$(*)($FFOM$,5\mathrm{.}29$ modified$)$ A stock is expected to pay a dividend of $$1$ per share

in one month and in four months. The stock price is $$50,$ and the risk$-$free rate

of interest is $8\%$ per annum with continuous compounding for all maturities. An

investor has just taken a short position in a six$-$month forward contract on the

stock.

$($a$)$ What are the forward price and the initial value of the forward contract?

$($b$)$ Three months later, the price of the stock is $$48$ and the risk$-$free rate of

interest is still $8\%$ per annum. What are the forward price and the value of

the short position in the forward contract?