A loan is to be returned in two equal yearly instalments. If the rate of interest is 10% p.a., compounded annually and each instalment is Rs. 6534, then the total interest charged (in Rs.) is:
1728
Understanding Loan Repayment with Equal Yearly Instalments
When a loan is repaid in equal yearly instalments, each instalment includes a portion of the principal loan amount and the interest accrued up to that point. To find the total interest charged, we first need to determine the original principal amount of the loan.
The key concept here is the present value of each future instalment. The sum of the present values of all future instalments, discounted at the given rate of interest, equals the original principal loan amount.
Calculating the Principal Loan Amount
Let the loan amount be \(P\). The rate of interest is \(r = 10\%\) or \(0.10\) per annum, compounded annually. Each equal yearly instalment is \(E = 6534\) Rs. There are two instalments.
The present value (PV) of an amount received at the end of \(n\) years with a discount rate \(r\) is given by the formula:
\( \text{PV} = \frac{\text{Future Value}}{(1+r)^n} \)
In this case, the Future Value for each instalment is the instalment amount itself, \(E\).
Present Value of the First Instalment
The first instalment of Rs. 6534 is paid at the end of the first year (\(n=1\)). Its present value (PV1) is:
\( \text{PV1} = \frac{E}{(1+r)^1} = \frac{6534}{(1+0.10)^1} = \frac{6534}{1.1} \)
Calculating PV1:
\( \text{PV1} = \frac{6534}{1.1} = 5940 \)
So, the present value of the first instalment is Rs. 5940.
Present Value of the Second Instalment
The second instalment of Rs. 6534 is paid at the end of the second year (\(n=2\)). Its present value (PV2) is:
\( \text{PV2} = \frac{E}{(1+r)^2} = \frac{6534}{(1+0.10)^2} = \frac{6534}{(1.1)^2} = \frac{6534}{1.21} \)
Calculating PV2:
\( \text{PV2} = \frac{6534}{1.21} = 5400 \)
So, the present value of the second instalment is Rs. 5400.
Total Principal Loan Amount
The total principal loan amount \(P\) is the sum of the present values of all the instalments:
\( P = \text{PV1} + \text{PV2} = 5940 + 5400 = 11340 \)
Thus, the original loan amount was Rs. 11340.
Calculating the Total Amount Repaid
The loan is returned in two equal yearly instalments, each of Rs. 6534.
Total amount repaid = Sum of instalments
Total amount repaid = \(2 \times 6534 = 13068\)
The total amount repaid over two years is Rs. 13068.
Calculating the Total Interest Charged
The total interest charged is the difference between the total amount repaid and the original principal loan amount.
Total Interest = Total Amount Repaid - Principal Loan Amount
Total Interest = \(13068 - 11340 = 1728\)
The total interest charged on the loan is Rs. 1728.
Summary of Calculations
| Item | Amount (Rs.) |
|---|---|
| Equal Yearly Instalment | 6534 |
| Number of Instalments | 2 |
| Total Amount Repaid | 13068 |
| Principal Loan Amount | 11340 |
| Total Interest Charged | 1728 |
The calculation shows that the total interest charged is Rs. 1728.
Revision Table: Loan Instalment Concepts
| Concept | Description |
|---|---|
| Equal Yearly Instalment | A fixed amount paid periodically (usually annually) to repay a loan, covering both principal and interest. |
| Compound Interest | Interest calculated on the initial principal and also on the accumulated interest of previous periods. |
| Present Value (PV) | The current value of a future sum of money or stream of cash flows, given a specified rate of return (discount rate). |
| Principal Loan Amount | The original amount of money borrowed. |
| Total Amount Repaid | The sum of all instalments paid to clear the loan. |
| Total Interest Charged | The difference between the Total Amount Repaid and the Principal Loan Amount. |
Additional Information: Loan Amortization
This problem relates to loan amortization, where the loan is paid off over time with regular payments. Each payment reduces the principal amount, and the interest component of the payment decreases over time as the principal outstanding decreases. The principal component of the payment increases over time.
For a loan \(P\) repaid over \(n\) periods at an interest rate \(r\) per period with equal instalments \(E\), the formula for the instalment amount can be derived from the sum of the present values of all instalments:
\( P = \frac{E}{(1+r)^1} + \frac{E}{(1+r)^2} + \dots + \frac{E}{(1+r)^n} \)
This is a geometric series, and the formula for \(P\) can be written as:
\( P = E \left[ \frac{1 - (1+r)^{-n}}{r} \right] \)
Alternatively, the instalment \(E\) can be calculated if \(P\), \(r\), and \(n\) are known:
\( E = P \left[ \frac{r}{1 - (1+r)^{-n}} \right] \)
In our problem, we were given \(E\), \(r\), and \(n\) and had to find \(P\), which we did by summing the individual present values. This method is equivalent to using the formula for \(P\) above.
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