A loan of Rs. 1,50,000 is availed with compound interest rate of 10% per annum for two years compounded annually. It is to be paid in equal yearly installments, and the installment is to be paid at the end of each year. The value of the equal yearly installment is : (Rounded off to two places of decimal)
Calculating Equal Yearly Loan Installments with Compound Interest
This problem involves calculating the value of equal yearly installments required to pay back a loan taken at a compound interest rate over a specific period. The loan is for Rs. 1,50,000 at a 10% annual compound interest rate for two years, with installments paid at the end of each year.
To find the equal yearly installment amount, we can use the formula for the present value of an annuity, which relates the loan amount (Present Value) to the series of equal payments (installments) over time, considering the interest rate.
Understanding the Loan Installment Calculation
The loan amount is essentially the present value of all future installments. The formula for the Present Value (PV) of an ordinary annuity (payments at the end of each period) is:
\(\text{PV} = \text{Installment} \times \frac{1 - (1 + r)^{-n}}{r}\)
Where:
- \(\text{PV}\) is the Present Value of the loan (initial loan amount).
- \(\text{Installment}\) is the value of the equal yearly payment we need to find.
- \(r\) is the annual interest rate (as a decimal).
- \(n\) is the total number of years (or periods).
We are given:
- Loan Amount (\(\text{PV}\)) = Rs. 1,50,000
- Annual Interest Rate (\(r\)) = 10% = 0.10
- Number of Years (\(n\)) = 2
We need to find the Installment amount. We can rearrange the formula to solve for the Installment:
\(\text{Installment} = \frac{\text{PV} \times r}{1 - (1 + r)^{-n}}\)
Step-by-Step Installment Calculation
Let's plug in the given values into the formula:
\(\text{Installment} = \frac{1,50,000 \times 0.10}{1 - (1 + 0.10)^{-2}}\)
\(\text{Installment} = \frac{15,000}{1 - (1.10)^{-2}}\)
\(\text{Installment} = \frac{15,000}{1 - \frac{1}{(1.10)^2}}\)
Calculate \((1.10)^2\):
\((1.10)^2 = 1.10 \times 1.10 = 1.21\)
Now substitute this back into the formula:
\(\text{Installment} = \frac{15,000}{1 - \frac{1}{1.21}}\)
Calculate the term in the denominator:
\(1 - \frac{1}{1.21} = 1 - 0.826446...\)
\(1 - 0.826446... \approx 0.173554...\)
Alternatively, work with fractions to maintain precision:
\(1 - \frac{1}{1.21} = \frac{1.21}{1.21} - \frac{1}{1.21} = \frac{1.21 - 1}{1.21} = \frac{0.21}{1.21}\)
Now substitute this back into the installment formula:
\(\text{Installment} = \frac{15,000}{\frac{0.21}{1.21}}\)
\(\text{Installment} = 15,000 \times \frac{1.21}{0.21}\)
\(\text{Installment} = \frac{15,000 \times 1.21}{0.21}\)
\(\text{Installment} = \frac{18,150}{0.21}\)
Perform the division:
\(\text{Installment} \approx 86,428.5714...\)
The question asks to round the value to two places of decimal.
\(\text{Installment} \approx 86,428.57\)
The value of the equal yearly installment is approximately Rs. 86,428.57.
Verification (Optional - Amortization Schedule)
We can verify this by creating a simple amortization schedule:
| Year | Opening Balance | Interest Due (10%) | Installment | Principal Paid | Closing Balance |
|---|---|---|---|---|---|
| 1 | 1,50,000.00 | \(1,50,000 \times 10\% = 15,000.00\) | 86,428.57 | \(86,428.57 - 15,000.00 = 71,428.57\) | \(1,50,000.00 - 71,428.57 = 78,571.43\) |
| 2 | 78,571.43 | \(78,571.43 \times 10\% = 7,857.14\) (approx) | 86,428.57 | \(86,428.57 - 7,857.14 = 78,571.43\) (approx) | \(78,571.43 - 78,571.43 = 0.00\) (approx) |
Due to rounding in the installment and intermediate calculations, the final closing balance might not be exactly zero, but it will be very close, confirming the installment amount is correct.
Based on the calculation, the value of the equal yearly installment is Rs. 86,428.57.
Revision Table: Key Concepts
| Term | Definition | Relevance to Problem |
|---|---|---|
| Loan Amortization | The process of paying off a debt over time through regular payments. Each payment covers interest and a portion of the principal. | The problem requires finding the fixed payment amount for loan amortization. |
| Compound Interest | Interest calculated on the initial principal and also on the accumulated interest from previous periods. | The interest rate is compounded annually, affecting how the balance grows each year. |
| Equal Yearly Installment | A fixed amount paid at regular intervals (here, annually) to cover interest and principal repayment. | This is the value we need to calculate using the loan amount, interest rate, and loan term. |
| Present Value of Annuity | The current worth of a series of future equal payments, discounted at a specific interest rate. | The initial loan amount represents the present value of the series of installment payments. |
Additional Information on Loan Repayments
When you take out a loan that is repaid with equal installments over time, it's commonly referred to as an amortizing loan. Each payment made towards an amortizing loan consists of two parts:
- Interest Payment: This part of the payment covers the interest accrued on the outstanding loan balance for that period. In the initial periods, a larger portion of the installment goes towards interest because the outstanding balance is higher.
- Principal Payment: This part reduces the actual amount of the loan owed. As the loan balance decreases over time, a larger portion of the equal installment goes towards the principal.
The schedule showing how each payment is broken down into interest and principal, and the resulting outstanding balance, is called an amortization schedule. While calculating the equal installment often uses a formula based on the present value of an annuity, an amortization schedule helps visualize the loan repayment process step-by-step and confirm the calculations.
Understanding how compound interest affects the outstanding balance and how equal installments are applied is crucial for managing loans and understanding financial products like mortgages and car loans.
Important Questions from Installments
A sum of Rs. P was borrowed and paid back in two equal yearly instalments, each of Rs. 35,280. If the rate of interest was 5% per annum and interest is compounding annually, then the value of P is ________.
A sum of Rs. 16400 is borrowed to be paid back in 2 years by equal payments allowing 5% compound interest. Find the annual payment.
A sum of Rs. 1100 was taken as a loan. This is to be paid in two equal installments. If the rate of interest is 20% per annum, compounded annually, find the amount payable in each installment.
The formula for finding the annual installment, when A is the amount taken on loan, where r% is the rate of interest, n is the number of installments, is:
A loan of Rs 15000 is to be repaid in 4 equal annual installments. If the compound interest rate is 10% per annum, what is the approximate amount of each installment?
