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Continuous Compounding Formula
The formula for continuous compounding is:
Example of Continuous Compounding Formula
Let us analyse 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 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 compute 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 it can be observed from the above continuous compounding 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?
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- 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%
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Explanation of Continuous Compounding Formula
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 continuous compounding example.
Use of Continuous Compounding Formula
The importance of continuous compounding formula is:
- Rather than continuous compounding of interest on a monthly, quarterly or annual basis, continuous compounding excel will effectively reinvest gains perpetually.
- The effect of allows the continuous compounding of 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 which will earn money but the continuous compounding of interest amount will also keep on multiplying.
Continuous Compounding Calculator
You can use the following Continuous Compounding 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)
Let us now do the same example of Continuous Compounding Excel.
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.
Continuous Compounding Example – 1
You can easily calculate the ratio in the template provided.
Let us compute the effects of the same on regular compounding:
As it 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.
Continuous Compounding Example – 2
Continuous Compounding Example – 3
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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 downloadable excel template.
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