LO$146.$ Calculating Annuities Due Suppose you are going to receive $$14,500$ per year for five years. The appropriate discount rate is

$7\mathrm{.}1$ percent.

a$.$ What is the present value of the payments if they are in the form of an ordinary annuity? What is the present value if the payments are an annuity due?

b$.$ Suppose you plan to invest the payments for five years. What is the future value if the payments are an ordinary annuity? What if the payments are an annuity due?

c$.$ Which has the higher present value, the ordinary annuity or annuity due? Which has the higher future value? Will this always be true?

LO$147.$ Annuity and Perpetuity Values Mary is going to receive a $30-$year annuity of $$11,500$ per year. Nancy is going to receive a perpetuity of $$11,500$ per year. If the appropriate discount rate is $4\mathrm{.}3$ percent, how much more is Nancy's cash flow worth?

LO$148.$ Calculating Present Values A six$-$year annuity of twelve $$8,250$ semiannual payments will begin nine years from now, with the first payment coming $9\mathrm{.}5$ years from now. If the discount rate is $8$ percent compounded semiannually, what is the value of this annuity five years from now? What is the value three years from now? What is the current value of the annuity?

LO$149.$ Present Value and Multiple Cash Flows What is the value today of $$5,100$ per year, at a discount rate of $7\mathrm{.}9$ percent, if the first payment is received six years from today and the last payment is received $20$ years from today?

LO$150.$ Variable Interest Rates A $15-$year annuity pays $$1,750$ per month, and payments are made at the end of each month. If the interest rate is $9$ percent compounded monthly for the first seven years, and $6$ percent compounded monthly thereafter, what is the value of the annuity today?

LO$151.$ Comparing Cash Flow Streams You have your choice of two investment accounts. Investment A is a $13-$year annuity that features end$-$of$-$month $$1,600$ payments and has an APR of $7\mathrm{.}8$ percent compounded monthly. Investment B is an annually compounded lump$-$sum investment with an interest rate of $9$ percent, also good for $13$ years. How much money would you need to invest in B today for it to be worth as much as Investment A $13$ years from now?

LO$152.$ Calculating Present Value of a Perpetuity Given a discount rate of $4\mathrm{.}6$ percent per year, what is the value at Date $t=7$ of a perpetual stream of $$7,300$ payments with the first payment at Date $t=15?$

LO$453.$ Calculating EAR A local finance company quotes an interest rate of $18\mathrm{.}4$ percent on one$-$year loans. So$,$ if you borrow $$20,000,$ the interest for the year will be $$3,680.$ Because you must repay a total of $$23,680$ in one year, the finance company requires you to pay $$23,\frac{680}{12},$ or $$1,973\mathrm{.}33$ per month over the next $12$ months. Is the interest rate on this loan $18\mathrm{.}4$ percent? What rate would legally have to be quoted? What is the effective annual rate?

LO$154.$ Calculating Future Values If today is Year $0,$ what is the future value of the following cash flows five years from now? What is the future value $10$ years from now? Assume an interest rate of $5\mathrm{.}7$ percent per year.