A father wanted to start saving for his daughter$$s university which she will start in $7$ years. He wants to pay for all $4$ years of university.

The current cost of college is $$25,000$ per year, and he expected this cost to increase by $5\%$ annually

Each year$$s college expenses would have to be paid in one installment at the beginning of that year so the first college payment will be due in $7$ years, and the last amount would be due in $10$ years.

He plans to make $8$ annual deposits to a savings account starting now and ending $10$ years from now and plans to increase his annual contribution by $6\%$ each year.

He wants the savings to cover the college expenses exactly, and expects a $8\%$ yearly return on his investments in a stock portfolio.

$1.$ How much should the initial investment be so that the savings in the account cover the college expenses exactly?

$2.$ Using the calculated initial investment,

Determine the NPV of $($i$)$ his annual investments and $($ii$)$ the college expenses over time, using the $8\%$ discount rate.

$3.$ Run a sensitivity for if the tuition were to increase between $2\%$ and $7\%$ per year