Question
A policyholder is given the choice to be paid the interest amount of R 22 000 in cash or in the form of an annuity at the end of each six months for 5 years. If interest is calculated at 16 % per annum compounded semi-annually, how much will the semi-annual payment of the annuity be?
Answers
GPT-4o mini
To find the semi-annual payment of the annuity, we first need to determine the number of periods and the interest rate per period derived from the annual interest rate.
The policyholder has a choice between receiving R 22,000 in cash or as an annuity over 5 years, with payments made every six months.
### Step 1: Determine the number of periods and the interest rate per period
1. **Annual interest rate (i)**: 16% per annum
2. **Interest rate per semi-annual period (r)**:
\[
r = \frac{16\%}{2} = 8\% = 0.08
\]
3. **Total number of semi-annual periods (n)**:
\[
n = 5 \text{ years} \times 2 = 10 \text{ periods}
\]
### Step 2: Use the formula for the Present Value of an Annuity
The formula for the Present Value (PV) of an annuity is:
\[
PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
\]
where:
- \(PV\) is the present value (R 22,000 in this case),
- \(P\) is the annuity payment per period,
- \(r\) is the interest rate per period (0.08), and
- \(n\) is the total number of periods (10).
We need to solve for \(P\):
\[
22000 = P \times \left( \frac{1 - (1 + 0.08)^{-10}}{0.08} \right)
\]
### Step 3: Calculate the factor
First, calculate the factor:
\[
(1 + 0.08)^{-10} = (1.08)^{-10}
\]
Now calculating \( (1.08)^{-10} \):
\[
(1.08)^{-10} \approx 0.4632 \quad (\text{using a calculator})
\]
Next, calculate:
\[
1 - (1.08)^{-10} \approx 1 - 0.4632 = 0.5368
\]
Now, calculate the entire fraction:
\[
\frac{0.5368}{0.08} \approx 6.71
\]
### Step 4: Solve for P (the semi-annual payment)
Substituting back into the equation:
\[
22000 = P \times 6.71
\]
To find \(P\), divide both sides by 6.71:
\[
P = \frac{22000}{6.71} \approx 3285.07
\]
### Conclusion
The semi-annual payment of the annuity will be approximately **R 3,285.07**.
The policyholder has a choice between receiving R 22,000 in cash or as an annuity over 5 years, with payments made every six months.
### Step 1: Determine the number of periods and the interest rate per period
1. **Annual interest rate (i)**: 16% per annum
2. **Interest rate per semi-annual period (r)**:
\[
r = \frac{16\%}{2} = 8\% = 0.08
\]
3. **Total number of semi-annual periods (n)**:
\[
n = 5 \text{ years} \times 2 = 10 \text{ periods}
\]
### Step 2: Use the formula for the Present Value of an Annuity
The formula for the Present Value (PV) of an annuity is:
\[
PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
\]
where:
- \(PV\) is the present value (R 22,000 in this case),
- \(P\) is the annuity payment per period,
- \(r\) is the interest rate per period (0.08), and
- \(n\) is the total number of periods (10).
We need to solve for \(P\):
\[
22000 = P \times \left( \frac{1 - (1 + 0.08)^{-10}}{0.08} \right)
\]
### Step 3: Calculate the factor
First, calculate the factor:
\[
(1 + 0.08)^{-10} = (1.08)^{-10}
\]
Now calculating \( (1.08)^{-10} \):
\[
(1.08)^{-10} \approx 0.4632 \quad (\text{using a calculator})
\]
Next, calculate:
\[
1 - (1.08)^{-10} \approx 1 - 0.4632 = 0.5368
\]
Now, calculate the entire fraction:
\[
\frac{0.5368}{0.08} \approx 6.71
\]
### Step 4: Solve for P (the semi-annual payment)
Substituting back into the equation:
\[
22000 = P \times 6.71
\]
To find \(P\), divide both sides by 6.71:
\[
P = \frac{22000}{6.71} \approx 3285.07
\]
### Conclusion
The semi-annual payment of the annuity will be approximately **R 3,285.07**.