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An investor in Treasury securities expects inflation to be 2.0%2.0\% in Year 11, 2.6%2.6\% in Year 22, and 3.75%3.75\% each year thereafter. Assume that the real risk-free rate is 1.95%1.95\% and that this rate will remain constant. Three-year Treasury securities yield 5.20%5.20\%, while 55-year Treasury securities yield 6.00%6.00\%. What is the difference in the maturity risk premiums (MRPs) on the two securities; that is, what is MRP5−MRP3\text{MRP}_5 - \text{MRP}_3?

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Q. An investor in Treasury securities expects inflation to be 2.0%2.0\% in Year 11, 2.6%2.6\% in Year 22, and 3.75%3.75\% each year thereafter. Assume that the real risk-free rate is 1.95%1.95\% and that this rate will remain constant. Three-year Treasury securities yield 5.20%5.20\%, while 55-year Treasury securities yield 6.00%6.00\%. What is the difference in the maturity risk premiums (MRPs) on the two securities; that is, what is MRP5−MRP3\text{MRP}_5 - \text{MRP}_3?
  1. Understand Fisher Equation: Understand the Fisher Equation which relates nominal interest rates, real interest rates, and expected inflation. The Fisher Equation is given by:\newlineNominal Interest Rate == Real Risk-Free Rate ++ Expected Inflation Rate ++ Maturity Risk Premium (MRP)(MRP)\newlineWe will use this equation to find the MRPs for the 33-year and 55-year Treasury securities.
  2. Calculate Expected Inflation: Calculate the expected inflation rate over the 33-year period. Since we have different expected inflation rates for each year, we need to find the average annual inflation rate for the first three years.\newlineExpected Inflation for 33 years = (Year 1 Inflation+Year 2 Inflation+Year 3 Inflation)/3(\text{Year 1 Inflation} + \text{Year 2 Inflation} + \text{Year 3 Inflation}) / 3\newlineExpected Inflation for 33 years = (2.0%+2.6%+3.75%)/3(2.0\% + 2.6\% + 3.75\%) / 3\newlineExpected Inflation for 33 years = 8.35%/38.35\% / 3\newlineExpected Inflation for 33 years = 2.7833%2.7833\%
  3. Calculate MRP for 33-year Treasury: Calculate the MRP for the 33-year Treasury security using the Fisher Equation and the 33-year nominal interest rate.\newlineNominal Interest Rate for 33 years == Real Risk-Free Rate ++ Expected Inflation for 33 years ++ MRP3_3\newline5.20%=1.95%+2.7833%+MRP35.20\% = 1.95\% + 2.7833\% + MRP_3\newlineMRP3=5.20%−1.95%−2.7833%_3 = 5.20\% - 1.95\% - 2.7833\%\newlineMRP3=0.4667%_3 = 0.4667\%
  4. Calculate Expected Inflation for 55 years: Calculate the expected inflation rate over the 55-year period. For the first two years, we have specific rates, and for the remaining three years, we have a constant rate.\newlineExpected Inflation for 55 years = (Year 1 Inflation+Year 2 Inflation+3×Year 3+ Inflation)/5(\text{Year 1 Inflation} + \text{Year 2 Inflation} + 3 \times \text{Year 3+ Inflation}) / 5\newlineExpected Inflation for 55 years = (2.0%+2.6%+3×3.75%)/5(2.0\% + 2.6\% + 3 \times 3.75\%) / 5\newlineExpected Inflation for 55 years = (2.0%+2.6%+11.25%)/5(2.0\% + 2.6\% + 11.25\%) / 5\newlineExpected Inflation for 55 years = 15.85%/515.85\% / 5\newlineExpected Inflation for 55 years = 3.17%3.17\%
  5. Calculate MRP for 55-year Treasury: Calculate the MRP for the 55-year Treasury security using the Fisher Equation and the 55-year nominal interest rate.\newlineNominal Interest Rate for 55 years == Real Risk-Free Rate ++ Expected Inflation for 55 years ++ MRP5MRP_5\newline6.00%=1.95%+3.17%+MRP56.00\% = 1.95\% + 3.17\% + MRP_5\newlineMRP5=6.00%−1.95%−3.17%MRP_5 = 6.00\% - 1.95\% - 3.17\%\newlineMRP5=0.88%MRP_5 = 0.88\%
  6. Calculate Difference in MRPs: Calculate the difference in the maturity risk premiums (MRPs) between the 55-year and 33-year Treasury securities.\newlineDifference in MRPs = MRP5−MRP3MRP_5 - MRP_3\newlineDifference in MRPs = 0.88%−0.4667%0.88\% - 0.4667\%\newlineDifference in MRPs = 0.4133%0.4133\%

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