Continuous Compounding Formula

What is Continuous Compounding?

Continuous Compounding calculates the Limit at which the Compounded interest can reach by constantly compounding for an indefinite period of time thereby increasing the Interest Component and ultimately the portfolio value of the Total Investments

Continuous Compounding Formula

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For eg:
Source: Continuous Compounding Formula (wallstreetmojo.com)

The continuous compounding formulaCompounding FormulaCompounding is a method of investing in which the income generated by an investment is reinvested, and the new principal amount is increased by the amount of income reinvested. Depending on the time period of deposit, interest is added to the principal amount.read more determines the interest earned, which is repeatedly compounded for an infinite time period.

where,

• P = Principal amount (Present Value)
• t = Time
• r = Interest Rate

The calculation assumes constant compounding over an infinite number of time periods. Since the time period is infinite, the exponent helps in a multiplication of the current investment. This is multiplied by the current rate and time. Despite a large number of investments, a difference in total interest earned through continuous compounding excel is less as compared to traditional compounding, which will be looked into through examples.

Example

Let us analyze some of the instances:

If an initial investment of $1,000 is invested at 8% interest per year with continuous compounding, how much would be in the account after five years?

• P = $1,000, r= 8%, n= 5 years
• FV = P * e ^rt  = 1,000 * e ^{(0.08) (5)} = 1,000 * e ^(0.40) [Exponent of 0.4 is 1.491] = 1,000 * 1.491                                    
• = $1,491.8

Let us calculate the effects of the same on regular compounding:

Annual Compounding:

• FV = 1,000 * (1 + 0.08) ^ 1 = $1,080

Semi-Annual Compounding:

• FV = 1,000 * [(1 + 0.08/2)] ^ 2   
• = 1,000 * (1.04) ^ 2   
• = 1,000 * 1.0816   =   $1,081.60

Quarterly Compounding:

• FV = 1,000 * [(1 + 0.08/4)] ^ 4
• = 1,000 * (1.02) ^ 4
• = 1,000 * 1.08243
• = $1,082.43

Monthly Compounding:

• FV = 1,000 * [(1 + 0.08/12)] ^ 12
• = 1,000 * (1.006) ^ 4
• = 1,000 * 1.083
• = $1,083

Continuous Compounding:

• FV = 1,000 * e ^0.08
• = 1,000 * 1.08328
• = $1,083.29

As can be observed from the above example, the interest earned from continuous compounding is $83.28, which is only $0.28 more than monthly compounding.

Another example can say a Savings Account pays 6% annual interest, compounded continuously. How much must be invested now to have $100,000 in the account 30 years from now?

• FV = PV * e^rt
• PV = FV * e – ^rt
• PV = 100,000 * e ^{– (0.06) (30)}
• PV = 100,000 * e ^{– (1.80)}
• PV = 100,000 * 0.1652988
• PV = $16,529.89

Thus, if an amount of $16,530 (rounded off) is invested today, it will yield $100,000 after 30 years at the given rate.

Another instance can be if a loan sharkLoan SharkA loan shark offers easy credit to borrowers at unreasonably high interest rates. Such lenders usually trap destitute borrowers who are desperate for immediate cash. They make profits out of exorbitant rates and unethical vehicles of debt recovery. read more charges 80% interest, compounded on a continuous basis, what will be the effective annual interest rate?

• Interest rate = e ^0.80 – 1
• = 2.2255 – 1 = 1.22.55 = 122.55%

Uses

1. Rather than continuous compounding of interest on a monthlyCompounding Of Interest On A MonthlyMonthly compound interest refers to the compounding of interest every month, which implies that the compounding interest is charged both on the principal and the accumulated interest.read more, quarterly, or annual basis, this will effectively reinvest gains perpetually.
2. The effect allows interest amount to be reinvested, thereby allowing an investor to earn at an exponential rate.
3. This determines that it is not only the principal amount that will earn money, but the continuous compounding of interest amount will also keep on multiplying.

Continuous Compounding Calculator

You can use the following Calculator

Continuous Compounding Formula in Excel (with excel template)

This is very simple. You need to provide the two inputs of Principle Amount, Time, and Interest rate.

You can easily calculate the ratio in the template provided.

Example – 1

You can easily calculate the ratio in the template provided.

Let us calculate the effects of the same on regular compounding:

As can be observed from the continuous compounding exampleCompounding ExampleCompounding is a method of investing in which the income generated by an investment is reinvested, and the new principal amount is increased by the amount of income reinvested. Depending on the time period of deposit, interest is added to the principal amount.read more, the interest earned from this compounding is $83.28, which is only $0.28 more than monthly compounding.

Example – 2

Example – 3

Recommended Articles:

This has been a guide to Continuous Compounding formula, its uses along with practical examples. Here we also provide you with Continuous Compounding Calculator with a downloadable excel template. You can refer the following articles as well –

• Compounding Quarterly Formula
• Matrix Multiplication in Excel
• Compound Journal Entry Definition
• Calculate Simple Interest Formula