Question

What is the future worth of a series of equal​ year-end deposits of $3,000 for 13 years in a savings account that earns 12​% annual interest if the following were​ true?

​(a) All deposits are made at the end of each​ year?
​(b) All deposits are made at the beginning of each​ year?

Answer 1

The future worth of a series of equal​ year-end deposits of $3,000 for 13 years in a savings account that earns 12​% annual interest if the following were​ true are as under:

(a) The future worth of a series of equal year-end deposits of $3,000 for 13 years at a 12% annual interest rate, compounded annually, is approximately $87,245.54.

(b) The future worth of a series of equal year-beginning deposits of $3,000 for 13 years at a 12% annual interest rate, compounded annually, is approximately $92,622.58.

Step-by-step explanation:

(a) To calculate the future worth of year-end deposits, we use the future value of an ordinary annuity formula:
`\( FV = P * \left( ((1 + r)^n - 1)/(r) \right) \)`, where
`\( P \)` is the annual deposit,
`\( r \)` is the interest rate per period, and
`\( n \)` is the number of periods. Substituting the values, we get
`\( FV = 3,000 * \left( ((1 + 0.12)^(13) - 1)/(0.12) \right) \)`, resulting in approximately $87,245.54.

(b) For year-beginning deposits, we adjust the formula by using the future value of an annuity due formula:
`\( FV_{\text{due}} = P * \left( ((1 + r)^n - 1)/(r) \right) * (1 + r) \)`, considering the additional compounding for deposits made at the beginning of each period. Substituting the values, we get
`\( FV_{\text{due}} = 3,000 * \left( ((1 + 0.12)^(13) - 1)/(0.12) \right) * (1 + 0.12) \)`, resulting in approximately $92,622.58.

In summary, the timing of deposits (end or beginning of the year) impacts the future worth due to the differences in compounding. Year-beginning deposits yield a slightly higher future value due to the additional compounding for the initial deposit at the beginning of the first year.