Loan Instalment Calculation Using Compound Interest

The problem asks us to find the value of each equal yearly instalment for a loan amount ($P$) of $3,825$, repaid over two years with a compound interest rate ($R$) of 4% per annum, compounded annually.

Problem Variables

• Principal Loan Amount ($P$): $3,825$
• Annual Interest Rate ($R$): $4\%$ or $0.04$
• Number of Instalments ($n$): $2$
• Instalment Amount: Let this be $x$

Instalment Formula

When a loan is repaid in equal instalments ($x$) over $n$ periods at a compound interest rate $R$, the principal amount $P$ is equal to the sum of the present values of all instalments. The formula is:

For $n=2$, the formula becomes:

Calculation Steps

1. Substitute the known values into the formula: $ 3825 = \frac{x}{1+0.04} + \frac{x}{(1+0.04)^2} $
2. Simplify the denominators: $ 3825 = \frac{x}{1.04} + \frac{x}{(1.04)^2} $ $ 3825 = \frac{x}{1.04} + \frac{x}{1.0816} $
3. Factor out $x$: $ 3825 = x \left( \frac{1}{1.04} + \frac{1}{1.0816} \right) $
4. Find a common denominator for the terms inside the parenthesis: $ 3825 = x \left( \frac{1.0816 + 1.04}{1.04 \times 1.0816} \right) $ $ 3825 = x \left( \frac{2.1216}{1.124864} \right) $ Alternatively, use the simplified formula derived from the annuity formula: $P = x \times \frac{1 - (1+R)^{-n}}{R}$ is for ordinary annuities, but for loan redemption it's often simpler to use the sum of present values. Let's stick to the sum of present values approach as it's more direct here. Re-calculating step 3's simplification: $ 3825 = x \left( \frac{1.04 + 1}{(1.04)^2} \right) $ $ 3825 = x \left( \frac{2.04}{1.0816} \right) $
5. Solve for $x$: $ x = 3825 \times \frac{1.0816}{2.04} $ $ x = 3825 \times 0.530196... $ $ x \approx 2028.00 $

Therefore, the value of each equal yearly instalment is approximately $2,028$.