example problem with solution of compound interest

Compound Interest Problems with Detailed Solutions

Compound interest problems with answers and solutions are presented.

Free Practice for SAT, ACT and Compass Maths tests

• A principal of $2000 is placed in a savings account at 3% per annum compounded annually. How much is in the account after one year, two years and three years?
• What would $1000 become in a saving account at 3% per year for 3 years when the interest is not compounded (simple interest)? What would the same amount become after 3 years with the same rate but compounded annually?
• $100 is the principal deposited in a 5% saving account not compounded (simple interest). The same amount of $100 is placed in a 5% saving account compounded annually. Find the total amount A after t years in each saving plan and graph both of them in the same system of rectangular axes. Use the graphs to approximate the time it takes each saving plan to double the initial amount.
• If $3000 is placed in an account at 5% and is compounded quarterly for 5 years. How much is in the account at the end of 5 years?
• $1200 is placed in an account at 4% compounded annually for 2 years. It is then withdrawn at the end of the two years and placed in another bank at the rate of 5% compounded annually for 4 years. What is the balance in the second account after the 4 years.
• $1,200 is placed in an account at 4% compounded daily for 2 years. It is then withdrawn and placed in another bank at the rate of 5% compounded daily for 4 years. What is the balance in the second account after the 4 years. (compare with the previous problem)
• $1200 is placed in an account at 4% compounded continuously for 2 years. It is then withdrawn and placed in another bank at the rate of 5% compounded continuously for 4 years. What is the balance in the second account after the 4 years. (compare with the two previous problem)
• What principal you have to deposit in a 4.5% saving account compounded monthly in order to have a total of $10,000 after 8 years?
• A principal of $120 is deposited in a 7% account and compounded continuously. At the same time a principal of $150 is deposited in a 5% account and compounded annually. How long does it take for the amounts in the two accounts to be equal?
• A first saving account pays 5% compounded annually. A second saving account pays 5% compounded continuously. Which of the two investments is better in the long term?
• What interest rate, compounded annually, is needed for a principal of $4,000 to increase to $4,500 in 10 year?
• A person deposited $1,000 in a 2% account compounded continuously. In a second account, he deposited $500 in a 8% account compounded continuously. When will the total amounts in both accounts be equal? When will the total amount in the second accounts be 50% more than the total amount in the second account?
• A bank saving account offers 4% compounded on a quarterly basis. A customer deposit $200, in this type of account, at the start of each quarter starting with the first deposit on the first of January and the fourth deposit on the first of October. What is the total amount in his account at the end of the year?
• An amount of $1,500 is invested for 5 years at the rates of 2% for the first two years, 5% for the third year and 6% for the fourth and fifth years all compounded continuously. What is the total amount at the end of the 5 years?

Solutions to the Above Questions

• Solution When interest is compounded annually, total amount A after t years is given by: A = P(1 + r) t , where P is the initial amount (principal), r is the rate and t is time in years. 1 year: A = 2000(1 + 0.03) 1 = $2060 2 years: A = 2000(1 + 0.03) 2 = $2121.80 3 years: A = 2000(1 + 0.03) 3 = $2185.45
• Solution Not compounded: A = P + P(1 + r t) = 1000 + 1000(1 + 0.03 · 3) = $1090 Compounded: A = P(1 + r) t = 1000(1 + 0.03) 3 = $1092.73 Higher return when compounded.
• Solution Compounded n times a year and after t years, the total amount is given by: A = P(1 + r/n) n t quarterly n = 4: Hence A = P(1 + r/4) 4 t = 3000(1 + 0.05/4) 4 × 5 = $3846.11
• Solution Annual compounding First two years: A = P(1 + r) t = 1200(1 + 0.04) 2 = $1297.92 Last four years : A = P(1 + r) t = 1297.92(1 + 0.05) 4 = $1577.63
• Solution Daily compounding (assuming 365 days per year) First two years: A = P(1 + r / 365) 365 t = 1200(1 + 0.04 / 365) 365 × 2 = $1299.94 Last four years : A = P(1 + r / 365) 365 t = 1299.94 (1 + 0.05 / 365) 365 ×4 = $1587.73 Higher final balances compared to annual compounding in last problem.
• Solution In continuous compounding, final balance after t years is given by: A = P e r t . First two years: A = P e r t = 1200 e 0.04 × 2 = $1299.94 Last four years : A = P e r t = $1299.94 e 0.05 × 4 = $1587.75 Same balances compared to daily compounding in last problem.
• Solution P initial balance to find and final balance A known and equal to $10,000. A = P(1 + 0.045 / 12) 12 × 8 = 10,000 P = 10,000 / ( (1 + 0.045 / 12) 12 × 8 ) = $6981.46
• Solution P initial balance is equal to $4,000 and final balance is equal to $4,500. A = P(1 + r) t = 4,500 4000(1 + r) 10 = 4,500 (1 + r) 10 = 4500 / 4000 Take ln of both sides. 10 ln(1 + r) = ln(4500 / 4000) ln(1 + r) = ln(4500 / 4000) / 10 1 + r = e 0.1 ln(4500 / 4000) r = e 0.1 ln(4500 / 4000) - 1 ? 0.012
• Solution A 1 = 1000 e 0.02 t A 2 = 500 e 0.08 t A 1 = A 2 gives 1000 e 0.02 t = 500 e 0.08 t Divide both sides by 500 e 0.02 t and simplify 100 / 500 = e 0.08 t - 0.02 t 2 = e 0.06 t Take ln of both sides. 0.06 t = ln 2 t = ln 2 / 0.06 ? 11.5 years A 2 is 50% more than A 1 gives the equation: A 2 = 1.5 A 1 500 e 0.08 t = 1.5 × 1000 e 0.02 t e 0.08 t - 0.02 t = 3 0.06 t = ln 3 t ? 18.5 years
• Solution A = P(1 + r/n) n t First quarter deposit, t = 1 year: A 1 = 200 (1 + 0.04 / 4) e 4 × 1 = $208.12 Second quarter deposit, t = 3/4 of 1 year : A 2 = 200 (1 + 0.04 / 4) e 4 × 3/4 = $206.06 Third quarter deposit , t = 1/2 of 1 year : A 3 = 200 (1 + 0.04 / 4) e 4 × 1/2 = $204.02 Fourth quarter deposit, t = 1/4 of 1 year : A 4 = 200 (1 + 0.04 / 4)e 4 × 1/4 = $202 Total amount = $208.12 + $206.06 + $204.02 + $202 = $820.2
• Solution A = P e r t End of first two years: A 1 = 1500 e 0.02 × 2 End of third year: A 2 = A 1 e 0.05 × 1 End of fifth year (last two years): A 3 = A 2 e 0.06 × 2 = 1500 e 0.02 × 2 + 0.05 × 1 + 0.06 × 2 = $1850.51

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Compound interest word problems

We will use the compound interest formula to solve these compound interest word problems.

Example #1 A deposit of $3000 earns 2% interest compounded semiannually. How much money is in the bank after for 4 years? Solution B  = P( 1 + r) n P = $3000 r = 2% annual interest rate / 2 interest periods = 1% semiannual interest rate n = number of payment periods = number of interest periods times number of years

n = 2 times 4 = 8

B  = 3000( 1 + 1%) 8   = 3000(1 + 0.01) 8 = 3000(1.01) 8 B = 3000(1.082856) B = 3248.57

After four years, there will be 3248.57 dollars in the bank account.

Example #2 A deposit of $2150 earns 6% interest compounded quarterly. How much money is in the bank after for 6 years? Solution B  = P( 1 + r) n P = $2150 r = 6% annual interest rate / 4 interest periods = 1.5% quarterly interest rate n = number of payment periods = number of interest periods times number of years

n = 4 times 6 = 24

B  = 2150( 1 + 1.5%) 24   = 2150(1 + 0.015) 24 = 2150(1.015) 24 B = 2150(1.4295) B = 3073.425

After 6 years, there will be 3073.425 dollars in the bank account.

Example #3 A deposit of $495 earns 3% interest compounded annually. How much money is in the bank after for 3 years? Solution B  = P( 1 + r) n P = $495 r = 3% annual interest rate / 1 interest period = 3% annual interest rate n = number of payment periods = number of interest periods times number of years

n = 1 times 3 = 3

B  = 495( 1 + 3%) 3   = 495(1 + 0.03) 3 = 495(1.03) 3 B = 495(1.092727) B = 540.89

After 3 years, there will be 540.89 dollars in the bank account.

Compound interest

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Compound Interest Formula

Compound interest is interest that is calculated on both the money deposited and the interest earned from that deposit.

The formula for compound interest is \(A=P(1+\frac{r}{n})^{nt}\), where \(A\) represents the final balance after the interest has been calculated for the time, \(t\), in years, on a principal amount, \(P\), at an annual interest rate, \(r\). The number of times in the year that the interest is compounded is \(n\).

Jasmine deposits $520 into a savings account that has a 3.5% interest rate compounded monthly. What will be the balance of Jasmine’s savings account after two years?

To find the balance after two years, \(A\), we need to use the formula, \(A=P(1+\frac{r}{n})^{nt}\). The principal, \(P\), in this situation is the amount Jasmine used to start her account, $520. The rate, \(r\), as stated in the problem, is 3.5% (or 0.35 as a decimal) and compounded monthly, so \(n=12\). Since we are looking for the balance of the account after two years, 2, is the time, \(t\).

\(A=520(1+\frac{0.035}{12})^{12(2)}\) \(A=557.65\)

The balance of Jasmine’s account after 2 years is $557.65.

Lex has $1,780.80 in his savings account that he opened 6 years ago. His account has an annual interest rate of 6.8% compounded annually. How much money did Lex use to open his savings account?

To find the principal, \(P\) we can use the same formula, \(A=P(1+\frac{r}{n})^{nt}\). We have the balance of the account, \(A\), after 6 years, which is $1,780.80. The interest rate, \(r\), is 6.8% (or 0.68 as a decimal) and is compounded annually, so \(n=1\). The time, \(t\), is 6, since we know he opened his account 6 years ago. Plug in the known values into the formula and solve for the missing variable, \(P\).

\(1{,}780.80=P(1+\frac{0.068}{1})^{1(6)}\) \(1{,}780.80=1.484P\) \(1{,}200=P\)

The principal amount Lex used to open his account 6 years ago is $1,200.

Compound Interest Sample Questions

  A teacher wants to invest $30,000 into an account that compounds annually. The interest rate at this bank is 1.8%. How much money will be in the account after 6 years?

Use the compound interest formula to solve this problem.

\(A=P(1+\frac{r}{n})^{nt}\)   From here, simply plug in each value and simplify in order to isolate the variable \(A\).

\(A=30{,}000(1+\frac{0.018}{1})^{1×6}\) \(A=30{,}000(1.018)^6\) \(A=$33{,}389.35\)

  An investment earns 3% each year and is compounded monthly. Calculate the total value after 6 years from an initial investment of $5,000.

Once again, use the compound interest formula to solve this problem.

\(A=5,000(1+\frac{0.03}{12}^{12×6}\) \(A=5,000(1.36)^{72}\) \(A=$5{,}984.74\)

  Kristen wants to have $2,000,000 for retirement in 45 years. She invests in a mutual fund and pays 8.5% each year, compounded quarterly. How much should she deposit into the mutual fund initially?

\(A=P(1+\frac{r}{n}^{nt}\)   From here, simply plug in each value and simplify in order to isolate the variable \(P\).

\(2,000,000=P(1+\frac{0.085}{4})^{4×45}\) \(2,000,000=P(1.02125)^{180}\) \(P=$45{,}421.08\)

  Sean invests $50,000 into an index annuity that averages 6.5% per year, compounded semi-annually. After 9 years how much will be in his account?

\(A=50{,}000(1+\frac{0.065}{2})^{2×9}\) \(A=50{,}000(1.0325)^{18}\) \(A=$88{,}918.29\)

  Calculate the interest rate for an account that started with $5,000 and now has $13,000 and has been compounded annually for the past 12 years.

\(A=P(1+\frac{r}{n})^{nt}\)   From here, simply plug in each value and simplify in order to isolate the variable \(r\).

\(13{,}000=5{,}000(1+\frac{r}{1})^{1×12}\) \(13{,}000=5{,}000(1+\frac{r}{1})^{12}\) \(2.6=(1+\frac{r}{1})^{12}\)

To get rid of the exponent 12, find the 12th root of both sides. This means raising both sides of the equation to the power of 112.

\(2.6^{\frac{1}{12}}=((1+\frac{r}{1})^{12})^{\frac{1}{12}}\) \(1.08288=1+r\)

Subtract 1 from both sides.

\(r=0.08288\) (as a decimal) \(r=8.288\%\) (as a percent)

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by Mometrix Test Preparation | This Page Last Updated: February 16, 2024

Compound Interest

Compound interest is an interest calculated on the principal and the existing interest together over a given time period. The interest accumulated on a principal over a period of time is also added to the principal and becomes the new principal amount for the next time period. Again, the interest for the next time period is calculated on the accumulated principal value. Compound interest is the method of calculation of interest used for all financial and business transactions across the world. The power of compounding is that it is always greater than or equal to the other methods like simple interest.

An amount of $1000 invested over a period of time at 10% rate would give a simple interest of $100, $100, $100... over successive time periods of 1 year, but would give a compound interest of $100, $210, $331, $464.10... Let us understand more about this, and the calculations of compound interest in the below content.

What is Compound Interest?

Compound interest is the interest paid on both principal and existing interest. Hence, it is usually termed " interest over the interest". Here, the interest so far accumulated is added to the principal and the resulting amount becomes the new principal for the next interval. i.e.,  Compound Interest = Interest on principal + Interest over existing interest.

The compound interest is calculated at regular intervals like annually(yearly), semi-annually (half-yearly), quarterly (4 times in a year), monthly (12 times in a year), etc; In case of compound interest, interest income from an investment makes the money grow faster over time! It is exactly what is done by the compound interest to money. Banks or any financial organization calculate the amount based on compound interest only.

Compound Interest Formula

Compound interest is the interest that is earned on an initial principal amount as well as the accumulated interest from previous periods. The compound interest is found after calculating the compounded amount over a period of time, based on the rate of interest , and the initial principal. Here are the formulas to find the compounded amount and compound interest.

Formula of Compound Amount

For an initial principal of P, rate of interest per annum of r (r%), time period t in years, frequency of the number of times the interest is compounded annually n, the formula to calculate the total compounded amount is as follows:

A = P (1 + r/n) nt

Formula of Compound Interest

The compound interest is obtained by subtracting the principal amount from the compound amount. Hence, the formula to find just the compound interest is as follows: CI = P (1 + r/n) nt  - P.

In the above expression,

• P is the principal amount
• r is the rate of interest(decimal obtained by dividing rate by 100)
• n is the number of times the interest is compounded annually
• t is the overall tenure.

If the given principal is compounded annually, then we have n = 1 and in this case, the above formulas turn into the following:

Compound amount, A = P(1 + r) t

Compound interest, C.I = P(1 + r) t - P .

Derivation of Compound Interest Formula

The compound interest formula is derived from the simple interest  formula. The formula for simple interest is the product of the principal, time period, and rate of interest (SI = Ptr/100). Before looking into to derivation of the formula for compound interest, let us understand the basic difference between simple interest and compound interest computation. The principal is constant over a period of time in case of simple interest computation, but in compound interest computation, the interest is added to the principal after every time period.

Derivation:

The compound interest formula is derived as follows:

• Let the principal be P and the rate of interest be R% per annum. Here, the interest is compounded annually, so the compounding period is 1 year. Note that the principal (P) will change after every 1 year. 
• Assume that the interest for the first year is I 1 . I 1 = R% of P = R/100 × P
• Then, the amount at the end of the first year is A 1 = P + I 1 = P + (R/100 × P). This will give A 1 = P (1 + R/100)
• Now, let us do the interest calculation for the second year. It is to be noted that the amount (principal + interest of the first year) of the first year will become the principal of the second year. Let this principal be P 2
• Now, P 2 = A 1 = P (1 + R/100)
• Now, the interest for the second year is I 2 = R% of P 2 = R/100 × P 2 = R/100 × P(1 + R/100)
• Now, the amount at the end of the second year will be A 2 = P 2 + I 2 = P (1 + R/100) + R/100 × P(1 + R/100)
• This expression can be written as A 2 = P (1 + R/100) (1 + R/100) = P (1 + R/100) 2
• Continuing in this manner for n compounding periods, the amount at the end of t years A = P(1 + R/100) t .

Instead of writing R/100 every time, we usually convert the rate into decimals by dividing by 100 to get r and substitute it in the formula P (1 + r) t .

Simple Interest and Compound Interest

From the above formulas and computations, we can observe that the compound interest is the same as the simple interest for the first interval. But, after a period of time, there is a noticeable difference in the total interest obtained.

The simple interest value for each time period is the same because the principal on which it is calculated is constant. But the compound interest varies and increases across the years. This is because the principal on which the compound interest is calculated each year is increasing. The principal for a particular year in case of compound interest is equal to the sum of the initial principal value, and the accumulated interest of the past years.

Example:  Assume that an amount of $10,000 is deposited at a rate of 10%. The below table explains the difference between simple interest and compound interest computation on this principal for a period of 5 years.

From the above table, we can understand the power of compounding. Compound interest is more profitable than simple interest if the amounts invested for more than 1 year. For more differences between simple and compound interests, click here .

Compound Interest Formula for Different Time Periods

If you want to calculate the compound interest for a different time period, you can adjust the values of n and t accordingly. The CI formulas are tabulated in the following table for different time periods. In all these formulas, P is the principal amount, r is the rate/100, and t is the number of years.

In these formulas, A is the total amount that includes both the compound interest and the principal. If we want to find just the compound interest then we need to subtract P from the formula. For example, the compound interest formula for compounded monthly would be CI = P (1 + r/12) 12t - P.

Continuous Compound Interest Formula

In all the above formulas of compound interest, the number of times the amount is compounded is finite. But if it is infinite, the compound interest formula turns into

A = lim n→∞ P (1 + r/n) nt

= lim n→∞ P [(1 + r/n) n ] t (by properties of exponents )

= P (e r ) t (by limit formulas )

This formula is known as the continuous compound interest formula and this gives the total amount after t years. Just the interest amount is calculated using the formula Pe rt - P as usual. Here is an example to understand this.

Example: If $5000 is invested in a bank where the amount is compounded continuously at a rate of 7% per year, then what is the resultant amount after 3 years?

Solution: Here, P = $5000, r = 7% = 0.07, and t = 3. Substituting these values in the continuous compound interest formula:

A = Pe rt = 5000(e 0.07 × 3 ) = $6168.39.

How to Calculate Compound Interest?

The compound interest is the total compounded amount minus the initial amount. Here are the steps to find the compound interest:

• Identify the principal amount (P), i.e., the amount that is invested.
• Identify the rate of interest (r%). Make sure to divide it by 100 while substituting it into the formula for the variable r.
• Identify the number of times the investment is compounded (n).
• Identify the number of years (t).

To find the compound interest:

• if the amount is compounded annually/half-yearly/quarterly/monthly/weekly/daily, then substitute all these values into the formula P(1+r/n) nt - P.
• if the amount is compounded continuously, then substitute these values in the formula Pe rt - P.

Important Notes on Compound Interest:

• Compound interest depends on the amount accumulated at the end of the previous tenure, not just on the original principal.
• Banks, insurance companies, etc. generally levy compound interest.
• The compound interest formula is A = P (1 + r/n) not . Here, if the amount is compounded annually, then n = 1 half-yearly, then n = 2 quarterly, then n = 4 monthly, then n = 12 daily, then n = 365
• If the amount is compounded continuously then we use the formula A = Pe rt .
• Note that while finding compound interest, each time period and the rate of interest must be of the same duration.

Tips & Tricks:

• The rule of 72: It is a quick method to know how long it will take for your money to double when the amount is compounded annually. It says two things: Doubling Time = 72/Interest Rate Interest Rate = 72/Doubling Time Using the rule of 72, we can find the number of years to double your money by simply dividing 72 by the rate of interest. For example, at an 8% compounded interest rate your money will double in 72 ÷ 8 = 9
• “Per annum” or “annual” or “per year” - all mean the same. All of these mean you’ll get the given rate of interest over a period of 1 year. Semi-annual is 6 months, while quarterly is 3 months in duration.

☛ Related Topics:

• Compound Interest Calculator
• Daily Compound Interest Calculator
• Monthly Compound Interest Calculator
• Compound Interest Calculator Quarterly

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Solved Examples of Compound Interest

Example 1: Noah lends $4000 to Emma at an interest rate of 10% per annum, compounded half-yearly for a period of 2 years. Find how much amount does he get after a period of 2 years from Emma without using the compound interest formula.

Let us identify the data given to us:

• The principal amount 'P' is $4000.
• The rate of interest, r' is 10% per annum. i.e., r = 10/100 = 0.1.
• Conversion period = Half-year,
• Rate of interest per half-year = 10/2 % = 5%.
• The time period 't' is 2 years.

We will calculate the amount without using the formula to understand what exactly is meant by compound interest.

The total interest to be paid over 2 years 200 + $210 + $220.5 + $231.53 = $862.03. Total Amount = P + I=$4000 + $862.03 = $4862.03. Therefore the total amount is $4862.03.

Answer: The required amount is $4862.03.

Example 2: Solve the above-given problem using the compound interest formula.

The principal amount 'P' is $4000. The rate of interest 'r' is 10% per annum. Conversion period = Half-year, Rate of interest per half-year = 10/2% = 5%. The time period 't' is 2 years. The compounding frequency 'n' is 2.

We will now use the formula and see whether the answer matches.

A = P (1 + r/2) 2t

= 4000 (1 + 0.1/2) 2(2)

Answer: Therefore the final amount is $4862.03, and the compound interest formula makes the solution simple.

Example 3: If $5500 is compounded annually at a rate of 3% per year, then how long it will take for the amount to double? Check your answer using the rule of 72.

Using the given information, P = $5500, r = 3/100 = 0.03, n = 1, and A = 2(5500) = 11000.

By substituting these values in the compound interest formula:

11000 = 5500 (1 + 0.03) t

Taking log on both sides:

log 2 = log 1.03 t

By properties of logarithms:

log 2 = t log 1.03

t = (log 2) / (log 1.03) = 23.5 years (approximately)

Checking with Rule of 72:

By using rule of 72, the doubling time = 72/3 = 24, and this is approximately what we had got earlier.

Answer: Doubling time = 23.5 years.

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FAQs on Compound Interest

What are compound interest formulas.

Here are the list of compound interest formulas  when the amount is compounded

• annually: A = P (1 + r) t
• half-yearly: A = P (1 + r/2) 2t
• quarterly: A = P (1 + r/4) 4t
• monthly: A = P (1 + r/12) 12t
• weekly: A = P (1 + r/52) 52t
• daily: A = P (1 + r/365) 365t

In all these formulas, P is the initial amount, 'r' is the rate of interest, and 't' is the time period.

How to Compute Compound Interest?

The compound interest is found using the formula: CI = P( 1 + r/n) nt - P. In this formula,

• P( 1 + r/n) nt  represents the compounded amount.
• the initial investment P should be subtracted from the compounded amount to get the compound interest.

What is the Important Difference Between Simple Interest and Compound Interest?

Simple interest is the interest calculated only on the principal (initial investment), but compound interest is the interest calculated on both principal and interest together. Thus, compound interest is more beneficial as compared to simple interest.

To calculate the compound interest, we just need to substitute the principal (P), rate r% (r/100), time (t), and the number of times the amount is compounded (n) in the formula P(1 + r/n) nt - P.

What Is the Monthly Compound Interest Formula?

The monthly compound interest formula is given as CI = P(1 + (r/12) ) 12t - P. Here, P is the principal (initial amount), r is the interest rate (for example if the rate is 12% then r = 12/100=0.12), n = 12 (as there are 12 months in a year), and t is the time.

What Is the Daily CI Formula?

The daily CI formula is given as A = P (1 + r / 365) 365 t , where P is the principal amount, r is the interest rate of interest in decimal form, n = 365 (it means that the amount compounded 365 times in a year), and t is the time. Here A gives the total amount (principal + interest).

What Is the Future Value Compound Interest Formula?

The future value compound interest formula is expressed as FV = PV (1 + r / n) n t . Here, PV = Present Value (Initial investment), r = rate of interest, n = number of times the amount is compounding, and t = time in years.

Is Interest Compounded Daily Better than Monthly?

If the amount is compounded daily then it gets compounded 365 a year. It will generate more money compared to interest compounded monthly, which has only 12 compounding cycles per year.

How Does Compound Interest Depend on Time Period?

The compound interest depends on the time period for which the amount is invested/borrowed. The time interval for the calculation of interest can be a day, a week, a month, quarterly, or half-yearly. The more the time interval is the less the compound interest. For example, we get more compound interest if the amount is compounded daily than it is compounded annually.

Which Interest is Greater? Compound Interest or Simple Interest?

Obviously, compound interest is greater than simple interest. This is because simple interest is calculated only on the principal in every tenure, whereas compound interest is calculated on the principal amount + interest so far.

Can Compound Interest be Greater than Principal?

The compound interest can be greater than the principal over a period of time.

How Do you Calculate Compound Interest for Half Year?

The formula for the calculation of compound interest for half year is CI = p(1 + r/2) 2t .- p. Here in this formula 'A' is the final amount, 'p' is the principal, and 't' is the time in years. In this formula, we have divided r by "2" as there are two half-years in a full year.

What Information is Required to Calculate Compound Interest?

To find the compound interest, we should know the principal (P), rate of interest (r%), time period (t), and the number of times the amount gets compounded in a year (n). 

What Are the Units of Compound Interest?

The unit of compound interest is the unit of currency and is the same as the unit used for the principal value. If the principal is in pounds or yen, the compound interest would also be in pounds or yen respectively.

MathBootCamps

Compound interest formula and examples.

Compound interest is when interest is earned not only on the initial amount invested, but also on any interest. In other words, interest is earned on top of interest and thus “compounds”. The compound interest formula can be used to calculate the value of such an investment after a given amount of time, or to calculate things like the doubling time of an investment. We will see examples of this below. [adsenseWide]

Examples of finding the future value with the compound interest formula

First, we will look at the simplest case where we are using the compound interest formula to calculate the value of an investment after some set amount of time. This is called the future value of the investment and is calculated with the following formula.

An investment earns 3% compounded monthly. Find the value of an initial investment of $5,000 after 6 years.

Determine what values are given and what values you need to find.

• Earns 3% compounded monthly: the rate is \(r = 0.03\) and the number of times compounded each year is \(m = 12\)
• Initial investment of $5,000: the initial amount is the principal, \(P = 5000\)

You are trying to find \(A\), the future value (the value after 6 years). Now apply the formula with the known values:

\(\begin{align}A &= P\left(1 + \dfrac{r}{m}\right)^{mt} \\ &= 5000\left(1 + \dfrac{0.03}{12}\right)^{12 \times 6} \\ &\approx \bbox[border: 1px solid black; padding: 2px]{5984.74}\end{align}\)

Answer: The value after 6 years will be $5,984.74.

Important! Be careful about rounding within the formula. You should do as much work as possible in your calculator and not round until the very end. Otherwise your answer may be off by a few dollars.

Let’s try one more example like this before we try some more difficult types of problems.

What is the value of an investment of $3,500 after 2 years if it earns 1.5% compounded quarterly?

As before, we are finding the future value, A. In this example, we are given:

• Value after 2 years: \(t = 2\)
• Earns 3% compounded quarterly: \(r = 0.015\) and \(m = 4\) since compounded quarterly means 4 times a year
• Principal: \(P = 3500\)

Applying the formula:

\(\begin{align}A &= P\left(1 + \dfrac{r}{m}\right)^{mt} \\ &= 3500\left(1 + \dfrac{0.015}{4}\right)^{4 \times 2}\\ &\approx \bbox[border: 1px solid black; padding: 2px]{3606.39}\end{align}\)

Answer : The value after 2 years will be $3,606.39.

There are other types of questions that can be answered using the compound interest formula. Most of these require some algebra, and the level of algebra required depends on which variable you need to solve for. We will look at some different possibilities below.

Example of finding the rate given other values

Suppose you were given the future value, the time, and the number of compounding periods, but you were asked to calculate the rate earned. This could be used in a situation where you are taking the amount of home sold for and determining the rate earned, if it is viewed as an investment. Consider the following example.

Mrs. Jefferson purchased an antique statue for $450. Ten years later, she sold this statue for $750. If the statue is viewed as an investment, what annual rate did she earn?

If we view this as an investment of \(P = $450\), then we know that the future value is \(A = $750\). This was after \(t = 10\) years. Finally, if we assume an annual rate, we will use \(m = 1\) and have:

\(A = P\left(1 + \dfrac{r}{m}\right)^{mt}\)

\(750 = 450\left(1 + \dfrac{r}{1}\right)^{1 \times 10}\)

This is the same as:

\(750 = 450\left(1 + r\right)^{10}\)

We are solving for the rate, \(r\). We will do this using the following steps.

Divide both sides by 450.

\(\dfrac{750}{450} = \left(1 + r\right)^{10}\)

Simplify on the left-hand side. But, we need to be careful about rounding, so we will keep the fraction for now.

\(\dfrac{5}{3} = \left(1 + r\right)^{10}\)

Take the left-hand side to the 1/10th power to clear the power of 10 on the right.

\(\left(\dfrac{5}{3}\right)^{\dfrac{1}{10}} = 1 + r\)

Calculate the value on the left and solve for \(r\).

\(\begin{align}1.0524 &= 1 + r \\1.0524 – 1 &= r \\ \bbox[border: 1px solid black; padding: 2px] {0.0524} &= r\end{align}\)

Therefore, Mrs. Jefferson earned an annual rate of 5.24%. Not bad! But there was definitely some more complicated algebra involved. In some cases, you may even have to make use of logarithms. A common situation where you might see this is when calculating the doubling time of an investment at a given rate.

Calculating the doubling time of an investment using the compound interest formula

Regardless of the amount initially invested, you can find the doubling time of an investment as long as you are given the rate and the number of compounding periods. Let’s look at an example and see how this could be done.

How many years will it take for an investment to double in value if it earns 5% compounded annually?

It may seem tough to decide where to start here, as we are only given the rate, \(r = 0.05\), and the number of compounding periods, \(m = 1\). Note that we are trying to find the time, \(t\).

Since we do not know the initial investment, we can simply call it \(P\). For this to double, its value would be \(2P\) and, using the compound interest formula, we would have:

\(2P = P\left(1 + \dfrac{0.05}{1}\right)^{t}\)

This could be written as:

\(2P = P\left(1.05\right)^{t}\)

Remember that this would only make sense if the amount invested is not zero, so we can divide both side by \(P\). This gives:

\(2 = \left(1.05\right)^{t}\)

To solve for t, we will take the natural log, ln, of both sides. By the laws of logarithms, this will allow us to bring the exponent to the front.

\(\ln(2) = t\ln\left(1.05\right)\)

Finally, we can divide and then use our calculators to find t.

\(\begin{align}t &= \dfrac{\ln(2)}{\ln\left(1.05\right)}\\ &\approx \bbox[border: 1px solid black; padding: 2px]{14.2 \text{ years}}\end{align}\)

Answer : It will take a little more than 14 years before the investment will double in value.

The same process could be used to determine when an investment would triple or even quadruple. You would just use a different multiple of \(P\) in the first part of the formula.

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The compound interest formula is used when an investment earns interest on the principal and the previously-earned interest. Investments like this grow quickly; how quickly depends on the rate and the number of compounding periods. When working with a compound interest formula question, always make note of what values are known and what values need to be found so that you stay organized with your work.

Now that you have studied compound interest, you should also review simple interest and how it is different .

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Compound Interest Formula With Examples

Compound interest, or 'interest on interest', is calculated using the compound interest formula A = P*(1+r/n)^(nt) , where P is the principal balance, r is the interest rate (as a decimal), n represents the number of times interest is compounded per year and t is the number of years.

• How to use the formula

To better understand the formula, let's create a simple example. We'll say that you invest $10,000 into a savings account for 20 years at an annual interest rate of 6%, compounded monthly . Here's how our calculation looks:

• r = 6/100 = 0.06 (decimal)

Plugging those figures into the formula, we can calculate as follows:

• A = 10000 × (1 + 0.06 / 12)^(12 × 20)
• A = 10000 × (1 + 0.005)^240
• A = 10000 × 3.310204
• A = 33,102.04

The investment balance after 20 years is therefore $33,102.04 . Checking this figure against our compound interest calculator , we can see that we have calculated correctly.

Keep scrolling to see variations of the formula for annual, quarterly, monthly and daily compounding, a step-by-step explanation of how to use the formula, and some information on how to integrate it within a spreadsheet...

• Table of contents:

Formula variations

Monthly compound interest formula, how to use the formula in excel or google sheets, example calculation, interactive compound interest formula.

• Formula for calculating interest rate (%)
• Formula for calculating principal
• Formula for calculating time factor

Monthly contributions formula

To assist those looking for a convenient formula reference, I've included a concise list of compound interest formula variations applicable to common compounding intervals. Later in the article, we will delve into each variation separately for a comprehensive understanding.

• A = future value of the investment/loan
• P = principal amount
• r = annual interest rate (decimal)
• R = annual interest rate (percentage)
• n = number of times interest is compounded per year
• t = time in years
• ^ = ... to the power of ...
• ln = the natural logarithm

How to use the compound interest formula

In order to use the compound interest formula you will require specific values for your initial balance (principal), annual interest rate (expressed as a decimal), the number of compounds per year and the number of years you wish to calculate for. Let's take a look at how we put these into our formula...

How to calculate compound interest using the formula

Start by multiply your initial balance by one plus the annual interest rate (expressed as a decimal) divided by the number of compounds per year. Next, raise the result to the power of the number of compounds per year multiplied by the number of years. Subtract the initial balance from the result if you want to see only the interest earned.

The above set out as a formula is:

A = P(1 + r/n)^nt

• P = principal investment or loan amount

It's worth noting that this formula gives you the future value of an investment or loan, which is compound interest plus the principal. Should you wish to calculate the compound interest only, you need to deduct the principal from the result. So, your formula looks like this:

Earned interest only (without principal) Interest = P(1 + r/n)^nt - P

Let's look at how we can use this formula for monthly compounding, and we can then go through an example calculation...

The formula for calculating compound interest with monthly compounding is:

A = P(1 + r/12)^12t

• A = future value of the investment
• P = principal investment amount

If you're using Excel, Google Sheets or Numbers, you can copy and paste the following into your spreadsheet and adjust your figures for the first four rows as you see fit. This example shows monthly compounding (12 compounds per year) with a 5% interest rate.

Here's how it will look in Excel or Google Sheets...

Now that we've looked at how to use the formula for calculations in Excel, let's go through a step-by-step example to demonstrate how to make a manual calculation using the formula...

If an amount of $10,000 is deposited into a savings account at an annual interest rate of 3%, compounded monthly , the value of the investment after 10 years can be calculated as follows...

• r = 3/100 = 0.03 (decimal)

If we plug those figures into the formula, we get the following:

A = 10000 × (1 + 0.03 / 12)^(12 * 10) = 13493.54.

So, the investment balance after 10 years is $13,493.54.

Formula methodology

Let's go through, step-by-step, how we get the 13493.54 result. Our methodology revolves around the PEMDAS order of operations . That is to say that we have to perform operations such as multiplication and addition in the right order.

Let's start off with our equation again:

A = 10000 (1 + 0.03 / 12) ^ (12(10))

(note that ^ means 'to the power of')

Using the order of operations we work out the totals in the brackets first.

Within the first set of brackets, you need to do the division first and then the addition (division and multiplication should be carried out before addition and subtraction). We can also work out the 12(10), which is 12 × 10. This gives us...

A = 10000 (1 + 0.0025)^120

A = 10000 (1.0025)^120

The exponent goes next. So, we calculate 1.0025^120.

This means we end up with:

10000 × 1.34935355

= 13,493.54.

The benefits of compound interest

I think it's worth taking a moment to mention the monetary gain that interest compounding can offer.

Looking back at our example, with simple interest (no compounding) , your investment balance at the end of the term would be $13,000, with $3,000 interest. With regular interest compounding, however, you would stand to gain an additional $493.54 on top.

Interest for $10,000 at 3% for 10 years:

• With simple interest : $3,000
• With compound interest : $3,493.54

Interest for $10,000 at 5% for 10 years:

• With simple interest : $5,000
• With compound interest : $6,470.09

I created the calculator below to show you the formula and resulting accrued investment/loan value (A) for the figures that you enter.

It may be that you want to manipulate the compound interest formula to work out the interest rate for IRR or CAGR , or a principal investment/loan figure. Here are the formulae you need.

Formula for calculating interest rate (r)

This formula can help you work out the yearly interest rate you're getting on your savings, investment or loan. Note that you should multiply your result by 100 to get a percentage figure (%) .

r = n[(A/P)^(1/nt)-1]

• r = interest rate (decimal)

Formula for calculating principal (P)

This formula is useful if you want to work backwards and calculate how much your starting balance would need to be in order to achieve a future monetary value.

P = A / (1 + r/n)^nt

Example: Let's say your goal is to end up with $10,000 in 5 years, and you can get an 8% interest rate on your savings, compounded monthly. Your calculation would be: P = 10000 / (1 + 0.08/12)^(12×5) = $6712.10. So, you would need to start off with $6712.10 to achieve your goal.

Formula for calculating time factor (t)

This variation of the formula works for calculating time (t), by using natural logarithms. You can use it to calculate how long it might take you to reach your savings target , based upon an initial balance and interest rate. You can see how this formula was worked out by reading this explanation on algebra.com .

t = ln(A/P) / n[ln(1 + r/n)]

• A = value of the accrued investment/loan

I've received a lot of requests over the years to provide a formula for compound interest with monthly contributions . So, let's go over how we do this.

In order to work out calculations involving regular contributions, you will need to combine two formulae: our original compound interest formula — P(1+r/n)^(nt) — plus the future value of a series formula for the monthly deposits.

These formulae assume that your frequency of compounding is the same as the periodic payment interval (monthly compounding, monthly contributions, etc).

If the additional deposits are made at the END of the period (end of month, year, etc), here are the two formulae you need:

Compound interest for principal:

P(1+r/n)^(nt)

Future value of a series:

PMT × {[(1 + r/n)^(nt) - 1] / (r/n)}

If the additional deposits are made at the BEGINNING of the period (beginning of year, etc), here are the two formulae you need:

PMT × {[(1 + r/n)^(nt) - 1] / (r/n)} × (1+r/n)

• PMT = monthly payment amount

If an amount of $5,000 is deposited into a savings account at an annual interest rate of 3%, compounded monthly, with additional deposits of $100 per month (made at the end of each month). The value of the investment after 10 years can be calculated as follows...

P = 5000. PMT = 100. r = 3/100 = 0.03 (decimal). n = 12. t = 10.

Let's plug those figures into our formulae and use our PEMDAS order of operations to create our calculation...

• Total = [ Compound interest for principal ] + [ Future value of a series ]
• Total = [ P(1+r/n)^(nt) ] + [ PMT × (((1 + r/n)^(nt) - 1) / (r/n)) ]
• Total = [ 5000 × (1 + 0.03 / 12)^(12 × 10) ] + [ 100 × (((1 + 0.03 / 12)^(12 × 10) - 1) / (0.03 / 12)) ]
• Total = [ 5000 × (1 + 0.0025)^(12 × 10) ] + [ 100 × (((1 + 0.0025)^(12 × 10) - 1) / (0.0025)) ]
• Total = [ 5000 × (1.0025)^120 ] + [ 100 × (((1.0025^120) - 1) / 0.0025) ]
• Total = [ 6746.77 ] + [ 100 × (0.34935354719 / 0.0025) ]
• Total = [ 6746.77 ] + [ 13974.14 ]
• Total = [ $20,720.91 ]

Our investment balance after 10 years therefore works out at $20,720.91.

To conclude

This article about the compound interest formula has expanded and evolved based upon your requests for adapted formulae and examples. So, I appreciate it's now quite long and detailed. That said, I hope you've found it useful. Please feel free to share any thoughts in the comments section below.

Should you need any help with checking your calculations, please make use of our popular compound interest calculator and daily compounding calculator .

Disclaimer: Whilst every effort has been made in building our calculator tools, we are not to be held liable for any damages or monetary losses arising out of or in connection with their use. Full disclaimer .

• Discounting and compounding . John A. Dutton e-Education Institute.

Your comments

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Calculate Compound Interest

Formula for Compound Interest

Compound interest is a great thing when you are earning it! Compound interest is when a bank pays interest on both the principal (the original amount of money)and the interest an account has already earned.

To calculate compound interest use the formula below. In the formula, A represents the final amount in the account after t years compounded 'n' times at interest rate 'r' with starting amount 'p' .

This page focuses on understanding the formula for compound interest ; if you're interested in taking a deeper dive into how compound interest works and exploring some real world examples, please read our article here .

Practice Problems

If you have a bank account whose principal = $1,000, and your bank compounds the interest twice a year at an interest rate of 5%, how much money do you have in your account at the year's end?

If you start a bank account with $10,000 and your bank compounds the interest quarterly at an interest rate of 8%, how much money do you have at the year's end? (assume that you do not add or withdraw any money from the account)

The first credit card that you got charges 12.49% interest to its customers and compounds that interest monthly. Within one day of getting your first credit card, you max out the credit limit by spending $1,200.00. If you do not buy anything else on the card and you do not make any payments, how much money would you owe the company after 6 months?

Note: since the duration of time is half of a year, the value of t is ½.

You win the lottery and get $1,000,000. You decide that you want to invest all of the money in a savings account. However, your bank has two different plans.

The bank gives you a 6% interest rate and compounds the interest each month.

The bank gives you a 12% interest rate and compounds the interest every 2 months .

Question: In 5 years from now, which plan will provide you with more money.

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How to Solve Compound Interest Problems?

Compound interest is an interest that is accumulated on the principal and interest together over a given time. In this article, let's familiarize ourselves with solving compound interest problems.

Simple interest is interest that is paid only on the principal of money, while compound interest is interest that is paid on principal and compound interest at regular intervals.

Related Topics

• How to Solve Simple Interest Problems

A step-by-step guide to solving compound interest problems

Compound interest is the interest paid on principal and interest that is combined at regular intervals. At regular intervals, the interest so far accumulated is added to the existing principal amount and then the interest is calculated for the new principal. The new principal is equal to the sum of the Initial principal, and the interest accumulated so far.

Compound interest \(=\) interest on principal \(+\) compounded interest at regular intervals

The compound interest is calculated at regular intervals like annually( yearly), semi-annually, quarterly, monthly, etc.

Compound interest formula

The compound interest is calculated after calculating the total amount over some time, based on the interest rate and the initial principle. For an initial principal of \(P\), rate of interest per annum of \(r\), period \(t\) in years, frequency of the number of times the interest is compounded annually \(n\), the formula for calculation of amount is as follows. 

\(\color{blue}{A=P\left(1+\frac{r}{n}\right)^{nt}}\)

The above formula represents the total amount at the end of the period and includes compound interest and principal. In addition, we can calculate the compound interest by subtracting the principal from this amount. The formula for calculating compound interest is as follows:

\(\color{blue}{CI=P\left(1+\frac{r}{n}\right)^{nt}-P}\)

In the above expression:

• \(P\) is the principal amount
• \(r\) is the rate of interest (decimal)
• \(n\) is the frequency or no. of  times the interest is compounded annually
• \(t\) is the overall tenure.

It should be noted that the above formula is a general formula when the principal is compounded \(n\) several times in a year. If the given principal is compounded annually, the amount after the period at the percent rate of interest, \(r\), is given as:

\(\color{blue}{A=P\left(1+\frac{r}{100}\right)^t}\)

\(\color{blue}{C.I.=P\left(1+\frac{r}{100}\right)^t-P}\)

Compound interest formula for different periods

The compound interest for a given principle can be calculated for different periods using different formulas.

Compound interest formula – quarterly

If the period for calculating the interest is quarterly, the profit is calculated every three months and the amount is combined \(4\) times a year. The formula to calculate the compound interest when the principal is compounded quarterly is given:

\(\color{blue}{C.I.=P\left(1+\frac{\frac{r}{4}}{100}\right)^{4t}-P}\)

Here the compound interest is calculated for the quarterly period, so the interest rate r is divided by \(4\) and the period is quadrupled. The formula to calculate the amount when the principal is compounded quarterly is given: 

\(\color{blue}{A=P\left(1+\frac{\frac{r}{4}}{100}\right)^{4t}}\)

In the above expression,

• \(A\) is the amount at the end of the period
• \(P\) is the initial principal value, \(r\) is the rate of interest per annum
• \(t\) is the period
• \(C.I.\) is the compound interest.

Compound interest formula – half-yearly

Profit for compound interest varies based on the calculation period. If the profit calculation period is half-yearly, the profit is calculated once every six months and is combined twice a year. The formula for calculating compound interest if the principal is compounded semi-annually or half-yearly is given as:

\(\color{blue}{C.I.=P\left(1+\frac{\frac{r}{2}}{100}\right)^{2t}-P}\)

Here the compound interest is calculated for six months, so the interest rate \(r\) is divided by \(2\) and the period is doubled. The formula to calculate the amount when the principal is compounded semi-annually or half-yearly is given: 

\(\color{blue}{A=P\left(1+\frac{\frac{r}{2}}{100}\right)^{2t}}\)

Compound Interest – Example 1:

Davide lends \($3,000\) to John at an interest rate of \(10%\) per annum, compounded half-yearly for \(2\) years. Can you help him find out how much amount he gets after \(2\) years from John?

The principal amount \(P\) is \($3,000\). The rate of interest \(r\) is \(10%\) per annum. Conversion period \(=\) Half-year,  Rate of interest per half-year \(= \frac{10%}{2}= 5%\). The time period \(t\) is \(2\) years. The compounding frequency \(n\) is \(2\). 

Now, substitute the given data in the compound interest formula: \(A\:=\:P\left(1+\frac{\frac{r}{2}}{100}\right)^{2n}\)

\(A=3,000\left(1+\frac{\frac{10}{2}}{100}\right)^{2\times 2}\)

\(=3646.51\)

Exercises for Solving Compound Interest Problems

• What interest rate do you need to turn \($1,000\) into \($5,000\) in \(20\) Years?
• If you deposit \($5,000\) into an account paying \(6%\) annual interest compounded monthly, how long until there is \($8,000\) in the account?
• \(\color{blue}{8.38%}\)
• \(\color{blue}{7.9}\)

by: Effortless Math Team about 2 years ago (category: Articles )

Effortless Math Team

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Compound Interest Formula

Compound interest formula is mentioned and explained here along with a solved example. To recall, compound interest can be defined as “An interest on interest to the principal sum of a loan or deposit.” In simple words, the compound interest is the interest that adds back to the principal sum, so that interest is earned during the next compounding period. Here, we will discuss maths compound interest questions with solutions and formulas in detail.

The formula for the Compound Interest is,

This is the total compound interest which is just the interest generated minus the principal amount. For the total accumulated wealth (or amount), the formula is given as:

Notations in Compound Interest Formula:

Note: If the compounding frequency per annum is 1 i.e. if the interest is compounded annually, the compound interest formula is given as:

Also Check: Simple Interest Formula

Maths Compound Interest Questions with solutions

A sum of Rs. 50,000 is borrowed and the rate of interest is 10% per annum. What is the compound interest for 5 years?

From the formula for compound interest, we know,

C.I = P(1+R⁄100) t – P

Here, P = 50,000 ; R = 10% ; T = 5 years ; C.I=?

So, Compound Interest will be-

C.I. = 50000(1+10⁄100) 5 – 50000 = 80525.50 – 50000

So, compound interest will be Rs. 30525.50 while the total accumulated wealth (A) will be Rs. 80525.50 at the end of 5 years with an interest rate of 10% per annum.

Additional Compound Interest Links:

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6.1: Simple and Compound Interest

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Discussing interest starts with the principal , or amount your account starts with. This could be a starting investment, or the starting amount of a loan. Interest, in its most simple form, is calculated as a percent of the principal. For example, if you borrowed $100 from a friend and agree to repay it with 5% interest, then the amount of interest you would pay would just be 5% of 100: \(\$ 100(0.05)=\$ 5\). The total amount you would repay would be $105, the original principal plus the interest.

Simple One-time Interest

\[A=P+I=P+P r=P(1+r)\]

• \(I\) is the interest
• \(A\) is the end amount: principal plus interest
• \(P\) is the principal (starting amount)
• \(r\) is the interest rate (in decimal form. Example: \(5\% = 0.05\))

A friend asks to borrow $300 and agrees to repay it in 30 days with 3% interest. How much interest will you earn?

\(\begin{array}{ll} P=\$ 300 & \text{the principal } \\ r=0.03 & 3 \%\text{ rate} \\ I=\$ 300(0.03)=\$ 9. & \text{You will earn }\$ 9 \text{ interest.}\end{array}\)

One-time simple interest is only common for extremely short-term loans. For longer term loans, it is common for interest to be paid on a daily, monthly, quarterly, or annual basis. In that case, interest would be earned regularly. For example, bonds are essentially a loan made to the bond issuer (a company or government) by you, the bond holder. In return for the loan, the issuer agrees to pay interest, often annually. Bonds have a maturity date, at which time the issuer pays back the original bond value.

Suppose your city is building a new park, and issues bonds to raise the money to build it. You obtain a $1,000 bond that pays 5% interest annually that matures in 5 years. How much interest will you earn?

Each year, you would earn 5% interest: \(\$ 1000(0.05)=\$ 50\) in interest. So over the course of five years, you would earn a total of $250 in interest. When the bond matures, you would receive back the $1,000 you originally paid, leaving you with a total of $1,250.

We can generalize this idea of simple interest over time.

Simple Interest over Time

\(I=P r t\)

\(A=P+I=P+P r t=P(1+r t)\)

• \(r\) is the interest rate in decimal form
• \(t\) is time

The units of measurement (years, months, etc.) for the time should match the time period for the interest rate.

APR – Annual Percentage Rate

Interest rates are usually given as an annual percentage rate (APR) – the total interest that will be paid in the year. If the interest is paid in smaller time increments, the APR will be divided up.

For example, a \(6 \%\) APR paid monthly would be divided into twelve \(0.5 \%\) payments. A \(4 \%\) annual rate paid quarterly would be divided into four \(1 \%\) payments.

Example 3: Treasury Notes

Treasury Notes (T-notes) are bonds issued by the federal government to cover its expenses. Suppose you obtain a $1,000 T-note with a 4% annual rate, paid semi-annually, with a maturity in 4 years. How much interest will you earn?

Since interest is being paid semi-annually (twice a year), the 4% interest will be divided into two 2% payments.

\(\begin{array}{ll} P=\$ 1000 & \text{the principal } \\ r=0.02 & 2 \%\text{ rate} \\ t = 8 & \text{4 years = 8 half-years} \\ I=\$ 1000(0.02)(8)=\$ 160. & \text{You will earn }\$ 160 \text{ interest total over the four years.}\end{array}\)

Excercies 1

A loan company charges $30 interest for a one month loan of $500. Find the annual interest rate they are charging.

\(I=\$ 30\) of interest

\(P=\$ 500\) principal

\(r=\) unknown

\(t=1\) month

Compound Interest

With simple interest, we were assuming that we pocketed the interest when we received it. In a standard bank account, any interest we earn is automatically added to our balance, and we earn interest on that interest in future years. This reinvestment of interest is called compounding . We looked at this situation earlier, in the chapter on exponential growth.

\(A=P\left(1+\frac{r}{k}\right)^{kt}\)

\(A\) is the balance in the account after t years.

\(P\) is the starting balance of the account (also called initial deposit, or principal)

\(r\) is the annual interest rate in decimal form

\(k\) is the number of compounding periods in one year.

• If the compounding is done annually (once a year), \(k = 1\).
• If the compounding is done quarterly, \(k = 4\).
• If the compounding is done monthly, \(k = 12\).
• If the compounding is done daily, \(k = 365\).

The most important thing to remember about using this formula is that it assumes that we put money in the account once and let it sit there earning interest.

A certificate of deposit (CD) is a savings instrument that many banks offer. It usually gives a higher interest rate, but you cannot access your investment for a specified length of time. Suppose you deposit $3000 in a CD paying 6% interest, compounded monthly. How much will you have in the account after 20 years?

In this example,

\(\begin{array} {ll} P=\$ 3000 & \text{the initial deposit} \\ r = 0.06 & 6\% \text{ annual rate} \\ k = 12 & \text{12 months in 1 year} \\ t = 20 & \text{since we’re looking for how much we’ll have after 20 years} \end{array}\)

So \(A=3000\left(1+\frac{0.06}{12}\right)^{20 \times 12}=\$ 9930.61\) (round your answer to the nearest penny)

Let us compare the amount of money earned from compounding against the amount you would earn from simple interest

As you can see, over a long period of time, compounding makes a large difference in the account balance. You may recognize this as the difference between linear growth and exponential growth.

Evaluating exponents on the calculator

When we need to calculate something like \(5^3\) it is easy enough to just multiply \(5 \cdot 5 \cdot 5=125\). But when we need to calculate something like \(1.005^{240}\), it would be very tedious to calculate this by multiplying 1.005 by itself 240 times! So to make things easier, we can harness the power of our scientific calculators.

Most scientific calculators have a button for exponents. It is typically either labeled like:

\([\wedge ]\), \([y^x]\), or \([x^y]\)

To evaluate \(1.005^{240}\) we would type 1.005 \([^]\) 240, or 1.005 \([y^x]\) 240. Try it out - you should get something around 3.3102044758.

You know that you will need $40,000 for your child’s education in 18 years. If your account earns 4% compounded quarterly, how much would you need to deposit now to reach your goal?

We’re looking for \(P\).

\(\begin{array} {ll} r = 0.04 & 4\% \\ k = 4 & \text{4 quarters in 1 year} \\ t = 18 & \text{Since we know the balance in 18 years} \\ A = \$40,000 & \text{The amount we have in 18 years} \end{array}\)

In this case, we’re going to have to set up the equation, and solve for \(P\).

\[\begin{align*} 40000 &=P\left(1+\frac{0.04}{4}\right)^{18 \times 4} \\ 40000 &=P(2.0471) \\ P &=\frac{40000}{2.0471}=\$ 19539.84\end{align*}\]

So you would need to deposit $19,539.84 now to have $40,000 in 18 years.

It is important to be very careful about rounding when calculating things with exponents. In general, you want to keep as many decimals during calculations as you can. Be sure to keep at least 3 significant digits (numbers after any leading zeros). Rounding 0.00012345 to 0.000123 will usually give you a “close enough” answer, but keeping more digits is always better.

To see why not over-rounding is so important, suppose you were investing $1000 at 5% interest compounded monthly for 30 years.

\(\begin{array} {ll} P = \$1000 & \text{the initial deposit} \\ r = 0.05 & 5\% \\ k = 12 & \text{12 months in 1 year} \\ t = 30 & \text{since we’re looking for the amount after 30 years} \end{array}\)

If we first compute \(\frac{r}{k}\), we find \(\frac{0.05}{12} = 0.00416666666667\)

Here is the effect of rounding this to different values:

If you are working in a bank, of course you wouldn’t round at all. For our purposes, the answer we got by rounding to 0.00417, three significant digits, is close enough - $5 off of $4500 isn’t too bad. Certainly, keeping that fourth decimal place would not have hurt.

Using your calculator

In many cases, you can avoid rounding completely by how you enter things in your calculator. For example, in the example above, we needed to calculate

\(A=1000\left(1+\frac{0.05}{12}\right)^{12 \times 30}\)

We can quickly calculate \(12 \times 30=360\), giving \(A=1000\left(1+\frac{0.05}{12}\right)^{360}\).

Now we can use the calculator.

\(\begin{array}{|c|c|} \hline \textbf { Type this } & \textbf { Calculator shows } \\ \hline 0.05 [\div] 12 [=] & 0.00416666666667 \\ \hline [+] 11 [=] & 1.00416666666667 \\ \hline [\mathrm{y}^{\mathrm{x}}] 360 [=] & 4.46774431400613 \\ \hline [\times] 1000 [=] & 4467.74431400613 \\ \hline \hline \end{array}\)

Using your calculator continued

The previous steps were assuming you have a “one operation at a time” calculator; a more advanced calculator will often allow you to type in the entire expression to be evaluated. If you have a calculator like this, you will probably just need to enter:

1000 \([\times]\) ( 1 \([+]\) 0.05 \([\div]\) 12 ) \([\text{y}^\text{x}]\) 360 \([=]\).

Exercise \(\PageIndex{2}\)

If $70,000 are invested at 7% compounded monthly for 25 years, find the end balance.

\[A = 70,000\left(1 + \frac{0.07}{12} \right)^{12(25)} = 400,779.27 \nonumber\]  

Because of compounding throughout the year, with compound interest the actual increase in a year is more than the annual percentage rate.  If $1,000 were invested at 10%, the table below shows the value after 1 year at different compounding frequencies:

If we were to compute the actual percentage increase for the daily compounding, there was an increase of $105.16 from an original amount of $1,000, for a percentage increase of \(\frac{105.16}{1000} = 0.10516= 10.516\% \) increase.  This quantity is called the annual percentage yield (APY) .

Notice that given any starting amount, the amount after 1 year would be

\[A = P\left(1 + \frac{r}{k} \right)^k\nonumber\].

To find the total change, we would subtract the original amount, then to find the percentage change we would divide that by the original amount:

\[\frac{P\left(1 + \frac{r}{k} \right)^k - P}{P} = \left(1 + \frac{r}{k} \right)^k - 1.\nonumber \]

Definition: Annual Percentage Yield

The annual percentage yield  is the actual percent a quantity increases in one year.  It can be calculated as

\[ APY = \left(1 + \frac{r}{k} \right)^k - 1\nonumber\]

Notice this is equivalent to finding the value of $1 after 1 year, and subtracting the original dollar.

Example \(\PageIndex{7}\)

Bank \(A\) offers an account paying 1.2% compounded quarterly.  Bank \(B\) offers an account paying 1.1% compounded monthly.  Which is offering a better rate?

We can compare these rates using the annual percentage yield – the actual percent increase in a year.

Bank \(A\):  \(APY = \left(1 + \frac{0.012}{4} \right)^4 - 1 = 0.012054 = 1.2054\% \)

Bank \(B\): \(APY = \left(1 + \frac{0.011}{12} \right)^{12} - 1 = 0.011056 = 1.1056\% \)

Bank \(B\)’s monthly compounding is not enough to catch up with Bank \(A\)’s better APR.  Bank \(A\) offers a better rate.

Example \(\PageIndex{8}\)

If you invest $2000 at 6% compounded monthly, how long will it take the account to double in value?

This is a compound interest problem, since we are depositing money once and allowing it to grow.  In this problem,

\(P = \$2000\)                 the initial deposit

\(r = 0.06\)                     6% annual rate

\(k = 12\)                       12 months in 1 year

So our general equation is \(A = 2000\left( 1 + \frac{0.06}{12} \right)^{12t}\).  We also know that we want our ending amount to be double of $2000, which is $4000, so we’re looking for \(t\) so that \(A = 4000\).  To solve this, we set our equation for \(A\) equal to 4000.

\(4000 = 2000\left(1 + \frac{0.06}{12} \right)^{12t}\)                  Divide both sides by 2000

\(2 = \left(1.005 \right)^{12t}\)                                   To solve for the exponent, take the log of both sides

\(\log \left( 2 \right) = \log \left( \left(1.005 \right)^{12t} \right)\)                     Use the exponent property of logs on the right side

\(\log \left( 2 \right) = 12t\log \left( {1.005} \right)\)                       Now we can divide both sides by \(12\log(1.005)\)

\(\frac{\log \left( 2 \right)}{12\log \left(1.005 \right)} = t\)                                    Approximating this to a decimal

\(t = 11.581\)

It will take about 11.581 years for the account to double in value.  Note that your answer may come out slightly differently if you had evaluated the logs to decimals and rounded during your calculations, but your answer should be close.  For example if you rounded \(\log(2)\) to 0.301 and \(\log(1.005)\) to 0.00217, then your final answer would have been about 11.577 years.

Important Topics  of this Section

Simple interest

Compound interest

Compounding frequency

Evaluating on a calculator

Compound Interest Word Problems

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