Question

A multinational company creates a sinking fund by setting a sum of Rs. 12,000 annually for 10 years to pay off a bond issue of Rs. 72,000. If the fund accumulates at\(5\% \)per annum compound interest, then the surplus after paying for bond is

• A. Rs. 78,900
• B. Rs. 68,500
• C. Rs. 72,000
• D. Rs. 1,44,000

Hint:

In problems involving sinking funds, you can calculate the accumulated value using the formula \( A = P \cdot \frac{(1 + r)^n - 1}{r} \), where \( P \) is the principal, \( r \) is the interest rate per period, and \( n \) is the number of periods. The surplus is the amount accumulated over and above the initial total investment, which is crucial for assessing the effectiveness of the sinking fund. Be sure to carefully substitute and simplify the values to calculate the final surplus.

Answer 1

The Correct Option is C

The accumulated value of the sinking fund is calculated using the formula:

\[ A = P \cdot \frac{(1 + r)^n - 1}{r}. \]

Here:

- \( P = 12,000 \),

- \( r = 0.05 \),

- \( n = 10 \),

- \( (1.05)^{10} \approx 1.6 \).

Substitute into the formula:

\[ A = 12,000 \cdot \frac{1.6 - 1}{0.05} = 12,000 \cdot \frac{0.6}{0.05} = 12,000 \cdot 12 = 1,44,000. \]

The surplus is:

\[ \text{Surplus} = A - 72,000 = 1,44,000 - 72,000 = Rs.72,000. \]

Answer 2

The accumulated value of a sinking fund is a crucial financial concept used to calculate how much money is needed to accumulate a certain amount over time, given a fixed interest rate. The formula for calculating the accumulated value \( A \) of a sinking fund is:

\[ A = P \cdot \frac{(1 + r)^n - 1}{r}. \]

Step 1: Identify the given values:

In this case, we are given: - \( P = 12,000 \) (the principal amount invested), - \( r = 0.05 \) (the interest rate per period), - \( n = 10 \) (the number of periods, which in this case could represent years or months depending on the context), - Additionally, we are provided that \( (1.05)^{10} \approx 1.6 \) (the factor by which the amount increases over 10 periods).

Step 2: Substitute the values into the formula:

Now, using the sinking fund formula, we substitute the given values: \[ A = 12,000 \cdot \frac{1.6 - 1}{0.05} = 12,000 \cdot \frac{0.6}{0.05} = 12,000 \cdot 12 = 1,44,000. \] This calculation shows that the total accumulated value after 10 periods will be Rs. 1,44,000.

Step 3: Calculate the surplus:

The surplus is simply the difference between the accumulated value and the initial total investment (which in this case is 72,000, i.e., \( 12,000 \times 6 \) due to the monthly payments): \[ \text{Surplus} = A - 72,000 = 1,44,000 - 72,000 = \text{Rs.} 72,000. \] Therefore, the surplus accumulated over the 10 periods is Rs. 72,000.

Conclusion: The surplus accumulated after 10 periods, given the provided interest rate and principal, is Rs. 72,000. This surplus is the additional amount gained through the interest earned on the regular deposits made into the sinking fund.