Consider an investor who, on January $1,2022,$ purchases a TIPS bond with an original principal of $$113,000,$ an $8$ percent annual $($or $4$ percent semiannual$)$ coupon rate, and $10$ years to maturity.

a$.$ If the semiannual inflation rate during the first six months is $0\mathrm{.}4$ percent, calculate the principal amount used to determine the first coupon payment and the first coupon payment $($paid on June $30,2022).$

b$.$ From your answer to part a$,$ calculate the inflation$-$adjusted principal at the beginning of the second six months.

c$.$ Suppose that the semiannual inflation rate for the second six$-$month period is $1\mathrm{.}2$ percent. Calculate the inflation$-$adjusted principal at the end of the second six months $($on December $31,2022)$ and the coupon payment to the investor for the second six$-$month period.

$($For all requirements, round your answers to $2$ decimal places. $($e$.$g$.,32\mathrm{.}16))$

$\backslash$table$[[,,],[$a$.,$Principal amount,$],[,$Coupon payment,$],[$b$.,$Inflation$-$adjusted principal,$],[$c$.,$Inflation$-$adjusted principal at the end of the second six months,$],[,$Coupon payment,$]]$