How to Calculate Profit

Almost everyone knows the concept of interest, but not everyone knows how to calculate interest. Interest is the value added to a loan or deposit to pay interest on one’s use of money over time. Interest can be calculated in 3 basic ways. Simple interest is the easiest calculation, usually applied to short-term loans. Compound interest is calculated slightly more complicated and has a slightly higher value. Finally, continuous compounding is the fastest growing interest rate and is the interest formula most banks use for mortgage loans. The necessary data used in the above calculations are generally the same, but the calculation is slightly different.

Table of Contents

Calculate simple interest

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Determine the amount of capital. The capital is the amount that you will use to calculate the interest. It could be money you put in a savings account or some kind of bond. In this case, you will be entitled to the amount of interest charged. On the other hand, if you take out a loan, such as a mortgage, the principal is the amount you borrow, and the calculated interest is the amount you have to pay.

In either case, whether you earn interest or pay interest, the amount of capital is usually denoted by the variable P. ^{[1] X Research Source}
For example, if you lend a friend $2,000, your principal will be $2,000.

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Determine interest rates. Before calculating how much the capital will grow, you need to know its rate of increase, i.e. the interest rate you will charge. The interest rate is usually announced or agreed upon by the two parties before the loan is made. ^{[2] X Research Source}

For example, suppose you lend money to a friend with an agreement that the borrower will repay you $2,000 plus interest at 1.5% at the end of the 6th month from the loan date. Before calculating the 1.5% interest rate, you will have to convert 1.5% to a decimal. To convert a percentage to a decimal, divide the number by 100:
- 1.5% ÷ 100 = 0.015.

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Determine the loan term. Loan term is the term for the loan period. In some cases, you will agree on a loan term when borrowing money. For example, most mortgage loans have a definite term. With many personal loans, borrowers and lenders can agree on an arbitrary loan term.

It is important that the loan term must match the interest rate, or at least have the same unit. For example, if interest is charged for one year, the loan term must also be calculated in years. If the interest rate is 3% a year but the loan term is only 6 months, you will charge annual interest with a term of 0.5 years.
As another example, if the agreed interest rate is 1% per month and you borrow money for 6 months, the loan term will be calculated as 6.

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Calculate profit. To calculate interest, we multiply the principal amount by the interest rate and loan term. This formula can be written in algebraic form as follows:

$I$ $=$ $P$ $*$ $r$ $*$ $t$ ${displaystyle I=P*r*t}$ ${displaystyle I=P*r*t}$
Using the example above, the capital you lend to a friend ( P{displaystyle P} $P$ ) is $2,000, and the interest rate ( r{displaystyle r} $r$ ) is 0.015 for 6 months, variable t{displaystyle t} $t$ in this case it is 1. The calculation of interest will be as follows:
- $I$ $=$ $P$ $r$ $t$ $=$ $($ $2000$ $)$ $($ $0$ $,$ $015$ $)$ $($ $first$ $)$ $=$ $30$ ${displaystyle I=Prt=(2000)(0.015)(1)=30}$ ${displaystyle I=Prt=(2000)(0.015)(1)=30}$ . Thus, the interest due will be $30.
If you want to calculate the total amount due to be received (A), with interest and principal return, we will use the formula A=P(first+rt){displaystyle A=P(1+rt)} ${displaystyle A=P(1+rt)}$ . The calculation is expressed as follows:
- $A$ $=$ $P$ $($ $first$ $+$ $r$ $t$ $)$ ${displaystyle A=P(1+rt)}$ ${displaystyle A=P(1+rt)}$
- $A$ $=$ $2000$ $($ $first$ $+$ $0$ $,$ $015$ $*$ $first$ $)$ ${displaystyle A=2000(1+0.015*1)}$ ${displaystyle A=2000(1+0.015*1)}$
- $A$ $=$ $2000$ $($ $first$ $,$ $015$ $)$ ${displaystyle A=2000(1,015)}$ ${displaystyle A=2000(1,015)}$
- $A$ $=$ $2030$ ${displaystyle A=2030}$ ${displaystyle A=2030}$

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Take another example. Let’s say you deposit $5,000 in a savings account with an annual interest rate of 3%; but only after 3 months, you withdraw the money, the interest will be due at that time.

$A$ $=$ $P$ $($ $first$ $+$ $r$ $t$ $)$ ${displaystyle A=P(1+rt)}$ ${displaystyle A=P(1+rt)}$
$A$ $=$ $5000$ $($ $first$ $+$ $0$ $,$ $03$ $*$ $0$ $,$ $25$ $)$ ${displaystyle A=5000(1+0.03*0.25)}$ ${displaystyle A=5000(1+0.03*0.25)}$
$A$ $=$ $5000$ $($ $first$ $,$ $0075$ $)$ ${displaystyle A=5000(1,0075)}$ ${displaystyle A=5000(1,0075)}$
$A$ $=$ $5037$ $,$ $5$ ${displaystyle A=5037.5}$ ${displaystyle A=5037.5}$
In 3 months, you will have a profit of $37.50.
Note, here t=0.25, because 3 months is a quarter (0.25) of the original one-year term.

Calculate compound interest

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Understand the meaning of compound interest. Compounding means that your interest is compounded into your principal account, and the amount of interest you begin to enjoy (or pay) will then increase. For a simple example, if you deposit $100 at 5% interest per year, after one year you will have a return of $5. If you compound that interest into your account, at the end of the second year you will enjoy 5% interest on the $105 and not just on the original $100. Over time, this amount can add up significantly. ^{[3] X Research Sources}

The formula for calculating the value (A) of compound interest is:
- $A$ $=$ $P$ $($ $first$ $+$ $r$ $n$ $)$ $n$ $t$ ${displaystyle A=P(1+{frac {r}{n}})^{nt}}$ ${displaystyle A=P(1+{frac {r}{n}})^{nt}}$

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Know the principal amount. As with simple interest, compounding starts with the principal amount. The way to calculate interest on loan or borrowing is the same. The amount of capital is usually expressed by the variable

P

{displaystyle P}

P

. ^{[4] X Research Sources}

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Calculate interest. The interest rate will be agreed from the beginning and expressed in decimals in the calculation. Remember that percentages are converted to decimals by dividing by 100 (or, to put it simply, moving the comma to the left 2 rows). You must also know the loan term for which interest is applied. Interest rates are expressed in letters

r

{displaystyle r}

r

. ^{[5] X Research Sources}

For example, a credit card has an interest rate of 15% a year. However, interest is usually paid on a monthly basis, so you may want to know what the monthly interest rate is. In this case, we would divide the annual interest rate by 12 to get a monthly rate of 1.25%. An interest rate of 15% a year is equivalent to 1.25% a month.

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Know when interest is compounded. Compound interest means that interest will be calculated over the term and added to the principal. With some loans, interest is compounded on the capital once a year. Some other loans may be charged on a monthly or quarterly basis. You need to know how many times the interest will be added each year. ^{[6] X Research Sources}

If compound interest is added annually, we have n=1.
If compound interest is added quarterly, we have n=4.

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Know the loan term. The loan term is the period for which interest is calculated. Loan terms are usually calculated on an annual basis. If you need to calculate interest over a certain period, you need to convert it to years. ^{[7] X Research Sources}

For example, with a 1-year loan, we have $t$ $=$ $first$ ${displaystyle t=1}$ ${displaystyle t=1}$ . However, for a duration of 18 months, we have $t$ $=$ $first$ $,$ $5$ ${displaystyle t=1.5}$ ${displaystyle t=1.5}$ .

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Identify the variables of the situation. Let’s say, you deposit $5,000 into a savings account that accumulates every month, what will your account be worth after 3 years? ^{[8] X Research Sources}

First, we will define the variables that you need to solve the calculation. In this field:
- $P$ $=$ $$$ $5,000 won$ ${displaystyle P=$5,000}$ ${displaystyle P=$5,000}$
- $r$ $=$ $0$ $,$ $05$ ${displaystyle r=0.05}$ ${displaystyle r=0.05}$
- $n$ $=$ $twelfth$ ${displaystyle n=12}$ ${displaystyle n=12}$
- $t$ $=$ $3$ ${displaystyle t=3}$ ${displaystyle t=3}$

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Apply the formula to calculate compound interest. After understanding the situation and determining the variables, we will substitute the formula to find the amount of profit.

With the above problem, the calculation will be expressed as follows:
- $A$ $=$ $P$ $($ $first$ $+$ $r$ $n$ $)$ $n$ $t$ ${displaystyle A=P(1+{frac {r}{n}})^{nt}}$ ${displaystyle A=P(1+{frac {r}{n}})^{nt}}$
- $A$ $=$ $5000$ $($ $first$ $+$ $0$ $,$ $05$ $twelfth$ $)$ $twelfth$ $*$ $3$ ${displaystyle A=5000(1+{frac {0.05}{12}})^{12*3}}$ ${displaystyle A=5000(1+{frac {0.05}{12}})^{12*3}}$
- $A$ $=$ $5000$ $($ $first$ $+$ $0$ $,$ $00417$ $)$ $36$ ${displaystyle A=5000(1+0.00417)^{36}}$ ${displaystyle A=5000(1+0.00417)^{36}}$
- $A$ $=$ $5000$ $($ $first$ $,$ $00417$ $)$ $36$ ${displaystyle A=5000(1,00417)^{36}}$ ${displaystyle A=5000(1,00417)^{36}}$
- $A$ $=$ $5000$ $($ $first$ $,$ $1616$ $)$ ${displaystyle A=5000(1,1616)}$ ${displaystyle A=5000(1,1616)}$
- $A$ $=$ $5808$ ${displaystyle A=5808}$ ${displaystyle A=5808}$
Thus, after 3 years, the interest will increase by $808, in addition to the original deposit of $5,000.

Calculating compound interest continuously

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Understand the concept of continuous compounding. As we have seen in the examples above, compound interest grows faster than simple interest because it adds the amount of interest to the capital at a certain point in time. Adding quarterly will help increase the value higher than adding yearly. Adding monthly also helps to increase the value higher than adding quarterly. The most valuable situation is the continuous compounding – that is, instantaneous. As soon as interest is calculated, it will be added to the account and added to the principal amount. Obviously this is just a theory. ^{[9] X Research Source}

Using a number of calculations, mathematicians developed a formula based on interest compounded and accrued to an account on a continuous stream. This formula is expressed as follows:
- $A$ $=$ $P$ $e$ $r$ $t$ ${displaystyle A=Pe^{rt}}$ ${displaystyle A=Pe^{rt}}$

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Know the variables for calculating interest. The formula for calculating continuous compound interest is similar in form to the previous situations but with a slight difference. The variables for this formula are: ^{[10] X Research Source}

$A$ ${displaystyle A}$ $A$ is the future value (or amount) of the loan after interest has been included.
$P$ ${displaystyle P}$ $P$ is the original capital.
e{displaystyle e} ${displaystyle e}$ . Although it looks like a variable, it is actually a constant. Alphabet e{displaystyle e} ${displaystyle e}$ is a special number called the “Eulerian constant,” named after the mathematician Leonard Euler, who discovered its properties.
- Most graphing calculators have a . button $e$ $x$ ${displaystyle e^{x}}$ ${displaystyle e^{x}}$ . If you press this button with the number 1 to indicate $e$ $first$ ${displaystyle e^{1}}$ ${displaystyle e^{1}}$ , you will find the value of $e$ ${displaystyle e}$ ${displaystyle e}$ is approximately 2,718.
$r$ ${displaystyle r}$ $r$ is the annual interest rate.
$t$ ${displaystyle t}$ $t$ is the loan term in years.

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Know the details of the loan. Banks often use continuous compounding for mortgage loans. Let’s say you want to take out a $200,000 mortgage with a 4.2% interest rate for 30 years. Accordingly, the variables that you will use in the calculation will be: ^{[11] X Research Source}

$P$ $=$ $200,000 yen$ ${displaystyle P=200,000}$ ${displaystyle P=200,000}$
$e$ ${displaystyle e}$ ${displaystyle e}$ , again, this is not a variable but the constant 2.718.
$r$ $=$ $0$ $,$ $042$ ${displaystyle r=0.042}$ ${displaystyle r=0.042}$
$t$ $=$ $30$ ${displaystyle t=30}$ ${displaystyle t=30}$

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Use the formula to calculate interest. Substitute the variables into the formula to calculate the interest you owe on a 30-year loan. ^{[12] X Research Source}

$A$ $=$ $P$ $e$ $r$ $t$ ${displaystyle A=Pe^{rt}}$ ${displaystyle A=Pe^{rt}}$
$A$ $=$ $200000$ $*$ $2$ $,$ $718$ $($ $0$ $,$ $042$ $)$ $($ $30$ $)$ ${displaystyle A=200000*2.718^{(0.042)(30)}}$ ${displaystyle A=200000*2.718^{(0.042)(30)}}$
$A$ $=$ $200000$ $*$ $2$ $,$ $718$ $first$ $,$ $26$ ${displaystyle A=200000*2.718^{1.26}}$ ${displaystyle A=200000*2.718^{1.26}}$
$A$ $=$ $200000$ $*$ $3$ $,$ $525$ ${displaystyle A=200000*3,525}$ ${displaystyle A=200000*3,525}$
$A$ $=$ $705000$ ${displaystyle A=705000}$ ${displaystyle A=705000}$
Note the huge value of continuous compounding.

Steps

Calculate simple interest

Calculate compound interest

Calculating compound interest continuously

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