How to Calculate the Sum Borrowed Using Compound Interest

When solving financial problems involving loans and payments, understanding the principles of compound interest and present value can be crucial. This article will walk you through the process of determining the sum borrowed when a loan is repaid with two annual installments, taking into account the rate of compound interest. This is a common scenario in many financial situations and can be applied to various lending scenarios, such as personal loans, mortgages, and business loans.

Understanding Compound Interest and Present Value

Compound interest is the interest calculated on the initial principal and also on the accumulated interest of previous periods of a deposit or loan. In this context, the initial amount borrowed grows over time with interest, and the amount to be repaid includes both the principal and the accrued interest.

The present value of a future payment is the current worth of a future sum of money or stream of cash flows given a specified rate of return. Essentially, this concept helps us understand how much a future payment is worth in today's dollars, accounting for the time value of money.

Problem Statement

Suppose a sum of money is borrowed and paid back in two annual installments of 5400 and 11664 respectively. The rate of interest compounded annually is 8%. The goal is to find the original sum borrowed.

Solution using Present Value

Using the formula for present value,
[ PV frac{P}{1 r^n} ]

where
- ( PV ) is the present value of the future payment - ( P ) is the future payment amount - ( r ) is the interest rate - ( n ) is the number of years

In this case, we have two payments:

The first payment of 5400 is made after 1 year. The second payment of 11664 is made after 2 years.

Given that the interest rate ( r 8% 0.08 ), we can calculate the present value of each payment as follows:

Present Value of the First Payment

[ PV_1 frac{5400}{1 0.08^1} frac{5400}{1.08} approx 5000 ]

Present Value of the Second Payment

[ PV_2 frac{11664}{1 0.08^2} frac{11664}{1.08^2} frac{11664}{1.1664} approx 10000 ]

Adding the present values of both payments gives us the total sum borrowed:

[ text{Total Present Value} PV_1 PV_2 approx 5000 10000 15000 ]

Therefore, the sum borrowed was approximately 15000.

Alternative Calculation Methods

There are several alternative methods to arrive at the same result:

1. Using Principal and Compound Interest Formula

Given that the amount borrowed was ( P ),

[ P(1 frac{8}{100})^2 5400 11664 ]

Simplifying,

[ P times 1.08 times 1.08 17064 ]

[ P frac{17064}{1.08 times 1.08} 14629.63 approx 14630 ]

Hence, the amount borrowed is approximately 14630.

2. Summing Present Values Directly

The sum borrowed can also be calculated by summing the present values directly:

[ text{Sum} frac{5400}{1 times frac{8}{100}} times frac{1}{100} frac{11664}{1 times frac{8}{100}^2} times frac{1}{100^2} ]

This simplifies to:

[ text{Sum} 5400 times frac{1}{0.08} times frac{1}{100} 11664 times frac{1}{0.08^2} times frac{1}{100^2} ]

Which further simplifies to:

[ text{Sum} 5400 times 100 times frac{1}{108} 11664 times frac{100 times 100}{108 times 108} 5400 times frac{100}{108} 11664 times frac{10000}{11664} 5000 10000 15000 ]

So, the sum borrowed is 15000.

3. Step-by-Step Calculation

Another method involves determining the principal balance step-by-step:

At the end of the second year, the amount paid is 11664. Since this includes interest, the capital outstanding at the beginning of the second year is 11664 / 1.08 10800. Subtract the first year's payment: 10800 - 5400 16200. The sum borrowed is then 16200 / 1.08 15000.

Hence, the sum borrowed is 15000.

Conclusion

The sum borrowed, as calculated using various methods, is 15000. Understanding compound interest and present value is crucial for managing loans and investments effectively. This article has provided multiple steps and methods to calculate the borrowed amount, which can be applied to similar financial scenarios.

Keywords:

compound interest, present value, loan calculation