This afternoon, you deposited $$ \$ 1,000 $$ into a retirement savings account. The account will compound interest at 5 percent annually, You will not withdraw any principal or interest until you retire in forty years. Which one of the following statements is correct? The interest you earn six years from now will equal the interest you earn ten years from now. The interest amount you earn will double in value every year. The total amount of interest you will earn will equal $$ \$ 1,000 imes .06 imes 40 $$. The present value of this investment is equal to $$ \$ 1,000 $$. The future value of this amount $$ =\$ 1,000 imes(1+40 y .06 $$.

Question

This afternoon, you deposited $$ \$ 1,000 $$ into a retirement savings account. The account will compound interest at 5 percent annually, You will not withdraw any principal or interest until you retire in forty years. Which one of the following statements is correct? The interest you earn six years from now will equal the interest you earn ten years from now. The interest amount you earn will double in value every year. The total amount of interest you will earn will equal $$ \$ 1,000 	imes .06 	imes 40 $$. The present value of this investment is equal to $$ \$ 1,000 $$. The future value of this amount $$ =\$ 1,000 	imes(1+40 y .06 $$.

UpStudy AI · Accepted Answer

1. We start with the fact that you deposit $$ \$1,000 $$ today. In financial terms, the amount deposited today is the present value (PV) of the investment.

2. The investment earns compound interest at an annual rate of $$ 5\% $$. When interest is compounded, the interest earned in later years will be higher than in earlier years because interest is earned on both the original principal and the accumulated interest. This means the interest earned six years from now will not equal the interest earned ten years from now.

3. The idea that the interest amount doubles each year is also incorrect. With compound interest growing at $$ 5\% $$ per year, the accumulation is exponential but does not simply double every year.

4. The total amount of interest earned is not calculated by simply multiplying the deposit by a fixed interest rate and time period (i.e., $$ \$1,000 \times 0.06 \times 40 $$). That would be the case for simple interest, not compound interest.

5. One of the statements is: "The present value of this investment is equal to $$ \$1,000 $$". This is correct because the present value is the amount of money you deposit today.

6. The last statement gives an incorrect formula for future value. The correct compound interest formula for future value (FV) is 
   $$
   \text{FV} = 1000 \times (1 + 0.05)^{40}
   $$
   not 
   $$
   1000 \times (1 + 40 \times 0.06).
   $$

7. Therefore, the correct statement is:  
   $$
   \text{"The present value of this investment is equal to } \$1,000\text{."}
   $$
The present value of the investment is \$1,000.

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