An investor in Treasury securities expects inflation to be $2.0\%$ in Year $1$ , $2.6\%$ in Year $2$ , and $3.75\%$ each year thereafter. Assume that the real risk-free rate is $1.95\%$ and that this rate will remain constant. Three-year Treasury securities yield $5.20\%$ , while $5$ -year Treasury securities yield $6.00\%$ . What is the difference in the maturity risk premiums (MRPs) on the two securities; that is, what is $\text{MRP}_5 - \text{MRP}_3$ ?

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Q. An investor in Treasury securities expects inflation to be

2.0\%

in Year

1

2.6\%

in Year

2

, and

3.75\%

each year thereafter. Assume that the real risk-free rate is

1.95\%

and that this rate will remain constant. Three-year Treasury securities yield

5.20\%

, while

5

-year Treasury securities yield

6.00\%

. What is the difference in the maturity risk premiums (MRPs) on the two securities; that is, what is

\text{MRP}_5 - \text{MRP}_3

Understand Fisher Equation: Understand the Fisher Equation which relates nominal interest rates, real interest rates, and expected inflation. The Fisher Equation is given by: $\newline$ Nominal Interest Rate $=$ Real Risk-Free Rate $+$ Expected Inflation Rate $+$ Maturity Risk Premium $(MRP)$ $\newline$ We will use this equation to find the MRPs for the $3$ -year and $5$ -year Treasury securities.
Calculate Expected Inflation: Calculate the expected inflation rate over the $3$ -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. $\newline$ Expected Inflation for $3$ years = $(\text{Year 1 Inflation} + \text{Year 2 Inflation} + \text{Year 3 Inflation}) / 3$ $\newline$ Expected Inflation for $3$ years = $(2.0\% + 2.6\% + 3.75\%) / 3$ $\newline$ Expected Inflation for $3$ years = $8.35\% / 3$ $\newline$ Expected Inflation for $3$ years = $2.7833\%$
Calculate MRP for $3$ -year Treasury: Calculate the MRP for the $3$ -year Treasury security using the Fisher Equation and the $3$ -year nominal interest rate. $\newline$ Nominal Interest Rate for $3$ years $=$ Real Risk-Free Rate $+$ Expected Inflation for $3$ years $+$ MRP $_3$ $\newline$ $5.20\% = 1.95\% + 2.7833\% + MRP_3$ $\newline$ MRP $_3 = 5.20\% - 1.95\% - 2.7833\%$ $\newline$ MRP $_3 = 0.4667\%$
Calculate Expected Inflation for $5$ years: Calculate the expected inflation rate over the $5$ -year period. For the first two years, we have specific rates, and for the remaining three years, we have a constant rate. $\newline$ Expected Inflation for $5$ years = $(\text{Year 1 Inflation} + \text{Year 2 Inflation} + 3 \times \text{Year 3+ Inflation}) / 5$ $\newline$ Expected Inflation for $5$ years = $(2.0\% + 2.6\% + 3 \times 3.75\%) / 5$ $\newline$ Expected Inflation for $5$ years = $(2.0\% + 2.6\% + 11.25\%) / 5$ $\newline$ Expected Inflation for $5$ years = $15.85\% / 5$ $\newline$ Expected Inflation for $5$ years = $3.17\%$
Calculate MRP for $5$ -year Treasury: Calculate the MRP for the $5$ -year Treasury security using the Fisher Equation and the $5$ -year nominal interest rate. $\newline$ Nominal Interest Rate for $5$ years $=$ Real Risk-Free Rate $+$ Expected Inflation for $5$ years $+$ $MRP_5$ $\newline$ $6.00\% = 1.95\% + 3.17\% + MRP_5$ $\newline$ $MRP_5 = 6.00\% - 1.95\% - 3.17\%$ $\newline$ $MRP_5 = 0.88\%$
Calculate Difference in MRPs: Calculate the difference in the maturity risk premiums (MRPs) between the $5$ -year and $3$ -year Treasury securities. $\newline$ Difference in MRPs = $MRP_5 - MRP_3$ $\newline$ Difference in MRPs = $0.88\% - 0.4667\%$ $\newline$ Difference in MRPs = $0.4133\%$