What is the relationship between discounting and$$ compounding?

b$.$ What is the relationship between the$$ present$-$value factor and the annuity$$ present$-$value factor?

c$.$ i$.$ What will

$$$$4$ comma $1004,100$

invested for

$1717$

years at

$1313$

percent compounded annually grow$$ to$?$

$$ii$.$ How many years will it take

$$$$380380$

to grow to

$$$$10$ comma $055\mathrm{.}5310,055\mathrm{.}53$

if it is invested at

$1414$

percent compounded$$ annually?

$$iii$.$ At what rate would

$$$$1$ comma $4001,400$

have to be invested to grow to

$$$$5$ comma $226\mathrm{.}845,226\mathrm{.}84$

in

$2727$

$$years$?$

d$.$ Calculate the future sum of

$$$$1$ comma $1001,100,$

given that it will be held in the bank for

$1111$

years and earn

$1717$

percent compounded semiannually.

e$.$ What is an annuity$$ due? How does this differ from an ordinary$$ annuity?

f$.$ What is the present value of an ordinary annuity of

$$$$2$ comma $7002,700$

per year for

$2525$

years discounted back to the present at

$88$

$$percent$?$ What would be the present value if it were an annuity$$ due?

g$.$ What is the future value of an ordinary annuity of

$$$$2$ comma $7002,700$

per year for

$2525$

years compounded at

$88$

$$percent$?$ What would be the future value if it were an annuity$$ due?

h$.$ You have just borrowed

$$$$280$ comma $000280,000,$

and you agree to pay it back over the next

$2525$

years in

$2525$

equal$$ end$-$of$-$year payments plus

$1313$

percent compound interest on the unpaid balance. What will be the size of these$$ payments?

i$.$ What is the present value of a perpetuity of

$$$$1$ comma $5001,500$

per year discounted back to the present at

$1414$

$$percent$?$

j$.$ What is the present value of an annuity of

$$$$1$ comma $9001,900$

per year for $10$ years, with the first payment occurring at the end of year $10($that is$,$ ten

$$$$1$ comma $9001,900$

payments occurring at the end of year $10$ through year$19),$ given a discount rate of

$1717$

$$percent$?$

k$.$ Given a discount rate of

$1818$

$$percent$,$ what is the present value of a perpetuity of

$$$$1$ comma $8001,800$

per year if the first payment does not begin until the end of year$10?$