7.i Sinking Fund¶

Introduction¶

A sinking fund is a fund established for the purpose of accumulating money over time to pay off a debt, replace a major asset, or meet a specific financial obligation in the future. It involves making regular contributions to the fund, which earns interest, allowing the fund to grow over time. Sinking funds are commonly used by corporations to repay bonds, governments to finance projects, or individuals to save for large future expenses.

The amount to be deposited regularly in a sinking fund depends on the desired future amount, the interest rate, and the number of compounding periods.

Formula for Sinking Fund¶

The formula to calculate the regular payment $ P $ required to achieve a future value $ A $ in a sinking fund is:

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

Where: - $ P $ = Regular payment amount - $ A $ = Future value (desired amount in the sinking fund) - $ r $ = Interest rate per period - $ n $ = Total number of payments (compounding periods)

Example 1: Creating a Sinking Fund to Repay a Loan¶

Problem¶

A company needs to accumulate $50,000 in 5 years to repay a loan. The sinking fund earns an annual interest rate of 6%, compounded annually. How much should the company deposit each year into the sinking fund?

Solution¶

Given: - $ A = 50000 $ (Future value) - $ r = 6\% = 0.06 $ (Annual interest rate) - $ n = 5 $ (Number of years)

Using the sinking fund formula:

\[ P = \frac{50000 \times 0.06}{(1 + 0.06)^5 - 1} \]

Calculating $ (1 + 0.06)^5 $:

\[ (1 + 0.06)^5 \approx 1.3382 \]

Now, substituting the values:

\[ P = \frac{50000 \times 0.06}{1.3382 - 1} \]

\[ P = \frac{3000}{0.3382} \approx 8867.79 \]

Explanation¶

The company needs to deposit approximately $8,867.79 each year for 5 years to accumulate $50,000 in the sinking fund at an annual interest rate of 6%. The interest earned by the sinking fund helps reduce the amount the company needs to deposit.

Example 2: Saving for a Future Purchase¶

Problem¶

An individual wants to save $30,000 in 10 years to purchase a new car. The sinking fund earns an annual interest rate of 4%, compounded semi-annually. How much should be deposited every six months?

Solution¶

Given: - $ A = 30000 $ (Future value) - $ r = \frac{4\%}{2} = 0.02 $ (Semi-annual interest rate) - $ n = 10 \times 2 = 20 $ (Total number of semi-annual periods)

Using the sinking fund formula:

\[ P = \frac{30000 \times 0.02}{(1 + 0.02)^{20} - 1} \]

Calculating $ (1 + 0.02)^{20} $:

\[ (1 + 0.02)^{20} \approx 1.4859 \]

Now, substituting the values:

\[ P = \frac{30000 \times 0.02}{1.4859 - 1} \]

\[ P = \frac{600}{0.4859} \approx 1235.13 \]

Explanation¶

The individual needs to deposit approximately $1,235.13 every six months for 10 years to accumulate $30,000 in the sinking fund at a 4% annual interest rate compounded semi-annually. The semi-annual compounding allows the fund to grow faster, reducing the required payment amount.

Understanding the Usefulness of Sinking Funds¶

Sinking funds help in systematically setting aside money for future obligations, reducing the financial burden when the obligation is due. By earning interest on the accumulated funds, the total amount required from regular contributions is minimized. This approach is particularly beneficial for organizations managing debt repayments or for individuals saving for significant future expenses.

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