Question

Due to a recession, expected inflation this year is only 2.75%. However, the inflation rate in Year 2 and thereafter is expected to be constant at some level above 2.75%. Assume that the expectations theory holds and the real risk-free rate (r*) is 1.5%. If the yield on 3-year Treasury bonds equals the 1-year yield plus 1.0%, what inflation rate is expected after Year 1?

Answer 1

Nominal risk free rate in year 1=2.75%+1%=3.75%=0.0375

Future Value of $1 at end of Year 1=(1+0.0375)=$1.0375

Assume,Inflation Rate in Year 2 and 3 =i

Nominal risk free rate in year2 and 3 =i+0.015

Future value at end of three years=1.0375*((1+i+0.015)^2)

Yield on 3 year Treasury Bond =1year yield+1%=3.75%+1%=0.0375+0.01=0.0475

Future Value of $1 at end of 3 years if one purchases 3 year Treasury Bond=(1+0.0475)^3

As per expectation theory:

1.0375*((1+i+0.015)^2)=(1+0.0475)^3

Log 1.0375+2 Log(i+1.015)=3Log1.0475

0.01599+2Log(i+1.015)=3*0.02105

2Log(i+1.015)=3*0.02105-0.01599=0.04447

Log(i+1.015)=0.04447/2=0.02224

i+1.015=1.05254

i=1.05254-1.015=0.03754

Expected inflation rate after 1 year=0.03754=3.75%(rounded to two decimal)