# 3.3 Savings Accounts And Short-Term Guaranteed Investment Certificates

## LEARNING OBJECTIVES

- Use simple interest in solving problems involving savings accounts and short-term GICs.

## Savings Accounts

A savings account is a deposit account that bears interest and has no stated maturity date. These accounts are found at most financial institutions, such as commercial banks (Royal Bank of Canada, TD Canada Trust, etc.), trusts (Royal Trust, Laurentian Trust, etc.), and credit unions (FirstOntario, Steinbach, Assiniboine, Servus, etc.). Owners of such accounts make deposits to and withdrawals from these accounts at any time, usually accessing the account at an automatic teller machine (ATM), at a bank teller, or through online banking.

A wide variety of types of savings accounts are available. This textbook focuses on the most common features of most savings accounts, including how interest is calculated, when interest is deposited, insurance against loss, and the interest rate amounts available.

**Calculation of Interest**. There are two common methods for calculating simple interest.- Accounts earn simple interest that is calculated based on the daily
*closing*balance of the account. The closing balance is the amount of money in the account at the end of the day. Therefore, any balances in the account throughout a single day do not matter. For example, if you start the day with $500 in the account and deposit $3,000 at 9:00 a.m., then withdraw the $3,000 at 4:00 p.m., your closing balance is $500. That is the principal on which interest is calculated, not the $3,500 in the account throughout the day. - Accounts earn simple interest based on a
*minimum*monthly balance in the account. For example, if in a single month you had a balance in the account of $900 except for one day, when the balance was $500, then only the $500 is used in calculating the entire month’s worth of interest.

- Accounts earn simple interest that is calculated based on the daily
**When Interest Is Deposited**. Interest is accumulated and deposited (paid) to the account once monthly, usually on the first day of the month. Thus, the interest earned on your account for the month of January appears as a deposit on February 1.**Insurance against Loss.**Canadian savings accounts at commercial banks are insured by the national Canada Deposit Insurance Corporation (CDIC), which guarantees up to $100,000 in savings. At credit unions, this insurance is usually provided provincially by institutions such as the Deposit Insurance Corporation of Ontario (DICO), which also guarantees up to $100,000. This means that if your bank were to fold, you could not lose your money (so long as your deposit was within the maximum limit). Therefore, savings accounts carry almost no risk.

**Interest Rate Amounts**. Interest rates are higher for investments that are riskier. Savings accounts carry virtually no risk, which means the interest rates on savings accounts tend to be among the lowest you can earn.

Although a wide range of savings accounts are available, these accounts generally follow one of two common structures when it comes to calculating interest. These structures are flat-rate savings accounts and tiered savings accounts. Only flat-rate savings accounts are discussed here.

A **flat-rate savings account** has a single interest rate that applies to the entire balance. The interest rate may fluctuate in sync with short-term interest rates in the financial markets.

## EXAMPLE

John invested $3,000 in a savings account with a simple interest rate of 4.25%. At the end of five years, how much interest did John earn?

**Solution:**

**Step 1:** The given information is

[latex]\begin{eqnarray*} P & = & \$3,000 \\ r & = & 4.25\% \mbox{ (per year)} \\ & = & 0.0425 \\ t & = & 5 \mbox{ years}\end{eqnarray*}[/latex]

**Step 2:** Solve for the amount of interest, [latex]I[/latex].

[latex]\begin{eqnarray*} I & = & P \times r \times t \\ & = & 3,000 \times 0.0425 \times 5 \\ & = & \$637.50 \end{eqnarray*}[/latex]

After five years, John earned $637.50 in interest.

## TRY IT

Lea invested $2,500 in a savings account earning 1.5% simple interest. After three years, Lea invested the maturity value from her savings account into a new savings account with a simple interest rate of 2.1%. Lea left the money in the second savings account for two years. What is the maturity value at the end of the entire five year time period?

**Click to see Solution**

Maturity value at the end of the first three years.

[latex]\begin{eqnarray*} S & = & P \times (1+r \times t) \\ & = & 2500 \times (1+0.015 \times 3) \\ & = & \$2,612.50 \end{eqnarray*}[/latex]

Maturity value at the end of the second two years.

[latex]\begin{eqnarray*} S & = & P \times (1+r \times t) \\ & = & 2612.50 \times (1+0.021 \times 2) \\ & = & \$2,722.23 \end{eqnarray*}[/latex]

At the end of five years, the maturity value is $2,722.23

To calculate the monthly interest for a flat-rate savings account:

- Identify the interest rate, opening balance, and the monthly transactions in the savings account.
- Set up a flat-rate table, as illustrated below. Create a number of rows equaling the number of monthly transactions (deposits or withdrawals) in the account plus one.

**Date****Closing Balance in Account****Number of Days****Simple Interest Earned**[latex]I=P \times r \times t[/latex] **Total Interest Earned** - For each row of the table, set up the date ranges for each transaction and calculate the balance in the account for each date range.
- Calculate the number of days that the closing balance is maintained for each row.
- Calculate the simple interest for each row of the table. Ensure that the rate and time are expressed in the same units. Do not round off the resulting interest amounts.
- Add up the Simple Interest Earned column and round off the answer to two decimals.

### NOTE

When you are calculating interest on any type of savings account, pay careful attention to the details on how interest is calculated and any restrictions or conditions on the balance that is eligible to earn the interest.

## EXAMPLE

The RBC High Interest Savings Account pays 0.75% simple interest on the daily closing balance in the account and the interest is paid on the first day of the following month. On March 1, 2023, the opening balance in the account was $2,400. On March 12, 2023, a deposit of $1,600 was made. On March 21, 2023, a withdrawal of $2,000 was made. Calculate the total simple interest earned for the month of March.

**Solution:**

**Step 1:** The given information is

[latex]\begin{eqnarray*} r & = & 0.75\% \end{eqnarray*}[/latex]

The following transactions dates are known.

- March 1 opening balance = $2,400
- March 12 deposit = $1,600
- March 21 withdrawal = $2,000

**Step 2: ** Set-up the flat-rate table.

- Determine the date ranges for each balance throughout the month and calculate the closing balances.
- For each row of the table, calculate the number of days between the two dates. (Note: you can use the built-in count days function on your financial calculator to count the number of days).
- For each row, calculate the amount of simple interest ([latex]I=P \times r \times t[/latex]).
- Add up the Total Interest Earned column.

Dates |
Closing Balance in Account |
Number of Days |
Simple Interest Earned |

March 1 to March 12 | $2400 | 11 | [latex]\begin{eqnarray*}I&= & 2400 \times 0.0075 \times \frac{11}{365}\\&= & \$0.542465\end{eqnarray*}[/latex] |

March 12 to March 21 | $2400 + $1600 = $4000 |
9 | [latex]\begin{eqnarray*}I&= & 4000 \times 0.0075 \times \frac{9}{365} \\ &= & \$0.739726\end{eqnarray*}[/latex] |

March 21 to April 1 | $4000 − $2000 = $2000 |
11 | [latex]\begin{eqnarray*}I&= & 2000 \times 0.0075 \times \frac{11}{365}\\ &= & \$0.452054\end{eqnarray*}[/latex] |

Total Interest Earned |
[latex]\begin{eqnarray*}I&= & 0.542465+0.739726+0.452054\\ &= &\$1.73\end{eqnarray*}[/latex] |

For the month of March, the savings account earned a total simple interest of $1.73, which was deposited to the account on April 1.

**TRY IT**

Michael has a savings account at Canadian Western Bank. The savings account pays 0.95% simple interest on the daily closing balance in the account. The interest is paid on the first day of the following month. On August 1, 2023, the opening balance in Michael’s account was $5,000. On August 10, 2023, Michael withdrew $1,100 from the account. On August 19, 2023, Michael deposited $900 into the account. On August 27, 2023, Michael withdrew $500 from the account. Calculate the total simple interest earned for the month of August.

**Click to see Solution**

Dates |
Closing Balance in Account |
Number of Days |
Simple Interest Earned |

Aug 1 to Aug 10 | $5000 | 9 | [latex]\begin{eqnarray*}I&= & 5000 \times 0.0095 \times \frac{9}{365}\\&= & \$1.171232877\end{eqnarray*}[/latex] |

Aug 10 to Aug 19 | $3900 | 9 | [latex]\begin{eqnarray*}I&= & 3900 \times 0.0095 \times \frac{9}{365} \\ &= & \$0.913561644\end{eqnarray*}[/latex] |

Aug 19 to Aug 27 | $4800 | 8 | [latex]\begin{eqnarray*}I&= & 4800 \times 0.0095 \times \frac{8}{365}\\ &= & \$0.999452055\end{eqnarray*}[/latex] |

Aug 27 to Sept 1 | $4300 | 5 | [latex]\begin{eqnarray*}I&= & 4300 \times 0.0095 \times \frac{5}{365}\\ &= & \$0.559589041\end{eqnarray*}[/latex] |

Total Interest Earned |
[latex]\begin{eqnarray*}I&= & 1.171232877+0.913561644+0.999452055+0.559589041\\ &= &\$3.64\end{eqnarray*}[/latex] |

## Short-Term Guaranteed Investment Certificates

A g**uaranteed investment certificate** (GIC) is an investment that offers a guaranteed rate of interest over a fixed period of time. GICs are found mostly at commercial banks, trust companies, and credit unions. In this section, you will deal only with short-term GICs, which are defined as those that have a term of less than one year.

Three factors determine the interest rate on a short-term GIC: principal, time, and redemption privileges.

**Principal**. Typically, a larger principal is able to realize a higher interest rate than a smaller principal.**Time**. The length of time that the principal is invested affects the interest rate. Short-term GICs range from 30 days to 364 days in length. A longer term usually realizes higher interest rates.**Redemption Privileges**. GICs come in two types: redeemable and nonredeemable. A**redeemable**GIC can be cashed in at any point before the maturity date, meaning that you can access your money any time you want it. A**nonredeemable**GIC “locks in” your money for the agreed-upon term. Accessing that money before the end of the term usually incurs a stiff financial penalty, either on the interest rate or in the form of a financial fee. Nonredeemable GICs carry a higher interest rate.

### NOTE

Short-term GICs involve a lump sum of money (the principal) invested for a fixed term (the time) at a guaranteed interest rate (the rate). Most commonly the only items of concern are the amount of interest earned and the maturity value. But, some instances may require the calculation of an interest rate or the term.

## EXAMPLE

Your parents have $10,000 to invest. They can either deposit the money into a 364-day nonredeemable GIC at Assiniboine Credit Union with a posted rate of 0.75% or they could put their money into back-to-back 182-day nonredeemable GICs with a posted rate of 0.7%. At the end of the first 182 days, they will reinvest both the principal and interest into the second GIC. The interest rate remains unchanged on the second GIC. Which option should they choose?

**Solution:**

**Option 1: 364-day GIC**

**Step 1:** The given information is

[latex]\begin{eqnarray*} P & = & \$10,000 \\ r & = & 0.75\% \mbox{ (per year)} \\ & = & 0.0075 \\ t & = & 364 \mbox{ days}\end{eqnarray*}[/latex]

**Step 2: ** Convert the time period from days to years.

[latex]\displaystyle{t=\frac{364}{365}}[/latex]

**Step 3:** Solve for the maturity value, [latex]S[/latex].

[latex]\begin{eqnarray*} S & = & P \times (1+r \times t) \\ & = & 10,000 \times \left(1+ 0.0075 \times \frac{364}{365} \right) \\ & = & \$10,074.79 \end{eqnarray*}[/latex]

**Option 2: Two consecutive 182-day GICs**

**Step 1:** The given information is

[latex]\begin{eqnarray*} P & = & \$10,000 \\ r & = & 0.75\% \mbox{ (per year)} \\ & = & 0.0075 \\ t & = & 182 \mbox{ days}\end{eqnarray*}[/latex]

**Step 2: ** Convert the time period from days to years.

[latex]\displaystyle{t=\frac{182}{365}}[/latex]

**Step 3:** Solve for the maturity value, [latex]S[/latex], for the first 182-day GIC.

[latex]\begin{eqnarray*} S & = & P \times (1+r \times t) \\ & = & 10,000 \times \left(1+ 0.0075 \times \frac{182}{365} \right) \\ & = & \$10,034.90 \end{eqnarray*}[/latex]

**Step 4:** Solve for the maturity value, [latex]S[/latex], for the second 182-day GIC.

[latex]\begin{eqnarray*} S & = & P \times (1+r \times t) \\ & = & 10,034.90 \times \left(1+ 0.0075 \times \frac{182}{365} \right) \\ & = & \$10,069.93 \end{eqnarray*}[/latex]

The 364-day GIC results in a maturity value of $10,074.79, while the two back-to-back 182-day GICs result in a maturity value of $10,069.93. Clearly, the 364-day GIC is the better option as it will earn $4.86 more in simple interest.

## TRY IT

If you place $25,500 into an 80-day short-term GIC at TD Canada Trust earning 0.55% simple interest, how much will you receive when the investment matures?

** Click to see Solution**

[latex]\begin{eqnarray*} S & = & 25,500 \times \left(1+ 0.0055 \times \frac{80}{365} \right) \\ & = & \$25,530.74 \end{eqnarray*}[/latex]

## TRY IT

Interest rates in the GIC markets are always fluctuating be cause of changes in the short-term financial markets. If you have $50,000 to invest today, you could place the money into a 180-day GIC at Canada Life earning a fixed rate of 0.4%, or you could take two consecutive 90-day GICs. The current posted fixed rate on 90-day GICs at Canada Life is 0.3%. Trends in the short-term financial markets suggest that within the next 90 days short-term GIC rates will be rising. What does the short-term 90-day rate need to be 90 days from now to arrive at the same maturity value as the 180-day GIC? Assume that the entire maturity value of the first 90-day GIC would be reinvested.

** Click to see Solution**

Maturity value of the 180-day GIC.

[latex]\begin{eqnarray*} S & = & 50,000 \times \left(1+0.004 \times \frac{180}{365} \right) \\ & = & \$50,098.63 \end{eqnarray*}[/latex]

Maturity value at the end of the first 90-day GIC.

[latex]\begin{eqnarray*} S & = & 50,000 \times \left(1+0.003 \times \frac{90}{365} \right) \\ & = & \$50,036.99\end{eqnarray*}[/latex]

Interest rate for the second 90-day GIC.

[latex]\begin{eqnarray*} 50,098.63 & = & 50,036.99 \times \left(1+r \times \frac{90}{365} \right) \\ 1.00123...& = & 1+r \times \frac{90}{365} \\ 0.00123... & = & r \times \frac{90}{365} \\ 0.005 & = & r \end{eqnarray*}[/latex]

The second 90-day GIC needs an interest rate of 0.5%

**Exercises**

- A savings account at your local credit union holds a balance of $5,894 for the entire month of September. The posted simple interest rate is 1.35%. Calculate the amount of interest earned for the month (30 days).

**Click to see Answer**$6.54

- If you place $25,500 into an 80-day short-term GIC at TD Canada Trust earning 0.55% simple interest, how much will you receive when the investment matures?

**Click to see Answer**$25,530.74

- In November of 2022, the opening balance on a savings account was $12,345. A deposit of $3,000 was made to the account on November 19, and a withdrawal of $3,345 was made from the account on November 8. If simple interest is paid at 0.95% based on the daily closing balance, how much interest for the month of November is deposited to the account on December 1?

**Click to see Answer**$8.57

- A 320-day short-term GIC earns 0.78% simple interest. If $4,500 is invested into the GIC, what is the maturity value and how much interest is earned?

**Click to see Answer**$4530.77, $30.77

- In February of a leap year, the opening balance on your savings account was $3,553. You made two deposits of $2,000 each on February 5 and February 21. You made a withdrawal of $3,500 on February 10, and another withdrawal of $750 on February 17. If simple interest is calculated on the daily closing balance at a rate of 1.45%, how much interest do you earn for the month of February?

**Click to see Answer**$3.63

- An investor places $30,500 into a short-term 120-day GIC at the Bank of Montreal earning 0.5% simple interest. The maturity value is then rolled into another short-term 181-day GIC earning 0.57% simple interest. Calculate the final maturity value.

**Click to see Answer**$30,636.49

- An investor with $75,000 is weighing options between a 200-day GIC or two back-to-back 100-day GICs. The 200-day GIC has a posted simple interest rate of 1.4%. The 100-day GICs have a posted simple interest rate of 1.35%. The maturity value of the first 100-day GIC would be reinvested in the second 100-day GIC (assume the same interest rate upon renewal). Which alternative is best and by how much?

**Click to see Answer**200 day GIC is better by $69.70

- Sun Life Financial Trust offers a 360-day short-term GIC at 0.65%. It also offers a 120-day short-term GIC at 0.58%. You are considering either the 360-day GIC or three consecutive 120-day GICs. For the 120-day GICs, the entire maturity value would be “rolled over” into the next GIC. Assume that the posted rate increases by 0.1% upon each renewal. If you have $115,000 to invest, which option should you pursue and how much more interest will it earn?

**Click to see Answer**Three back-to-back 120 GICs are better by $35.75

- Interest rates in the GIC markets are always fluctuating because of changes in the short-term financial markets. If you have $50,000 to invest today, you could place the money into a 180-day GIC at Canada Life earning a fixed rate of 0.4%, or you could take two consecutive 90-day GICs. The current posted fixed rate on 90-day GICs at Canada Life is 0.3%. Trends in the short-term financial markets suggest that within the next 90 days short-term GIC rates will be rising. What does the short-term 90-day rate need to be 90 days from now to arrive at the same maturity value as the 180-day GIC? Assume that the entire maturity value of the first 90-day GIC would be reinvested.

**Click to see Answer**0.4995%

- You are considering the following short-term GIC investment options for an amount of $90,000.
- Option 1: One 360-day GIC at 0.8% simple interest.
- Option 2: Two 180-day GICs at 0.75% simple interest.
- Option 3: Three 120-day GICs at 0.72% simple interest.
- Option 4: Four 90-day GICs at 0.715% simple interest.

In all cases, assume that the posted rates remain unchanged and that the entire maturity value will be reinvested in the next short-term GIC. Calculate the total maturity value for each option at the end of 360 days.

**Click to see Answer**$90,710.14, $90,666.99, $90,640.64, $90,636.36

#### Attribution

“8.3: Savings Accounts and Short-Term GICs” from Business Math: A Step-by-Step Handbook Abridged by Sanja Krajisnik; Carol Leppinen; and Jelena Loncar-Vines is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

“8.3: Application: Savings Accounts and Short-Term GICs” from Business Math: A Step-by-Step Handbook (2021B) by J. Olivier and Lyryx Learning Inc. through a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License unless otherwise noted.