MCR3U Simple and Compound Interest: Notes and Practice

When you borrow or invest money, interest is the cost (or reward) for using it. A savings account pays you interest for letting the bank use your money; a car loan, student loan or credit card charges you interest for using the lender's money. There are two completely different ways interest can be calculated, and they connect directly back to the two sequence types you just learned: simple interest grows ARITHMETICALLY (same dollar amount every period), while compound interest grows GEOMETRICALLY (interest earns interest, so it compounds). Over a few months the difference is pocket change, but over the decades of a retirement savings plan or a mortgage it can amount to tens of thousands of dollars, which is why every financial decision you make later will rest on the formulas in this lesson.

Simple interest

Simple interest is calculated only on the original amount (the principal, $P$ ), never on interest already earned. It's a flat, constant amount added every period — an arithmetic pattern. Invest 1000 dollars at 6% per year simple interest and you earn $1000 (0.06) = 60$ dollars every year, so the balances $1000, 1060, 1120, 1180, \dots$ form an arithmetic sequence with $d = P r = 60$ . Two details cost marks. First, $r$ must be a decimal (6% is 0.06). Second, $t$ must be in YEARS to match an annual rate: 9 months is $t = \frac{9}{12} = 0.75$ , and 146 days is $t = \frac{146}{365} = 0.4$ . The formula can also be rearranged to find any missing value, such as $r = \frac{I}{Pt}$ or $t = \frac{I}{P r}$ . Simple interest is used mainly for short-term loans and some investments such as short-term deposits.

I = P r t A = P + I = P (1 + r t)

P = principal, r = annual interest rate as a decimal, t = time in years, I = interest earned, A = final amount.

Compound interest

Compound interest is recalculated every period based on the CURRENT balance — so the interest you earned last period now earns interest too. After one period the balance is $P + P i = P (1 + i)$ . The next period's interest is calculated on that whole amount, so after two periods the balance is $P (1 + i) (1 + i) = P (1 + i)^{2}$ , after three it is $P (1 + i)^{3}$ , and so on. Each period multiplies the balance by the same factor, so this is exactly a geometric sequence, with common ratio $(1 + i)$ , where $i$ is the interest rate PER COMPOUNDING PERIOD. The exponent $n$ counts how many times interest has been added. The final amount $A$ is also called the future value (FV), and the principal $P$ is also called the present value (PV).

A = P (1 + i)^{n}

P = principal (present value), i = interest rate per compounding period, n = number of compounding periods, A = amount (future value).

Fig 1Compound growth pulls away from simple (linear) growth more and more sharply as time goes on.

Years	Simple interest, 6%	Compounded annually, 6%	Gap (dollars)
0	1000.00	1000.00	0.00
1	1060.00	1060.00	0.00
2	1120.00	1123.60	3.60
5	1300.00	1338.23	38.23
10	1600.00	1790.85	190.85
20	2200.00	3207.14	1007.14

1000 dollars invested both ways. The two agree after one year, then compounding pulls ahead faster and faster.

Compounding periods

Interest rates are almost always quoted as an ANNUAL rate, $r$ . If compounding happens more often than once a year, you must convert: divide $r$ by the number of compounding periods per year ( $m$ ) to get $i$ , and multiply the number of years by $m$ to get $n$ . So 6% per year compounded quarterly for 5 years means $i = \frac{0.06}{4} = 0.015$ per quarter and $n = 4 (5) = 20$ quarters. More frequent compounding earns slightly more, because interest is added sooner and starts earning its own interest sooner. A nominal 12% compounded monthly grows each dollar to $(1.01)^{12} \approx 1.1268$ in a year, which is the same as about 12.68% compounded annually. The difference is small in one year but it compounds too. For a quick estimate of doubling time, the Rule of 72 says money doubles in about $72 \div (annual rate in %)$ years: about 12 years at 6%.

Compounding	Periods/year ( $m$ )	$i$ per period	$n$ for $t$ years
Annually	1	$r /1$	$t$
Semi-annually	2	$r /2$	$2 t$
Quarterly	4	$r /4$	$4 t$
Monthly	12	$r /12$	$12 t$

Present value

Sometimes you know the future amount you want and need to find how much to invest today. Just solve $A = P (1 + i)^{n}$ for $P$ — this is called finding the present value. Dividing by $(1 + i)^{n}$ "undoes" $n$ periods of growth, which is why present value is sometimes called discounting: 5000 dollars four years from now is worth less than 5000 dollars today, because money in hand today could be earning interest. The same formula answers "what is a future payment worth now?" and "how much must I deposit to reach a goal?". You can also solve for the rate: if 2000 dollars grows to 2500 dollars in 5 years compounded annually, then $(1 + i)^{5} = 1.25$ , so $1 + i = 5 1.25 \approx 1.0456$ and the rate is about 4.56% per year. To find how LONG it takes, try values of $n$ on a calculator or graph $y = P (1 + i)^{x}$ and read where it crosses the target: at 6% compounded quarterly, $(1.015)^{n}$ first reaches 2 at $n = 47$ quarters, just under 12 years.

P = \frac{A}{( 1 + i ) ^{n}} = A (1 + i)^{- n}

Key idea

Always match

i

and

n

to the SAME compounding period. If interest compounds monthly,

i

must be the monthly rate and

n

must be the number of months — never mix a monthly

i

with a yearly

n

. To recognise which formula a question needs: the words simple interest mean

I = P r t

with no exponent; compounded, compound interest or a compounding frequency mean

A = P (1 + i)^{n}

; and if the unknown is the amount needed TODAY (invest now, worth today, present value), rearrange to

P = A (1 + i)^{- n}

. Keep every decimal of

i

in your calculator and round money to the cent only at the end.

Simple and Compound Interest

Key things to remember

Simple interest

Compound interest

Compounding periods

Present value

Worked examples

Where marks get lost

Flashcards 9

Practice