Question

Indicom cell-phone is available for Rs. 2500 cash or Rs. 520 cash down payments followed by 4 equal monthly instalments. If the rate of interest charged is \(25 \%\) per annum, calculate the monthly instalment: (a) 520 (b) 480 (c) 550 (d) none of these

Short answer

Answer: (a) 520

Step-by-step solution

Identifying the given information

We are given the following information: - Cash price of the cell-phone: Rs. 2500 - Down payment: Rs. 520 - Number of monthly installments: 4 - Annual interest rate: \(25 \%\)

Calculate the present value of the annuity

First, we need to form the present value annuity (PVA) equation. The PVA represents the initial loan amount: PVA = Down Payment - Cash Price PVA = 520 - 2500 PVA = -1980 Since the value is negative, we need to change it to positive to indicate the loan amount: PVA = 1980

Convert the annual interest rate to a monthly interest rate

The annual interest rate is given as \(25 \%\). We need to convert this to a monthly interest rate: Monthly Interest Rate = (1 + Annual Interest Rate)^(1/12) - 1 Monthly Interest Rate = (1 + 0.25)^(1/12) - 1 Monthly Interest Rate ≈ 0.01898

Calculate the monthly installment using the present value annuity formula

The present value annuity formula is: PVA = (PMT * (1 - (1 + i)^-n)) / i Where: - PVA = Present Value Annuity - PMT = Monthly Installment - i = Monthly Interest Rate - n = Number of Installments Plugging in the values: 1980 = PMT*(1-(1+0.01898)^(-4))/0.01898 Now, we'll solve for PMT: PMT ≈ 519.97

Choose the correct answer

Since the calculated installment is very close to 520, we can conclude that the correct answer is: (a) 520

Key concepts

present value annuity

In simple terms, the present value annuity (PVA) is the total amount that a series of future payments is worth right now. Think of it as the current value of a loan. When you're given an option to make equal payments over a period of time, you'd need to know what that series of payments is worth in today's money to ensure you're making a financially sound decision. In our problem with the cell-phone, we took the cash price and subtracted the upfront payment (often called a down payment) to arrive at the present value annuity.

• PVA helps in understanding if an installment plan is truly beneficial or not.
• It's used to calculate loan amounts or investments.

Understanding how to compute the present value annuity allows you to determine fair pricing for installment plans, making your financial decisions better informed.

monthly installment

Monthly installments are the backbone of financing offers. It’s the fixed amount paid each month that reduces the loan over time. Calculating the right monthly installment depends on several factors such as the interest rate and the loan duration. In the exercise, once we found out the present value annuity, the next step was to find out what you would need to pay monthly to eventually pay off the full loan plus the interest added. The mathematical formula combines both the annuity's present value and the monthly interest rate to compute this amount.

• Gives clarity on future monthly financial commitments.
• It’s an essential part of personal budgeting.

By calculating the monthly installment accurately, one can avoid taking on an unaffordable financial commitment.

interest rate conversion

Interest rate conversion is an important part when dealing with financial agreements spread over a period. Since interest rates are usually given on an annual basis, for monthly calculations, we need to convert these to monthly rates. In the exercise, we learned that converting a 25% annual rate to a monthly rate involved using a formula for precision: \[ \text{Monthly Interest Rate} = (1 + \text{Annual Interest Rate})^{\frac{1}{12}} - 1 \]By doing so, we ensure that the monthly figures we are calculating align with the actual cost of the financing arrangement.

• Facilitates understanding how interest affects monthly payments.
• Ensures precision in financial calculations.

This step is crucial as miscalculated interest rates can lead to significantly different payment outcomes.

financial mathematics

Financial mathematics encompasses the techniques used to solve various time value of money problems. It's the fusion of mathematics with financial knowledge like interest rates, annuities, and periodic payments. The principles of financial mathematics were applied in the exercise to help us break down the problem into manageable steps.

• Helps in evaluating investment options and financing plans.
• Provides a framework for making sound financial decisions.

Mastering financial mathematics ensures you can approach decisions involving loans, annuities, or investments with confidence, knowing you have an analytical method for evaluating options.