Understanding Continuously Compounded Interest & Key Financial Ratios

Continuously compounded interestTimes Interest EarnedThe Times Interest Earned (TIE) ratio measures a company's ability to meet its debt obligations on a periodic basis. This ratio can be calculated by dividing a company's EBIT by its periodic interest expense. The ratio shows the number of times that a company can make its periodic interest payments is interest that is computed on the initial principalPrincipal PaymentA principal payment is a payment toward the original amount of a loan that is owed. In other words, a principal payment is a payment made on a loan that reduces the remaining loan amount due, rather than applying to the payment of interest charged on the loan., as well as all interest other interest earned. The idea is that the principal will receive interest at all points in time, rather than in a discrete way at certain points in time.

The continuous payment of interest leads to exponential growth and is many times used as an argument for wealth creation. Albert Einstein is credited with the phrase “compound interest is the most powerful force in the universe.” While it is undetermined if he actually said it, it says a lot about the importance of the concept.

To understand continuously compounded interest, we will quickly review simple interest and compound interest.

Consider the following example: An investor invests $1,000 in a 5-year term deposit that pays a continuously compounded interest of 6%.

What is Simple Interest?

Simple interestSimple InterestSimple interest formula, definition and example. Simple interest is a calculation of interest that doesn't take into account the effect of compounding.  In many cases, interest compounds with each designated period of a loan, but in the case of simple interest, it does not.  The calculation of simple interest is equal to the principal amount multiplied by the interest rate, multiplied by the number of periods. is only computed on the initial principal and not on any interest earned by the initial principal amount. Consider the following example: An investor invests $1,000 in a 5-year term deposit paying a simple interest of 6%.

Total Interest Earned = Principal * Interest * Time

Total Interest Earned = $1,000 * .06 * 5 = $300

Average Annual Interest = Total Interest Earned / Time

Average Annual Interest = $300 / 5 = $60

What is Compound Interest?

Compound interestCompound InterestCompound interest refers to interest payments that are made on the sum of the original principal and the previously paid interest. An easier way to think of compound interest is that is it "interest on interest," where the amount of the interest payment is based on changes in each period, rather than being fixed at the original principal amount. is computed on the initial principal as well as on the interest earned by the principal over a specified period of time. Consider the following example: An investor invests $1,000 in a 5-year term deposit with an interest rate of 8% with the interest compounded annually.

Therefore, at the end of each year, the interest amount generated in that year is added to the principal amount. It is the new principal amount and the interest for the next year is generated based on the principal amount.

Total Interest Earned = Principal * [(1 + Interest Rate)^Time – 1]

Total Interest Earned = $1,000 * [(1 + .06)⁵ – 1 = $338.23

Average Annual Interest = Total Interest Earned / Time

Average Annual Interest = $338.23 / 5 = $67.65

Formula for Compounded Interest

General compound interest takes into account interest earned over some previous interval of time.

General Compound Interest = Principal * [(1 + Annual Interest Rate/N)^N*Time

Where:

• N is the number of times interest is compounded in a year.

Consider the following example: An investor is given the option of investing $1,000 for 5 years in two deposit options.

• Deposit A pays 6% interest with the interest compounded annually.
• Deposit B pays 6% interest with the interest compounded quarterly.

Clearly, Deposit B is a better option as it provides a higher return.

Continuously Compounded Interest Formula

Continuously compounded interest is the mathematical limit of the general compound interest formula, with the interest compounded an infinitely many times each year. Or in other words, you are paid every possible time increment. Mathematicians, have derived a way to approximate the value such a sum would converge to, and it is given by the following formula:

Where:

• N is the number of times interest is compounded in a year.

Continuously compounded interest is the mathematical limit of the general compound interest formula with the interest compounded an infinitely many times each year. Consider the example described below.

• Initial principal amount is $1,000.
• Rate of interest is 6%.
• The deposit is for 5 years.

Total Interest Earned = Principal * [(e^{Interest Rate*Time}) – 1]

Total Interest Earned = $1,000 * [e^.06*5 – 1] = $349.86

Average Annual Interest = Total Interest Earned / Time

Average Annual Interest = $349.86 / 5 = $69.97

Table of Interest Payments and Total Return

Consider the example described above.

• Initial principal amount is $1,000.
• Rate of interest is 6%.
• The deposit is for 5 years.

No. of Compounding Periods Each YearInterest AmountReturn (in %)1338.2256
33.822562343.916434.391643345.868334.586834346.85534.68555347.450534.745056347.848934.784897348.134234.813428348.348634.834869348.515634.8515610348.649334.8649311348.758834.8758812348.850234.8850213348.927534.8927514348.993834.8993815349.051334.9051316349.101634.9101617349.14634.914618349.185534.9185519349.220934.9220920349.252734.92527100349.737434.973741,000349.846734.9846710,000349.857634.98576100,000349.858734.98587

More Resources

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To keep learning and developing your knowledge of financial analysis, we highly recommend the additional CFI resources below:

• Math for Corporate Finance Course
• Continuously Compounded ReturnContinuously Compounded ReturnContinuously compounded return is what happens when the interest earned on an investment is calculated and reinvested back into the account for an infinite number of periods. The interest is calculated on the principal amount and the interest accumulated over the given periods
• Effective Annual Interest RateEffective Annual Interest RateThe Effective Annual Interest Rate (EAR) is the interest rate that is adjusted for compounding over a given period. Simply put, the effective
• Interest IncomeInterest IncomeInterest income is the amount paid to an entity for lending its money or letting another entity use its funds. On a larger scale, interest income is the amount earned by an investor’s money that he places in an investment or project.