Final answer:
An account earning interest compounded daily will have the greatest accumulated value at the end of one year compared to the others, as it compounds interest on an ongoing basis using the principal plus the previously earned interest. Simple interest does not compound, and accounts compounding annually do so less frequently than daily, resulting in lower accumulation.
Step-by-step explanation:
The question asks which of the following accounts would have the greatest accumulated value at the end of one year when the same amount of principal is invested at the same interest rate: an account earning no interest, simple interest, interest compounded annually, or interest compounded daily. Since simple interest is calculated only on the principal, and compound interest is calculated on the principal plus the accumulated interest, any form of compound interest will accumulate more than simple interest if the rates are the same.
Among compound interest, the more frequently interest is compounded within a year, the greater the accumulated value. This is because each time interest is added to the principal, the next interest calculation includes the previously earned interest. Therefore, an account earning interest compounded daily will have the greatest accumulated value at the end of one year, compared to an account compounding annually or one earning simple interest.
For example, if one were to consider a $1,000 investment at a 5% interest rate compounded annually, this would yield a total of $1,000(1+0.05)^1 = $1,050 at the end of the year. In contrast, if that interest is compounded daily, the formula would be $1,000(1+0.05/365)^(365*1), which results in a slightly higher amount due to daily compounding.