A couple, both $30,$ is buying a car for their growing family. They've saved $300,000$ for a down payment and budgeted $$30,000$ per month for car loan payments and their child's savings. The car's price is $950,000.$ They plan to invest in a mutual fund for their child's education with an average return of $45\%$ per year compounded monthly over $18$ years.

Considering potential salary increases and changes in their financial situation as they embrace parenthood, the couple is open to revisiting and adjusting their budget periodically. They aim to increment their monthly savings by $10\%$ every year to ensure that they stay on track with their financial goals and adapt to any unforeseen circumstances.

Three car financing options:

Option $1$: $18$ year fixed rate at $30\%$ per year compounded monthly, monthly payments, minimum $6\%$ down payment, $2$ point closing costs

Option $2$: $10$ year fixed rate at $36\%$ per year compounded monthly, monthly payments, minimum $4\%$ down payment, $1$ point closing costs

Option $3$: $5$ year fixed rate at $40\%$ per year compounded monthly, monthly payments, minimum $1\%$ down payment, $0$ point closing cost

Questions:

$1.$ Estimate a probability mass function of the long$-$term annual inflation rate of USA $($with at least three values$)$ using economic sources. Utilize this function in your calculations.

$2.$ For each credit option, calculate the expected savings account balance the couple will have attained for their child$$s education. If any excess amount will be left from $300,000$ after initial payments are made, decide on how this money should be used. To maximize their account balance, should the couple use the remaining money to make a larger down payment, or should they invest it to their savings account?

$3.$ Evaluate each credit option in two ways:

i$.$ using the expected balances,

ii$.$ using the balances determined by the expected values of the inflation rate.

Which approach provides a better estimate of the present worth of the balances? Discuss. Find the best credit option for each method.

$4.$ How sensitive are your findings against the salary increases the couple expects? Make the same calculations and comparison with $0\%$ and $20\%$ as the $3-$year increments in their savings. Select the best credit options in each case. Was your selection different in Q$3?$