Continuous Compounding Effective Interest Rate Formula

Nominal, Period, and Effective Interest Rates

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Nominal, Period and Effective Interest Rates Based on Discrete Compounding of Interest

Usually, financial agencies report the interest rate on a nominal annual basis with a specified compounding period that shows the number of times interest is compounded per year. This is called simple interest, nominal interest, or annual interest rate. If the interest rate is compounded annually, it means interest is compounded once per year and you receive the interest at the end of the year. For example, if you deposit 100 dollars in a bank account with an annual interest rate of 6% compounded annually, you will receive $100\ast \left(1+0.06\right)=106$ dollars at the end of the year.

But, the compounding period can be smaller than a year (it can be quarterly, monthly, or daily). In that case, the interest rate would be compounded more than once a year. For example, if the financial agency reports quarterly compounding interest, it means interest will be compounded four times per year and you would receive the interest at the end of each quarter. If the interest is compounding monthly, then the interest is compounded 12 times per year and you would receive the interest at the end of the month.

For example: assume you deposit 100 dollars in a bank account and the bank pays you 6% interest compounded monthly. This means the nominal annual interest rate is 6%, interest is compounded each month (12 times per year) with the rate of 6/12 = 0.005 per month, and you receive the interest at the end of each month. In this case, at the end of the year, you will receive $100\ast {\left(1+0.005\right)}^{12}=106.17$ dollars, which is larger than if it is compounded once per year: $100\ast {\left(1+0.06\right)}^1=106$ dollars. Consequently, the more compounding periods per year, the greater total amount of interest paid.

Please watch the following video, Nominal and Period Interest Rates (Time 3:52).

Nominal and Period Interest Rates

Click for the transcript of "Nominal and Period Interest Rates" video.

PRESENTER: In this video, I'm going to explain nominal, period, and effective interest rates. Financial agencies usually report the interest rate on an annual base. The interest rate can be compounded once or more per year. If the interest rate is compounded annually, it means the interest rate is compounded once per year. If the interest rate is compounded quarterly, then interest rate is compounded four times a year. And if interest rate is compounded monthly, it means the interest rate is compounded 12 times a year.

Let's work on an example. Assume you deposit $100 in an imaginary bank account that gives you 6% interest rate, compounded annually. So nominal interest rate is 6%, compounded annually. The interest rate of 6% is compounded once a year, and you will receive interest and the principal of your money in the end of year one. So you will receive $100 multiplied by 1 plus 6% power of 1 in the end of year one, which equals $106.

Now let's assume the bank pays you 6% interest, compounded quarterly. So it means nominal interest rate is 6% quarterly, or interest rate will be compounded four times a year, and interest rate is calculated at the end of each quarter. In order to calculate the amount of money that you will receive in the end of year one, we need to calculate the period interest rate, which is going to be 6% divided by 4 and it equals 1.5%. You deposit your $100 at present time, and the bank calculates the interest with a rate of 1.5% per quarter. There are four quarters in a year, so the interest will be compounded four times per year at the rate of 1.5% per quarter. Then, at the end of the year, you will receive $100 multiplied by 1 plus 0.15 power 4, which equals $106 plus $0.14. As you can see, if bank considers interest rate which is compounded quarterly, it will give you slightly higher interest comparing to the case that interest rate was compounded annually.

Now let's assume bank pays you 6% interest compounded monthly, which means interest rate is compounded 12 times a year. In this case, bank calculates the interest every month. And similar to the previous example, period interest rate is going to be 6% divided by 12, which is going to be 0.5% per month. And you will receive $100 multiplied by 1 plus 0.005 power 12, which equals $106 plus $0.17. Because there are 12 compounding periods, and per period interest is 0.5%. As you can see here, interest rate is compounded monthly, so you will receive slightly higher money in the end of the year. The more compounding per year you have, the higher interest you will receive in the end of the year.

Credit: Farid Tayari

Period interest rate i = r/m
Where m = number of compounding periods per year
r = nominal interest rate = mi

"An effective interest rate is the interest rate that when applied once per year to a principal sum will give the same amount of interest equal to a nominal rate of r percent per year compounded m times per year. Annual Percentage Yield (APY) is the standard term used by the banking industry to identify an effective interest rate."

The future value, F1, of investing P at i% per period for m period after one year:

P	_	_	_	_	_	F1 = P(F/Pi,m) = P(1+i)m
	0	1	2	...	m periods per year

And if the effective interest rate, E, is applied once a year, then future value, F2, of investing P at E% per year:

P	_				_	F2 = P(F/PE,1) = P(1+E)1
	0				1 period per year

Then:

$\begin{array}{l}F1=F2\\ P{\left(1+i\right)}^m=P{\left(1+E\right)}^1\end{array}$

Since P the same in both sides:${\left(1+i\right)}^m=E+1$

Then:

$$\text{Effective Annual Interest}:E={\left(1+i\right)}^m-1$$

(Equation 2-1)

If the effective Annual Interest, E, is known and equivalent period interest rate i is unknown, the equation 2-1 can be written as:

$$i={\left(E+1\right)}^{1/m}-1$$

(Equation 2-2)

Going back to the previous example, $$\begin{array}{l}i=6/12=0.005\\ \text{so,}E={\left(1+0.005\right)}^{12}-1=1.0617-1=0.0617\text{ or}6.17\%\end{array}$$

Please watch the following video, Effective Interest Rate (Time 4:02).

Effective Interest Rate

Click for the transcript of "Effective Interest Rate" video.

PRESENTER: In this video, I'm going to explain how to calculate the effective interest rate. In the previous video, we learn how to calculate the period interest rate, which is nominal interest rate, r, divided by the number of compounding period per year, m. So to calculate the future value, you will need to know the number of period from present time and desired future and also period interest rate. For example, f, future value at the end of year one equals p, multiply 1 plus i power m, where m is the number of compounding period per year.

An effective interest rate is the interest rate that when applied once per year, it will give you the same amount of interest equal to a nominal rate of r. Annual percentage yield, or APY, is the term that is used in the banking industry for effective interest rate. You can see here, when you read somewhere, that for example interest rate is 6% compounded monthly, it is a bit confusing. Because it doesn't tell you what would be the actual interest rate per year. Effective interest rate is the rate that helps us here. Effective interest rate is the per year rate that gives you exactly the same interest equal to using nominal rate that is compounded multiple times a year.

Going back to the example in the previous video, you saw that if you deposit $100 in a bank account, that gives you 6% interest rate compounded monthly, you will receive $106 plus $0.17 per year. So you can guess effective interest rate here can be 6.17%. Now let's see if we can find a general equation. In previous slide, I explained how we calculate the F1 future value at the end of year one from period interest rate, i, and number of compounding periods per year, m.

If you want to calculate the future value at the end of year one using effective interest rate, here we show it, we have to we will have F2 equal P multiply 1 plus E power 1. Effective interest rate is E And we want to calculate the future value in the end of year one. The future value of money at the end of year one using per period interest rate and effective interest rate should be equal. So F1 should be equal to F2.

And we have an Equation 2-1. This equation can be written for i. E is the effective interest rate. m is the number of compounding periods per year, and i is period interest rate. Going back to the example in the previous video, we deposited $800 in a bank account that gives us 6% of interest compounded monthly. To calculate the effective interest rate, we need to calculate the period interest rate first and then we use the equation that we just extracted. So effective interest rate would be 6.17%, which means if we apply 6.17% interest rate per year, it will give us exactly the same future value as applying interest rate of 6% compounded monthly.

Credit: Farid Tayari

Example 2-1:

Assume an investment that pays you 2000 dollars in the end of the first, second, and third year for an annual interest rate of 12% compounded quarterly. Calculate the time zero present value and future value of these payments after three years.

P=?	_	_	_	_	2000	_	_	_	2000	_	_	_	2000	F=?
	0	1	2	3	4	5	6	7	8	9	10	11	12

Quarterly period interest rate i = 12/4 = 3%

$$\begin{array}{l}P=2,000*\left(P/F_{3\%,4}\right)+2,000*\left(P/F_{3\%,8}\right)+2,000*\left(P/F_{3\%,12}\right)\\ =2000\left[1/{\left(1+0.03\right)}^4\right]+2000\left[{\left(1+1/0.03\right)}^8\right]+2000\left[1/{\left(1+0.03\right)}^{12}\right]\\ =\$4,758.55\end{array}$$
$$\begin{array}{l}F=2,000*\left(F/P_{3\%,(12-4)}\right)+2,000*(F/P_{3\%,(12-8)})+2,000*\left(F/P_{3\%,(12-12)}\right)\\ =2,000*\left(F/P_{3\%,8}\right)+2,000*\left(F/P_{3\%,4}\right)+2,000\\ =2000*{\left(1+0.03\right)}^8+2000*{\left(1+0.03\right)}^4+2000\\ =\text{ \$}6,784.56\end{array}$$

Please note that since the interest rate is compounded quarterly, we have to structure the calculations in a quarterly base. So there will be 12 quarters (three years and 4 quarters per each year) on the time line.
The 2000 dollars interest is paid at the end of the first, second, and third year, which are going to be the last quarters of each year (4^th quarter, 8^th quarter, and 12^th quarter).

Please watch the following video, Nominal and Period Interest Rates Example (Time 3:45).

Nominal and Period Interest Rates

Click for the transcript of "Nominal and Period Interest Rates Example" video.

PRESENTER: Let's work on an example. Assume there is an investment that pays you $2,000 in the end of the year one, year two, and year three, for an annual interest rate of 12% compounded quarterly. And we want to calculate the present value at time zero and a future value in the end of year three of these payments.

The first thing that we need to do is to draw the timeline and locate the payments on the line. The smallest interval in the timeline should be compounding period, which is quarter in this example. The project lifetime is three years. So we should have 12 quarters or time interval on the timeline.

Then we place the payments. First payment is at the end of the year one, which will be 4th quarter. Second payment of $2,000 will be at the end of second year, which will be 8th quarter. And third payment at the end of the third year, which is going to be twelfth quarter.

Now, we have to calculate the present value of these payments. But first we need to calculate the period interest rate, which is going to be 12 divided by 4 equals 3, because we have 4 quarters in a year. It is very important to note that we have to use the period interest rate, because our time intervals are quarter.

Then we calculate the present value of these payments. First payment is in the end of the first year, which is going to be 4th quarter, with 3% interest per quarter. Second payment is in the 8th quarter with 3% interest rate per quarter. And the third $2,000 is in the 12th quarter, with 3% interest rate. And the result which shows the present value of these three payments.

Now, future value. Again, first we have to calculate the period interest rate and it is going to be 3%. Then we calculate the future value of these three payments. By future value we mean at the end of the project lifetime, which is at the end of third year or 12th quarter. In order to calculate the present value of the first payment we need to know how many time periods are between this time and the future time.

The first $2,000 is paid at the 4th quarter, which is 8 quarters away from the future time, because future time is at 12th period. So we need to write 12 minus 4 as the time period here in the factor, because the future time is in 12th period. The second $2,000 is paid at the end of the second year or 8th quarter, which is 4 quarters away from the future time. And the last $2,000 is paid at the end of the third year or 12th period. This is the same time as our desired future time. And N or time difference would be zero.

Credit: Farid Tayari

Continuous Compounding of Interest

If an annual interest rate compounds annually, then it should be compounded once a year.
If an annual interest rate compounds semi-annual, then it should be compounded twice a year.
If an annual interest rate compounds quarterly, then it should be compounded 4 times per year.
If an annual interest rate compounds monthly, then it should be compounded 12 times per year.
If an annual interest rate compounds daily, then it should be compounded 365 times per year.
And if the compounding period becomes smaller, then the number of compoundings per year, m, becomes larger. In the limit as m goes to infinity, period interest, i, approaches zero. This case is called Continues Compounding of Interest. Using differential calculus, Continues Interest Single Discrete Payment Compound-Amount Factor (F/P_r,n ) can be calculated as:

$$F/P_{r,n}=e^{rn}$$

(Equation 2-3)

And, Continues Interest Single Discrete Payment Present Worth Factor (P/Fr,n)

$$P/F_{r,n}=1/e^{rn}$$

(Equation 2-4)

r is nominal interest rate compounded continuously
n is number of discrete valuation periods
e is base of natural log (ln) = 2.7183

Example 2-2:

Lets recalculate example 2-1 considering continues compound interest rate of 12%:

$$\begin{array}{l}P=2,000*\left(P/F_{12\%,1}\right)+2,000*\left(P/F_{12\%,2}\right)+2,000*\left(P/F_{12\%,3}\right)=2000\left[1/e^{0.12*1}\right]+2000\left[1/e^{0.12*2}\right]+2000\left[1/e^{0.12*3}\right]\\ =\text{ \$}4,742.45\\ F=2,000*\left(F/P_{12\%,2}\right)+2,000*\left(F/P_{12\%,1}\right)+2,000=2000*e^{0.12*2}+2000*e^{0.12*1}+2000=\text{ \$}6,797.49\end{array}$$

Note:The following links explains how to use the excel function (EXP) to calculate e raised to the power of number:

Link 1: EXP Function in Excel
Link 2: Excel Functions

Please watch the following video, Continuous Compounding of Interest (Time 4:54).

Continuous Compounding of Interest

Click for the transcript of "Continuous Compounding of Interest" video.

In this video, I'm going to explain continuous compounding interest, and I will show you how to calculate the future and present value in case of continuous compounding.

If we have more and more compounding period per year, then compounding period becomes smaller and smaller. Then number of compounding period per year, m, becomes larger and larger. So in this case, future value can be calculated as present time, multiply 1 plus i power n multiply m. M is the number of compounding period per year. I is the period interest rate, which equals r divided by m, and r is the nominal interest rate, which is m multiply i.

In the limit as m goes to infinity, period interest rate i, which is r divided by m, approaches to 0. In this case, it is called continuous compounding of interest.

Now, let's calculate compound-amount factor, F over P, or future value factor for continuous interest. So this factor equals 1 plus i power n multiply m, and we can rewrite i as r over m.

Now, we need to calculate the limit as m goes to infinity. In this case, this term approaches to 0, and this term approaches to infinity. So we can extract an e term here, and we calculate the limit as e power rn.

So compound-amount factor, or future value factor, for continuous interest will be e power rn, or future value can be calculated as P multiply by e power rn. F is the future value for continuous compounding interest. R is the nominal interest rate compounded continuously, n, number of discrete valuation periods, which can be one year, two year, three years, and so on. And e is the base of natural log.

Similarly, we can calculate the present value in case of continuous compounding interest. The present value factor equals the inverse of future value factor. So present value can be calculated as P equals F divided by e power r,n. P is the present value for continuous compounding interest.

Now, let's work on an example. It is a previous example, but we are going to consider the continuous compounding interest rate. Assume there is an investment that pays you $2,000 in the end of year one, year two and year three, and you want to calculate the present value at the present time and the future value in the end of the year three. And we have to consider continuous compounding interest rate of 12%.

First, we draw the time line. We are going to have three $2,000 payments at the end of year one, year two, and year three, and we want to calculate the present value of these three payments.

The first payment is going to be at the end of year one. So we need to discount that for one year with the 12% of continuous interest. The second payment is at the end of year two, so n is going to be 2. And the last payment is going to be at year three, so n equals 3.

And now, we substitute the factor, which is going to be 1 over e power 12% multiplied by 1 and so on, and the result.

Now, we are going to calculate the future value of these three payments. The first payment is happening at the end of the year one, which is two years away from future time. So n equals 2. The second payment is one year away from future time, so n equals 1. And the last payment is exactly at the same time as the future time, so n is 0 and we write the $2,000, and we don't need any compounding. And then we replace the factors. E power 12% multiply by 2 for the first payment and so on. And we have the result.

Credit: Farid Tayari

"Flat" or "Add-on" Interest Rate

A flat or add-on interest rate is applied to the initial investment principal each interest compounding period. This means total interest received for the investment on a flat interest is calculated linearly and simply is the summation of interest on all periods. For example, if you invest 1000 dollars at the present time in a project with flat interest rate of 12% per annum for 100 days, you will receive 32.88 dollars after 100 days:
$$1000*0.12*\left(100/365\right)=32.88\text{ dollars interest}$$
The flat interest rate is usually applied when interest is calculated for a portion of a year or period.

Note: In engineering economics, the term "simple interest" is usually used as "add-on" or "flat" interest rate as defined here.

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Source: https://www.e-education.psu.edu/eme460/node/655

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