What is the present value of $5000 to be received in 4 years, if the interest rate is 15% p.a., compounding semi-annually?

Question 3Answer

a.

$2861

b.

$2768

c.

$2804

d.

$2787

To calculate the present value (PV) of $5000 to be received in 4 years with an interest rate of 15% per annum, compounded semi-annually, we use the present value formula:

\[
PV = \frac{FV}{(1 + \frac{r}{n})^{nt}}
\]

Where:
- $FV$ is the future value, which is $5000.
- $r$ is the annual interest rate, 15% or 0.15.
- $n$ is the number of times the interest is compounded per year. Since it's compounded semi-annually, $n = 2$.
- $t$ is the number of years, 4 years.

### Step 1: Break down the formula components
- $r = 0.15$
- $n = 2$
- $t = 4$
- $FV = 5000$

### Step 2: Plug the values into the formula

\[
PV = \frac{5000}{\left(1 + \frac{0.15}{2}\right)^{2 \times 4}}
\]

\[
PV = \frac{5000}{\left(1 + 0.075\right)^8}
\]

\[
PV = \frac{5000}{(1.075)^8}
\]

### Step 3: Calculate $ (1.075)^8 $
\[
(1.075)^8 \approx 1.718186
\]

### Step 4: Calculate the present value
\[
PV = \frac{5000}{1.718186} \approx 2909.12
\]

### Step 5: Round to the nearest value in the options
None of the options matches exactly, but the closest value is approximately **$2861**, which corresponds to **option (a)**.

### Final Answer: 
**(a) $2861**

Would you like more details or have any questions about this?

Here are some related questions you might explore:
1. What is the effect of increasing the interest rate on the present value?
2. How does the present value change if compounding is quarterly instead of semi-annually?
3. How do you calculate the future value if you know the present value and interest rate?
4. What is the formula for calculating compound interest?
5. How does continuous compounding differ from semi-annual compounding?

**Tip:** The more frequent the compounding periods, the lower the present value will be for the same future amount!

What is the present value of $5000 to be received in 4 years, if the interest rate is 15% p.a., compounding semi-annually?

Discover how to calculate the present value of $5000 to be received in 4 years with a 15% annual interest rate compounded semi-annually. This problem involves key financial math concepts like present value and compound interest. Suitable for Grades 10-12.

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