SOLUTION: You plan to invest $2,100 in an individual retirement arrangement (IRA) today at a nominal annual rate of 9%, which is expected to apply to all future years. a. (1)

Question 1149961: You plan to invest $2,100 in an individual retirement arrangement (IRA) today at a nominal annual rate of 9%, which is expected to apply to all future years.
a. (1) The amount you will have in the account at the end of 11 years if interest is compounded annually is $
. (Round to two decimal places.)
(2) The amount you will have in the account at the end of 11 years if interest is compounded semiannually is $
. (Round to two decimal places.)
(3) The amount you will have in the account at the end of 11 years if interest is compounded daily is $
. (Round to two decimal places.)
(4) The amount you will have in the account at the end of 11 years if interest is compounded continuously is $
. (Round to two decimal places.)
b. (1) If the 9% nominal rate is compounded annually, the EAR is
%. (Round to two decimal places.)
(2) If the 9% nominal rate is compounded semiannually, the EAR is
%. (Round to two decimal places.)
(3) If the 9% nominal rate is compounded daily, what is the EAR is
%. (Round to two decimal places.)
(4) If the 9% nominal rate is compounded continuously, what is the EAR is
%. (Round to two decimal places.)
c. If interest is compounded continuously rather than annually, at the end of 11 years your IRA balance will be $
greater. (Round to two decimal places.)
d. The more frequent the compounding the
(enter either 'larger' or 'smaller') the future value. This result is shown in part a by the fact that the future value becomes
(enter either 'larger' or 'smaller') as the compounding period moves from annually to continuously. Since the future value is
(enter either 'larger' or 'smaller') for a given fixed amount invested, the effective return also
(enter either 'increases' or 'decreases') directly with the frequency of compounding.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
the basic equation for future value of a present value is:

f = p * (1 + r) ^ n

f = the future value
p = the present value
r = the interest rate (not the percent) per time period.
n = the number of time period.

if the interest rate is given as the annual percentage rate, you need to do the following:

apr / 100 = nominal annual interest rate.

if you have compounding, you need to:

divide the annual interest rate by the number of compounding periods per year.
multiply the number of years by the number of compounding periods per year.

in your problem:
p = 2100.
r = 9% / 100 = .09 per year.
n = 11 years.
f = what you want to find.

with annual compounding, you do the following:

f = 2100 * (1 + .09) ^ 11 = 5418.895451.

with semi-annual compounding, you do the following:

f = 2100 * (1 + .09/2) ^ (11 * 2) = 5530.669217.
you divided the annual interest rate by 2 to get a semi-annual interest rate of .09/2.
you multiplied the number of years by 2 to get 22 semi-annual time periods.

with monthly compounding, you do the following:

f = 2100 * (1 + .09/12) ^ (11 * 12) = 5630.753689.

with daily compounding, you do the following, assuming 365 days in a year:

f = 2100 * (1 + .09/365) ^ (11 * 365) = 5650.902744.

with continuous compounding, you use a different formula.
that formula is:

f = p * e ^ (r * t)

f is the future value.
p is the present value.
r is the interest rate per time period.
t is the number of time periods.

with the continuous compounding formula, you can specify any time period and get the same answer, so just leave it as .09 per year for 11 years and your answer will be good.

your problem becomes f = 2100 * e ^ (.09 * 11) = 5651.592392.

if you had done the same problem using semi-annual compounding, then the formula would have been:

f = 2100 * e ^ (.09/2 * 11 * 2) = 5651.592392.

same answer, because 2 / 2 = 1 (dividing interest rate by 2 and multiplying number of years by 2 effectively means multiplying by 1.

.09/2 = .09 * 1/2
11 * 2 = 11 * 2
.09/2 * 11 * 2 = 1/2 * .09 * 2 * 11 = 1/2 * 2 * .09 * 11 = .09 * 11 because 1/2 * 2 = 1.

the EAR (effective annual rate) is found as follows:

find the interest rate per time period and then add 1 to it and then raise it to the power of the number of compounding periods and then subtract 1 from it.

with with annual compounding, you get (1 + .09) ^ 1 = 1.09 - 1 = .09

with semi-annual compounding, you get (1 + .09/2) ^ 2 = 1.092025 - 1 = .092025.

with daily compounding, you get (1 + .09/265) ^ 365 = 1.094162145 - 1 = .094162145.

note that 2100 * (1 + .09/365) ^ (11 * 365) is equivalent to:
2100 * ((1 + .09/365) ^ (365)) ^ 11 which is equivalent to:
2100 * 1.094162145 ^ 11 which is equal to 5650.902744.
the 1.094162145 is the EAR.

the more compounding periods per year at the same nominal interest rate, the higher the future value.

continuous compounding gives you the highest future value you can attain.

you can get close to this by using a very high number of compounding periods per year.

for example, if you have 5000 compounding periods per year, you will get:

f = 2100 * (1 + .09/5000) ^ (11 * 5000) = 5651.542037.
with continuous compounding, you got f = 5651.592392.
that's pretty close, but not equal to what you can get with continuous compounding.

the higher the number of compounding periods per year, the closer you will get to continuous compounding, but you will never be equal, unless you round your answer to a specified number of decimal digits.

for example, i used 1,000,000,000 compounding periods per year in my TI-84 Plus calculator.
i got f = 2100 * (1 + .09/1,000,000,000) ^ (11 * 1,000,000,000) = 5651.592392.
i then stored the result into one of the memory locations in my calculator.
using the continuous compounding formula, i also got 5651.592392 in the display.
just using the calculsator display, the numbers look the same.
however, when i subtract the value using the discrete compounding from the value using the continuous compounding, i show a difference of 2.518 * 10 ^ -7.
it's a very small difference, but it's still a difference.
the value using continuous compounding is still larger than the value using discrete compounding, even when the number of compounding periods per year is 1,000,000,000.

bottom line.
the higher the number of compounding periods per year, the higher the future value.
continuous compounding gives you the highest future value you can get.

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SOLUTION: You plan to invest ​$2,100 in an individual retirement arrangement​ (IRA) today at a nominal annual rate of 9​%, which is expected to apply to all future years. a.​ (1)

SOLUTION: You plan to invest $2,100 in an individual retirement arrangement (IRA) today at a nominal annual rate of 9%, which is expected to apply to all future years. a. (1)