9.2 Simple Interest

Discussing interest starts with the principal, or amount your account starts with. This could be a starting investment, or the starting amount of a loan. Interest, in its most simple form, is calculated as a percent of the principal. For example, if you borrowed $100 from a friend and agree to repay it with 5% interest, then the amount of interest you would pay would just be 5% of 100: $$ 100 open parenthesis 0.05 close parenthesis equals $ 5$ . The total amount you would repay would be $105, the original principal plus the interest.

Note

Simple One-time Interest

$I equals P sub 0 r$

$A = P_{0} + I = P_{0} + P_{0} r = P_{0} (1 + r)$

where

$I$ is the interest
$A$ is the end amount: principal plus interest
$P sub 0$ is the principal (starting amount)
$r$ is the interest rate (in decimal form. Example: $5 % equals 0.05$ )

Example 1

A friend asks to borrow $300 and agrees to repay it in 30 days with 3% interest. How much interest will you earn?

Show solution

$\begin{matrix} P_{0} = $ 300 & the principal \\ r = 0.03 & 3 % rate \\ I = $ 300 (0.03) = $ 9 . & You will earn $ 9 interest. \end{matrix}$

One-time simple interest is only common for extremely short-term loans. For longer term loans, it is common for interest to be paid on a daily, monthly, quarterly, or annual basis. In that case, interest would be earned regularly. For example, bonds are essentially a loan made to the bond issuer (a company or government) by you, the bond holder. In return for the loan, the issuer agrees to pay interest, often annually. Bonds have a maturity date, at which time the issuer pays back the original bond value.

# Ch 9.2 - simple interest. The interest is always a percent of the ORIGINAL
# principal, so it never earns interest of its own.
def simple(P0, r, t):
    """r is the rate PER PERIOD, t is the number of periods."""
    interest = P0 * r * t
    return interest, P0 + interest

P0, APR, YEARS, PER_YEAR = 1000, 0.04, 4, 2   # the book's $1000 T-note
# TRY IT: PER_YEAR = 12 (monthly). Does more frequent payment earn more?
# TRY IT: APR = 0.72 - the 72% rate the payday lender in Try it Now 1 charges.

rate = APR / PER_YEAR
periods = YEARS * PER_YEAR
interest, total = simple(P0, rate, periods)
print(f"${P0:,} at {100 * APR:g}% APR paid {PER_YEAR}x a year for {YEARS} years")
print(f"   rate per period = {100 * APR:g}% / {PER_YEAR} = {100 * rate:g}%")
print(f"   periods         = {YEARS} x {PER_YEAR} = {periods}")
print(f"   I = P0 * r * t  = {P0} x {rate} x {periods} = ${interest:,.2f}")
print(f"   you end with A  = ${total:,.2f}\n")

print("period    interest that period    total interest    balance")
for n in range(1, periods + 1):
    print(f"{n:>6}    {P0 * rate:>19,.2f}    {P0 * rate * n:>14,.2f}    {P0 + P0 * rate * n:>10,.2f}")
print("Every row of the interest column is IDENTICAL. That is what makes it")
print("'simple' - and it is exactly the linear growth of chapter 8.\n")

# Turning a fee back into a rate: $30 of interest on a $500 one month loan.
FEE, LOAN, MONTHS = 30, 500, 1     # TRY IT: FEE = 75 on a two week loan
monthly = FEE / LOAN / MONTHS
print(f"a ${FEE} charge on a ${LOAN} loan for {MONTHS} month(s)")
print(f"   rate per month = {FEE}/{LOAN}/{MONTHS} = {monthly:.4f} = {100 * monthly:.2f}%")
print(f"   annual rate    = {100 * monthly:.2f}% x 12 = {100 * monthly * 12:.1f}% APR")

We can generalize this idea of simple interest over time.

Note

Simple Interest over Time

$I equals P sub 0 r t$

$A = P_{0} + I = P_{0} + P_{0} r t = P_{0} (1 + r t)$

where

$I$ is the interest
$A$ is the end amount: principal plus interest
$P sub 0$ is the principal (starting amount)
$r$ is the interest rate in decimal form
$t$ is time

The units of measurement (years, months, etc.) for the time should match the time period for the interest rate.

Example 3 Treasury Notes

Treasury Notes (T-notes) are bonds issued by the federal government to cover its expenses. Suppose you obtain a $1,000 T-note with a 4% annual rate, paid semi-annually, with a maturity in 4 years. How much interest will you earn?

Show solution

Since interest is being paid semi-annually (twice a year), the 4% interest will be divided into two 2% payments.

$\begin{matrix} P_{0} = $ 1000 & the principal \\ r = 0.02 & 2 % rate per half-year \\ t = 8 & 4 years = 8 half-years \\ I = $ 1000 (0.02) (8) = $ 160 . & You will earn $ 160 interest total over the four years. \end{matrix}$

Adapted from Math in Society by David Lippman, hosted on LibreTexts (math.libretexts.org) and licensed under CC BY-SA 3.0. Changes were made. License: CC-BY-SA-3.0.