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Gilberto opened a savings account for his daughter and deposited $1,000 on the day she was born....

Question:

Gilberto opened a savings account for his daughter and deposited $1,000 on the day she was born. Each year on her birthday, he deposited another $1,000. If the account pays 8% interest, compounded annually, how much is in the account at the end of the day on her 9th birthday?

Future Value of the Ordinary Annuity:

If the payment of an annuity is paid at the end of the period, then it is called an ordinary annuity. The future value {eq}FV {/eq} is referred to as the amount in the account after the {eq}n {/eq} number of periods, which includes the payment for each annuity {eq}C {/eq} and compounded interest amount at rate {eq}i. {/eq} The formula is given by:

$$FV = \dfrac{C \left[ \left( 1+ i \right)^n - 1 \right] }{i} $$

Answer and Explanation: 1

Given Data:

  • Payment amount: {eq}C = \$ 1000 {/eq}
  • Rate of interest: {eq}i = 8 \% = \dfrac{8}{100} = 0.08 {/eq}
  • Number of periods: {eq}n = 9 \ \text{years} {/eq}

We can find the amount by using the future value formula for ordinary Annuity:

$$\begin{align*} FV &= \dfrac{C \left[ \left( 1+ i \right)^n - 1 \right] }{i} \\[0.3cm] &= \dfrac{1000 \left[ \left( 1 + 0.08 \right)^9 - 1 \right]}{0.08} \\[0.3cm] &= \dfrac{1000 \left[ \left( 1.08 \right)^9 - 1 \right]}{0.08} \\[0.3cm] &= \dfrac{1000 \left[1.99900 - 1 \right]}{0.08} \\[0.3cm] &= \dfrac{1000 \left[0.99900 \right]}{0.08} \\[0.3cm] &= \dfrac{999}{0.08} \\[0.3cm] \therefore FV &= 12487.5 \end{align*} $$

Hence, the future value is {eq}\color{blue}{\$ 12487.5} {/eq}.


Learn more about this topic:

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How to Find the Value of an Annuity

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Chapter 21 / Lesson 15
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An annuity is a type of savings account that pays back the investor in the future. Learn the formula used to calculate an annuity's value, and understand the importance of labeling specific numbers to calculate an output over time.


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