Question

Due to a recession, expected inflation this year is only 3.75%. However, the inflation rate in Year 2 and thereafter is expected to be constant at some level above 3.75%. Assume that the expectations theory holds and the real risk-free rate (r*) is 3.5%. If the yield on 3-year Treasury bonds equals the 1-year yield plus 1.5%, what inflation rate is expected after Year 1? Round your answer to two decimal places.

Answer 1

Inflation for year 1 = 3.75%, Real risk free rate = 3.5%

Nominal risk free rate for year 1 = (1 + Real risk free rate)(1+ Inflation) - 1 = (1 + 3.75%)(1 + 3.5%) - 1 = 1.0375 x 1.035 - 1 = 1.073812 - 1 = 0.073812 = 7.3812%

Since Treasury bond are considered to be risk free, Yield of 1 year treasury bond = Nominal risk free rate for 1 year = 7.3812%

Yield of 3 year treasury bond = 1 year yield + 1.5%

Yield for 3 year treasury bond = 7.3812% + 1.5% = 8.8812%

Now according to expectations theory

(1 + Yield of 3 year treasury bond)³ = (1 + yield of 1 year treasury)(1 + Forward rate for a 2 year treasury bond 1 year from now)²

(1 + 8.8812%)³ = (1 + 7.3812%)(1 + Forward rate for a 2 year treasury bond 1 year from now)²

(1 + Forward rate for a 2 year treasury bond 1 year from now)² = [(1.088812)³ / (1.073812)]

(1 + Forward rate for a 2 year treasury bond 1 year from now)² = (1.08812 x 1.088812 x 1.088812) / 1.073812 = 1.202071

Now

(1 + Forward rate for a 2 year treasury bond 1 year from now)² = (1 + Real risk free rate)²(1 + Constant Annual Forward rate for inflation over 2 years 1 year from now)²

1.202071 = (1 + 3.5%)²(1 + Constant Annual Forward rate for inflation over 2 years 1 year from now)²

(1 + Constant Annual Forward rate for inflation over 2 years 1 year from now)² = 1.202071 / (1 + 3.5%)² = 1.202071 / (1.035)² = 1.122146

Constant Annual forward rate for inflation over 2 year 1 year from now = (1.122146)^1/2 - 1 = 1.059314 - 1 = 5.9314% = 5.93%

Hence Expected inflation rate after year 1 = 5.93%