Ganado's subsidiary in Mexico City needs to borrow $90$ million Mexican pesos or its $ equivalency for one year. The current spot exchange rate is MXN$9/$$$.$ A Houston$-$based bank is willing to lend $ at an annual rate of $5\mathrm{.}5\%,$ while a Mexico City bank is quoting a rate of $7\mathrm{.}5\%$ for a peso loan. The inflation rates in the US and Mexico are expected to be $2\%$ and $3\mathrm{.}5\%$ for the coming year, respectively.

a$)$ Assuming the relative PPP holds, what will the spot exchange rate be at the end of one year? what would be the cost in terms of peso to the firm of the dollar loan? Will this be lower or higher than the peso loan?

b$)$ If the actual spot rate at the end of one year turned out to be MXN$8/$$$,$ what would be the actual peso$-$denominated interest cost of the $ loan?