Compounding is the effect where an investment earns interest not only on the principal component but also gives interest on interest. So compounding is basically Interest on interest. When we say that the investment will be compounded annually, it means that we will earn interest on the annual interest along with the principal. Daily compounding is basically when our daily interest/return will get the compounding effect. The concept is such that it assumes that the interest earned every day is reinvested at the same rate and will get increased as the time passes. That is the reason that if we annualized the daily compound interest, it will be always higher than the simple interest rate.

Formula For daily compound interest:

Generally, the rate of interest on investment is quoted on per annum basis. So the formula for an ending investment is given by:

**Ending Investment = Start Amount * (1 + Interest Rate) ^ n**

Where n – Number of years of investment

This formula is applicable if the investment is getting compounded annually, means that we are reinvesting the money on an annual basis. For daily compounding, the interest rate will be divided by 365 and n will be multiplied by 365, assuming 365 days in a year.

So

**Ending Investment = Start Amount * (1 + Interest Rate / 365 ) ^ (n * 365)**

Mục Lục

#### Daily Compound Interest Formula – Example #1

**Let say you have $1000 to invest and you can leave that amount for 5 years. Financial institution in which you are depositing the money is offering you 10% interest rate which will be compounded daily. Calculate the Daily Compound Interest.**

**Solution:**

**For Daily Compounding:**

Ending Investment is calculated using the formula given below

**Ending Investment = Start Amount * (1 + Interest Rate / 365 ) ^ (n * 365)**

- Ending Investment = $1,000 * (1 + (10% / 365)) ^ (5 * 365)
- Ending Investment =
**$1,648.61**

Daily Compound Interest is calculated using the formula given below

**Daily Compound Interest = Ending Investment – Start Amount**

- Daily Compound Interest =$1,648.61 – $1,000
- Daily Compound Interest =
**$648.61**

Daily compound interest which you have earned $648.60.

**If the given rate is compounded annually, then**

**For Annual Compounding**

Ending Investment is calculated using the formula given below

**Ending Investment = Start Amount * (1 + Interest Rate) ^ n**

- Ending Investment = $1,000 * (1 + 10%) ^ 5
- Ending Investment =
**$1,610.51**

Daily Compound Interest is calculated using the formula given below

**Daily Compound Interest = Ending Investment – Start Amount**

- Daily Compound Interest =$1,610.51 – $1,000
- Daily Compound Interest =
**$610.51**

So you can see that in daily compounding, the interest earned is more than annual compounding.

#### Daily Compound Interest Formula – Example #2

**Let say you have got a sum of amount $10,000 from a lottery and you want to invest that to earn more income. You do not need that funds for another 20 years. You approached 2 banks which gave you different rates:**

**Bank 1:**Interest Rate: 12.5% Compounding Daily**Bank 2:**Interest Rate: 12.5% Compounding Annually

**Solution:**

**Interest Rate: 12.5% Compounding Daily**

Ending Investment is calculated using the formula given below

**Ending Investment = Start Amount * (1 + Interest Rate / 365 ) ^ (n * 365)**

- Ending Investment = $10,000 * (1 + (12.5% / 365)) ^ (20 * 365)
- Ending Investment =
**$121,772.81**

Daily Compound Interest is calculated using the formula given below

**Daily Compound Interest = Ending Investment – Start Amount**

- Daily Compound Interest = $121,772.81 – $10,000
- Daily Compound Interest =
**$111,772.81**

**Interest Rate: 12.5 % Compounding Annually**

Ending Investment is calculated using the formula given below

**Ending Investment = Start Amount * (1 + Interest Rate) ^ n**

- Ending Investment = $10,000 * (1 + 12.5%) ^ 20
- Ending Investment =
**$105,450.94**

Daily Compound Interest is calculated using the formula given below

**Daily Compound Interest = Ending Investment – Start Amount**

- Daily Compound Interest = $105,450.94 – $10,000
- Daily Compound Interest =
**$95,450.94**

So If we see, effectively, you are earning more if you choose to invest in Bank 1 due to daily compounding.

### Explanation

Compounding is a very intriguing concept in finance but there is some assumption which sometimes does not make much practical sense. Like in daily compounding, it is assumed that all the interest amount will be reinvested at the same rate for the investment period but actually, the interest rate never remains the same and varies. Because of which we might not be able to invest our money at the same rate and our effective return might differ. So basically, this is kind of theoretical representation which tells us that what we might end up with if all the money is reinvested at the end of each day at that rate.

### Relevance and Uses of Daily Compound Interest Formula

Compounding as a whole is helpful in earning interest on interest, which makes logical sense. In simple interest, you earn interest on the same principal for the investment term and you basically lose out income which you can earn on that additional amount. So for example: if you have $100 and the simple interest rate is 10%, for 2 years, you will have 10%*2*100 = $20 as interest. But if you invest that only for 1 year, then you will earn $10 and then again you invest $110 at 10% for a year, you will have $11 interest in 2^{nd} year. So in total, you have $21 interest and you were losing out on $1 interest in case of simple interest. For daily compounding, we can say that, the more the merrier. As you increase the compounding frequency, you will effectively earn more money since your money will go through more rounds of compounding.

## Daily Compound Interest Formula

In compound interest, the interest for every period is calculated on the amount for the previous period. This means the amount for the previous time period becomes the principal for the current time period. “Compounding” means adding interest to the current principal amount.

## What is Daily Compound Interest Formula?

The daily compound interest formula is where the interest is calculated 365 times in a year, hence the value of n is 365. By the given explanation, the daily compound interest formula is:

A = P (1 + r / n)^{n t}

Here,

- P = the principal amount
- r = rate of interest
- t = time in years
- n = number of times the amount is compounding.

When the amount compounds daily, it means that the amount compounds 365 times in a year. i.e., n = 365.

Where

- A is the total amount (A = Principal + Interest).

### Solved Examples Using Daily Compound Interest Formula

**Example 1: **You have invested $1000 in a bank where your amount gets compounded daily at an interest rate of 5%. Then what is the amount you get after 10 years? Calculate this by using the daily compound interest formula.

**Solution:**

To find: The amount after 10 years.

The principal amount is, P = $1000.

The rate of interest is, r = 5% =5/100 = 0.05.

The time in years is, t = 10.

Using the daily compound interest formula is:

A = P (1 + r / 365)^{365 t}

A = 1000 ( 1+ 0.05/365)^{365×10}

A =$1648.66

**Answer: The amount after 10 years = $1648.66.**

**Example 2: **How long does it take for $15000 to double if the amount is compounded daily at 10% annual interest? Calculate this by using the daily compound interest formula and round your answer to the nearest integer.

**Solution:**

To find: The time taken for $15000 to double.

The principal amount, P = $15000.

The rate of interest is, r = 10% =10/100 = 0.1.

The final amount is, A = 15000 x 2 = $30000

Let us assume that the required time in years is t.

Using the daily compound interest formula is:

A = P (1 + r / 365)^{365 t}

30000 = 15000 (1 + 0.1 / 365)^{365 t}

Dividing both sides by 15000,

2 =(1.0002739)^{365 t}

Taking ln on both sides

ln 2 = 365 × t × ln 1.0002739

t = ln 2/(365 ln 1.0002739)

t= 7

**Answer: It takes 7 years for $15000 to become double.**

### Compound Interest Formulas Used in This Calculator

The basic compound interest formula A = P(1 + r/n)^{nt} can be used to find any of the other variables. The tables below show the compound interest formula rewritten so the unknown variable is isolated on the left side of the equation.

### How to Use the Compound Interest Calculator: Example

Say you have an investment account that increased from $30,000 to $33,000 over 30 months. If your local bank offers a savings account with daily compounding (365 times per year), what annual interest rate do you need to get to match the rate of return in your investment account?

In the calculator above select “Calculate Rate (R)”. The calculator will use the equations: r = n((A/P)^{1/nt} – 1) and R = r*100.

Enter:

- Total P+I (A): $33,000
- Principal (P): $30,000
- Compound (n): Daily (365)
- Time (t in years): 2.5 years (30 months equals 2.5 years)

Showing the work with the formula r = n((A/P)^{1/nt} – 1):

Your Answer: R = 3.813% per year

So you’d need to put $30,000 into a savings account that pays a **rate of 3.813% per year** and compounds interest daily in order to get the same return as the investment account.

## How to Derive A = Pe^{rt} the Continuous Compound Interest Formula

A common definition of the constant *e* is that:

With continuous compounding, the number of times compounding occurs per period approaches infinity or n → ∞. Then using our original equation to solve for A as n → ∞ we want to solve:

This equation looks a little like the equation for *e*. To make it look more similar so we can do a substitution we introduce a variable m such that m = n/r then we also have n = mr. Note that as n approaches infinity so does m.

Replacing n in our equation with mr and cancelling r in the numerator of r/n we get:

## How Compound Interest Works

Compound interest is calculated by multiplying the initial principal amount by one plus the annual interest rate raised to the number of compound periods minus one. The total initial amount of the loan is then subtracted from the resulting value.

The formula for calculating the amount of compound interest is as follows:

**Compound interest**= total amount of principal and interest in future (or future value)*less*principal amount at present (or present value)

= [P (1 +i)^{n}] – P

= P [(1 +i)^{n }– 1]

Where:

P = principal

i= nominal annual interest rate in percentage terms

n = number of compounding periods

Take a three-year loan of $10,000 at an interest rate of 5% that compounds annually. What would be the amount of interest? In this case, it would be:

$10,000 [(1 + 0.05)

^{3}– 1] = $10,000 [1.157625 – 1] = $1,576.25

### How Compound Interest Grows

Because compound interest includes interest accumulated in previous periods, it grows at an ever-accelerating rate. In the example above, though the total interest payable over the three-year period of this loan is $1,576.25, the interest amount is not the same for all three years, as it would be with simple interest. The interest payable at the end of each year is shown in the table below.

## How Compound Interest Grows Over Time

If you invested $10,000 which compounded annually at 5%, it would be worth over $40,000 after 30 years, accruing over $30,000 in compounded interest.

Compound interest can significantly boost investment returns over the long term. While a $100,000 deposit that receives 5% simple annual interest would earn $50,000 in total interest over 10 years, the annual compound interest of 5% on $10,000 would amount to $62,889.46 over the same period. If the compounding period were instead paid monthly over the same 10-year period at 5% compound interest, the total interest would instead grow to $64,700.95.

## Compound Interest Schedules

Interest can be compounded on any given frequency schedule, from daily to annually. There are standard compounding frequency schedules that are usually applied to financial instruments.

The commonly used compounding schedule for savings accounts at banks is daily. For a certificate of deposit (CD), typical compounding frequency schedules are daily, monthly, or semiannually; for money market accounts, it’s often daily. For home mortgage loans, home equity loans, personal business loans, or credit card accounts, the most commonly applied compounding schedule is monthly.

There can also be variations in the time frame in which the accrued interest is actually credited to the existing balance. Interest on an account may be compounded daily but only credited monthly. It is only when the interest is actually credited, or added to the existing balance, that it begins to earn additional interest in the account.

Some banks also offer something called continuously compounding interest, which adds interest to the principal at every possible instant. For practical purposes, it doesn’t accrue that much more than daily compounding interest unless you want to put money in and take it out the same day.

More frequent compounding of interest is beneficial to the investor or creditor. For a borrower, the opposite is true.

### Compounding Periods

When calculating compound interest, the number of compounding periods makes a significant difference. The basic rule is that the higher the number of compounding periods, the greater the amount of compound interest.

The following table demonstrates the difference that the number of compounding periods can make for a $10,000 loan with an annual 10% interest rate over a 10-year period.

### Time Value of Money Consideration

Understanding the time value of money and the exponential growth created by compounding is essential for investors looking to optimize their income and wealth allocation.

The formula for obtaining the future value (FV) and present value (PV) are as follows:

FV = PV (1 +i)^{n }and PV = FV / (1 + i)^{ n}

For example, the future value of $10,000 compounded at 5% annually for three years:

= $10,000 (1 + 0.05)

^{3}

= $10,000 (1.157625)

= $11,576.25

The present value of $11,576.25 discounted at 5% for three years:

= $11,576.25 / (1 + 0.05)

^{3}

= $11,576.25 / 1.157625

= $10,000

The reciprocal of 1.157625, which equals 0.8638376, is the discount factor in this instance.

### Rule of 72 Consideration

The so-called Rule of 72 calculates the approximate time over which an investment will double at a given rate of return or interest “i,” and is given by (72/i). It can only be used for annual compounding.

As an example, an investment that has a 6% annual rate of return will double in 12 years. An investment with an 8% annual rate of return will thus double in nine years.

## Compound Annual Growth Rate (CAGR)

The compound annual growth rate (CAGR) is used for most financial applications that require the calculation of a single growth rate over a period of time.

Let’s say your investment portfolio has grown from $10,000 to $16,000 over five years; what is the CAGR? Essentially, this means that PV = -$10,000, FV = $16,000, and t = 5, so the variable “i” has to be calculated. Using a financial calculator or Excel, it can be shown that i = 9.86%.

*According to the cash-flow convention, your initial investment (PV) of $10,000 is shown with a negative sign because it represents an outflow of funds. PV and FV must necessarily have opposite signs to solve for “i” in the above equation.*

### CAGR Real-Life Applications

The CAGR is extensively used to calculate returns over periods of time for stock, mutual funds, and investment portfolios. The CAGR is also used to ascertain whether a mutual fund manager or portfolio manager has exceeded the market’s rate of return over a period of time. If, for example, a market index has provided total returns of 10% over a five-year period, but a fund manager has only generated annual returns of 9% over the same period, the manager has underperformed the market.

The CAGR can also be used to calculate the expected growth rate of investment portfolios over long periods of time, which is useful for purposes such as saving for retirement. Consider the following examples:

**Example 1:** A risk-averse investor is happy with a modest 3% annual rate of return on her portfolio. Her present $100,000 portfolio would, therefore, grow to $180,611 after 20 years. In contrast, a risk-tolerant investor who expects an annual return of 6% on her portfolio would see $100,000 grow to $320,714 after 20 years.

**Example 2:** The CAGR can be used to estimate how much needs to be stowed away to save for a specific objective. A couple who would like to save $50,000 over 10 years toward a down payment on a condo would need to save $4,165 per year if they assume an annual return (CAGR) of 4% on their savings. If they are prepared to take a little extra risk and expect a CAGR of 5%, they would need to save $3,975 annually.

**Example 3:** The CAGR can also demonstrate the virtues of investing earlier rather than later in life. If the objective is to save $1 million by retirement at age 65, based on a CAGR of 6%, a 25-year old would need to save $6,462 per year to attain this goal. A 40-year old, on the other hand, would need to save $18,227, or almost three times that amount, to attain the same goal.

CAGRs also crop up frequently in economic data. Here is an example: China’s per capita GDP increased from $193 in 1980 to $6,091 in 2012. What is the annual growth in per capita GDP over this 32-year period? The growth rate “i” in this case works out to be an impressive 11.4%.

## Pros and Cons of Compounding

Though the miracle of compounding has led to the apocryphal story of Albert Einstein calling it the eighth wonder of the world or man’s greatest invention, compounding can also work against consumers who have loans that carry very high interest rates, such as credit card debt. A credit card balance of $20,000 carried at an interest rate of 20% compounded monthly would result in a total compound interest of $4,388 over one year or about $365 per month.

On the positive side, compounding can work to your advantage when it comes to your investments and can be a potent factor in wealth creation. Exponential growth from compounding interest is also important in mitigating wealth-eroding factors, such as increases in the cost of living, inflation, and reduced purchasing power.

Mutual funds offer one of the easiest ways for investors to reap the benefits of compound interest. Opting to reinvest dividends derived from the mutual fund results in purchasing more shares of the fund. More compound interest accumulates over time, and the cycle of purchasing more shares will continue to help the investment in the fund grow in value.

Consider a mutual fund investment opened with an initial $5,000 and an annual addition of $2,400. With an average annual return of 12% over 30 years, the future value of the fund is $798,500. The compound interest is the difference between the cash contributed to an investment and the actual future value of the investment. In this case, by contributing $77,000, or a cumulative contribution of just $200 per month, over 30 years, compound interest is $721,500 of the future balance.

Of course, earnings from compound interest are taxable, unless the money is in a tax-sheltered account; it’s ordinarily taxed at the standard rate associated with the taxpayer’s tax bracket.

## Compound Interest Investments

An investor who opts for a reinvestment plan within a brokerage account is essentially using the power of compounding in whatever they invest.

Investors can also experience compounding interest with the purchase of a zero-coupon bond. Traditional bond issues provide investors with periodic interest payments based on the original terms of the bond issue, and because these are paid out to the investor in the form of a check, interest does not compound. Zero-coupon bonds do not send interest checks to investors; instead, this type of bond is purchased at a discount to its original value and grows over time. Zero-coupon bond issuers use the power of compounding to increase the value of the bond so it reaches its full price at maturity.

Compounding can also work for you when making loan repayments. Making half your mortgage payment twice a month, for example, rather than making the full payment once a month, will end up cutting down your amortization period and saving you a substantial amount of interest.

## How to Calculate Compound Interest

If it’s been a while since your math class days, fear not: There are handy tools for figuring out compounding. Many calculators (both handheld and computer-based) have exponent functions you can utilize for these purposes.

### Calculating Compound Interest in Excel

If more complicated compounding tasks arise, you can perform them in Microsoft Excel—in three different ways.

- The first way to calculate compound interest is to multiply each year’s new balance by the interest rate. Suppose you deposit $1,000 into a savings account with a 5% interest rate that compounds annually, and you want to calculate the balance in five years. In Microsoft Excel, enter “Year” into cell A1 and “Balance” into cell B1. Enter years 0 to 5 into cells A2 through A7. The balance for year 0 is $1,000, so you would enter “1000” into cell B2. Next, enter “=B2*1.05” into cell B3. Then enter “=B3*1.05” into cell B4 and continue to do this until you get to cell B7. In cell B7, the calculation is “=B6*1.05”. Finally, the calculated value in cell B7—$1,276.28—is the balance in your savings account after five years. To find the compound interest value, subtract $1,000 from $1,276.28; this gives you a value of $276.28.
- The second way to calculate compound interest is to use a fixed formula. The compound interest formula is ((P*(1+i)^n) – P), where P is the principal, i is the annual interest rate, and n is the number of periods. Using the same information above, enter “Principal value” into cell A1 and 1000 into cell B1. Next, enter “Interest rate” into cell A2 and “.05” into cell B2. Enter “Compound periods” into cell A3 and “5” into cell B3. Now you can calculate the compound interest in cell B4 by entering “=(B1*(1+B2)^B3)-B1”, which gives you $276.28.
- A third way to calculate compound interest is to create a macro function. First start the Visual Basic Editor, which is located in the developer tab. Click the Insert menu, and click on Module. Then type “Function Compound_Interest (P As Double, I As Double, N As Double) As Double” in the first line. On the second line, hit the tab key and type in “Compound_Interest = (P*(1+i)^n) – P.” On the third line of the module, enter “End Function.” You have created a function macro to calculate the compound interest rate. Continuing from the same Excel worksheet above, enter “Compound interest” into cell A6 and enter “=Compound_Interest(B1, B2, B3).” This gives you a value of $276.28, which is consistent with the first two values.

### Other Compound Interest Calculators

A number of free compound interest calculators are offered online, and many handheld calculators can carry out these tasks as well.

- The free compound interest calculator offered through Financial-Calculators.com is simple to operate and offers to compound frequency choices from daily through annually. It includes an option to select continuous compounding and also allows input of actual calendar start and end dates. After inputting the necessary calculation data, the results show interest earned, future value, annual percentage yield or APY) (a measure that includes compounding), and daily interest.
^{2} - Investor.gov, a website operated by the U.S. Securities and Exchange Commission (SEC), offers a free online compound interest calculator. The calculator is fairly simple, but it does allow inputs of monthly additional deposits to the principal, which is helpful for calculating earnings where additional monthly savings are being deposited.
^{3} - A free online interest calculator with a few more features is available at TheCalculatorSite.com. This calculator allows calculations for different currencies, the ability to factor in monthly deposits or withdrawals, and the option to have inflation-adjusted increases to monthly deposits or withdrawals automatically calculated as well.

## How Can I Tell if Interest Is Compounded?

The Truth in Lending Act (TILA) requires that lenders disclose loan terms to potential borrowers, including the total dollar amount of interest to be repaid over the life of the loan and whether interest accrues simply or is compounded.

Another method is to compare a loan’s interest rate to its annual percentage rate (APR), which the TILA also requires lenders to disclose. The APR converts the finance charges of your loan, which include all interest and fees, to a simple interest rate. A substantial difference between the interest rate and APR means one or both of two scenarios: Your loan uses compound interest, or it includes hefty loan fees in addition to interest. Even when it comes to the same type of loan, the APR range can vary wildly between lenders depending on the financial institution’s fees and other costs.

You’ll note that the interest rate you are charged also depends on your credit. Loans offered to those with excellent credit carry significantly lower interest rates than those charged to borrowers with poor credit.

## What is a Simple Definition of Compound Interest?

Compound interest refers to the phenomenon whereby the interest associated with a bank account, loan, or investment increases exponentially—rather than linearly—over time. The key to understanding the concept is the word “compound.”

Suppose you make a $100 investment in a business that pays you a 10% dividend every year. You have the choice of either pocketing those dividend payments like cash or reinvesting those payments into additional shares. If you choose the second option, reinvesting the dividends and compounding them together with your initial $100 investment, then the returns you generate will start to grow over time.

## Who Benefits From Compound Interest?

Simply put, compound interest benefits investors, but the meaning of “investors” can be quite broad. Banks, for instance, benefit from compound interest when they lend money and reinvest the interest they receive into giving out additional loans. Depositors also benefit from compound interest when they receive interest on their bank accounts, bonds, or other investments.

It is important to note that although the term “compound interest” includes the word “interest,” the concept applies beyond situations for which the word interest is typically used, such as bank accounts and loans.

## Can Compound Interest Make You Rich?

Yes. In fact, compound interest is arguably the most powerful force for generating wealth ever conceived. There are records of merchants, lenders, and various businesspeople using compound interest to become rich for literally thousands of years. In the ancient city of Babylon, for example, clay tablets were used over 4,000 years ago to instruct students on the mathematics of compound interest.

In modern times, Warren Buffett became one of the richest people in the world through a business strategy that involved diligently and patiently compounding his investment returns over long periods of time. It is likely that, in one form or another, people will be using compound interest to generate wealth for the foreseeable future.

### Examples

#### Example #1

**A sum of $4000 is borrowed from the bank where the interest rate is 8%, and the amount is borrowed for two years. Let us determine how much will be daily compounded interest calculation by the bank on loan provided.**

**Solution:**

= ($4000(1+8/365)^(365*2))-$4000

#### Example #2

Daily compounding is practically applicable for credit card spending, which is charged by the banks on the individuals who use credit cards. Credit cards generally have a cycle of 60 days, during which time the bank does not charge any interest, but interest is charged when the interest does not pay back within 60 days. If a sum of $4000 is used using a credit card by an individual for its spending. And the interest rate is 15% per annum as the interest charged for a credit card is generally very high. And the amount is repaid by the individual after 120 days that is 60 days after the grace period is over. So the individual needs to pay the bank interest for 60 days, and he is charged at a daily compounding rate.

**Solution:**

= $4000(1+15/365)^(365*(12/60))-$4000

#### Example #3

**A sum of $35000 is borrowed from the bank as a car loan where the interest rate is 7% per annum, and the amount is borrowed for five years. Let us determine how much will be daily compounded interest calculation by the bank on loan provided.**

**Solution:**

✅ Math Formulas ⭐️⭐️⭐️⭐️⭐

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