Alternative Format — Lesson 2: Banking and Simple Interest

Let's Start Thinking

Quotes About Money

In this lesson, we will begin looking at financial applications of sequences. Connected with that, let's take a moment to reflect on some quotes about money and our relationship to it from a wide range of historical figures.

Epictetus was a Greek philosopher who lived around 2000 years ago. He said:

“Wealth consists not in having great possessions, but in having few wants.”
-Epictetus (Berger, 2014)

And around the late 1600s or early 1700s, Jonathan Swift, who you may know as the author of Gulliver's Travels, said:

“A wise person should have money in their head, but not in their heart.”

-Jonathan Swift (Berger, 2014)

This means that we need to be conscious about what we do with our money, but not emotionally consumed by it.

And in the 1900s, author Ayn Rand said:

“Money is only a tool. It will take you wherever you wish, but it will not replace you as the driver.”
-Ayn Rand (Rand, 1957)

And perhaps circus legend PT Barnum summed it up best:

“Money is a terrible master but an excellent servant.”
-PT Barnum (Berger, 2014)

Money is the topic of an enormous number of books. Canadian author David Chilton wrote a book called The Wealthy Barber Returns, in which he writes about spending and investing. And a quote from Chilton is:

“Sometimes when people ask you to do something, you’ll have to reply 'I can’t afford it.'”
-David Chilton (Chilton, 2011)

Perhaps you don't agree with every detail in these five quotes. But the overall message is clear. It is important to develop habits of spending that keep you in control of your money. Your future self will thank you for it.

In this lesson, we will look at different ways that we can save money, and we will look at calculating simple interest.

References

1. Berger, R. (2014, April 30). Top 100 Money Quotes of All Time. Forbes. Retrieved from https://www.forbes.com/sites/robertberger/2014/04/30/top-100-money-quotes-of-all-time/#55e9b364998d
2. Berger, R. (2014, April 30). Top 100 Money Quotes of All Time. Forbes. Retrieved from https://www.forbes.com/sites/robertberger/2014/04/30/top-100-money-quotes-of-all-time/#55e9b364998d
3. Berger, R. (2014, April 30). Top 100 Money Quotes of All Time. Forbes. Retrieved from https://www.forbes.com/sites/robertberger/2014/04/30/top-100-money-quotes-of-all-time/#55e9b364998d
4. Chilton, D. (2011). The Wealthy Barber Returns. Kitchener: Financial Awareness Corporation.
5. Rand, A. (1957). Atlas Shrugged. New York: New American Library.

Banking and Investing Choices

An Introduction to Banking

In the past decades, banks have seen several changes in how they operate, most notably with respect to the banks' locations and how customers access their accounts.

Locations:

• Bricks and mortar: Banks that have physical locations ("branches") where you do your banking are called "bricks and mortar" banks. Traditionally, all banks used to operate in this way.

• Online: Some banks have no or few physical branches. Customers do all of their banking online or at automated kiosks.

In the remainder of this lesson part, we will look at several topics related to banking:

• Chequing accounts
• Savings accounts
• Guaranteed Investment Certificates
• Mutual Funds
• Tax-saving investments

Chequing Accounts

A chequing account is a convenient and safe place to store your money with a bank. You can access your money in several ways, usually including:

• making withdrawals, where you take cash out of your account from a teller or ATM;
• writing cheques, where you give a cheque to another person who can then redeem that at their bank for the amount indicated; or,
• sending e-transfers, where you send a link to another person's email who can then have the amount indicated deposited into their bank account.

Source: Cheque Book - DNY59/iStock/Getty Images

Chequing accounts often have fees associated with them (especially with bricks and mortar banks) and earn little to no interest. 

Chequing accounts are used for spending, not saving. Other options are better for money that you will not need for awhile.

Savings Accounts

A savings account offers higher interest than a chequing account, but it also has more strict rules about how you can access your money. For instance, fees to access your money may be higher and cheques cannot be written against a savings account. Further, the interest rates are still very low so that the amount of interest earned even in savings accounts is usually negligible.

Savings accounts are good for achieving some short-term financial goals or for keeping cash for emergencies. There are other investment opportunities, though, that pay better interest rates if you know that you will not need the money for a longer period of time. 

Alternatives to Savings Accounts

Guaranteed Investment Certificates (GICs)

A guaranteed investment certificate (or GIC) is an investment that typically pays higher interest than a savings account, but requires you to have no access to the money in the GIC for a set amount of time. For instance, you may have a GIC that guarantees that you will earn \(3\%\) interest each year, but only if you do not access that money for, say, two years.

A GIC is a good investment tool for short-term financial goals with no risk of losing money. But, investors need to be careful that they do not lock their money away for too long. For instance, if someone invests too much of their money in a two-year GIC and then goes to university or college after a year and half, they will not be able to use the money in their GIC to pay for their tuition.

By the way, this is profitable for banks because the bank takes the money that customers deposit and uses it to grant loans to other customers — at a higher interest rate.

A GIC is a very conservative investment; there is no risk to the investment since it is a Guaranteed Investment Certificate, but there are other investment tools — like mutual funds — that normally earn higher rates of return.

Mutual Funds

Tax-Saving Investments

Let's now consider two types of investments that have tax-saving benefits: Tax-Free Savings Accounts (TFSAs) and Registered Retirement Savings Plans (RRSPs).

First off, these are not different types of investments like an additional option to GICs and mutual funds. Both TFSAs and RRSPs can contain GICs or mutual funds; TFSAs and RRSPs are classifications of your investment that have special benefits and restrictions.

TFSAs and RRSPs are investments that help you to pay less income tax. Income tax is a charge by the federal and provincial governments to fund things like health care and education. The government has put rules in place to encourage people to save money in TFSAs or RRSPs by giving income tax breaks to investors.

Tax-Free Savings Accounts (TFSA)

Normally, income tax is charged on all of your earnings: your salary or wage; tips; commission; scholarships; investment interest; etc. In a TFSA, there is no income tax charged to the investment interest.

So while you will pay taxes on the money you use to invest in a TFSA, when you take the money out at a later date, you will not pay taxes on it or on any interest you received. This can be useful in financial planning because your money can increase in value tax-free.

The government sets a limit as to how much can be invested in a TFSA each year. Each person's annual limit is the same.

Registered Retirement Savings Plans (RRSP)

As the name indicates, an RRSP generally holds money that investors are saving for their retirement. It operates as a tax shelter. This means that the income tax that would normally have been paid on the amount invested in your RRSP is deferred until the money is withdrawn from the RRSP. The tax on the growth in the value of the RRSP (because of interest or other means of growth) is also deferred until the money is withdrawn from the RRSP.

Retirement - FatCamera/iStock/Getty Images

For example, suppose you earn \($50~000\) in a year. As a way to help low-income earners, the government does not charge income tax on approximately the first \($10~000\) you earn. Thus, your income tax would be a percentage of \($40~000\) instead of the full amount you earned. Suppose you invest \($5000\) in an RRSP for your retirement. This money is legally "hidden" or sheltered from the collection of income tax: the government will calculate your income tax based on \($35~000\) instead of \($40~000\). Eventually you will pay the tax on the \($5000\) you sheltered in your RRSP: when you retire and withdraw from your RRSP, the government will tax the amount you withdraw.

So an RRSP allows you to defer paying taxes until a later date. Further, those who have a higher income are charged a higher tax rate; we say that they are in a higher "tax bracket." By putting money in an RRSP, investors can try to remain in a lower tax bracket.

The government sets a limit as to how much can be invested in an RRSP each year. Each person's annual limit can be different as it is based on your previous years' incomes and RRSP deposits.

Additional Considerations

Inflation

In general, costs to purchase goods increase over time. This increase is measured by the inflation rate. For example, if someone buys a basket of goods at the grocery store today, that exact same basket of goods purchased next year will likely cost a little bit more, usually around \(2\%\) more. The inflation rate changes from year to year, and in some years it may even be negative (so deflation is experienced, where the average cost of items actually decreases).

Shopping Cart - RapidEye/iStock/Getty Images

Because the value of money is not constant over time, we often speak of the "purchasing power" of money. This is important when considering interest rates. If a GIC offers an interest rate of, say, \(3\%\) and inflation in that year is \(2.5\%\), the investor has only seen a real purchasing power increase of \(0.5\%\). And if the interest rate you are earning is lower than the inflation rate (as is normally the case in savings accounts and almost certainly the case in chequing accounts), then the investor actually sees a decrease in their purchasing power. The value of their money has decreased.

Fees

Further, bank fees need to be carefully assessed. Fees vary from one bank to another and should be studied before opening an account. Many banks offer no-fee children's accounts until the investor turns eighteen years old, after which time monthly or per transaction fees are charged. Typically, "bricks and mortar" banks have higher fees than online banks.

Other Types of Investments

This lesson has only presented an introduction to banking and types of accounts. There are many other investment options available: stocks, bonds, exchange-traded funds, etc.

There are also Registered Education Savings Plans (RESPs) that offer generous government incentives for children and youth to save for university or college education.

Simple Interest

Percent

In this lesson part, we will look at solving simple interest problems and see how simple interest relates to arithmetic sequences. Before we do so, here is a quick reminder of working with percentages.

The word percent (symbolized as \(\%\)) means "out of one hundred."
We can see that from the word itself:

• per means "out of" (think of a phrase like kilometres per hour), and
• cent means "hundred" (think of a word like century, meaning one hundred years).

Since percent is a measure out of \(100\), something like \(27\) percent can be written as

• a percent: \(27\%\)
• a fraction: \(\dfrac{27}{100}\)
• a decimal: \(0.27\)

We can calculate the percentage of a number by multiplying. For example, we can calculate \(25\% \) of \(32\):

\(\begin{align*} 25\% \text{ of } 32 &= 25\% \times 32\\ &=0.25 \times 32\\ &= 8 \end{align*}\)

We will make use of calculations involving percentages as we study simple interest.

Example 1

Let's take a look at an example of earning interest. 

Solution

Let's start by calculating the interest earned in one year. The yearly interest rate is \(3.2\%\), and Malik invests \($750\). Thus, the interest earned in one year is

\( 3.2\% \text{ of } 750 \)

And in math, "of" means multiply. So this is 

\( 3.2\% \text{ of } 750 = 3.2\% \times 750 \)

Next, we change the percent to a decimal by dividing by \(100\), which is the same as moving the decimal two places to the left. So we get

\(\begin{align*} 3.2\% \text{ of } 750 &= 3.2\% \times 750 \\ &= 0.032 \times 750\\ &=24 \end{align*}\)

Now the bank deposits this interest in a separate account. So in the next year, Malik still has \($750\) in the GIC. So he'll earn the same amount of interest in each year. Thus, the interest earned in four years is

\(4 \times 24 = 96\)

Therefore, Malik earns \($96\) in interest in four years.

Further Comments

Before we move on from this example, let's consider an alternate way that the interest could have been paid. This will illustrate the difference between two ways that interest is calculated — simple interest or compound interest.

Simple vs. Compound Interest

The example we just did is an example of simple interest. This is the case because, in this example

• interest is earned on the original amount invested only.

Suppose though that the bank would take Malik's annual interest and deposit it back into his GIC, not into a separate account. This would then become a compound interest situation because, in that case

• interest is earned on the original amount invested and on previously-earned interest.

Compounded interest will be a topic in a future lesson.

Definitions

Let's summarize with a couple of definitions.

Simple Interest Formula

We can now state the simple interest formula. 

For example, if the time of the investment is measured in years, \(r\) must be given as a yearly interest rate.

Accumulated Amount

The simple interest formula only calculates the interest earned on an investment or loan.

We can also speak about the accumulated amount.

Example 2

Solution

The word annual is commonly used in financial settings; it means yearly.

1. To determine the value of the account after \(5\) years, we first determine the interest earned in \(5\) years.

   We are given the following:

   • The principal invested, in dollars, is \(P=1500\).
   • The annual interest rate is \(r=4\%\) or \(r=0.04\).
   • The time in years is \(t=5\).

   This is enough information to determine \(I\) and then \(A\):

   \(\begin{align*} I&=Prt\\ &= 1500(0.04)(5)\\ &= 300 \end{align*}\)

   The accumulated amount after \(5\) years is:

   \(\begin{align*} A&= P+I\\ &= 1500+300\\ &=1800 \end{align*}\)

   Therefore, the accumulated amount in the account after \(5 \) years is \($1800\).

2. To show that the annual balances form an arithmetic sequence, we first calculate the interest earned in one year:

   \(\begin{align*} 4\% \text{ of } 1500 &= 0.04 \times 1500\\ &= 60 \end{align*}\)

   Since the account pays simple interest, the amount of interest earned in each year is the same.

   Thus, the sequence of balances is defined recursively as:

   \(\begin{align*} &t_1=1500\\ &t_n=t_{n-1}+60 \end{align*}\)

   for \(n \ge 2\).

   This leads to the sequence of balances:

   \(1500,~1560,~1620,~1680, \ldots\)

   This sequence is arithmetic with common difference \(60\).

3. To show that the annual balances grow linearly, consider this table of annual balances:

   Year	Balance at End of Year (in dollars)
   \(0\)	\(1500\)
   \(1\)	\(1560\)
   \(2\)	\(1620\)
   \(3\)	\(1680\)
   \(4\)	\(1740\)
   \(5\)	\(1800\)

   Since the balances increase by \($60\) each year, the first differences are all \(60\).

   Therefore, the balances grow linearly with time.

4. If \(f(x)\) is the balance in the account at time \(x\), in years, then \(f(x)\) is the accumulated amount after \(x\) years:

   \(\begin{align*} f(x) &= P+I\\ &= P+Prt\\ &= 1500+1500(0.04)(x)\\ &= 1500+60x \end{align*}\)

   Notice that we again have shown that this is a linear relationship.

Simple Interest, Arithmetic Sequences, and Linear Relations

Since simple interest always grows at a constant rate, we can make the following statements connecting simple interest, arithmetic sequences, and linear relationships.

Check Your Understanding 1

Question

Determine the value of \($1200\) invested for \(5\) years in a simple interest account that pays \(4.3\%\) annually. Round your answer to the nearest cent.

Answer

The value after \(5\) years is \($1458.00\).

Feedback

We are given the following: 

• The principal invested, in dollars, is \(P=1200\). 
• The annual interest rate is \(r=4.3\%\) or \(r=0.043\). 
• The time in years is \(t=5\). 

This is enough information to determine \(I\) and then \(A\): 

Step 1: Calculate the value of the interest, \(I\)

\(\begin{align*} I&=Prt \\ &= 1200(0.043)(5) \\ &= 258.00 \end{align*}\)

Therefore, the interest earned in \(5\) years is \($258.00\).

Step 2: Calculate the value of the accumulated amount, \(A\)

\(\begin{align*} A&=P+I \\ &= 1200+258.00 \\ &= 1458.00 \end{align*}\)

Therefore, the accumulated amount in the account after \(5\) years is \($1458.00\).

Example 3

Solution

The accumulated amount, \(A\) or \(f(t)\), is given by:

\(\begin{align*} A &= P+I\\ &= P+Prt\\ &= P(1+rt) \end{align*}\)

Comparing \(f(t)=1300(1+0.03t)\) and \(A=P(1+rt)\), we see that:

• \(P=1300\), and
• \(r=0.03\) or \(3\%\).

Therefore, \(f(t)=1300(1+0.03t)\) represents the accumulated amount after \(t\) years when \($1300\) is invested in a simple interest account paying \(3\%\) per year.

Example 4

Solution — Part A

We are given the following:

• The interest owing, in dollars, is \(I=150\).
• The principal borrowed, in dollars, is \(P=1250\).
• The annual interest rate is \(r=0.04\).

This is enough information to determine the time of the loan, \(t\):

\(\begin{align*} I&=Prt\\ t &= \dfrac{I}{Pr}\\ &= \dfrac{150}{1250(0.04)}\\ &= 3 \end{align*}\)

Therefore, the money is borrowed for \(3\) years.

Solution — Part B

We are given the following:

• The interest earned, in dollars, is \(I=153\).
• The monthly interest rate is \(r=0.015\).
• The investment lasts for \(1\) year. Since \(r\) is a monthly interest rate, we require time in months, so \(t=12\).

This is enough information to determine the principal, \(P\):

\(\begin{align*} I&=Prt\\ P &= \dfrac{I}{rt}\\ &= \dfrac{153}{0.015(12)}\\ &= 850 \end{align*}\)

Therefore, the principal is \($850\).

Check Your Understanding 2

Question — Version 1

The interest on a simple interest investment is \($234.00\). The simple interest rate is \(1.3\%\) per month. Determinte the time (in years) if the principal is \($1500\).

Answer — Version 1

The number of years is \(1\).

Feedback — Version 1

We are given the following information:

• The interest earned, in dollars, is \(I=$234.00\).
• The principal, in dollars, is \(P=$1500\). 
• The monthly interest rate is \(r=1.3/100=0.013\). 

This is enough information to determinte the time, \(t\): 

\(\begin{align*} I &= Prt \\ t&= \dfrac{I}{Pr} \\ &= \dfrac{234.00}{1500(0.013)} \\ &=12 \end{align*}\)

Since \(r\) is a monthly interest rate, this is the time in months. We divide by \(12\) to get the time in years. Therefore, the time is \(1\) year.

Question — Version 2

The interest on a simple interest loan is \($752.40\). The simple interest rate is \(1.9\%\) per month. Determine the principal if the time is \(3\) years. Round to the nearest dollar.

Answer — Version 2

The principal is \($1100\).

Feedback — Version 2

We are given the following information:

• The interest owing, in dollars, is \(I=$752.40\). 
• The monthly interest rate is \(r=1.9/100=0.019\).
• The investment lasts for \(3\) years. Since \(r\) is a monthly interest rate, we require time in months, so \(t=12(3)=36\). 

This is enough information to determinte the principal, \(P\):

\(\begin{align*} I &= Prt \\ P &= \dfrac{I}{rt} \\ &= \dfrac{752.40}{0.019(36)} \\ &=1100 \end{align*}\)

Therefore, the principal is \($1100\).

Example 5

Solution

Vijay already has a principal of \($1200\); to reach his goal, he requires an accumulated amount of \($1500\). Thus, he requires the interest earned to be \(I=300\).

Therefore, we are given the following:

• The interest earned, in dollars, is \(I=300\).
• The principal, in dollars, is \(P=1200\).
• Since he needs this money in two and a half years, the time of the investment in years is \(t=2.5\).

This is enough information to determine the rate, \(r\):

\(\begin{align*} I&=Prt\\ r &= \dfrac{I}{Pt}\\ &= \dfrac{300}{1200(2.5)}\\ &= 0.1 \end{align*}\)

Expressing \(r=0.1\) as a percentage, we see that Vijay requires an annual simple interest rate of \(10\%\). Vijay's financial goal seems unreasonable since it is unlikely that he will be able to invest money in a short-term investment that will yield such a high interest rate. 

Check Your Understanding 3

Question

Determine the annual simple interest rate, expressed as a percent, that is required to grow an investment from \($800\) to \($833.60\) in \(2\) years.

Answer

The annual simple interest rate is \(2.1\%\).

Feedback

The account starts at \($800\); to reach a value of \($833.60\), a total of \($33.60\) of interest is required.

Therefore, we are given the following:

• The interest earned, in dollars, is \(I=$33.60\).
• The principal, in dollars, is \(P=$800\).
• The time of the investment in years is \(t=2\).

This is enough information to determine the annual rate, \(r\):

\(\begin{align*} I&=Prt \\ r &= \dfrac{I}{Pt} \\ &= \dfrac{33.60}{800(2)} \\ &= 0.021 \end{align*}\)

Expressing \(r=0.021\) as a percentage, we see that the required annual simple interest rate is \(2.1\%\).

Wrap-Up

Take It With You

Previously in this lesson, we considered a situation in which Malik invested \($750\) in a \(4\)-year GIC that paid \(3.2\%\) interest per year. Suppose Malik was able to re-invest the interest each year back into the GIC and earn interest on the interest. Can you modify what you learned in this lesson to determine how much Malik would have in his account if he chose this option?