This afternoon, you deposited $1,000 into a retirement savings account. The account will compound interest at 6 percent annually. You will not withdraw any principal or interest until you retire in 40 years. Which one of the following statements is correct? A. The interest you earn six years from now will equal the interest you earn ten years from now.
B. The interest amount you earn will double in value every year.
C. The total amount of interest you will earn will equal $1,000.

You need to calculate the interest amount you earn in six and ten years from now, how much the interest grow every year, and the total amount of interest during the 40 years.

A. Interest you earn six and ten years from now.

Six years from now:

Formula:

[tex]Future\text{ }value=deposit\times(1+rate)^t[/tex]

[tex]Interest=Future\text{ }value-deposit[/tex]

Where:

rate = 6% / 12 = 0.06/12 = 0.005 (montly interest rate)

t = 6 years × 12 months/year = 72 (number of periods)

Thus:

Calculation

[tex]Future\text{ }value=\$ 1,000\times (1+0.005)^{72}=\$ 1,432.04[/tex]

[tex]Interest=\$ 1,432.04-\$ 1,000.00=\$ 432.04[/tex]

Ten years from now:

Formula:

[tex]Future\text{ }value=deposit\times(1+rate)^t[/tex]

[tex]Interest=Future\text{ }value-deposit[/tex]

Where:

rate = 6% / 12 = 0.06/12 = 0.005 (montly interest rate)

t = 10 years × 12 months/year = 120 (number of periods)

Thus:

Calculation

[tex]Future\text{ }value=\$ 1,000\times (1+0.005)^{120}=\$ 1,819.40[/tex]

[tex]Interest=\$ 1,819.40-\$ 1,000.00=\$ 819.40[/tex]

Conclusion: the interests are different.

B. The interest amount you earn will double in value every year.

Calculate the interest amount one year from now and two years from now and check how they compare:

Interest t years from now:

[tex]I_t=P-P(1+r)^t\\\\I_t=P[1-(1+r)^t][/tex]

Interest t+1 years from now:

[tex]I_{t+1}=P[1-(1+r)^{t+1}][/tex]

Ratio:

[tex]I_{t+1}/I_t=[1-(1+r)^{t+1}]/[1-(1+r)^t][/tex]

In general, the right hand side is not equal to 2 but it changes every year, thus this option is not true.

C. The total amount of interest you will earn will equal $1,000 (0.06)(40)

The equation used to obtain this $1,000(0.06)(40) is for simple interest, not compounded interest.

This option is wrong.

D. The present value of this investment is equal to $1,000.

This is TRUE . The present value is the value of the deposit today without any interest.

E. The future value of this amount is equal to $1,000 (1 + 40)(0.06)

This equation is wrong because it is confusing the terms.