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# Continuous and Discrete Compounding

When you invest a dollar today, you expect to receive more than a dollar after a period of time. The additional amount earned on your investment is the time value of money and is calculated based on the interest rate.

There are primarily two ways of calculating interest:

1. Discrete (Includes simple and compound interest)
2. Continuous compounding

Let us look at each of the above methods in detail:

#### Discrete compounding

Simple Interest: Simple interest is interest paid only on the “principal” or the amount originally borrowed, and not on the interest owed on the loan.

For example, the simple interest due at the end of three years on a loan of \$100 at a 5% annual interest rate is \$15 (5% of \$100, or \$5, for each of the three years). No interest is calculated in the second year on the \$5 interest that was due after the first year, and no interest is calculated in the third year on the interest that was due after two years.

The future value (FV) using simple interest is calculated using the following formula:

FV = P (1+rt)

Where:
P is the principal
r is the interest rate
t is the time period

Compound interest: Compound interest is interest calculated, not only on the principal, or the amount originally borrowed, but also on the interest that has accrued, or built up, at the time of the calculation.

Here’s how the amount owed on a three-year loan at an interest rate of 5% would differ, depending on whether simple interest or compound interest was charged:

 Simple Interest Compound Interest Amount of Loan \$100 \$100 Amount Owed After One Year 105 105 Amount Owed After Two Years 110 110.25 (\$105 plus 5% of \$105) Amount Owed After Three Years 115 115.7625 \$110.25 plus 5% of \$110.25)

The future value (FV) using compound interest is calculated using the following formula:

FV = P (1+r)^n

Where:
P is the principal
r is the interest rate
t is the number of periods

Instead of yearly interest calculations, the compounding can also be at more frequent intervals, for example, semi-annually, quarterly, or monthly. In such a situation, the future value is calculated as follows:

FV = P (1+r/m)^mt

Where:
P is the principal
r is the interest rate
t is the term of the loan or investment in years
m is the number of compounding periods in a year.

Let us take the same example above but this time let’s assume that the interest is compounded quarterly instead of annually. The future value of the principal will be:

FV = \$100 (1+5%/4)^(4*3) = 116.0755

#### Continuous compounding

In case of continuous compounding, the interest is compounded continuously. This means that the time periods for compounding are so small that they literally equal zero.

The future value of the principal with continuous compounding is given as follows:

FV = P*e^(rt)

In our example, the future value using continuous compounding will be:

FV = \$100*exp(5%*3) = 116.1834

In practice, no one compounds interest continuously but it is used extensively for pricing options, forwards and other derivatives.

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