Sara Sanders purchased $40$ shares of Apple stock at $ $190\mathrm{.}68$ per share using the prevailing minimum initial margin requirement of $57\%.$ She held the stock for exactly $5$ months and sold it without any brokerage costs at the end of that period. During the $5-$month holding$$ period, the stock paid $ $1\mathrm{.}36$ per share in cash dividends. Sara was charged $4\mathrm{.}7\%$ annual interest on the margin loan. The minimum maintenance margin was $25\%.$

a$.$ Calculate the initial value of the$$ transaction, the debit balance$,$ and the equity position on$$ Sara's transaction.

b$.$ For each of the following share$$ prices, calculate the actual margin$$ percentage, and indicate whether$$ Sara's margin account would have excess$$ equity, would be$$ restricted, or would be subject to a margin$$ call:

$(1)$ $ $174\mathrm{.}43,(2)$ $ $206\mathrm{.}33,$ and$(3)$ $ $122\mathrm{.}48.$

c$.$ Calculate the dollar amount of$(1)$ dividends received and$(2)$ interest paid on the margin loan during the $5-$month holding period.

d$.$ Use each of the following sale prices at the end of the $5-$month holding period to calculate$$ Sara's annualized rate of return on the Apple stock$$ transaction:

$(1)$ $ $184\mathrm{.}17,(2)$ $ $194\mathrm{.}24,$ and$(3)$ $ $205\mathrm{.}25.$