Housing versus Saving as Investments. In the text, we found that at a $9\%$ annual rate of interest, we could buy a house worth about $$125,000$ with a monthly payment $($not including taxes and insurance$)$ of $$1000.$ W$$ also estimated that this house, located in South Dakota, would be worth about $$440,000$ in $30$ years. In this activity, you will investigate what will happen if you decide to spend less on the house and invest the difference.

For this activity, your instructor will give you an amount $$M$ that will be the monthly payment on a house and an amount $S that you will save per month in a savings plan.

$($a$)$ Look up the current interest rates on a $30-$year mortgage, and use that rate to calculate the amount you can afford to borrow at a monthly payment of $$M.$ Be sure to give the value of the rate you are using and the website or other source for that rate.

$($b$)$ Assuming you buy a house for the loan amount you found in $($a$),$ and assuming the same $4\mathrm{.}287\%$ percent growth rate in the value of your house as in the text, how much will your house be worth in $30$ years?

$($c$)$ Now, if you invest $$S$ per month at $9\%$ compounded monthly for $30$ years, how much will the total balance be at the end of the $30$ years?

$($d$)$ Combining parts $($b$)$ and $($c$),$ what are your total assets between the house and the savings plan after the $30$ years?

$($e$)$ Now suppose instead of spending $$M$ per month on a house and saving $$S$ per month, you had spent the entire $$(M+S)$ per month on housing. Find the value of the house you could buy at current mortgage rates and how much this house would be worth after $30$ years at the same $4\mathrm{.}287\%$ growth rate. How does this compare with your total in part d$).$

$($f$)$ Can you think of any other information or assumptions that we haven't included here that might be included to make our comparison more realistic?

Reducing Debt. In the text, we calculated that at a $9\%$ annual rate of interest, we could buy a house worth about $$125,000$ with a monthly payment $($not including taxes and insurance$)$ of $$1000.$ Interest rates in $2009$ have been around $6\%$ or even lower for $30$ year mortgages.

$($a$)$ If you want to borrow $$125,000$ for $30$ years at an annual percentage interest rate of $6\%,$ what would your monthly interest payment be$.$

$($b$)$ In the text, we developed a formula which determined the balance $B(k)$ remaining on your mortgage after $k$ payments. Use this formula to determine your remaining balance after $5$ years, $10$ years, and $15$ years. What percent of your loan have you paid off after half of your $30$ year payment schedule is complete?

$($c$)$ If the answer to the previous question is discouraging, you might want to consider how you could pay off the loan more quickly. There are basically two ways to do this. One is increase your monthly payment, and the other is to reduce your interest rate while keeping the payment the same.

i$.$ Suppose that after you have made $10$ years worth of payments $(120$ payments$)$ you find that you can afford to put an additional $$100$ per month towards your house payment. If you do this, you can consider this as a new mortgage with a higher monthly payment and the loan amount is now the remaining balance you have left at the time you start the larger payment. This will also change how many payments you will make, as you would end up paying off the loan before the full $30$ years is up$.$ Using the mortgage formula and your new values for the monthly payment and the loan amount $($and the same interest rate$),$ determine how long it will take to pay off the loan.

ii$.$ How much money did you save in interest by paying off the loan more quickly compared to the original $30-$year payment schedule?