$1.$ Suppose you deposit $$1,000$ today $($t $=0)$ in a bank account that pays an interest rate of $7\%$ per year. If you keep the account for $5$ years before you withdraw all the money, how much will you be able to withdraw after $5$ years?

$$

a$.$ Calculate using formula.

b$.$ Calculate using year$-$by$-$year approach.

$$

$2.$ Find the present value of a security that will pay $$2,500$ in $4$ years. The opportunity cost $($interest rate that you could earn from alternative investments$)$ is $5\%.$

$$

a$.$ Calculate using the formula.

b$.$ Calculate using year$-$by$-$year discounting approach.

$$

$3.$ Solve for the unknown in each of the following:

$$

Present value

Years

Interest rate

Future value

$$50,000$

$12$

$?$

$$152,184$

$$21,400$

$30$

$?$

$$575,000$

$$16,500$

$?$

$14\%$

$$238,830$

$$21,400$

$?$

$9\%$

$$213,000$

$$

$4.$ Suppose you enter into a monthly deposit scheme with Chase, where you have your salary account. The bank will deduct $$25$ from your salary account every month and the first payment $($deduction$)$ will be made one month from now $($so$,$ what type of annuity is this?$).$ If you are planning to maintain the account for the next $3$ years, how much money will you have when you close your account $3$ years from now? The appropriate interest rate is $6\%.$

$$

a$.$ Use formula to calculate the future value of this annuity.

b$.$ Use financial calculator to compute the future value $($clearly identify N$,$ I$/$Y$,$ PV$,$ and PMT$).$

c$.$ What will be the future value if the first payment was made immediately? Use both approaches separately for this part. $[$Hint: what type of annuity is it now?$]$

$$

$5.$ Assume that you will pay $$60$ a month at the end of each of the next $24$ months. The future value of the annuity is $$3000.$ Compute the appropriate rate of interest. You can use a financial calculator to solve this problem.

$$

$6.$ The future value of an annuity is $$3,000.$ There will be quarterly payments for the next $5$ years. If the interest rate is $4\%,$ how much money will you have to put every quarter so that you can receive $$3,000$ in five years from now?

$$

$7.$ Given a discount rate of $8\%$ per year, what is the value at year $5$ of a perpetual stream of $$2,350$ annual payments that begins at year $15?$

$$

$8.$ You want to buy a new car from Executive Machines for $$75,000.$ The contract is in the form of a $72-$month annuity due at an APR of $8\mathrm{.}5\%.$ How much will your monthly payment be$?$

$$

$9.$ Suppose you will receive $$100,$ $$265,$ and $$315$ in $1,2,$ and $3$ years from now, respectively. If the appropriate discount rate is $10\%,$ what is the net present value of these cash flows?

$$

$10.$ You have $30$ years left until retirement and want to retire with $$3,000,000.$ Your salary is paid annually, and you will receive $$110,000$ at the end of the current year. Your salary will increase at $2\%$ per year and you can earn a return of $10\mathrm{.}4\%$ on the money you invest. If you save a constant percentage of your salary, what percentage of your salary must you save each year?