How to Calculate the Interest Rate of Continuous Compounding Given Future Value

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What is Continuous Compounding?

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.

What is Continuous Compounding?
- Continuous Compounding Formula
- Example
- Uses
- Continuous Compounding Calculator
- Continuous Compounding Formula in Excel (with excel template)
- Recommended Articles:

Continuous Compounding Formula

Continuous Compounding Formula = P * e^rf

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

The continuous compounding formula Compounding 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 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.

Example

Let us analyze some of the instances:

You can download this Continuous Compounding Excel Template here – Continuous Compounding Excel Template

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 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 A 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 continuously, 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 continuously compounding of interest on a monthly Monthly 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, this will effectively reinvest gains perpetually.
The effect allows interest amount to be reinvested, thereby allowing an investor to earn exponentially.
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

Continuous Compounding Formula in Excel (with excel template)

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 Compounding 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

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