The present value of an immediate annuity of Rs. 10,000 paid each quarter for four quarters at 16% p.a. compounded quarterly is

Options

A. 40000
B. 36300
C. 36286.75
D. 36289.25

Answer

To find the present value of an immediate annuity of Rs. 10,000 paid each quarter for four quarters at an interest rate of 16% per annum compounded quarterly, we can follow these steps:

### Step 1: Identify the parameters
- **C** (cash flow per period) = Rs. 10,000
- **R** (annual interest rate) = 16%
- **n** (number of periods) = 4 (since it is paid quarterly for 4 quarters)

### Step 2: Calculate the interest rate per period
Since the interest is compounded quarterly, we need to convert the annual interest rate to a quarterly rate:
- **r** (quarterly interest rate) = R / 4 = 16% / 4 = 4% = 0.04 (in decimal)

### Step 3: Use the present value formula for an annuity
The present value \( PV \) of an annuity can be calculated using the formula:
\[ PV = C \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \]
Substituting the values we have:
\[ PV = 10,000 \times \left( \frac{1 - (1 + 0.04)^{-4}}{0.04} \right) \]

### Step 4: Calculate \( (1 + r)^{-n} \)
First, calculate \( (1 + 0.04)^{-4} \):
\[ (1 + 0.04)^{-4} = (1.04)^{-4} \]
Calculating \( 1.04^4 \):
\[ 1.04^4 = 1.16985856 \quad \text{(approximately)} \]
Thus,
\[ (1.04)^{-4} = \frac{1}{1.16985856} \approx 0.854804 \]

### Step 5: Substitute back into the formula
Now substitute this value back into the present value formula:
\[ PV = 10,000 \times \left( \frac{1 - 0.854804}{0.04} \right) \]
Calculating \( 1 - 0.854804 \):
\[ 1 - 0.854804 = 0.145196 \]
Now, substituting this value:
\[ PV = 10,000 \times \left( \frac{0.145196}{0.04} \right) \]

### Step 6: Calculate the final present value
Calculating \( \frac{0.145196}{0.04} \):
\[ \frac{0.145196}{0.04} = 3.6299 \]
Now multiply by 10,000:
\[ PV = 10,000 \times 3.6299 \approx 36299 \]

### Final Answer
The present value of the immediate annuity is approximately Rs. 36,299.

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