6.3%. a. What will be the 1-year spot interest rate in three years if the expectations theory of term structure is correct? (Do not round intermediate calculations. Enter your answer as a percent rounded to 1 decimal place.) 1-year spot in 3 years % b. If investing in long-term bonds carries additional risks, then how would the risk equivalent of a 1-year spot rate in three years relate to your answer to part (a)?

Answer 1

Part (a)

• Invest say $ 100 in a three year instrument. Maturity amount after three years, A₃ = 100 x (1 + r₃)³ = 100 x (1 + 6.5%)³ = $  120.79
• Invest $ 100 in a four year instrument. Maturity amount after four years, A₄ = 100 x (1 + r₄)⁴ = 100 x (1 + 6.3%)⁴ = $  127.68
• If expectation theory holds true then, the 1-year spot interest rate in three years = A₄ / A₃ - 1 = 127.68 / 120.79 - 1 = 5.70%

Part (b)

If investing in long-term bonds carries additional risks, then risk equivalent of 1 year spot rate in three years will be lower than what we have calculated as answer to part (a).

This is because 5.70% will then be made of maturity risk premium + risk equivalent of 1 year spot rate in three years

Hence, 5.70% = Maturity risk premium + the risk equivalent of a 1-year spot rate in three years

Hence, the risk equivalent of a 1-year spot rate in three years = 5.70% - maturity risk premium < 5.70%