A stock is expected to pay a dividend of $$1$ per share in two months and in five months. The stock price is $$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? Explain.