. (TCO 3) Mark deposited $1,000 today, in an account that pays eight percent interest, compounded semi-annually. Which one of the following statements is correct concerning this investment? (Points : 3)
Mark will earn more interest in year 4 than he will in year 3.
Mark will receive equal interest payments every six months over the life of the investment.
Mark would have earned more interest if he had invested in an account paying 8 percent simple interest.
Mark would have earned more interest if he had invested in an account paying annual interest.
Mark will earn less and less interest each year over the life of the investment.
Mark will earn more int. in yr 4 than
in yr 3.
To determine the correct statement concerning Mark's investment, we need to calculate the interest earned in each year.
The formula to calculate compound interest is:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal amount (initial deposit)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = number of years
In this case:
P = $1,000
r = 8% = 0.08
n = 2 (compounded semi-annually)
t = 4 (for year 4)
Let's calculate the interest earned in year 3 and year 4 to determine the correct statement.
For Year 3:
A = $1,000(1 + 0.08/2)^(2*3)
A = $1,000(1 + 0.04)^6
A = $1,000(1.04)^6
A ≈ $1,291.02
The interest earned in Year 3 is:
Interest Year 3 = A - P = $1,291.02 - $1,000 = $291.02
For Year 4:
A = $1,000(1 + 0.08/2)^(2*4)
A = $1,000(1 + 0.04)^8
A = $1,000(1.04)^8
A ≈ $1,367.62
The interest earned in Year 4 is:
Interest Year 4 = A - P = $1,367.62 - $1,000 = $367.62
From the calculations:
- Mark will earn more interest in year 4 than he will in year 3. Therefore, statement 1 is correct.
- Mark will not receive equal interest payments every six months over the life of the investment because the interest is compounded semi-annually. Therefore, statement 2 is incorrect.
- Mark would not have earned more interest if he had invested in an account paying 8 percent simple interest because compound interest generally yields higher returns. Therefore, statement 3 is incorrect.
- Mark would not have earned more interest if he had invested in an account paying annual interest because compound interest provides higher returns when compounded more frequently. Therefore, statement 4 is incorrect.
- Mark will earn more interest each year over the life of the investment, as shown by the calculations. Therefore, statement 5 is incorrect.
In conclusion, the correct statement concerning this investment is: "Mark will earn more interest in year 4 than he will in year 3."