Interest Rate Quotes and Adjustments

Your bank is offering you an account that will pay $13\%$ interest in total for a two$-$year deposit. Determine the equivalent discount rate for a period length of

a$.$ Six months.

b$.$ One year.

c$.$ One month.

X $2.$ Which do you prefer: a bank account that pays $6\%$ per year $($EAR$)$ for three years or

a$.$ An account that pays $3\%$ every six months for three years?

b$.$ An account that pays $9\%$ every $18$ months for three years?

c$.$ An account that pays $0\mathrm{.}6\%$ per month for three years?

hapter $5$ Interest Rates

X$3.$ Many academic institutions offer a sabbatical policy. Every seven years a professor is given a year free of teaching and other administrative responsibilities at full pay. For a professor earning $$100,000$ per year who works for a total of $42$ years, what is the present value of the amount she will earn while on sabbatical if the interest rate is $6\%($EAR$)?$

$4.$ You have found three investment choices for a one$-$year deposit: $9\%$ APR compounded monthly, $10\%$ APR compounded annually, and $8\%$ APR compounded daily. Compute the EAR for each investment choice. $($Assume that there are $365$ days in the year.$)$

$5.$ You are considering moving your money to new bank offering a one$-$year CD that pays an APR of $2\%$ with monthly compounding. Your current bank's manager offers to match the rate you have been offered. The account at your current bank would pay interest every six months. How much interest will you need to earn every six months to match the CD$?$

$6.$ Your bank account pays interest with an EAR of $5\%.$ What is the APR quote for this account based on semiannual compounding? What is the APR with monthly compounding?

$7.$ Suppose the interest rate is $6\mathrm{.}8\%$ APR with monthly compounding. What is the present value of an annuity that pays $$108$ every six months for six years?

$8.$ You can earn $$38$ in interest on a $$1000$ deposit for eight months. If the EAR is the same regardless of the length of the investment, determine how much interest will you earn on a $$1000$ deposit for

a$.9$ months.

b$.1$ year.

c$.1\mathrm{.}6$ years.

$9.$ Suppose you invest $$101$ in a bank account, and five years later it has grown to $$136\mathrm{.}14.$

a$.$ What APR did you receive if the interest was compounded semiannually?

b$.$ What APR did you receive if the interest was compounded monthly?

X$$

$10.$ Your son has been accepted into college. This college guarantees that your son's tuition will not increase for the four years he attends college. The first $$8500$ tuition payment is due in six months. After that, the same payment is due every six months until you have made a total of eight payments. The college offers a bank account that allows you to withdraw money every six months and has a fixed APR of $8\%($semiannual$)$ guaranteed to remain the same over the next four years. How much money must you deposit today if you intend to make no further deposits and would like to make all the tuition payments from this account, leaving the account empty when the last payment is made?

$11.$ You make monthly payments on your mortgage. It has a quoted APR of $10\%($monthly compounding$).$ What percentage of the outstanding principal do you pay in interest each month?

Application: Discount Rates and Loans

$12.$ Capital One is advertising a $60-$month, $6\mathrm{.}61\%$ APR motorcycle loan. If you need to borrow $$11,000$ to purchase your dream Harley Davidson, what will your monthly payment be$?$

$13.$ Oppenheimer Bank is offering a $30-$year mortgage with an EAR of $5\mathrm{.}5\%.$ If you plan to borrow $$170,000,$ what will your monthly payment be$?$

$14.$ You have decided to refinance your mortgage. You plan to borrow whatever is outstanding on your current mortgage. The current monthly payment is $$1991$ and you have made every payment on time. The original term of the mortgage was $30$ years, and the mortgage is exactly four years and eight months old. You have just made your monthly payment. The mortgage interest rate is $5\mathrm{.}537\%($APR$).$ How much do you owe on the mortgage today?

$15.$ You have just sold your house for $$1,100,000$ in cash. Your mortgage was originally a $30-$year mortgage with monthly payments and an initial balance of $$750,000.$ The mort