The following three defense stocks are to be combined into a stock index in January $2022($perhaps a portfolio manager believes these stocks are an appropriate benchmark for his or her performance$).$ Assume the index is scaled by a factor of $10$ million; that is$,$ if the total value of all firms in the market is $$5$ billion, the index would be quoted as $500.$

$\backslash$table$[[,,$Price$],[$Douglas McDonnell,Shares $($millions$),\frac{1}{1}\frac{?}{22},\frac{1}{1}\frac{?}{23},\frac{1}{1}\frac{?}{24},],[$Dynamics General,$210,$$$94,$$$97,$$$111,],[$International$,450,36,31,45,],[$Rockwell$,350,65,54,68,]]$

a$.$ Calculate the initial value of the index if a value$-$weighting scheme is used.

Note: Round your answer to $2$ decimal places.

Index value

b$.$ What is the rate of return on this index for the year ending December $31,2022?$ For the year ending December $31,2023?$ Note: A negative value should be indicated by a minus sign. Do not round intermediate calculations. Enter your answers as a percent rounded to $2$ decimal places.

$\backslash$table$[[2022$ return,$\%$M