5.3: Savings Accounts And Short-Term GICs
Formula & Symbol Hub
For this section you will need the following:
Symbols Used
- [latex]I=[/latex] Simple interest
- [latex]P=[/latex] Present value or principal
- [latex]r=[/latex] Interest rate
- [latex]S=[/latex] Maturity or future value
- [latex]t=[/latex] time period of transaction
Formulas Used
-
Formula 5.1 – Simple Interest
[latex]I=Prt[/latex]
-
Formula 5.2a – Simple Interest Future Value
[latex]S=P(1+rt)[/latex]
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.
- How Interest Is Calculated: 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 [latex]\$500[/latex] in the account and deposit [latex]\$3,000[/latex] at 9:00 a.m., then withdraw the [latex]\$3,000[/latex] at 4:00 p.m., your closing balance is [latex]\$500[/latex]. That is the principal on which interest is calculated, not the [latex]\$3,500[/latex] 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 [latex]\$900[/latex] except for one day, when the balance was [latex]\$500[/latex], then only the [latex]\$500[/latex] is used in calculating the entire month’s worth of interest.
- 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 [latex]\$100,000[/latex] 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 [latex]\$100,000[/latex]. 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. At the time of writing, interest rates on savings accounts ranged from a low of [latex]0.05\%[/latex] to a high of [latex]1.95\%[/latex]. Though this is not much, it is better than nothing and certainly better than losing money!
While 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. Each of these is discussed separately.
HOW TO
Calculate Interest for Flat-Rate Savings Accounts
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.
Step 1: Identify the interest rate, opening balance, and the monthly transactions in the savings account.
Step 2: Set up a flat-rate table as illustrated here. Create a number of rows equaling the number of monthly transactions (deposits or withdrawals) in the account plus one.
| Date | Closing Balance in Account | # of Days | Simple Interest Earned |
| [latex]I=Prt[/latex] | |||
| Total Interest earned | |||
Step 3: 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.
Step 4: Calculate the number of days that the closing balance is maintained for each row.
Step 5: Apply Formula 5.1[latex]I=Prt[/latex] (Simple Interest) to each row in the table. Ensure that rate and time are expressed in the same units. Do not round off the resulting interest amounts ([latex]I[/latex]).
Step 6: Sum the Simple Interest Earned column and round off to two decimals.
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 5.3.1
The RBC High Interest Savings Account pays [latex]0.75\%[/latex] 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, the opening balance in the account was [latex]\$2,400[/latex]. On March 12, a deposit of [latex]\$1,600[/latex] was made. On March 21, a withdrawal of [latex]\$2,000[/latex] was made. Calculate the total simple interest earned for the month of March.
Solution
Calculate the total interest amount ([latex]I[/latex]) for the month.
Step 1: Given variables:
The following transactions dates are known:
[latex]\begin{align*}\text{March 1 opening balance}&=\$2,400\\[2ex]\text{March 12 deposit}&=\$1,600\\[2ex]\text{March 21 withdrawal}&=\$2,000\end{align*}[/latex]
Step 2: Set up a flat-rate table (see table below).
Step 3: Determine the date ranges for each balance throughout the month and calculate the closing balances (see table below).
Step 4: For each row of the table, calculate the number of days involved.
Step 5: Apply simple interest formula [latex]I=Prt[/latex] to calculate simple interest on each row.
Step 6: Sum the Simple Interest Earned.
| Dates (Step 2) | Closing Balance in Account (Step 3) | # of Days (Step 4) | Simple Interest Earned ([latex]I=Prt[/latex]) (Step 5) |
|---|---|---|---|
| March 1 to March 12 | [latex]\$2,400[/latex] | [latex]12−1=11[/latex] | [latex]\begin{eqnarray*}I&=&\$2,400(0.0075)\left(\frac{11}{365}\right)\\I&=&\$0.542465\end{eqnarray*}[/latex] |
| March 12 to March 21 | [latex]\$2,400+\$1,600=\$4,000[/latex] | [latex]21−12=9[/latex] | [latex]\begin{eqnarray}I&=&\$4,000(0.0075)\left(\frac9{365}\right)\\I&=&\$0.739726\end{eqnarray}[/latex] |
| March 21 to April 1 | [latex]\$4,000−\$2,000=\$2,000[/latex] | [latex]31+1−21=11[/latex] | [latex]\begin{eqnarray}I&=&\$2,000(0.0075)\left(\frac{11}{365}\right)\\I&=&\$0.452054\end{eqnarray}[/latex] |
| Step 6: Total Monthly Interest Earned. | [latex]\begin{eqnarray}I&=&\$0.542465+\$0.739726+\$0.452054\\I&=&\$1.73\end{eqnarray}[/latex] | ||
Step 7: Write as a statement.
For the month of March, the savings account earned a total simple interest of [latex]\$1.73[/latex], which was deposited to the account on April 1.
Try It
1) Canadian Western Bank offers a Summit Savings Account with posted interest rates as indicated in the table below. Only each tier is subject to the posted rate, and interest is calculated daily based on the closing balance.
| Balance | Interest Rate |
|---|---|
| [latex]\$0[/latex] – [latex]\$5,000.00[/latex] | [latex]0\%[/latex] |
| [latex]\$5,000.01[/latex] – [latex]\$1,000,000.00[/latex] | [latex]1.05\%[/latex] |
| [latex]\$1,000,000.01[/latex] and up | [latex]0.80\%[/latex] |
December’s opening balance was [latex]\$550,000[/latex]. Two deposits in the amount of [latex]\$600,000[/latex] each were made on December 3 and December 21. Two withdrawals in the amount of [latex]\$400,000[/latex] and [latex]\$300,000[/latex] were made on December 13 and December 24, respectively. What interest for the month of December will be deposited to the account on January 1?
Solution
Step 1: Interest rates as per table in question.
[latex]\begin{align*}\text{December opening balance}&=\$550,000\\[2ex]\text{December 3 Deposit}&=\$600,000\\[2ex]\text{December 13 Withdrawal}&=\$400,000\\[2ex]\text{December 21 Deposit}&=\$600,000\\[2ex]\text{December 24 Withdrawal}&=\$300,000\end{align*}[/latex]
Step 2: See the information below as per the question.
Putting this information into a table can be helpful for visualizing what values to use.
Date: Dec 3 to Dec 13
Closing Balance In Account: [latex]\$550,000[/latex]
# of Days: [latex]3 - 1 = 2[/latex]
0% $0 to $5,000 (This portion only): [latex]\begin{eqnarray*}P&=&\$5,000\\I&=&\$0\end{eqnarray*}[/latex]
1.05% $5,000.01 to $1,000,000 (This portion only): [latex]\begin{eqnarray*}P&=&\$545,000\\I&=&\$545,000\left(0.0105\right)\left(\frac2{365}\right)\\I&=&\$31.356164\end{eqnarray*}[/latex]
Date: Dec 1 to Dec 3
Closing Balance In Account: [latex]\$550,000 + \$600,000=\$1,150,000[/latex]
# of Days: [latex]13 − 3 = 10[/latex]
0% $0 to $5,000 (This portion only): [latex]\begin{eqnarray*}P&=&\$5,000\\I&=&\$0\end{eqnarray*}[/latex]
1.05% $5,000.01 to $1,000,000 (This portion only):[latex]\begin{eqnarray*}P&=&\$995,000\\I&=&\$955,000\left(0.0105\right)\left(\frac{10}{365}\right)\\I&=&\$286.232876\end{eqnarray*}[/latex]
0.8% $1,000,000.01 and up (This portion only): [latex]\begin{eqnarray*}P&=&\$150,000\\I&=&\$150,000\left(0.008\right)\left(\frac{10}{365}\right)\\I&=&\$32.876712\end{eqnarray*}[/latex]
Date: Dec 13 to Dec 21
Closing Balance In Account: [latex]\$1,150,000 − \$400,000 = \$750,000[/latex]
# of Days: [latex]21 - 13 = 8[/latex]
0% $0 to $5,000 (This portion only): [latex]\begin{eqnarray*}P&=&\$5,000\\I&=&\$0\end{eqnarray*}[/latex]
1.05% $5,000.01 to $1,000,000 (This portion only): [latex]\begin{eqnarray*}P&=&\$745,000\\I&=&\$745,000\left(0.0105\right)\left(\frac8{365}\right)\\I&=&\$171.452054\end{eqnarray*}[/latex]
Date: Dec 21 to Dec 24
Closing Balance In Account: [latex]\$750,000 + \$600,000 = \$1,350,000[/latex]
# of Days: [latex]24 - 21 = 3[/latex]
0% $0 to $5,000 (This portion only): [latex]\begin{eqnarray*}P&=&\$5,000\\I&=&\$0\end{eqnarray*}[/latex]
1.05% $5,000.01 to $1,000,000 (This portion only): [latex]\begin{eqnarray*}P&=&\$995,000\\I&=&\$995,000\left(0.0105\right)\left(\frac3{365}\right)\\I&=&\$85.869863\end{eqnarray*}[/latex]
0.8% $1,000,000.01 and up (This portion only): [latex]\begin{eqnarray*}P&=&\$350,000\\I&=&\$350,000\left(0.008\right)\left(\frac3{365}\right)\\I&=&\$23.013698\end{eqnarray*}[/latex]
Date: Dec 24 to Jan 1
Closing Balance In Account: [latex]\$1,350,000 − \$300,000 = \$1,050,000[/latex]
# of Days: [latex]31 + 1 - 24 = 8[/latex]
0% $0 to $5,000 (This portion only): [latex]\begin{eqnarray*}P&=&\$5,000\\I&=&\$0\end{eqnarray*}[/latex]
1.05% $5,000.01 to $1,000,000 (This portion only): [latex]\begin{eqnarray*}P&=&\$995,000\\I&=&\$995,000\left(0.0105\right)\left(\frac8{365}\right)\\I&=&\$228.986301\end{eqnarray*}[/latex]
0.8% $1,000,000.01 and up (This portion only): [latex]\begin{eqnarray*}P&=&\$50,000\\I&=&\$50,000\left(0.008\right)\left(\frac8{365}\right)\\I&=&\$8.767123\end{eqnarray*}[/latex]
Step 3: Total Monthly Interest Earned [latex]I[/latex]:
[latex]\begin{eqnarray*}I&=&\$31.356164+\$286.232876+\$32.876712+\$171.452054+\$85.869863+\$23.013698+\$228.986301+\$8.767123\\I&=&\$868.55\end{eqnarray*}[/latex]
Step 4: Write as a statement.
The total monthly interest earned is [latex]\$868.55[/latex].
Tiered Savings Accounts
A tiered savings account pays higher rates of interest on higher balances in the account. This is very much like a graduated commission on gross earnings. For example, you might earn [latex]0.25\%[/latex] interest on the first [latex]\$1,000[/latex] in your account and [latex]0.35\%[/latex] for balances over [latex]\$1,000[/latex]. Most of these tiered savings accounts use a portioning system. This means that if the account has [latex]\$2,500[/latex], the first [latex]\$1,000[/latex] earns the [latex]0.25\%[/latex] interest rate and it is only the portion above the first [latex]\$1,000[/latex] (hence, [latex]\$1,500[/latex]) that earns the higher interest rate.
HOW TO
Calculate Monthly Interest for a Tiered Savings Account
Follow these steps to calculate the monthly interest for a tiered savings account:
Step 1: Identify the interest rate, opening balance, and the monthly transactions in the savings account.
Step 2: Set up a tiered interest rate table as illustrated below. Create a number of rows equaling the number of monthly transactions (deposits or withdrawals) in the account plus one. Adjust the number of columns to suit the number of tiered rates. Fill in the headers for each tiered rate with the balance requirements and interest rate for which the balance is eligible.
| Dates | Closing Balance in Account | # of Days | Tier Rate #1 Balance Requirements and Interest Rate | Tier Rate #2 Balance Requirements and Interest Rate | Tier Rate #3 Balance Requirements and Interest Rate |
|---|---|---|---|---|---|
|
|
|
|||
| Total Monthly Interest Earned | |||||
Step 3: 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.
Step 4: For each row, calculate the number of days that the closing balance is maintained.
Step 5: Assign the closing balance to the different tiers, paying attention to whether portioning is being used. In each cell with a balance, apply simple interest formula [latex]I=Prt[/latex]. Ensure that rate and time are expressed in the same units. Do not round off the resulting interest amounts ([latex]I[/latex]).
Step 6: To calculate the Total Monthly Interest Earned, sum all interest earned amounts from all tier columns and round off to two decimals.
Example 5.3.2
The Rate Builder savings account at your local credit union pays simple interest on the daily closing balance as indicated in the table below:
| Balance | Interest Rate |
|---|---|
| [latex]\$0.00[/latex] to [latex]\$500.00[/latex] | [latex]0\%[/latex] on entire balance |
| [latex]\$500.01[/latex] to [latex]\$2,500.00[/latex] | [latex]0.5\%[/latex] on entire balance |
| [latex]\$2,500.01[/latex] to [latex]\$5,000.00[/latex] | [latex]0.95\%[/latex] on this portion of balance only |
| [latex]\$5,000.01[/latex] and up | [latex]1.35\%[/latex] on this portion of balance only |
In the month of August, the opening balance on an account was [latex]\$2,150.00[/latex]. Deposits were made to the account on August 5 and August 15 in the amounts of [latex]\$3,850.00[/latex] and [latex]\$3,500.00[/latex]. Withdrawals were made from the account on August 12 and August 29 in the amounts of [latex]\$5,750.00[/latex] and [latex]\$3,000.00[/latex]. Calculate the simple interest earned for the month of August.
Solution
Calculate the total interest amount ([latex]I[/latex]) for the month of August.
Step 1: The interest rate structure is in the table above.
The transactions and dates are also known:
[latex]\text{August 1 opening balance}=\$2,150.00[/latex]
[latex]\text{August 5 deposit}=\$3,850.00[/latex]
[latex]\text{August 12 withdrawal}=\$5,750.00[/latex]
[latex]\text{August 15 deposit}=\$3,500.00[/latex]
[latex]\text{August 29 withdrawal}=\$3,000.00[/latex]
Step 2: Set up a tiered interest rate table with four columns for the tiered rates (see table below).
Step 3: Determine the date ranges for each balance throughout the month and calculate the closing balances (see table below).
Step 4: Calculate the number of days involved on each row of the table.
Step 5: Assign the closing balance to each tier accordingly. Apply Formula 5.1[latex]I=Prt[/latex] to any cell containing a balance (see table below).
Step 6: Total up all of the interest from all cells of the table.
| Calculations of Interest Based on Date with Total Interest Earned at the Bottom |
|---|
| Dates: Aug 1 to Aug 5
Closing Balance in Account: [latex]\$2,150[/latex] # of Days: [latex]5-1=4[/latex] 0.5% $500.01 to $2,500 (Entire balance): [latex]\begin{eqnarray*}P&=&\$2,150\\I&=&\$2,150(0.005)\left(\frac{4}{365}\right)\\I&=&\$0.117808\end{eqnarray*}[/latex] |
| Dates: Aug 5 to Aug 12
Closing Balance in Account: [latex]\$2,150+\$3,850=\$6,000[/latex] # of Days: [latex]12-5=7[/latex] 0.5% $500.01 to $2,500 (Entire balance): [latex]\begin{eqnarray*}P&=&\$2,500\\I&=&\$2,500(0.005)\left(\frac{7}{365}\right)\\I&=&\$0.239726\end{eqnarray*}[/latex] 0.95% $2,500.01 to $5,000 (This portion only): [latex]\begin{eqnarray*}P&=&\$2,500\\I&=&\$2,500(0.0095)\left(\frac{7}{365}\right)\\I&=&\$0.455479\end{eqnarray*}[/latex] 1.35% $5,000.01 and up (This portion only): [latex]\begin{eqnarray*}P&=&\$1,000\\I&=&\$1,000(0.0135)\left(\frac{7}{365}\right)\\I&=&\$0.258904\end{eqnarray*}[/latex] |
| Dates: Aug 12 to Aug 15
Closing Balance in Account: [latex]\$6,000-\$5,750=\$250[/latex] # of Days: [latex]15-12=3[/latex] 0% $0 to $500 (Entire balance): [latex]\begin{eqnarray*}P&=&\$250.00\\I&=&\$0.00\end{eqnarray*}[/latex] |
| Dates: Aug 15 to Aug 29
Closing Balance in Account: [latex]\$250+\$3,500=\$3,750[/latex] # of Days: [latex]29-15=14[/latex] 0.5% $500.01 to $2,500 (Entire balance): [latex]\begin{eqnarray*}P&=&\$2,500\\I&=&\$2,500(0.005)\left(\frac{14}{365}\right)\\I&=&\$0.479452\end{eqnarray*}[/latex] 0.95% $2,500.01 to $5,000 (This portion only): [latex]\begin{eqnarray*}P&=&\$1,250\\I&=&\$1,250(0.005)\left(\frac{14}{365}\right)\\I&=&\$0.455479\end{eqnarray*}[/latex] |
| Dates: Aug 29 to Sep 1
Closing Balance in Account: [latex]\$3,750-\$3,000=\$750[/latex] # of Days: [latex]31-29+1=3[/latex] 0.5% $500.01 to $2,500 (Entire balance): [latex]\begin{eqnarray*}P&=&\$750\\I&=&\$750(0.005)\left(\frac{3}{365}\right)\\I&=&\$0.030821\end{eqnarray*}[/latex] |
| Total Interest Earned:
[latex]\begin{eqnarray}I&=&\$0.117808+\$0.239726+\$0.455479+\$0.258904+\$0.00+\$0.479452+\$0.455479+\$0.030821\\ I&=&\$2.04\end{eqnarray}[/latex] |
Step 7: Write as a statement.
For the month of August, the tiered savings account earned a total simple interest of [latex]\$2.04[/latex], which was deposited to the account on September 1.
Short-Term Guaranteed Investment Certificates (GICs)
A guaranteed 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, defined as those that have a time frame of less than one year.
The table below summarizes three factors that determine the interest rate on a short-term GIC: principal, time, and redemption privileges.
| Factors Determining Interest Rate | Higher Interest Rates | Lower Interest Rates |
|---|---|---|
| Principal Amount | Large | Small |
| Time | Longer | Shorter |
| Redemption Privileges | Nonredeemable | Redeemable |
- Amount of 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 [latex]30[/latex] days to [latex]364[/latex] days in length. A longer term usually realizes higher interest rates.
- Redemption Privileges: The two types of GICs are known as 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.
To summarize, if you want to receive the most interest it is best to invest a large sum for a long time in a nonredeemable short-term GIC.
HOW TO
Calculate Interest of a Short-Term GIC
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. Therefore, you need the same four steps as for single payments involving simple interest shown in Section 5.2.
Example 5.3.3
Your parents have [latex]\$10,000[/latex] to invest. They can either deposit the money into a [latex]364[/latex]-day nonredeemable GIC at Assiniboine Credit Union with a posted rate of [latex]0.75\%[/latex], or they could put their money into back-to-back [latex]182[/latex]-day nonredeemable GICs with a posted rate of [latex]0.7\%[/latex]. At the end of the first [latex]182[/latex] 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
For both options, calculate the future value ([latex]S[/latex]), of the investment after [latex]364[/latex] days. The one with the higher future value is your parents’ better option.
Step 1: Given variables:
For the first GIC investment option:
[latex]P=\$10,000[/latex]; [latex]r=0.75\%\;\text{per year}[/latex]; [latex]t=364\;\text{days}[/latex]
For the second GIC investment option:
[latex]\text{Initial}\;P=\$10,000[/latex]; [latex]r= 0.7\%\;\text{per year}[/latex]; [latex]t=182\;\text{days each}[/latex]
Step 2: The rate is annual, the time is in days. Convert the time to an annual number. Transforming both time variables, [latex]t=\frac{364}{365}[/latex] and [latex]t=\frac{182}{365}[/latex]
Step 3: (1st GIC option): Calculate the maturity value [latex]S_1[/latex] of the first GIC option after its [latex]364[/latex]-day term.
[latex]\begin{eqnarray*}S_{1}&=&\$10,000\left(1+(0.0075)\left(\frac{364}{365}\right)\right)\\[1ex]S_{1}&=&\$10,074.79\end{eqnarray*}[/latex]
Step 4: (2nd GIC option, 1st GIC): Calculate the maturity value [latex]S_2[/latex]after the first [latex]182[/latex]-day term.
[latex]\begin{eqnarray*}S_{2}&=&\$10,000\left(1+(0.007)\left(\frac{182}{365}\right)\right)\\[1ex]S_{2}&=&\$10,\!034.90\end{eqnarray*}[/latex]
Step 5: (2nd GIC option, 2nd GIC): Reinvest the first maturity value as principal for another term of [latex]182[/latex] days and calculate the final future value [latex]S_3[/latex].
[latex]\begin{eqnarray*}S_{3}&=&\$10,034.90\left(1+(0.007)\left(\frac{182}{365}\right)\right)\\[1ex]S_{3}&=&\$10,069.93\end{eqnarray*}[/latex]
Step 6: Write as a statement.
The [latex]364[/latex]-day GIC results in a maturity value of [latex]\$10,074.79[/latex], while the two back-to-back [latex]182[/latex]-day GICs result in a maturity value of [latex]\$10,069.93[/latex]. Clearly, the [latex]364[/latex]-day GIC is the better option as it will earn [latex]\$4.86[/latex] more in simple interest.
Section 5.3 Exercises
In the exercise that follow, try it on your own. Full solution is available should you get stuck.
- If you place [latex]\$25,500[/latex] into an [latex]80[/latex]-day short-term GIC at TD Canada Trust earning [latex]0.55\%[/latex] simple interest, how much will you receive when the investment
matures?
Solution
Step 1: Given information:
[latex]P=\$25,500[/latex]; [latex]t=80\;\text{days}[/latex]; [latex]r=0.55\%\;\text{annually}[/latex]
Step 2: Convert daily [latex]t[/latex] to match annual [latex]r[/latex]:
[latex]\begin{align*}t=\frac{80}{365}\end{align*}[/latex]
Step 3: Solve for [latex]S[/latex].
[latex]\begin{eqnarray*}S&=&P\left(1+rt\right)\\S&=&\$25,5000\times\left(1+0.0055\times\frac{80}{365}\right)\\S&=&\$25,530.74\end{eqnarray*}[/latex]
Step 4: Write as a statement.
When the investment matures I will receive [latex]\$25,530.74[/latex].
- Interest rates in the GIC markets are always fluctuating be cause of changes in the short-term financial markets. If you have [latex]\$50,000[/latex] to invest today, you could place the money into a [latex]180[/latex]-day GIC at Canada Life earning a fixed rate of [latex]0.4\%[/latex], or you could take two consecutive [latex]90[/latex]-day GICs. The current posted fixed rate on [latex]90[/latex]-day GICs at Canada Life is [latex]0.3\%[/latex]. Trends in the short-term financial markets suggest that within the next [latex]90[/latex] days short-term GIC rates will be rising. What does the short-term [latex]90[/latex]-day rate need to be [latex]90[/latex] days from now to arrive at the same maturity value as the [latex]180[/latex]-day GIC? Assume that the entire maturity value of the first [latex]90[/latex]-day GIC would be reinvested.
Solution
Step 1: Given information:
For the first GIC investment option:
[latex]P=\$50,000[/latex]; [latex]r=0.4\%\;\text{per year}[/latex]; [latex]t=180\;\text{days}[/latex]
For the second GIC investment option:
[latex]\text{Initial}\;P=\$50,000[/latex]; [latex]r=0.3\%\;\text{per year}[/latex]; [latex]t=90\;\text{days}[/latex]
then,
[latex]P=S\;\text{of first 90-day GIC}[/latex]; [latex]S=\text{maturity value of 180-day GIC}[/latex]; [latex]t=90\;\text{days}[/latex]
Step 2: Transforming both time variables:
[latex]\begin{eqnarray*}t&=&\frac{180}{365}\;\text{and}\;t&=&\frac{90}{365}\end{eqnarray*}[/latex]
Step 3: (1st GIC option):
[latex]\begin{eqnarray*}S_1&=&\$50,000\left(1+0.004\times\frac{180}{365}\right)\\[1ex]S_1&=&\$50,098.63\end{eqnarray*}[/latex]
Step 3: (2nd GIC option, 1st GIC):
[latex]\begin{eqnarray*}S_2&=&\$50,000\left(1+0.003\times\frac{90}{365}\right)\\[1ex]S_2&=&\$50,036.99\end{eqnarray*}[/latex]
Step 3: (2nd GIC option, 2nd GIC):
[latex]\begin{eqnarray*}I&=&\left(S\;\text{of 180-day GIC}\right)-\left(S\;\text{of 90-day GIC}\right)\\I&=&$50,098.63-\$50,036.99\\I&=&\$61.64\end{eqnarray*}[/latex]
[latex]\$61.64[/latex] is what the second [latex]90[/latex]-day GIC must earn in interest.
[latex]\begin{eqnarray*}r&=&\frac{I}{Pt}\\[1ex]r&=&\frac{\$61.64}{\$50,036.99\left(\frac{90}{365}\right)}\\[1ex]r&=&0.004995\;\text{or}\;0.4995\%\end{eqnarray*}[/latex]
Step 4: Write as a statement.
The short-term [latex]90[/latex]-day rate needs to be [latex]0.50\%[/latex] in [latex]90[/latex] days from now to arrive at the same maturity value as the [latex]180[/latex]-day GIC.
THE FOLLOWING LATEX CODE IS FOR FORMULA TOOLTIP ACCESSIBILITY. NEITHER THE CODE NOR THIS MESSAGE WILL DISPLAY IN BROWSER.[latex]I=Prt[/latex]
Attribution
“8.3 Savings Accounts and Short Term GIC’s” 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.
