A company's stock is trading at a P$/$E of $15.$ It announces that it expects that sales of a new product will be much more robust than expected. As a result,

management believes that future earnings should grow at a significantly higher rate than previously expected.

$($i$.)$ What would be the expected impact on the company's P$/$E ratio? Higher, lower, no change?

$($ii$.)$ What is your reasoning for your answer in $($i$.)?$

$($iii$.)$ What would be the expected impact on the stock price?

$($iv$.)$ If the P$/$E and stock price react differently than you had expected, why might that be$?$