Find the annual payments for an ordinary annuity and an annuity due for 8 years with a PV of $1,000 and an interest rate of 9%. Round your answers to the nearest cent.

To find the annual payments for both an ordinary annuity and an annuity due, we'll use the following formulas:

### 1. **Ordinary Annuity**
For an ordinary annuity, the payment $ P $ is calculated using the formula:
\[
P = \frac{PV \times r}{1 - (1 + r)^{-n}}
\]
where:
- $ PV $ is the present value of the annuity.
- $ r $ is the interest rate per period.
- $ n $ is the number of periods.

### 2. **Annuity Due**
For an annuity due, the payment $ P $ is calculated using the formula:
\[
P = \frac{PV \times r}{\left[1 - (1 + r)^{-n}\right] \times (1 + r)}
\]
This formula is essentially the same as the ordinary annuity formula but multiplied by an additional factor of $ (1 + r) $ since payments are made at the beginning of each period.

### Given Data:
- Present Value ($ PV $) = $1,000
- Interest Rate ($ r $) = 9% = 0.09
- Number of Periods ($ n $) = 8 years

Now let's calculate the payments for both the ordinary annuity and the annuity due.

### Step 1: Calculate the Ordinary Annuity Payment
\[
P_{\text{ordinary}} = \frac{1000 \times 0.09}{1 - (1 + 0.09)^{-8}}
\]

### Step 2: Calculate the Annuity Due Payment
\[
P_{\text{due}} = \frac{1000 \times 0.09}{\left[1 - (1 + 0.09)^{-8}\right] \times (1 + 0.09)}
\]

I'll calculate these now.The annual payments are as follows:

1. **Ordinary Annuity**: $180.67
2. **Annuity Due**: $165.76

These values are rounded to the nearest cent.

Would you like further details on how these were calculated or any additional questions? 

Here are some related questions you might find useful:

1. How do these payments change if the interest rate increases to 10%?
2. What is the total amount paid over the 8 years for both types of annuities?
3. How does the timing of payments affect the present value of an annuity due versus an ordinary annuity?
4. How would the payments change if the annuity term were 10 years instead of 8 years?
5. What is the future value of these annuities at the end of the 8 years?

**Tip:** Remember, for an annuity due, payments are made at the beginning of each period, which generally results in a lower payment amount compared to an ordinary annuity for the same present value and interest rate.

Learn how to calculate the annual payments for an ordinary annuity and an annuity due over 8 years, with a present value of $1,000 and a 9% interest rate. This step-by-step solution uses key annuity formulas, perfect for finance students and learners interested in the time value of money.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation