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

The continuous compounding formula 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:

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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 5 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:

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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 it 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 shark 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

Rather than continuous compounding of interest on a monthly, quarterly or annual basis, this will effectively reinvest gains perpetually.
The effect of allows interest amount to be reinvested thereby allowing an investor to earn at an exponential rate.
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

P
r
t
Continuous Compounding Formula =

Continuous Compounding Formula =	P x e^{(r x t)} =
	0 * e^{(0 * 0) =}	0