Question

You plan to invest

​$2 comma 1002,100

in an individual retirement arrangement​ (IRA) today at a nominal annual rate of

88​%,

which is expected to apply to all future years.a. How much will you have in the account after

99

years if interest is compounded​ (1) annually,​ (2) semiannually,​ (3) daily​ (assume a​ 365-day year), and​ (4) continuously?b. What is the effective annual​ rate, EAR, for each compounding period in part

a​?

c. How much greater will your IRA balance be in

99

years if interest is compounded continuously rather than​ annually?d. How does the compounding frequency affect the future value and effective annual rate for a given​ deposit? Explain in terms of your findings in parts a through

c.

a.​ (1) The amount you will have in the account at the end of

99

years if interest is compounded annually is

​$nothing.

​(Round to the nearest​ cent.) ​(2) The amount you will have in the account at the end of

99

years if interest is compounded semiannually is

​$nothing.

​(Round to the nearest​ cent.)​(3) The amount you will have in the account at the end of

99

years if interest is compounded daily is

​$nothing.

​ (Round to the nearest​ cent.) ​(4) The amount you will have in the account at the end of

99

years if interest is compounded continously is

​$nothing.

​(Round to the nearest​ cent.) b.​ (1) If the

88​%

nominal rate is compounded​ annually, the EAR is

nothing​%.

​(Round to two decimal​ places.)​(2) If the

88​%

nominal rate is compounded​ semiannually, the EAR is

nothing​%.

​(Round to two decimal​ places.)​(3) If the

88​%

nominal rate is compounded​ daily, what is the EAR is

nothing​%.

​(Round to two decimal​ places.)​(4) If the

88​%

nominal rate is compounded​ continously, what is the EAR is

nothing​%.

​(Round to two decimal​ places.)c.  If interest is compounded continuously rather than​ annually, at the end of

99

years your IRA balance will be

​$nothing

greater.  ​(Round to the nearest​ cent.)d.  The more frequent the compounding the

▼

smaller

larger

the future value. This result is shown in part a by the fact that the future value becomes

▼

larger

smaller

as the compounding period moves from annually to continuously. Since the future value is

▼

larger

smaller

for a given fixed amount​ invested, the effective return also

▼

increases

decreases

directly with the frequency of compounding.  ​(Select from the​ drop-down menus.)

Answer 1

a]

future value = present value * (1 + (r/n))^n*t

where r =  nominal annual rate

n = number of compounding periods per year

t = number of years

With continuous compounding, future value = present value * e^rt

b]

EAR = (1 + (r/n))ⁿ - 1

With continuous compounding, EAR = e^r - 1

c]

With annual compounding, future value = $4,197.91

With continuous compounding, future value = $4,314.31

Difference = $4,314.31 - $4,197.91 = $116.40

If interest is compounded continuously rather than​ annually, at the end of 9 years your IRA balance will be ​$116.40 greater. 

d]

The more frequent the compounding the larger the future value. This result is shown in part a by the fact that the future value becomes larger as the compounding period moves from annually to continuously. Since the future value is larger for a given fixed amount​ invested, the effective return also increases decreases directly with the frequency of compounding.