Question

Delphine deposits $900 at the END of each year for 9 years in a savings account. The account pays 8% interest, compounded annually. Lidia calculates that the future value of the ordinary annuity is $11,238.80. What would be the future value if deposits are made at the BEGINNING of each period rather than the END?

Group of answer choices

$12,317.90

$12,960.00

$13,037.91

$12,137.90

Answer 1

If Delphine makes $900 deposits at the beginning of each year with 8% interest, compounded annually, the future value of the annuity due is calculated by multiplying the future value of the ordinary annuity by (1 + r), resulting in a future value of $12,137.90.

Step-by-step explanation:

When Delphine deposits $900 at the beginning of each year into an account offering 8% interest, compounded annually, we're dealing with an annuity due as opposed to an ordinary annuity (where deposits are made at the end of each period). To find the future value of an annuity due, we can use the future value formula for an ordinary annuity and then multiply the result by (1 + r), where 'r' is the interest rate.

The formula for the future value of an ordinary annuity is:

FV = P * (((1 + r)n - 1) / r)

Where:

• P = periodic payment amount
• r = interest rate per period
• n = number of periods

Given the previous calculation by Lidia of $11,238.80 as the future value of the ordinary annuity, we'd simply multiply this amount by (1 + 0.08) to account for the additional year of growth on each payment since they are made at the beginning of the period.

FV due = FV ordinary * (1 + r)

Therefore, the future value for the annuity due would be:
FV due = $11,238.80 * (1 + 0.08)

FV due = $11,238.80 * 1.08

FV due = $12,137.90

So, the future value if deposits are made at the beginning of each period will be $12,137.90.

Answer 2

Certainly! To calculate the future value of the annuity with deposits made at the beginning of each period, you can use the future value of an annuity due formula. The formula is:

\[ FV_{\text{due}} = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \times (1 + r) \]

where:

- \( FV_{\text{due}} \) is the future value with deposits at the beginning,

- \( P \) is the periodic payment (deposit),

- \( r \) is the interest rate per period, and

- \( n \) is the total number of periods.

Given that \( P = $900 \), \( r = 0.08 \), and \( n = 9 \), substitute these values into the formula:

\[ FV_{\text{due}} = $900 \times \left( \frac{(1 + 0.08)^9 - 1}{0.08} \right) \times (1 + 0.08) \]

Now, calculate each part of the expression:

\[ FV_{\text{due}} = $900 \times \left( \frac{(1.08)^9 - 1}{0.08} \right) \times (1.08) \]

\[ FV_{\text{due}} \approx $12,160.31 \]

Therefore, the future value with deposits made at the beginning of each period is approximately $12,160.31.