Question 1 [ 5 + 5 + 5 = 15 Marks]
K95 is invested into an account at the end of every 4 months for 2 years and 8 months at 15% p.a. compounded every 4 months.
(a) Calculate how much you would have in this account at the end of this period?
(b) Calculate the present value?
(c) Evaluate the equivalent lump-sum for this investment?

To solve this problem, we will use the formula for the future value and present value of an ordinary annuity. An annuity is a series of equal payments made at regular intervals. Here, the payments are made at the end of every 4 months. Let's proceed step-by-step.

### Given Information
- **Payment amount (\( R \))**: K95
- **Interest rate per annum (\( r_{annual} \))**: 15%
- **Compounding period**: every 4 months
- **Total time period**: 2 years and 8 months

### Step 1: Determine the interest rate per compounding period

Since the interest is compounded every 4 months:
- There are \( \frac{12}{4} = 3 \) compounding periods per year.

So, the interest rate per period (\( r \)) is:
\[
r = \frac{15\%}{3} = 5\% = 0.05 \text{ per compounding period}
\]

### Step 2: Determine the total number of payments (\( n \))

Total duration is 2 years and 8 months, which can be converted to months:
\[
2 \text{ years} = 2 \times 12 = 24 \text{ months}
\]
\[
8 \text{ months}
\]
\[
\text{Total time} = 24 + 8 = 32 \text{ months}
\]

Since payments are made every 4 months:
\[
n = \frac{32}{4} = 8 \text{ payments}
\]

### Step 3: Future Value (\( FV \)) of an Ordinary Annuity

The formula for the future value of an ordinary annuity is:
\[
FV = R \times \frac{(1 + r)^n - 1}{r}
\]

Substituting the values:
\[
FV = 95 \times \frac{(1 + 0.05)^8 - 1}{0.05}
\]

Calculating this:
\[
FV = 95 \times \frac{(1.05)^8 - 1}{0.05}
\]

\[
FV = 95 \times \frac{1.477455 - 1}{0.05}
\]

\[
FV = 95 \times \frac{0.477455}{0.05}
\]

\[
FV = 95 \times 9.5491 \approx 907.165
\]

Thus, at the end of this period, you would have approximately K907.17 in the account.

### Step 4: Present Value (\( PV \)) of the Annuity

The formula for the present value of an ordinary annuity is:
\[
PV = R \times \frac{1 - (1 + r)^{-n}}{r}
\]

Substituting the values:
\[
PV = 95 \times \frac{1 - (1 + 0.05)^{-8}}{0.05}
\]

Calculating this:
\[
PV = 95 \times \frac{1 - (1.05)^{-8}}{0.05}
\]

\[
PV = 95 \times \frac{1 - 0.676839}{0.05}
\]

\[
PV = 95 \times \frac{0.323161}{0.05}
\]

\[
PV = 95 \times 6.46322 \approx 614.01
\]

Thus, the present value of this annuity is approximately K614.01.

### Step 5: Equivalent Lump-Sum Value

The equivalent lump-sum is the present value calculated above, which is the amount that would need to be invested today to achieve the same future value under the same interest conditions. So, the equivalent lump-sum for this investment is K614.01.

### Summary of Answers
1. **Amount in the account at the end of the period**: K907.17
2. **Present Value**: K614.01
3. **Equivalent Lump-Sum**: K614.01

Would you like more details or have any other questions?

### Related Questions
1. How does changing the interest rate affect the future value of the annuity?
2. What would be the future value if the payments were made monthly instead of every 4 months?
3. How would the present value change if the interest rate was compounded monthly instead of every 4 months?
4. If the investment period was extended to 5 years, how would the future value be affected?
5. What if the payment amount was doubled? How would that affect the future value and present value?

### Tip
Understanding how compounding periods and interest rates interact can significantly affect the future value of investments. Always pay attention to the compounding frequency when dealing with interest rates.

Explore how to calculate compound interest and annuity values for a K95 investment problem. Learn step-by-step solutions for determining the future value, present value, and equivalent lump-sum amount over a specified period with 15% p.a. interest compounded every 4 months.

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