# When she is 30, Sue starts making annual deposits of $2150 into a bond fund that pays 5 percent...

## Question:

When she is 30, Sue starts making annual deposits of $2150 into a bond fund that pays 5 percent annual interest compounded continuously. Assuming that her deposits are made as a continuous income flow, how much money will be in her account if she retires at age 55?

## Continuously Payable Investment:

It is an investment in which the equal amounts are payable continuously for a certain time period and the interest is also compounded continuously.

The formula for the accumulated value is {eq}AV = P{\left( {1 + i} \right)^n}{\overline a _{\left. {\overline {\, n \,}}\! \right| }} {/eq} where {eq}{\overline a _{\left. {\overline {\, n \,}}\! \right| }} = \dfrac{{1 - {{\left( {1 + i} \right)}^{ - n}}}}{{\ln \left( {1 + i} \right)}} {/eq}.

Here, {eq}P{\overline a _{\left. {\overline {\, n \,}}\! \right| }} {/eq} is the present value of the payments and {eq}{\left( {1 + i} \right)^n} {/eq} is the accumulation factor.

## Answer and Explanation:

**Given**

- Now, the payments of 2,150 per year are paid for 25 years and her age is 30 and the interest rate is 5% compounded continuously.

Use the formula {eq}AV = P{\left( {1 + i} \right)^n}{\overline a _{\left. {\overline {\, n \,}}\! \right| }} {/eq} where {eq}{\overline a _{\left. {\overline {\, n \,}}\! \right| }} = \dfrac{{1 - {{\left( {1 + i} \right)}^{ - n}}}}{{\ln \left( {1 + i} \right)}}. {/eq}.

Firstly, calculate {eq}{\overline a _{\left. {\overline {\, {25} \,}}\! \right| }} {/eq}.

Put the values {eq}{\overline a _{\left. {\overline {\, n \,}}\! \right| }} = \dfrac{{1 - {{\left( {1 + i} \right)}^{ - n}}}}{{\ln \left( {1 + i} \right)}} {/eq}.

{eq}\begin{align*} {\overline a _{\left. {\overline {\, {25} \,}}\! \right| }} &= \dfrac{{1 - {{\left( {1 + 0.05} \right)}^{ - 25}}}}{{\ln \left( {1 + 0.05} \right)}}\\ &= \dfrac{{1 - {{\left( {1.05} \right)}^{ - 25}}}}{{\ln \left( {1.05} \right)}}\\ &= 14.44342 \end{align*} {/eq}

Now, substitute the values in {eq}AV = P{\left( {1 + i} \right)^n}{\overline a _{\left. {\overline {\, n \,}}\! \right| }} {/eq}.

{eq}\begin{align*} AV &= 2150{\left( {1.05} \right)^{25}}\left( {14.44342} \right)\\ &= 105,157.6754 \end{align*} {/eq}

Therefore, the accumulated amount is 105,157.6754.

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from Finance 101: Principles of Finance

Chapter 6 / Lesson 1