Simple interest and short-term financing

Interest is the price of money over time. A business that borrows to buy inventory pays interest; a business that leaves cash on deposit earns it. For short periods, typically less than a year, interest is usually calculated as simple interest, where the interest is a fixed percentage of the original amount for each unit of time and is not added to the balance to earn more interest. Simple interest governs lines of credit, short-term loans, promissory notes, treasury bills, late-payment charges and many everyday financial contracts. It is also the foundation for compound interest in the next lesson.

The simple interest formula

Here I is the interest in dollars, P the principal (the amount borrowed or invested), r the annual interest rate as a decimal, and t the time in years. Because rates are almost always quoted per year, time must be expressed in years too: 9 months is 9/12 = 0.75 years, and 146 days is 146/365 years.

The formula can be rearranged to find any one of the four quantities when the other three are known.

Counting days

When a loan runs between two dates, the time is the exact number of days between them. By convention the start date is counted and the end date is not (or, equivalently, count the end date and not the start). Subtracting dates in a spreadsheet does this automatically: in most spreadsheets =DATE(2026,8,9)−DATE(2026,3,15) gives 147. By hand, count the days left in the first month, add the full months, and add the days in the last month.

In Canada, interest on loans and deposits is normally calculated on the actual number of days divided by 365, even in a leap year. The federal Interest Act also requires that when a rate is stated for a period shorter than a year, such as a daily or monthly rate, the contract must disclose the equivalent annual rate; otherwise the lender may be limited to 5 % a year.

Maturity value and present value

The maturity value (or future value) of a simple interest loan or investment is the principal plus the interest. Turning the formula around gives the present value: the amount that must be invested today at a given rate to grow to a stated amount by a future date.

Equivalent payments and the focal date

Money due at different times cannot simply be added. To compare or combine payments, move each one to a common date, called the focal date, using \( S = P(1 + rt) \) to move money forward in time and \( P = S/(1 + rt) \) to move it back. The payments are equivalent when their values at the focal date are equal. This is how lenders and borrowers reschedule debts.

Lines of credit and changing balances

An operating line of credit lets a business borrow up to a limit and repay whenever it likes, paying interest only on the balance used. The rate is usually floating, set as the bank's prime rate plus a margin, and interest is calculated daily and charged monthly. When the balance or the rate changes during a month, split the month into periods, calculate the interest for each period, and add them.

Promissory notes

A promissory note is a written promise by one party (the maker) to pay a stated sum to another (the payee) on a stated date or on demand. Businesses use notes to formalize overdue accounts, to finance large purchases from suppliers and to document loans between related companies. An interest-bearing note states a face value and a rate; its maturity value is found with \( S = P(1 + rt) \). Because a note is a written, signed obligation, the payee can sell it to a bank or finance company before maturity. The buyer pays the present value of the maturity value at its own required rate, so the price falls when interest rates rise and rises when they fall, a first glimpse of how bond prices behave.

Treasury bills and other discounted instruments

Government of Canada treasury bills (T-bills) are short-term debts with terms of up to one year. They pay no interest directly. Instead, they are sold at a discount to their face value and repaid at face value at maturity, and the difference is the investor's return. Canadian T-bill yields are quoted as simple annual rates on a 365-day basis, calculated on the price paid.

The high cost of short-term consumer credit

Simple interest makes it easy to compare the cost of short-term borrowing. A credit card at 20.99 % charges about $31 a month on a balance of $1,800 (\( 1{,}800 \times 0.2099 \times 30/365 \approx 31.05 \)). A fee that looks small relative to a short loan can mean a very high annual rate.

Payday loans are a striking example. Under federal rules in force since January 1, 2025, the cost of a payday loan is capped at $14 for every $100 borrowed. A $14 fee on a 14-day loan of $100 is equivalent to a simple annual rate of \( 0.14 \times 365/14 = 3.65 \), or 365 %. The Criminal Code's criminal interest rate was also lowered to 35 % APR from the same date, with payday loans regulated by the provinces as an exception. These are the rules at the time of writing (October 2026).

# Simple interest helpers and a line-of-credit month
from datetime import date

def days(a, b): return (b - a).days
def interest(P, r, d, basis=365): return P * r * d / basis

d = days(date(2026, 4, 4), date(2026, 9, 30))
print(d, "days; maturity value", round(8500 + interest(8500, 0.0725, d), 2))
print("365 vs 360:", round(interest(12000, .065, 146), 2), round(interest(12000, .065, 146, 360), 2))
pieces = [(18000, .072, 11), (25500, .072, 3), (25500, .0695, 5), (15500, .0695, 11)]
print("June interest", round(sum(round(interest(*p), 2) for p in pieces), 2))
print("PV of 15,000 in 200 days at 5%:", round(15000 / (1 + .05 * 200 / 365), 2))

179 days; maturity value 8802.22
365 vs 360: 312.0 316.33
June interest 110.9
PV of 15,000 in 200 days at 5%: 14600.0