Question

How do I calculate the present value of a series of weekly payments with a 10.4% nominal interest rate compounded weekly?

Answer 1

To calculate the present value of a series of weekly payments with a nominal interest rate compounded weekly, you can use the formula for the present value of an annuity. Here’s how to do it step-by-step:

### Step 1: Determine the Variables
1. **Nominal interest rate (r)**: The nominal interest rate is 10.4% per year, which must be converted to a weekly rate since payments are made weekly.
\[
r_{\text{weekly}} = \frac{0.104}{52} \approx 0.002; \text{ (per week)}
\]

2. **Number of payments (n)**: Determine how many weekly payments you will be making.

3. **Payment amount (PMT)**: This is the amount you will pay each week.

### Step 2: Present Value of an Annuity Formula
The present value \( PV \) of an annuity (a series of equal payments made at regular intervals) can be calculated using the following formula:

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

where:
- \( PMT \) is the payment per period,
- \( r \) is the interest rate per period (weekly in this case),
- \( n \) is the total number of periods (weeks).

### Step 3: Plug in the Values
1. Substitute the weekly interest rate \( r_{\text{weekly}} \), payment amount \( PMT\), and number of payments \( n \) into the formula to calculate the present value of the annuity.

### Example Calculation
Assume:
- You have weekly payments of $100,
- You will make 52 payments (1 year).

Using the formula:
1. \( PMT = 100 \)
2. \( r = 0.002 \)
3. \( n = 52 \)

Now plug these into the formula:

\[
PV = 100 \times \left( \frac{1 - (1 + 0.002)^{-52}}{0.002} \right)
\]

### Step 4: Calculate
1. First, calculate \( (1 + 0.002)^{-52} \):
\[
(1 + 0.002)^{-52} \approx (1.002)^{-52} \approx 0.8765
\]

2. Then calculate \( 1 - (1 + 0.002)^{-52} \):
\[
1 - 0.8765 = 0.1235
\]

3. Finally, calculate the present value:
\[
PV = 100 \times \left( \frac{0.1235}{0.002} \right) \approx 100 \times 61.75 \approx 6175
\]

Thus, the present value of the series of weekly payments is approximately **$6,175.**

### Note
Be sure to adjust any values as needed based on your specific payment amount and number of periods.