8.5: Application - Loans

Almost everybody needs a little financial help from time to time. Perhaps you want to purchase a big-ticket item like a big-screen 3D HDTV without having the cash in full today to pay for it up front. However, the television is drastically on sale this week and if you wait to purchase it you will lose out and potentially have to pay full price next month. Or maybe as you flip through all of your bills you notice that though you will eventually have enough income to cover them, some of your payments fall due a few days before you will actually receive your income. What will you do for those few days until the income is deposited into your account?

Businesses also find themselves in similar situations. Maybe a supplier is offering a special deal on a product line, but the business does not have the cash to stock up. Also typical of business operations is that they make purchases in advance of sales, so they need to spend the money before they can receive the revenue to pay for their expenses. How can a business get access to short-term financing?

In this section, short-term financing in the form of demand loans is explored. This will include looking at loans where payments are variable as well as loans possessing fixed payments. The section ends by looking briefly at a loan you may already have—a student loan.

Demand Loans and Characteristics

A demand loan is a short-term loan that generally has no specific maturity date, can be paid at any time without any interest penalty, and where the lender can demand repayment in full at any time. It allows borrowing when needed and repayment when money permits, subject to the following six characteristics:

1. Credit Limit. This establishes the maximum amount that can be borrowed.
2. Variable Interest Rate. Almost all demand loans use variable simple interest rates based on the prime rate. Only the best, most secure customers can receive prime, while others usually get "prime plus" some additional amount.
3. Fixed Interest Payment Date. Interest is always payable on the same date each and every month. For simplicity, the payment is usually tied to a chequing or savings account, allowing the interest payment to occur automatically.
4. Interest Calculation Procedure. Interest is always calculated using a simple interest procedure based on the daily closing balance in the account. This means the first day but not the last day is counted.
5. Security. Loans can be secured or unsecured. Secured loans are those loans that are guaranteed by an asset such as a building or a vehicle. In the event that the loan defaults, the asset can be seized by the lender to pay the debt. Unsecured loans are those loans backed up by the general goodwill and nature of the borrower. Usually a good credit history or working relationship is needed for these types of loans. A secured loan typically enables access to a higher credit limit than an unsecured loan.
6. Repayment Structure. The repayment of the loan is either variable or fixed.
   1. A variable repayment structure allows the borrower to repay any amount at any time, although a minimum requirement may have to be met such as "at least 2% of the current balance each month." A current balance is the balance in an account plus any accrued interest. Accrued interest is any interest amount that has been calculated but not yet placed (charged or earned) into an account.
   2. A fixed repayment structure requires a fixed payment amount toward the current balance on the same date each and every month.

Types of Simple Interest Financing

While many types of financial tools use simple interest, these are the four most common:

1. Personal Line of Credit (LOC). A demand loan for individuals is generally unsecured and is granted to those individuals who have high credit ratings and an established relationship with a financial institution. Since it is unsecured, the credit limit is usually a small amount, such as $10,000. Repayment is variable and usually has a minimum monthly requirement based on the current balance.
2. Home Equity Line of Credit (HELOC). This is a special type of line of credit for individuals that is secured by residential homeownership. Typically, an amount not exceeding 80% of the equity in a home is used to establish the credit limit, thus enabling an individual access to a large amount of money. The interest rates tend to follow mortgage interest rates and are lower than personal lines of credit. Repayment is variable, usually involving only the accrued interest every month.
3. Operating Loans. An operating loan is the business version of a line of credit. It may or may not be secured, depending on the nature of the business and the strength of the relationship the business has with the financial institution. Repayment can be either variable or fixed.
4. Student Loans. A loan available to students to pursue educational opportunities. Although these are long-term in nature, the calculation of interest on a student loan uses simple interest techniques. These loans are not true demand loans since a student loan cannot be called in at any time. Repayment is fixed monthly.

Repayment Schedules

When you work with short-term loans, regardless of the repayment structure, you should always set up a repayment schedule. A repayment schedule is a table that details the financial transactions in an account including the balance, interest amounts, and payments. The table below presents the table structure used for setting up a repayment schedule.

Date	Balance before Transaction (\(P\))	Annual Interest Rate (\(r\))	Number of Days (\(t\))	Interest Charged (\(I\))	Accrued Interest	Payment (+) or Advance (−)	Principal Amount	Balance after Transaction
Start Date								(1)
(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)

Each number in the table corresponds to an entry below that explains how to use each column or row.

1. The first row appears if the schedule has an opening balance. In these instances, list the start date in the first column and the opening balance in the last column.
2. List the date of any transaction. This could include a payment, advance, interest rate change, or an accrued interest payment.
3. Carry this number forward from (1).
4. Record the interest rate that applies to the date interval in the first column.
5. Calculate the number of days between the date on the previous row and the current row. Remember to count the first day but not the last day. Express the number annually to match the interest rate.
6. Compute the interest charges for the date interval using Formula 8.1, \(I = Prt\). Use (3), (4), and (5) as your values for the formula.
7. Enter the cumulative total of any unpaid or accrued interest as of the current row's date. This amount is the sum of (6) from the current row plus any number recorded in this column from the row above.
8. Enter the amount of the transaction occurring for this row of the table. Payments should be recorded as positives (credits) as they will decrease the balance, and advances should be recorded as negatives (debits) as they will increase the balance. If this is an interest payment date, one of the following two events will happen:
   1. Only the interest payment occurs on this date. Copy the accrued interest from (7) into this column.
   2. An additional payment or advance is made on this date. Add the accrued interest to the advance or payment, placing the sum into this column.

Regardless of which event happens, cross out the accrued interest amount in (7) as it is considered paid, so the accrued interest balance is reduced to zero once again.

9. One of two events will happen in this column:
   1. On an interest payment date, subtract the accrued interest from the amount in (8).
   2. On a date other than an interest payment date, any payment or advance will have its total amount applied to the principal. Therefore, carry the amount in (8) across to this column.
10. Subtract the amount in (9) from the amount in this column on the previous row to create the new balance in the account. Copy this amount to the next row in (3).

How It Works

Although the calculations of a repayment schedule are relatively straightforward, the complexity of the repayment schedule sometimes causes grief. When a repayment schedule is required, follow these steps:

Step 1: Set up the repayment schedule as per the example table.

Step 2: Record the start date and the opening balance for the loan.

Step 3: In chronological order, make new row entries in the schedule by filling in the details provided in the question. You require a new row in the table whenever one of the following three events occurs:

1. A payment or advance is made.
2. An interest payment date occurs.
3. The interest rate changes.

Step 4: Starting with the first row, work left to right across the table, filling in all information. Pay particular attention to the nuances of the "Payment or Advance" and "Principal Amount" columns as discussed previously. Once a row is complete, move to the next row until you fill in the entire table.

Step 5: Calculate any totals requested such as total interest or total principal paid.

Important Notes

For simplicity in writing the numbers into repayment schedules and performing calculations, it is this textbook's practice to round all interest calculations to two decimals throughout the table. In real-world applications, you must keep track of all decimals in the account at all times.

Things To Watch Out For

Because of the size of the repayment schedule and the large amount of information involved in the calculations, the number-one error is what most people call a “silly” error. It means that a wrong date is recorded, a wrong amount is written down, a payment is recorded as an advance or vice versa, or simply the wrong button is pressed on a calculator. Take the time to ensure you read and record the correct numbers and that you pause when performing calculations. For example, advances mean the balance should get bigger, while payments mean the balance should get smaller. Just by thinking for a second about the basic principles of debt you should be able to catch those silly errors.

Example \(\PageIndex{1}\): Operating Loan, Fixed Repayment

On July 15, when the prime rate was set at 4%, Canadian Footwear took out an operating loan from CIBC for $8,000 at prime plus 1.25%. The terms of the loan require a fixed payment of $1,500 on the 15th of every month until the loan is repaid. The prime rate climbed by 0.5% on September 29. Create a repayment schedule for the loan and calculate the total interest paid.

Solution

Create the repayment schedule for the loan and sum the interest charges for the entire operating loan.

What You Already Know

For a fixed repayment loan, capture three important categories of information:

Opening Balance	Payments	Dates of Interest Rate Changes†
July 15: $8,000	August 15: $1,500 September 15: $1,500 October 15: $1,500 November 15: $1,500 December 15: $1,500 January 15: $?????*	July 15: 5.25% September 29: 5.75%

*Note: You can presume there is a payment on January 15 since the first five payments total up to $7,500, which falls short of repaying the loan. The last payment, on January 15, is still unknown as it will be based on the remaining balance in the account plus any accrued interest.

† Calculated as prime plus 1.25%.

How You Will Get There

Step 1:

Set up a repayment table.

Step 2:

Fill in the start date and opening balance.

Step 3:

Chronologically fill in all information, with one transaction per row of the table.

Step 4:

Work left to right and top to bottom throughout the table.

Step 5:

Once you have completed this step, sum the interest charges paid on the 15th of every month.

Perform

Steps 1-4:

Date	Balance before Transaction (\(P\))	Annual Interest Rate (\(r\))	Number of Days (\(t\))	Interest Charged (\(I=Prt\))	Accrued Interest	Payment (+) or Advance (−)	Principal Amount	Balance after Transaction
July 15								$8,000.00
Aug 15	$8,000.00	5.25%	31/365 (1)	$35.67 (2)	$35.67	$1,500.00	$1,464.33 (3)	$6,535.67 (4)
Sept 15	$6,535.67	5.25%	31/365	$29.14	$29.14	$1,500.00	$1,470.86	$5,064.81
Sept 29	$5,064.81	5.25%	14/365	$10.20	$10.20	$0.00	$0.00	$5,064.81
Oct 15	$5,064.81	5.75%	16/365	$12.77	$22.97 (5)	$1,500.00	$1,477.03	$3,587.88
Nov 15	$3,587.88	5.75%	31/365	$17.52	$17.52	$1,500.00	$1,482.48	$2,105.30
Dec 15	$2,105.30	5.75%	30/365	$9.95	$9.95	$1,500.00	$1,490.05	$615.25
Jan 15	$615.25	5.75%	31/365	$3.00	$3.00	$618.25 (6)	$615.25 (7)	$0.00

Select calculations and comments from the above table:

(1) This is the number of days from the date on the previous row to the current row, or July 15 to August 15. Count the first day, but not the last.

(2) \(I = Prt = \$8,000(0.0525)(31/365) = \$35.67\)

(3) $1,500.00 − $35.67 = $1,464.33. The accrued interest of $35.67 is now paid and can be crossed out.

(4) $8,000.00 − $1,464.33 = $6,535.67

(5) $10.20 + $12.77 = $22.97

(6) The last payment must clear the balance owing and the accrued interest: $615.25 + $3.00 = $618.25

(7) $618.25 − $3.00 = $615.25

Step 5:

Total interest charges = $35.67 + $29.14 + $22.97 + $17.52 + $9.95 + $3.00 = $118.25

To clear the operating loan it takes six payments: five of $1,500 each and a final payment of $618.25. The loan incurs total interest paid of $118.25.

Example \(\PageIndex{2}\): HELOC, Variable Repayment, No Minimum Requirement

Lynne has access to a HELOC that requires only the payment of accrued interest on the first of every month. On March 1, the opening balance on her HELOC was $15,000. She took advances of $6,000 and $10,000 on March 21 and May 4, respectively. She made additional payments of $11,000 and $15,000 on April 15 and June 17. The interest rate on her HELOC sits at prime plus 2%. On March 1, the prime rate was 3%. On April 26, it rose by 0.5%. Determine the total interest paid on her HELOC from March 1 to July 1.

Solution

This is a HELOC using simple interest. Calculate the total interest that she had to pay in March, April, May, and June and then sum the total interest paid.

What You Already Know

Capture five important categories of information:

Opening Balance	Advances	Payments
March 1: $15,000	March 21: $6,000 May 4: $10,000	April 15: $11,000 June 17: $15,000

Interest Payment	Dates Dates of Interest Rate Changes*
April 1 May 1 June 1 July 1	March 1: 5% April 26: 5.5%

*Calculated as prime plus 2%.

How You Will Get There

Step 1:

Set up a repayment table.

Step 2:

Fill in the start date and opening balance.

Step 3:

Chronologically fill in all information, with one transaction per row of the table.

Step 4:

Work left to right and top to bottom throughout the table.

Step 5:

Once you have completed this step, sum the accrued interest payments on April 1, May 1, June 1, and July 1.

Perform

Steps 1- 4

Date	Balance before Transaction (\(P\))	Annual Interest Rate (\(r\))	Number of Days (\(t\))	Interest Charged (\(I=Prt\))	Accrued Interest	Payment (+) or Advance (−)	Principal Amount	Balance after Transaction
Mar 1		5%						$15,000.00
Mar 21	$15,000 (1)	5%	20/365 (2)	$41.10 (3)	$41.10 (4)	-$6000	-$6000 (5)	$21,000 (6)
Apr 1	$21,000	5%	11/365	$31.64	$72.74 (7)	$72.74 (8)	$0.00 (9)	$21,000
Apr 15	$21,000	5%	14/365	$40.27	$40.27	$11,000	$11,000	$10,000
Apr 26	$10,000	5%	11/365	$15.07	$55.34	$0.00	$0.00	$10,000
May 1	$10,000	5.5%	5/365	$7.53	$62.87	$62.87	$0.00	$10,000
May 4	$10,000	5.5%	3/365	$4.52	$4.52	-$10,000	-$10,000	$20,000
June 1	$20,000	5.5%	28/365	$84.38	$88.90	$88.90	$0.00	$20,000
June 17	$20,000	5.5%	16/365	$48.22	$48.22	$15,000	$15,000	$5,000
July 1	$5,000	5.5%	14/365	$10.55	$58.77	$58.77	$0.00	$5,000

Select calculations and comments from the above table:

(1) This is the balance from the last column in the row above carried forward = $15,000.

(2) The date interval from the previous row to this row is March 1 to March 21: \(t = 21 − 1 = 20\) days.

(3) \(I = Prt = \$15,000(0.05)(20/365) = \$41.10\)

(4) The accrued interest from the row above plus (3). $0 + $41.10 = $41.10

(5) Not an interest payment date, so the full payment is applied to principal = −$6,000

(6) Previous balance in the column above minus the principal portion from (5). $15,000 − (−$6,000) = $21,000

(7) The accrued interest from the row above plus interest from this row. $41.10 + $31.64 = $72.74

(8) This is an interest payment date. Carry (7) across and cross it out, reducing the accrued interest balance to zero.

(9) This is an interest payment date. Take (8) and subtract (7). $72.74 − $72.74 = $0.00

Step 5:

April 1 + May 1 + June 1 + July 1 = $72.74 + $62.87 + $88.90 + $58.77 = $283.28.

The total interest paid on the HELOC from March 1 to July 1 is $283.28.

riable Repayment, with Minimum Requirement

Everlyne has a personal LOC with her bank with a maximum credit limit of $10,000. The interest rate is prime plus 3.5%, and the current prime rate is 4.5%. Regardless of any account transaction activity, her bank requires on the first of every month for her to pay "the greater of 5% of the current balance or $100" from her chequing account. She is allowed to exceed her maximum credit limit, but if she does the entire balance is subject to 21% interest until such time as the balance is restored below the credit limit. On October 1, the opening balance on her LOC was $2,000. She took advances of $5,000, $6,000, and $1,000 on October 21, November 13, and December 1, respectively. She made payments of $1,000, $3,000, and $8,500 on November 1, November 20, and December 15, respectively. The prime rate decreased by 1/4% on November 5. Calculate her total required payments (not her voluntary payments) and the portion of those payments that went toward interest for the months of October, November, and December.

Solution

Calculate a personal LOC using simple interest to determine the total required payments on November 1, December 1, and January 1 and the amount of interest included in those payments.

What You Already Know

Capture five important categories of information:

Opening Balance	Advances	Payments
Oct 1: $2,000	October 21: $5,000 Nov. 13: $6,000 December 1: $1,000	November 1: $1,000 November 20: $3,000 December 15: $8,500

Interest Payment	Dates Dates of Interest Rate Changes*
November 1 December 1 January 1	October 1: 8% November 5: 7.75% Interest rate jumps to 21% if credit limit is exceeded

*Calculated as prime plus 4.5%

Perform

Steps 1- 4

Date	Balance before Transaction (\(P\))	Annual Interest Rate (\(r\))	Number of Days (\(t\))	Interest Charged (\(I=Prt\))	Accrued Interest	Payment (+) or Advance (−)	Principal Amount	Balance after Transaction
Oct 1								$2,000
Oct 21	$2,000	8%	20/365	$8.77	$8.77	-$5,000	-$5,000	$7,000
Nov 1	$7,000	8%	11/365	$16.88	$25.65	$1,000 + $351.28 = $1,351.28 (1)	$1,325.63 (2)	$5,674.37
Nov 5	$5,674.37	8%	4/365	$4.97	$4.97	$0.00	$0.00	$5,674.37
Nov 13	$5,674.37	7.75%	8/365	$9.64	$14.61	-$6,000	−$6,000	$11,674.37
Nov 20	$11,674.37	21% (3)	7/365	$47.02	$61.63	$3,000	$3,000	$8,674.37
Dec 1	$8,674.37	7.75%	11/365	$20.26	$81.89	−$1,000 + $437.81 = −$562.19 (4)	−$644.08 (5)	$9,318.45
Dec 15	$9,318.45	7.75%	14/365	$27.70	$27.70	$8,500	$8,500	$818.45
Jan 1	$818.45	7.75%	17/365	$2.95	$30.65	$100 (6)	$69.35 (7)	$749.10

Select calculations and comments from the table:

(1) The required payment is the greater of 5% of the current balance or $100 regardless of her monthly account activity. The current balance is $7,000 + $25.65 = $7,025.65. Five percent of that number is $351.28. The $1,000 extra payment plus the required payment of $351.28 totals $1,351.28.

(2) The $1,351.28 − $25.65 = $1,325.63 toward the principal. The accrued interest is now paid and is crossed out.

(3) Note that by exceeding the credit limit of $10,000, she changes the interest rate to the penalty rate.

(4) The current balance is $8,674.37 + $81.89 = $8,756.26, of which 5% is $437.81. Therefore, with an advance of $1,000 and a payment of $437.81 on the same date, the net daily result is an advance of $562.19.

(5) −$562.19−$81.89=−$644.08, which is added to the principal. The accrued interest is paid and is crossed out.

(6) The current balance is $818.45 + $30.65 = $849.10, of which 5% is $42.46, so pay $100 - the greater amount.

(7) $100 − $30.65 = $69.35. The accrued interest is now paid and is crossed out.

Step 5:

Total Required Payments = Nov 1 + Dec 1+ Jan 1 = $351.28 + $437.81 + $100.00 = $889.09

Total Interest Charges = Nov 1 + Dec 1 + Jan 1 = $25.65 + $81.89 + $30.65 = $138.19

Throughout the months of October, November, and December, the total required payments on the LOC came to $889.09. Of that amount, $138.19 went toward the interest charges.

Student Loans

A student loan is a special type of loan designed to help students pay for the costs of tuition, books, and living expenses while pursuing postsecondary education. The Canada Student Loans Program (CSLP) is administered by the federal government and run by the National Student Loan Service Centre (NSLSC) under contract to Human Resources and Skills Development Canada (HRSDC).

If and when you take out a student loan, the following characteristics are in place for full-time students (defined as having a 60% workload or more):

2. Fixed interest rate of prime + 5% (where prime is determined on the day that your grace period ends).