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. 5,808, then the total interest charged in this scheme is:
Rs. 1,536
Understanding Loan Instalment Calculation
This problem involves calculating the total interest paid on a loan that is repaid in equal annual instalments. We are given the instalment amount, the interest rate, and the number of instalments. To find the total interest, we first need to determine the original principal amount of the loan. The principal amount of the loan is the sum of the present values of all future instalments.
Calculating the Present Value of Each Instalment
The present value (PV) of a future payment is the amount of money that would need to be invested today at a given interest rate to equal the future payment. The formula for the present value of a single amount is:
$$PV = \frac{FV}{(1 + r)^n}$$
Where:
- $FV$ is the Future Value (the instalment amount)
- $r$ is the annual interest rate (as a decimal)
- $n$ is the number of years from now until the payment is made
In this case, the instalment amount ($FV$) is Rs. 5,808, and the annual interest rate ($r$) is 10% or 0.10.
First Instalment
The first instalment is paid after 1 year. So, its present value ($PV_1$) is:
$$PV_1 = \frac{5808}{(1 + 0.10)^1} = \frac{5808}{1.1} = 5280$$
The present value of the first instalment is Rs. 5,280.
Second Instalment
The second instalment is paid after 2 years. So, its present value ($PV_2$) is:
$$PV_2 = \frac{5808}{(1 + 0.10)^2} = \frac{5808}{(1.1)^2} = \frac{5808}{1.21} = 4800$$
The present value of the second instalment is Rs. 4,800.
Determining the Original Principal Loan Amount
The total principal amount of the loan is the sum of the present values of all the instalments:
$$\text{Principal Loan Amount} = PV_1 + PV_2$$
$$\text{Principal Loan Amount} = 5280 + 4800 = 10080$$
The original principal amount of the loan was Rs. 10,080.
Calculating the Total Amount Paid
The loan is paid back in two equal yearly instalments of Rs. 5,808 each. The total amount paid over the two years is:
$$\text{Total Amount Paid} = \text{Number of Instalments} \times \text{Instalment Amount}$$
$$\text{Total Amount Paid} = 2 \times 5808 = 11616$$
The total amount paid is Rs. 11,616.
Calculating the Total Interest Charged
The total interest charged on the loan is the difference between the total amount paid back and the original principal loan amount:
$$\text{Total Interest} = \text{Total Amount Paid} - \text{Principal Loan Amount}$$
$$\text{Total Interest} = 11616 - 10080 = 1536$$
The total interest charged in this scheme is Rs. 1,536.
Summary of Calculations
| Item | Calculation / Value |
|---|---|
| Instalment Amount | Rs. 5,808 |
| Interest Rate (r) | 10% or 0.10 |
| Number of Instalments | 2 |
| PV of 1st Instalment | Rs. 5,280 ($5808 / (1.1)^1$) |
| PV of 2nd Instalment | Rs. 4,800 ($5808 / (1.1)^2$) |
| Total Principal Loan Amount | Rs. 10,080 ($5280 + 4800$) |
| Total Amount Paid | Rs. 11,616 ($2 \times 5808$) |
| Total Interest Charged | Rs. 1,536 ($11616 - 10080$) |
The total interest charged is Rs. 1,536, which corresponds to the first option.
Revision Table: Loan Instalment Concepts
| Concept | Description | Importance in Loan Calculations |
|---|---|---|
| Instalment (EMI) | A fixed amount paid by the borrower to the lender on a specified date each month or year. | Includes both principal and interest components. |
| Principal Amount | The original amount of money borrowed. | Basis for calculating interest and total repayment. |
| Interest Rate | The cost of borrowing money, expressed as a percentage of the principal. | Determines the interest portion of each instalment and the total interest paid. |
| Compounding | Interest earned on both the initial principal and the accumulated interest from previous periods. | Affects the total amount repaid over the loan term. |
| Present Value | The current worth of a future sum of money or stream of cash flows given a specified rate of return. | Used to determine the original loan principal from known future instalment payments. |
Additional Information: Loan Repayment Mechanics
When a loan is repaid through equal instalments, each instalment payment consists of two parts: a portion that covers the interest due for the period and a portion that reduces the outstanding principal balance. In the early periods of the loan, a larger portion of the instalment typically goes towards interest, while a smaller portion reduces the principal. As the loan matures, a larger portion of the instalment goes towards reducing the principal.
Calculating the present value of future cash flows (like instalments) is a fundamental concept in finance used to determine the fair value of a loan or investment today, given future payments and a required rate of return (the interest rate). This process is essentially discounting future amounts back to the present using the interest rate as the discount rate.
Important Questions from Installments
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