This section contains problems $5$ to $7$ out of $10$ problems. Each problem is worth $7$ points. Please show your work for each problem for credit. Ignore transaction costs and taxes for the problems. Please fit all your work for the problems on one side of the paper, write your name on the sheet, make a digital file of the sheet, and upload the file.

$5.$ An annuity pays $$40,000$ per quarter for $30$ years and the payments are made at the end of each quarter. The first payment is made at the end of the first quarter. If the annual interest rate is $8$ percent compounded quarterly for the first $10$ years, and $12$ percent compounded quarterly thereafter, what is the present value of the annuity $($i$.$e$,$ value of the annuity now$)?$

$6.$ Your objective is to have $$8,000,000$ in an account that earns $8\%$ annual return when you retire in $40$ years.

a$)$ If you make an equal deposit at the end of every$\_$year to the account for the next $40$ years, what is the annual deposit? Assume that your first deposit occurs at the end of next year.

b$)$ You retire after $40$ years and have $$8,000,000$ in the account as planned. You expect to live for another $25$ years after retirement. Assume that you leave the $$8,000,000$ in the account that continues to earn $8\%$ annual return. You plan to make an equal withdrawal from the account every$\_$year for the next $25$ years. The first withdrawal is made at the end of the first year after your retirement and the account balance will be depleted after you make the ${25}^{th}$ withdrawal. How much can you withdraw at the end of each year for the next $25$ years?

$7.$ You have found your dream home. The selling price is $$500,000.$ You will put $$100,000$ as down payment and obtain a $30-$year fixed$-$rate mortgage loan at $6$ percent annual interest rate from a bank for the rest.

a$)$ You are required to make an equal payment every month for $360$ months to pay off the balance on the loan. Assume that the first payment begins in one month after you have obtained the loan. What will each monthly payment be$?$

b$)$ If you want to pay off the remaining principal on your mortgage loan after $10$ years $($i$.$e$.,120$ months$),$ how much will you have to pay? Assume that you have never missed your payments during the first ten years after you have obtained the loan. The bank imposes no charges for early payoff of the loan.