Many persons prepare for retirement by making monthly contributions to a savings program. Suppose that 

​$1 comma 9001,900 

is set aside each year and invested in a savings account that pays 

2020​% 

interest per​ year, compounded continuously.

**a.** Determine the accumulated savings in this account at the end of 

2626 

years.

**b.** In Part​ (a), suppose that an annuity will be withdrawn from savings that have been accumulated at the EOY 

2626. 

The annuity will extend from the EOY 

2727 

to the EOY 

3636. 

What is the value of this annuity if the interest rate and compounding

frequency in Part​ (a) do not​ change?

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Click the icon to view the interest and annuity table for continuous compounding when 

*i*equals=2020​% 

per year.

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Part 1

**a.** The accumulated savings amount at the end of 

2626 

years will be 

​$enter your response here. 

​(Round to the nearest​ dollar.)

Let's break this down step by step:

### Part a: Accumulated Savings after 26 Years

For continuously compounded interest, the formula to calculate the future value $ A $ of an investment is:

\[
A = P \cdot e^{rt}
\]

Where:
- $ P $ is the annual contribution (investment per year),
- $ r $ is the annual interest rate (expressed as a decimal),
- $ t $ is the time period in years, and
- $ e $ is Euler's number (approximately 2.71828).

Given:
- $ P = 1,900 $ (annual contribution),
- $ r = 20\% = 0.20 $ (annual interest rate),
- $ t = 26 $ years.

However, since contributions are made annually, the future value must consider the contributions made over time. This is modeled as a series of continuous compounding for each contribution, which follows the formula for the future value of a continuously compounded annuity:

\[
A = P \cdot \left( \frac{e^{rt} - 1}{r} \right)
\]

Plugging the values into the formula:

\[
A = 1,900 \cdot \left( \frac{e^{0.20 \cdot 26} - 1}{0.20} \right)
\]

I will now calculate this value.The accumulated savings at the end of 26 years will be **$1,712,586**.

### Part b: Value of the Annuity from Year 27 to Year 36

Now, let's determine the value of an annuity withdrawn from the accumulated savings at the end of 26 years. The annuity will be withdrawn over a 10-year period (from the end of year 27 to the end of year 36), with the same 20% interest rate, compounded continuously.

To calculate the annuity's value, we'll use the formula for the present value of an annuity, considering continuous compounding:

\[
A_{\text{annuity}} = \frac{PV \cdot r}{1 - e^{-r \cdot t}}
\]

Where:
- $ PV $ is the accumulated amount at the end of year 26 (\$1,712,586),
- $ r $ is the interest rate (0.20),
- $ t $ is the time period (10 years).

Let me calculate this for you.The value of the annuity, withdrawn from year 27 to year 36, will be **$396,127** annually.

Would you like any further details or clarifications?

Here are five related questions to deepen your understanding:
1. How does continuous compounding differ from other types of compounding, like monthly or yearly?
2. How would the accumulated savings change if the interest rate was reduced to 15%?
3. What is the impact of increasing the contribution amount to $2,500 annually?
4. How does the length of the annuity period (e.g., 15 years instead of 10) affect the annual withdrawal?
5. What is the future value of a one-time lump sum investment with continuous compounding?

**Tip:** Continuous compounding assumes that interest is being added to the principal at every possible instant, leading to higher returns compared to periodic compounding methods.

Many persons prepare for retirement by making monthly contributions to a savings program. Suppose that $1,900 is set aside each year and invested in a savings account that pays 20% interest per year, compounded continuously.

**a.** Determine the accumulated savings in this account at the end of 26 years.

**b.** In Part​ (a), suppose that an annuity will be withdrawn from savings that have been accumulated at the EOY 26. The annuity will extend from the EOY 27 to the EOY 36. What is the value of this annuity if the interest rate and compounding frequency in Part​ (a) do not​ change?

Learn how to calculate the future value of a retirement savings plan with continuous compounding at 20% interest over 26 years. Additionally, understand how to determine the value of an annuity withdrawn over 10 years after the savings period. This detailed guide provides step-by-step solutions with key financial formulas for continuous compounding.

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