5.8: Compound Interest
Introduction
A Question of Banking
“Wow! $4320 is a terrific amount!” Jeremy exclaimed when Candice filled him in on the balance in the savings account.
“It is great,” Marcus commented, “But I think we would have made out better if the interest had been compounded monthly.”
“Really, what does that mean?” Jeremy asked.
“It means that the interest is earned and then reinvested and you earn interest on the earned interest,” Marcus explained.
“Really?” Candice asked.
“Yes, let me explain,” Marcus said.
Before Marcus explains, it is time to learn about compound interest. Once you have learned the information in this lesson you will be ready to figure out if Marcus is correct.
What You Will Learn
By the end of this lesson, you will be able to demonstrate the following skills.
- Use the compound interest formula \begin{align*}A = p(1 + r)^t\end{align*} to find the new balance after a given time.
- Compare balances after a given time at the same rate of simple interest and compound interest.
- Compare amounts earned at the same rate compounded at different periods over the same time.
- Solve real – world problems involving compound interest.
Teaching Time
I. Use the Compound Interest Formula \begin{align*}\underline{A = p(1 + r)^t}\end{align*} to find the New Balance After a Given Time
In the last section, we discussed the importance of interest. It is not only big business for banks and investors but it is a way of securing a comfortable retirement and reaching financial goals. Understanding interest helps you to make the best decisions. The last section dealt with simple interest which illustrates the basic idea of interest.
In most cases in the real world, however, interest is calculated not with the simple interest formula \begin{align*}I = prt\end{align*} but with the compound interest formula \begin{align*}A = P(1+r)^t\end{align*}. Basically, this formula accounts for the fact that as you invest and earn interest, your balance grows. You are not only due interest, then, on your original balance, but on the new balance which includes the first installment(s) of interest. You get paid interest on the interest.
Let’s look at an example.
Example
You invest $100 for 3 years at 10% interest. After one year, you’d have the original amount plus the 10% interest. You’d have $110. Then, in the second year, you’d get paid the same 10% interest but not on just $100 but on $110. In the second year you’d earn $11 in interest and you’d have $121. In the third year you’d earn $12.10 and your ending balance would be $133.10. You got paid interest on interest and this made a difference in your final balance.
The example above shows compounding every year. That means that you get paid interest once a year on your balance. In the real world, interest is oftentimes compounded monthly or daily. For compound interest, we’ll use the formula \begin{align*}A = P(1+r)^t\end{align*} where \begin{align*}A\end{align*} is your final balance, \begin{align*}P\end{align*} is the principal amount, \begin{align*}r\end{align*} is the interest rate for the period (daily, monthly, semi-annually, annually, etc.) and \begin{align*}t\end{align*} is the number of time periods for which the money is invested. If compounding occurs monthly, there will be 12 periods per year.
In the example above, \begin{align*}P\end{align*} was $100, \begin{align*}r\end{align*} was 10%, and \begin{align*}t\end{align*} was 3 periods since the interest was compounded yearly. We would substitute these values in the compound interest formula:
\begin{align*}A &= P(1+r)^t\\ A &= 100(1+.10)^3\\ A &= 100(1.10)^3\\ A &= 100(1.331)\\ A &= 133.10\end{align*}
Using the formula we arrive at the same quantity discussed in the example.
Exactly! We use the formula to figure out the interest in a systematic way!
Example
What is the final balance on a savings account that earns 4.5% interest per year that is compounded annually (once per year) for 9 years if the beginning balance is $2,500?
Let’s start by taking the information that we have been given and substitute it into the formula.
\begin{align*}P &= 2500, r = .045, t = 9\\ A &= 2500(1+.045)^9\\ A &= 2500(1.045)^9\\ A &= 2500(1.486)\\ A &= 3715\end{align*}
The final balance is $3,715.
Oftentimes, the interest is compounded more frequently. For example, it could be compounded monthly instead of annually. In this case, the time period is only one month and the rate must be computed for each month, too.
Example
A fireman invests $40,000 in a retirement account for 2 years. The interest rate is 6%. The interest is compounded monthly. What will his final balance be?
Notice that we divide by 12 because there are twelve months in a year and the interest is compounded monthly.
\begin{align*}r &= \frac{.06}{12} = .005\\ P &= 40000, t = 24\\ A &= 40000(1+.005)^{24}\\ A &= 40000(1.005)^{24}\\ A &= 40000(1.127)\\ A &= 45080\end{align*}
Final balance is $45,080.
II. Compare Balances After a Given Time at the Same Rate of Simple Interest and Compound Interest
You may be wondering what the big difference is between simple interest and compound interest. At first glance, it doesn’t seem like there is a big difference, however, if you figure out the math, the difference will become clear.
Let’s see how different it might be using the same principal for the same amount of time at the same rate. However, one account will be paid simple interest and the other will be compounded yearly.
Account 1—Simple Interest | Account 2—Compound Interest |
---|---|
Principal Amount: $20,000 | Principal Amount: $20,000 |
Interest Rate: 8% per year | Interest Rate: 8% per year |
Time Frame: 20 years | Time Frame: 20 years |
Simple Interest formula \begin{align*}I = prt\end{align*} | Compound Interest formula \begin{align*}A = P(1+r)^t\end{align*} |
\begin{align*}p = 20000, r = .08, t = 20\end{align*} | \begin{align*}P = 20000, r = .08, t = 20\end{align*} |
\begin{align*}I = prt\end{align*} | \begin{align*}A = P(1+r)^t\end{align*} |
\begin{align*}I = 20000 \cdot .08 \cdot 20\end{align*} | \begin{align*}A = 20000(1+.08)^{20}\end{align*} |
\begin{align*}I = 32000\end{align*} | \begin{align*}A = 20000(1.08)^{20}\end{align*} |
\begin{align*}20000+32000 = 52000\end{align*} | \begin{align*}A = 20000(4.661)\end{align*} |
\begin{align*}A = 93220\end{align*} |
Simple interest gives you a balance of $52,000...not bad. Compound interest gives you a final balance of $93,220. That’s a huge difference! The total is over $40,000 more by compounding the interest.
Example
Compare the final balances on a principal of $10,000 paid simple interest of 5% for a year or compound interest of 5% per year compound monthly. Then compare after 5 years and 10 years.
Years | Simple Interest | Compound Interest |
---|---|---|
1 | $10,500 | $10,512 |
5 | $12,500 | $12,834 |
10 | $15,000 | $16,470 |
So you can see that there is big difference between the simple interest calculation and the compound interest calculation.
III. Compare Amounts Earned at the Same Rate Compounded at Different Periods over the Same Time
Compounding interest certainly makes a big difference compared to simple interest. What about the frequency of compounding? In some of the examples above, we saw annual compounding, quarterly compounding, and monthly compounding. With the ease of modern technology, most banks actually compound daily. Let’s see an example.
Compare the same principal invested for the same amount of time at the same rate with compounding annually and compounding daily.
Example
George invests $15,000 for 10 years in a bank that pays 6% and compounds annually.
David invests $15,000 for 10 years in a bank that pays 6% and compounds daily.
What will the difference be in their ending balances?
Let’s substitute the given information into the formulas and then calculate the balances.
George: | David: |
---|---|
\begin{align*}r = .06\end{align*} | \begin{align*}r = \frac{.06}{365} = .0001643\end{align*} |
\begin{align*}P = 55000, t = 120\end{align*} | \begin{align*}P = 15000, t = 3650\end{align*} |
\begin{align*}A = 15000(1+.06)^{10}\end{align*} | \begin{align*}A = 15000(1+.0001643)^{3650}\end{align*} |
\begin{align*}A = 15000(1.06)^{10}\end{align*} | \begin{align*}A = 15000(1.0001643)^{3650}\end{align*} |
\begin{align*}A = 15000(1.79085)\end{align*} | \begin{align*}A = 15000(1.8215)\end{align*} |
\begin{align*}A = 26862.75\end{align*} | \begin{align*}A = 27322.50\end{align*} |
George’s ending balance is $26,862.75 while David’s is $27,322.50, a difference of $459.75 because of daily compounding.
You can see that amounts that are compounded at different periods create different amounts of income.
III. Solve Real – World Problems Involving Compound Interest
When you consider real-world problems, be sure to understand the situation completely. Answer all parts of the question that is being posed. Also, you may have noticed in the examples that rounding occurred at different decimal place levels. It is important to choose the rounding based on the situation. In other words, if you are working with money, it is appropriate to round to the hundredths place because that is the place to which we have coins and make prices. In some cases, more decimal places are needed to compare numbers accurately while in yet other cases having too many decimal places is cumbersome and unnecessary.
Example
Mr. Tomkins wants to invest a $10,000 bonus that he received from work. He will be investing it for 10 years and has two options: 1) a 6% interest rate compounded monthly or 2) an 8% simple interest rate. Which one will yield him a greater ending balance?
\begin{align*}&\underline{\text{Option} \ 1} \qquad \qquad \qquad \qquad \qquad \quad \underline{\text{Option} \ 2}\\ & A = P(1+r)^t\\ & A = 10000(1+.005)^{120} \qquad \qquad \quad \ \ I = prt\\ & A = 10000(1.005)^{120} \qquad \qquad \qquad \quad I = 10000 \cdot .08 \cdot 10\\ & A = 10000(1.8194) \qquad \qquad \qquad \quad \ \ I = 8000\\ & A = 18194 \qquad \qquad \qquad 10000+8000 = 18000\end{align*}
Option 1 is better because he would have an ending balance of $18,194. In option 2, he would have $194 less.
Now let’s go back to the problem from the introduction and work on figuring it out using what we have learned about compound interest.
Real-Life Example Completed
A Question of Banking
Here is the original problem once again. Reread it and then solve it for the balance if the interest had been compound interest. Then compare it with the simple interest balance to see which method is more profitable.
“Wow! $4320 is a terrific amount!” Jeremy exclaimed when Candice filled him in on the balance in the savings account.
“It is great,” Marcus commented, “But I think we would have made out better if the interest had been compounded monthly.”
“Really, what does that mean?” Jeremy asked.
“It means that the interest is earned and then reinvested and you earn interest on the earned interest,” Marcus explained.
“Really?” Candice asked.
“Yes, let me explain,” Marcus said.
There are two parts to your answer.
Solution to Real – Life Example
First, let’s figure out the balance if the interest had been compound interest.
\begin{align*}A &= P(r + 1)^t\\ A &= 4000(.04 + 1)^{24}\\ A &= 4000 (1.04)^{24}\\ A &= 4000(2.56)\\ A &= \$10,240\end{align*}
In this case, the amount of the balance with compounded interest would have been more than double the balance with simple interest. Marcus was correct after all!
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Simple Interest
- interest calculated by only considering the principal times the rate times the time.
- Compound Interest
- when interest is earned, it is reinvested and you earn interest on the interest which is then added to the balance.
Time to Practice
Directions: Calculate the simple interest by using \begin{align*}I = PRT\end{align*}.
- Principal = $2000, Rate = 5%, Time = 3 years
- Principal = $12,000, Rate = 4%, Time = 2 years
- Principal = $10,000, Rate = 5%, Time = 5 years
- Principal = $30,000, Rate = 2.5%, Time = 10 years
- Principal = $12,500, Rate = 3%, Time = 8 years
- Principal = $34,500, Rate = 4%, Time = 10 years
- Principal = $16,000, Rate = 3%, Time = 5 years
- Principal = $120,000, Rate = 5%, Time = 4 years
Directions: Calculate the following compound interest calculated yearly.
- Principal = $3000, Rate = 4%
- Principal = $5000, Rate = 3%
- Principal = $12,000, Rate = 2%
- Principal = $34,000, Rate = 5%
- Principal = $18,000, Rate = 3%
- Principal = $7800, Rate = 4%
- Principal = $8500, Rate = 3%
Directions: Calculate the following compound interest calculated quarterly. You may round when necessary.
- Principal = $300, Rate = 3%
- Principal = $500, Rate = 4%
- Principal = $7000, Rate = 3%
- Principal = $2000, Rate = 2%
- Principal = $4000, Rate = 3%
Directions: Calculate the following compound interest calculated monthly.
- Principal = $2000, Rate = 2%
- Principal = $300, Rate = 3%
- Principal = $4000, Rate = 2%
- Principal = $1600, Rate = 3%
- Principal = $5000, Rate = 4%
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