You wish to retire in 15 years, at which time you want to have accumulated enough money to receive an annual annuity of $18,000 for 20 years after retirement. During the period before retirement you can earn 9 percent annually, while after retirement you can earn 11 percent on your money.

What annual contributions to the retirement fund will allow you to receive the $18,000 annuity? 

To solve this problem, we need to break it into two phases: the accumulation phase before retirement and the distribution phase after retirement.

### Step 1: Determine the present value of the annuity required at retirement

You want to receive an annuity of $18,000 per year for 20 years after retirement. During this time, the interest rate is 11% annually. We first calculate how much money you need at the time of retirement to support this annuity.

The present value (PV) of an annuity can be calculated using the formula:

\[
PV = P \times \left(1 - \frac{1}{(1 + r)^n}\right) \div r
\]

Where:
- $P = 18,000$ (the annual payment),
- $r = 0.11$ (the annual interest rate during retirement),
- $n = 20$ (the number of years of annuity payments).

Let's calculate the present value of the annuity.

\[
PV = 18,000 \times \left(1 - \frac{1}{(1 + 0.11)^{20}}\right) \div 0.11
\]

### Step 2: Determine the annual contributions before retirement

After determining the present value of the annuity at retirement, we need to calculate how much you need to contribute annually to accumulate that amount. You have 15 years until retirement, and during this period, you can earn 9% annually. The formula for the future value of an annuity (FVA) is:

\[
FVA = C \times \frac{(1 + r)^n - 1}{r}
\]

Where:
- $C$ is the annual contribution,
- $r = 0.09$ (the annual interest rate before retirement),
- $n = 15$ (the number of years of contribution).

The future value of your contributions at retirement must equal the present value of the annuity from Step 1. Therefore, we solve for $C$ using:

\[
FVA = PV
\]

Let's calculate the two parts: first the PV of the annuity and then the annual contribution necessary to reach that amount. I'll solve this step by step.The present value of the annuity you need at retirement is approximately \$143,339.91.

### Step 2: Calculate the annual contributions required to accumulate this amount in 15 years

We now calculate the annual contributions ($C$) required to accumulate this amount, using the formula for the future value of an annuity:

\[
FVA = C \times \frac{(1 + r)^n - 1}{r}
\]

Where:
- $FVA = 143,339.91$,
- $r = 0.09$,
- $n = 15$.

Let's calculate the necessary annual contribution $C$.It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can calculate the annual contribution required with this formula:

\[
C = \frac{PV}{\frac{(1 + r)^n - 1}{r}}
\]

Where:
- $PV = 143,339.91$ (the present value of the annuity you calculated),
- $r = 0.09$ (interest rate before retirement),
- $n = 15$ (years to retirement).

This formula will give you the annual contribution necessary to meet your retirement goals.

Feel free to ask if you need more details or have questions! Here are 5 related questions:

1. How would changing the interest rates affect the required contributions?
2. What is the impact of increasing the retirement age on the annuity?
3. Can you explain how inflation would affect your retirement annuity?
4. How does compounding frequency (monthly vs annually) influence retirement savings?
5. What if you wanted to leave some principal to heirs at the end of the 20-year annuity?

**Tip**: Always re-evaluate your retirement plan periodically to adjust for life changes and market conditions.

You wish to retire in 15 years, at which time you want to have accumulated enough money to receive an annual annuity of $18,000 for 20 years after retirement. During the period before retirement you can earn 9 percent annually, while after retirement you can earn 11 percent on your money. What annual contributions to the retirement fund will allow you to receive the $18,000 annuity?

Learn how to calculate the necessary annual contributions to accumulate enough money for a $18,000 annuity over 20 years, starting 15 years from now. This financial planning problem involves calculating the present and future value of annuities with interest rates of 9% and 11%. Suitable for finance students and individuals planning for retirement.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation