Question
Find the future value of a series of payments of R100 each, payable for 1 year at the beginning of each week, assuming a nominal interest rate of 10.4% compounded weekly and 52 weeks in a year.
Answers
GPT-4o mini
To find the future value of a series of payments (also known as an annuity), we can use the future value of an annuity formula. When payments are made at the beginning of each period, it is termed an annuity due.
The future value (FV) of an annuity due can be calculated using the formula:
\[
FV = P \times \left( (1 + r)^n - 1 \right) \times \frac{(1 + r)}{r}
\]
Where:
- \(P\) = payment amount per period (R100)
- \(r\) = interest rate per period
- \(n\) = total number of payments
### Step 1: Determine the weekly interest rate and the number of payments
Given:
- Nominal annual interest rate = 10.4%
- Compounding frequency = weekly
- Total number of weeks in a year = 52
The weekly interest rate is calculated as:
\[
r = \frac{10.4\%}{52} = \frac{0.104}{52} \approx 0.002
\]
### Step 2: Calculate the total number of payments
Since payments are made weekly for one year, the total number of payments is:
\[
n = 52
\]
### Step 3: Substitute values into the future value formula
Now, we can substitute the values into the formula:
\[
FV = 100 \times \left( (1 + 0.002)^{52} - 1 \right) \times \frac{(1 + 0.002)}{0.002}
\]
### Step 4: Calculate
1. Calculate \( (1 + 0.002)^{52} \):
\[
(1 + 0.002)^{52} \approx 1.104188
\]
2. Then, calculate \( (1 + 0.002)^{52} - 1 \):
\[
1.104188 - 1 \approx 0.104188
\]
3. Now, calculate \( \frac{(1 + 0.002)}{0.002} \):
\[
\frac{(1 + 0.002)}{0.002} \approx \frac{1.002}{0.002} \approx 501
\]
4. Substitute these values back into the FV formula:
\[
FV = 100 \times 0.104188 \times 501
\]
\[
FV \approx 100 \times 52.828188
\]
\[
FV \approx 5282.82
\]
### Final Result
The future value of a series of payments of R100 each, payable at the beginning of each week for 1 year at a nominal interest rate of 10.4% compounded weekly, is approximately **R5282.82**.
The future value (FV) of an annuity due can be calculated using the formula:
\[
FV = P \times \left( (1 + r)^n - 1 \right) \times \frac{(1 + r)}{r}
\]
Where:
- \(P\) = payment amount per period (R100)
- \(r\) = interest rate per period
- \(n\) = total number of payments
### Step 1: Determine the weekly interest rate and the number of payments
Given:
- Nominal annual interest rate = 10.4%
- Compounding frequency = weekly
- Total number of weeks in a year = 52
The weekly interest rate is calculated as:
\[
r = \frac{10.4\%}{52} = \frac{0.104}{52} \approx 0.002
\]
### Step 2: Calculate the total number of payments
Since payments are made weekly for one year, the total number of payments is:
\[
n = 52
\]
### Step 3: Substitute values into the future value formula
Now, we can substitute the values into the formula:
\[
FV = 100 \times \left( (1 + 0.002)^{52} - 1 \right) \times \frac{(1 + 0.002)}{0.002}
\]
### Step 4: Calculate
1. Calculate \( (1 + 0.002)^{52} \):
\[
(1 + 0.002)^{52} \approx 1.104188
\]
2. Then, calculate \( (1 + 0.002)^{52} - 1 \):
\[
1.104188 - 1 \approx 0.104188
\]
3. Now, calculate \( \frac{(1 + 0.002)}{0.002} \):
\[
\frac{(1 + 0.002)}{0.002} \approx \frac{1.002}{0.002} \approx 501
\]
4. Substitute these values back into the FV formula:
\[
FV = 100 \times 0.104188 \times 501
\]
\[
FV \approx 100 \times 52.828188
\]
\[
FV \approx 5282.82
\]
### Final Result
The future value of a series of payments of R100 each, payable at the beginning of each week for 1 year at a nominal interest rate of 10.4% compounded weekly, is approximately **R5282.82**.