Many persons prepare for retirement by making monthly contributions to a savings program. Suppose that ​$1 comma 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b. In Part​ (a), suppose that an annuity will be withdrawn from savings that have been accumulated at the EOY 26.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?

To solve this problem, we'll break it down into two parts.

### Part (a): Accumulated Savings after 26 Years

We need to determine the accumulated value of the savings program after 26 years with continuous compounding. This involves applying the formula for the future value of a continuously compounded savings plan:

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

where:
- $ P = 1,900 $ is the annual contribution,
- $ r = 0.20 $ is the continuous compounding interest rate (20%),
- $ t = 26 $ years is the time period,
- $ e $ is Euler's number (approximately 2.718).

### Step 1: Calculate the future value of the savings

Using the formula, we calculate the accumulated value of the account after 26 years.

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

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

The accumulated value at the end of year 26 will be used to fund an annuity. The annuity will span 10 years (from the EOY 27 to the EOY 36). The value of the annuity can be determined using the present value formula for an annuity with continuous compounding:

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

where:
- $ P $ is the accumulated savings from Part (a),
- $ r = 0.20 $,
- $ t = 10 $ years (for the annuity).

Now, let me calculate the accumulated savings for Part (a), and then use that to find the annuity in Part (b).The accumulated savings after 26 years is approximately \$1,712,586.30.

### Part (b): Annuity Value from Year 27 to Year 36

Now, we use this amount to calculate the annuity over the next 10 years with the same interest rate and continuous compounding.

The formula for the annuity is:

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

where $ P = 1,712,586.30 $, $ r = 0.20 $, and $ t = 10 $. Let me calculate this next.The value of the annuity from the end of year 27 to the end of year 36 is approximately \$7,404,064.73.

### Summary:
- **Part (a)**: The accumulated savings at the end of 26 years is \$1,712,586.30.
- **Part (b)**: The value of the annuity over the next 10 years (from EOY 27 to EOY 36) is \$7,404,064.73.

Would you like further details or have any questions about these calculations?

### Related Questions:
1. How would the results change if the interest rate were 15% instead of 20%?
2. What would the future value be if contributions were made quarterly rather than annually?
3. How much interest is earned in total over the 26 years?
4. How would the annuity value change if it spanned 15 years instead of 10?
5. What happens if contributions are increased by 3% each year?

### Tip:
When saving for retirement, compounding interest and regular contributions significantly boost the final amount, especially with continuous compounding.

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?

Calculate the accumulated savings from a $1,900 annual contribution to a savings account with 20% continuous compounding over 26 years. Learn how to find the value of an annuity spanning 10 years using these savings. Detailed formulas and step-by-step solutions provided.

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