Question

suridhi borrowed Rs. 10815 , which is to be paid back in 3 half yearly instalments. If the interest is compounded half yearly at \(\frac{40}{3} \%\) per annum, how much is each instalment? (a) 2048 (b) 3150 (c) 4096 (d) 5052

Short answer

Answer: (b) 3150

Step-by-step solution

Calculate the Effective Interest Rate

To calculate the effective interest rate per half-year, use the given annual interest rate and divide it by 2. This is because interest is compounded half-yearly, so the effective interest rate will also be for a half-year period. Effective Interest Rate per half-year = \(\frac{Annual Interest Rate}{2}\) Here, the annual interest rate is \(\frac{40}{3} \%\). So the effective interest rate per half-year is: Effective Interest Rate per half-year = \(\frac{\frac{40}{3}}{2}\) Effective Interest Rate per half-year = \(\frac{20}{3}\%\)

Calculate the Future Value

The future value of each instalment can be calculated using the compound interest formula. Let's assume the present value of each instalment is 'P', and the number of half-yearly periods is 'n' (which is 3 in this case). The future value 'FV' can be calculated as: \(FV = P \times (1 + \frac{Interest \thinspace rate}{100})^n\) Using the effective interest rate calculated in Step 1, we have: \(FV = P\times(1 + \frac{\frac{20}{3}}{100})^3\) \(FV = P\times(1 + \frac{1}{15})^3\)

Determine the Total Future Value

Since there are 3 instalments, the total future value should be equal to the total borrowed amount, i.e., 10815. Total Future Value = 3 * FV 10815 = 3 * \(P \times (1 + \frac{1}{15})^3\) Now, solve for P.

Solve for the Present Value of Each Instalment

We'll now solve for 'P' using the equation obtained in Step 3. 10815 = 3 * \(P \times (1 + \frac{1}{15})^3\) \(P = \frac{10815}{3\times(1 + \frac{1}{15})^3}\) Calculating the value of P, we get approximately: P ≈ 3150 So, each instalment amounts to Rs. 3150. The correct answer is (b) 3150.

Key concepts

Effective Interest Rate

Understanding the effective interest rate is essential when handling loans with compound interest, especially when compounding is not annual. In this case, the interest is compounded half-yearly, meaning it compounds twice each year. The effective interest rate is the rate applied to each compounding period. To find this, take the annual interest rate and divide it by the number of compounding periods per year. For a rate given annually, here's the formula:

• Effective Interest Rate per period = \( \frac{Annual\ Interest\ Rate}{Number\ of\ periods\ per\ year} \)

For this exercise, the annual interest rate is \( \frac{40}{3} \% \). With two periods per year, the calculation becomes:

• Effective Interest Rate per half-year = \( \frac{\frac{40}{3}}{2} = \frac{20}{3}\% \)

This effective rate indicates how much the loan grows in each six-month period. Understanding this rate is crucial for calculating other aspects like future value or instalments.

Future Value

The concept of future value is central to understanding how money grows over time under compound interest. Future value (FV) refers to how much a current sum of money will be worth at a future date considering the compounding interest. The formula to calculate future value when interest is compounded is:

• \( FV = P \times (1 + \frac{Interest\ Rate}{100})^n \)

Where:

• \( P \) is the present value or initial amount.
• \( n \) is the number of compounding periods.
• Interest rate is the effective rate for each period.

In the exercise provided, each instalment grows according to this equation. For three half-yearly periods, the calculation uses the effective interest rate of \( \frac{20}{3}\% \). By applying this formula, you can determine how each instalment contributes to reaching the total future value of Rs. 10815.

Instalment Calculation

Calculating the instalments for a loan involves determining how much needs to be paid at each period to repay the total debt by the last period. Here, we're splitting the total amount owed, Rs. 10815, into three equal payments.To calculate each instalment, we use the future value formula mentioned, where the total amount borrowed equals the total future value of all instalments.Given by:

• Total Borrowed = Total Future Value = 3 * \( P \times (1 + \frac{1}{15})^3 \)

By rearranging to solve for \( P \), the present value of each instalment:

• \( P = \frac{10815}{3 \times (1 + \frac{1}{15})^3} \)

After performing the calculations, \( P \approx 3150 \). This means each instalment is Rs. 3150. This step ensures you distribute the payment evenly considering the interest accrued.

Half-yearly Compounding

The technique of half-yearly compounding affects how interest accumulates. Compounding happens more frequently than with annual compounding, which can lead to more total interest accrued over the same principal and annual rate.When a loan or investment compounds semi-annually, the annual interest rate is divided by two to find the rate per period. Here's why it's impactful:

• Interest is added to the principal more often, leading to interest on interest.
• As compounding frequency increases, the total amount of interest paid or earned also increases.

In our exercise, the half-yearly compounding means the interest effective rate was \( \frac{20}{3}\% \) per period, compounding twice a year. This impacts how the instalments accumulate to repay the full borrowed amount of Rs. 10815, demonstrating a realistic scenario many borrowers face.