Continuous Compounding Formula

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

The continuous compounding formula determines the interest earned, which is repeatedly compounded for an infinite period.

where,

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

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

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)  = 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 * ^ 2   
• = 1,000 * (1.04) ^ 2   
• = 1,000 * 1.0816   =   $1,081.60

Quarterly Compounding:

• FV = 1,000 * ^ 4
• = 1,000 * (1.02) ^ 4
• = 1,000 * 1.08243
• = $1,082.43

Monthly Compounding:

• FV = 1,000 * ^ 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 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 shark charges 80% interest, compounded continuously, what will be the effective annual interest rate?

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

1. Rather than continuously compounding of interest on a monthly, quarterly, or annual, this will effectively reinvest gains perpetually.
2. The effect allows interest amount to be reinvested, thereby allowing an investor to earn exponentially.
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.

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

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 example, the interest earned from this compounding is $83.28, which is only $0.28 more than monthly compounding.

Example - 2

Example - 3