HKC Co is concerned about exceeding its overdraft limit of $$2$ million in the next two periods. It has been experiencing considerable volatility in cash flows in recent periods because of trading difficulties experienced by its customers, who have often settled their accounts after the agreed credit period of $60$ days. HKC has also experienced an increase in bad debts due to a small number of customers going into liquidation.The company has prepared the following forecasts of net cash flows for the next two periods, together with their associated probabilities, in an attempt to anticipate liquidity and financing problems. These probabilities have been produced by a computer model which simulates a number of possible future economic scenarios. The computer model has been built with the aid of a firm of financial consultants. Period $1$ cash flow Probability Period $2$ cash flow Probability $$000$ $$0008,00020\%7,00030\%4,00050\%3,00050\%(2,000)30\%(9,000)20\%$ HKC Co expects to be overdrawn at the start of period one by $$500,000.$ Required: $($a$)$ Calculate the following values $(24$ marks, each sub$-$question worth eight marks$)$: $(1)$ the expected value of the period one closing balance.$($ii$)$ the expected value of the period two closing balance.$($iii$)$ the probability of a negative cash balance at the end of period $2.($iv$)$ the probability of exceeding the overdraft limit at the end of period $2.$Discuss whether the above analysis can assist the company in managing its cash flows. $($b$)$ Discuss whether profitability or liquidity is the primary objective of working capital management.