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# Cumulative Interest Definition, Formulas and Uses

## What Is Cumulative Interest?

Cumulative interest is the sum of all interest payments made on a loan over a certain period. On an amortizing loan, cumulative interest will increase at a decreasing rate, as each subsequent periodic payment on the loan is a higher percentage of the loan’s principal and a lower percentage of its interest.

## Using Cumulative Interest

Cumulative interest is sometimes used to determine which loan in a series is most economical. However, cumulative interest alone does not account for other important factors, such as initial loan costs (if a lender pays these costs out of pocket as opposed to rolling them over into the loan's balance) and the time value of money.

The time value of money (TVM), also known as the present discount rate, is a core concept in finance. It centers on the notion that money available at present is worth more than the same amount in the future, due to its potential earning capacity. Provided money can earn interest, any amount is worth more the sooner it is received.

The general formula for TVM is: FV = PV x (1 + (i / n)) ^ (n x t)

FV = Future value of money

PV = Present value of money

i = interest rate

n = number of compounding periods per year

t = number of years

## Cumulative Interest vs. Compound Interest

While cumulative interest is additive, compound interest can be thought of as “interest on interest.” The formula is as follows:

Compound Interest = Total amount of Principal and Interest in future (or Future Value) less Principal amount at present (or Present Value)

= [P (1 + i)n] – P

= P [(1 + i)n – 1]

(Where P = Principal, i = nominal annual interest rate in percentage terms, and n = number of compounding periods.)

For example, what would the amount of interest be on a five-year loan of \$10,000 at an interest rate of 5% that compounds annually? In this case, it would be: \$10,000 [(1 + 0.05)5] – 1 = \$10,000 [1.27628 – 1] = \$2,762.82.

Selecting compound interest will make a sum grow at a faster rate than simple interest, which is calculated only on the principal amount. This happens because, when interest is compounded, the money earned through interest is added to the principal periodically, so that more interest is earned in the next period. This process repeats itself, leading to larger gains due to interest.

## Cumulative Interest and Measures of Bond Performance

While cumulative interest is one method of calculating how well a bond investment is performing, the following are more comprehensive yield methods: nominal yieldcurrent yieldeffective annual yield, and yield to maturity.

## Cumulative Interest Example

Cumulative interest refers to all of the interest earned or paid over the life of a security or loan, added together. If you borrowed \$10,000 at an interest rate of 3% annually, you’d pay \$300 in interest in the first year. If you paid off \$1200 in the first year and only owed \$8,800 in year two, your interest for year two would be \$264. Your cumulative interest for years one and two would be \$564.

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