Barry heard in his Personal Finance class that he should start investing as soon as possible. He had always thought that it would be smart to start investing after he finishes college and his salary is high enough to pay the bills and to have money left over. He projects that will 50-10 years from now. Barry wants to compare the difference between investing now and investing later. A financial advisor who spoke to Barry suggested that a Roth IRA (Individual Retirement Account) would be a more profitable investment over long term than a regular IRA, so Barry wants to seriously consider the Roth IRA. When table values do not include the information you need use the FV= $1(1 +R)^N where R is the period rate and N is the number of periods.

Question

Barry heard in his Personal Finance class that he should start investing as soon as possible. He had always thought that it would be smart to start investing after he finishes college and his salary is high enough to pay the bills and to have money left over. He projects that will 50-10 years from now. Barry wants to compare the difference between investing now and investing later. A financial advisor who spoke to Barry suggested that a Roth IRA (Individual Retirement Account) would be a more profitable investment over long term than a regular IRA, so Barry wants to seriously consider the Roth IRA. When table values do not include the information you need use the FV= $1(1 +R)^N where R is the period rate and N is the number of periods.

1. If Barry purchases a $2,000 Roth IRA when he is 25. Years old and expects to earn an average of 6% per year compounded annually over 35 years (until he is 60), how much will accumulate in the investment?

2. If Barry doesn’t put the money in the IRA until he is 35 years old, how much money will accumulate in the account by the time he is 60 years old? How much less will he earn because he invested 10 years later?
3. Interest rate is critical to the speed at which your investment grows. If $1 is invested at 2%, it takes approximately 34.9 years to double. If $ 1 is invested at 5%, it takes approximately 14.2 years to double. Use table 13-1 to determine how many years it takes $1 to double if invested at 10; at 12%.

4. at what interest rate would you need to invest to have your money double in 10 years?

Answer 1

1. P1 = Po(1+r)^n

Po = $2,000

r = 0.06

n = 35yrs * 1comp./yr. = 35 Compounding
periods.

P1 = 2000(1.06)^35 = $15,372.17

2. Use same Eq as above with a 25-yr.
investment(60-35).

P2 = Po(1+r)^n

Difference = P1-P2.

3. Use your table and compare with calculated values below.

P = 1(1.10)^n = 2
n*Log1.1 = Log 2
n = Log 2/Log1.1 = 7.2725409 Compounding
periods.

T=7.273comp. periods * 1yr/comp. period = 7.273 years.

P = 1(1.12)^n = 2
n*Log 1.12 = Log 2
n = Log 2/Log 1.12 = 6.11 compounding
periods. T = 6.11 years.

4. P = $1(1+r)^10 == 2
10*Log(1+r) = Log 2
Log(1+r) = Log 2/10 = 0.03010
1+r = 10^0.0301
1+r = 1.07177
r = 0.07177 or 7.18%.

Answer 2

1. To calculate the amount that will accumulate in the Roth IRA, we can use the formula for future value of a single sum compounded annually:

FV = P(1 + R)^N

Where:
FV = Future value
P = Principal (initial investment)
R = Rate of return (as a decimal)
N = Number of periods (years in this case)

In this case:
P = $2,000
R = 6% = 0.06
N = 35 years

Substituting these values into the formula:
FV = $2,000(1 + 0.06)^35

Now, let's calculate the result.

2. If Barry doesn't start investing until he is 35 years old, he will have 25 years to accumulate funds in the account. The calculation will be similar to the previous one, but the number of years (N) will be different:

N = 25 years

Using the same formula as before:
FV = $2,000(1 + 0.06)^25

Now, we need to calculate the value.

To find out how much less Barry will earn because he invested 10 years later, we can subtract the value calculated in question 2 from the value calculated in question 1.

3. To determine how many years it takes for an investment to double at a certain interest rate, we can use the rule of 72. The rule of 72 states that you can estimate the number of years it takes for an investment to double by dividing 72 by the interest rate.

For example:
If the interest rate is 2%, it takes approximately 72/2 = 36 years to double.
If the interest rate is 5%, it takes approximately 72/5 = 14.4 years to double.

Now, let's use this rule to determine how long it takes for an investment to double at 10% and 12% interest rates.

For a 10% interest rate:
Number of years to double = 72/10

For a 12% interest rate:
Number of years to double = 72/12

4. To determine the interest rate needed to double your money in 10 years, we can rearrange the formula for future value (FV) to solve for the interest rate (R):

FV = P(1 + R)^N
R = (FV/P)^(1/N) - 1

In this case, we want to solve for R when N = 10 (years) and the future value (FV) is 2 times the principal (P):

R = (2/1)^(1/10) - 1

Now, let's calculate the answer.