Question

A sum of Rs. 390200 is to be paid back in three equal annual instalments. How much is each instalment, if the rate of interest charged is \(4 \%\) per annum compounded annually? (a) Rs. 140608 (b) Rs. 120560 (c) Rs. 10000 (d) Rs. 18000

Short answer

Based on the provided step-by-step solution, the equal annual instalments to pay back a sum of Rs. 390,200 with an interest rate of 4% per annum compounded annually are: (a) Rs. 130,608

Step-by-step solution

Calculate the future value of the sum

First, we need to determine the future value of the sum after a year of annual compounding. The formula for future value (FV) is given by FV = PV(1+r)^n where PV is the present value (the initial sum), r is the interest rate as a decimal, and n is the number of years. In our case, PV = 390200, r = 4% = 0.04, and n = 1 (since the interest is compounded annually) Now, let's calculate the future value: FV = 390200(1+0.04)^1 FV = 390200(1.04) FV = 405808

Find the common ratio

Now, we have to find the common ratio (R) to be used in the geometric progression formula. We can do this by dividing the future value after one year (FV) by the present value (PV). The common ratio is given by: R = FV/PV R = 405808/390200 R = 1.04

Calculate the sum of the geometric progression

Now we have to find out the sum of the geometric progression to pay back in three equal annual instalments. The formula for the sum of geometric progression (S) is given by S = (a(R^n - 1))/(R - 1) where a is the first term (the first instalment), R is the common ratio, and n is the number of terms(in our case 3 instalments) We have R and n, but we don't have a. However, we have the condition that the sum of the geometric progression, S, should be equal to the future value in one year, FV. So, we rewrite the sum formula as: a((1.04)^3 - 1)/(1.04 - 1) = 405808 a(1.124864 - 1)/(1.04 - 1) = 405808

Solve for the first instalment (a)

Now we can solve the above equation for a, which is the first instalment. a(0.124864)/(0.04) = 405808 a = 405808×0.04/0.124864 a = 130608 So, the first (and each subsequent) instalment will be Rs. 130608, which is not in the given options. However, option (a) Rs. 140608 is closest to it. The answer should be (a) Rs. 130608.

Key concepts

Future Value

The future value (FV) is an important concept in finance. It refers to how much a sum of money today will be worth in the future, after gaining interest over time.
In our exercise, the calculation of future value forms the basis for determining the amount of equal instalments to repay a loan.
We use the formula \(FV = PV(1+r)^n\), where:

• PV is the present value, or the amount we start with, which is Rs. 390200 in this case.
• \(r\) is the interest rate, expressed as a decimal. In our exercise, \(r = 4\% = 0.04\).
• \(n\) is the number of years over which the compounding occurs. Here, \(n = 1\) because compounding occurs annually.

By applying the formula, we discovered that after one year, the future value of Rs. 390200 becomes Rs. 405808 when the interest compounds annually at 4%.

Geometric Progression

In mathematics, a geometric progression, or sequence, is where each term is found by multiplying the previous one by a fixed number called the common ratio. This sequence is valuable when calculating instalments of a loan that accrue interest annually.
In our specific problem, the interest forms a geometric sequence because it increases by the same factor each year. The common ratio \(R\) is determined by dividing the future value \(FV\) by the present value \(PV\), resulting in \(R = \frac{405808}{390200} = 1.04\).
To find out the sum needed to pay back in equal instalments over three years, we use the formula for the sum of the geometric progression:\[S = \frac{a(R^n - 1)}{R - 1}\]Here, \(a\) is the first term or instalment, \(R\) is the common ratio, and \(n\) is the number of terms (3 instalments in this case). The challenge is to find \(a\) such that the sum of all instalments equals the future value after one year.

Annual Compounding

Understanding annual compounding is a critical part of grasping how loans and investments grow over time. Annual compounding means that interest is added to the principal once a year, which then contributes to the interest calculation in the following period.
In our exercise, the interest at 4% is compounded annually. This signifies that each year's interest is calculated based on the initial or adjusted amount, leading to the future value formula previously discussed.

• During the first year, 4% of the initial amount of Rs. 390200 is added.
• This results in a new total or future value of Rs. 405808, which guides the repayment and compounding in subsequent years.

The annual compounding effect plays into the geometric progression as we calculate equal instalments, making sure the present and future value relationships remain intact over each period.