Grade by Grade
C. Discrete Functions · Financial Applications: Interest, Present Value and Annuities

The Compound Interest Formula: Compounding Periods and Future Value

C3.3About 70 minutes
Learning goals
  • I can find the interest rate per compounding period, i, and the number of compounding periods, n, from a stated annual rate.
  • I can use A = P(1 + i)ⁿ and a scientific calculator to find the future value (amount) and the interest earned.
  • I can compare investments or debts with different rates and compounding frequencies.
  • I can explain why compounding more often gives more interest, but only a little more.

Words to know

future value (A)
The amount an investment or debt will be worth at a later date, including interest.
i
The interest rate per compounding period, as a decimal: i = annual rate ÷ number of compounding periods per year.
n
The total number of compounding periods: n = years × compounding periods per year.
compounding frequency
How many times per year interest is compounded: annually (1), semi-annually (2), quarterly (4), monthly (12), weekly (52), daily (365).
nominal (stated) rate
The annual rate as advertised, for example "6% compounded monthly".

Same rate, different calendar

Banks rarely add interest just once a year. A savings account might compound monthly, a GIC semi-annually, and some accounts daily. The advertised rate is always an annual rate, so before using the formula you must cut it into the right-sized pieces.

Today you will use the full compound interest formula with any compounding period and a scientific calculator.

Remember from Grade 9

In Grade 9 you learned that the rate per period is i = annual rate ÷ periods per year and that n = years × periods per year, and you compared the effective annual rate of different offers. Here you will use the formula with more confidence, connect it to exponential functions, and use it to solve for any of its parts over the next few lessons.

The compound interest formula

A = P(1 + i)ⁿ

  • A = amount (future value)
  • P = principal (present value)
  • i = interest rate per compounding period (as a decimal)
  • n = number of compounding periods

The interest earned is I = A − P.

For an annual rate r compounded k times a year for t years: i = r ÷ k and n = kt. So as a function of time, A(t) = P(1 + r/k)ᵏᵗ, an exponential function of t.

CompoundingPeriods per year (k)i for 4.8%n for 6 yearsAmount of $5 000Interest
annually10.0486$6 624.27$1 624.27
semi-annually20.048 ÷ 212$6 646.14$1 646.14
quarterly40.048 ÷ 424$6 657.36$1 657.36
monthly120.048 ÷ 1272$6 664.96$1 664.96
weekly520.048 ÷ 52312$6 667.90$1 667.90
daily3650.048 ÷ 3652190$6 668.66$1 668.66
The same principal, the same stated rate and the same 6 years. Only the compounding frequency changes.
Interest earned on $5 000 at 4.8% for 6 years05001 0001 5002 000annual$1 624.27semi$1 646.14quarter$1 657.36month$1 664.96week$1 667.90day$1 668.66compounding frequency
More frequent compounding earns more, but the gains shrink: going from annual to semi-annual adds about $21.87, while going from weekly to daily adds less than $1.
Worked example

Example 1: Finding a future value

Find the amount and the interest earned when $2 500 is invested for 4 years at 3.6% compounded monthly.

  1. i = 0.036 ÷ 12 = 0.003 and n = 4 × 12 = 48.
  2. A = 2500(1.003)⁴⁸.
  3. Calculator: 2500 × 1.003 ^ 48 = 2886.5879… so A ≈ $2 886.59.
  4. I = A − P ≈ 2886.59 − 2500 = $386.59.

Amount $2 886.59; interest $386.59.

$1 000 at 8% per year: when is interest added?1 0001 0501 1001 1501 20000.250.50.7511.251.51.752time (years)balance ($)quarterly (i = 0.02): $1 171.66 after 2 yearsannually (i = 0.08): $1 166.40 after 2 years
Interest is added in jumps at the end of each compounding period. With quarterly compounding (solid line) the jumps come four times a year and each one earns interest sooner, so the balance ends higher than with annual compounding (dashed line). The vertical axis starts at $1 000.
Worked example

Example 2: Is a higher rate always better?

Compare $1 000 invested for 3 years at (a) 4.5% compounded annually and (b) 4.4% compounded monthly. Then compare $4 000 for 5 years at (c) 5% compounded annually and (d) 4.9% compounded monthly.

  1. (a) 1000(1.045)³ ≈ $1 141.17.
  2. (b) 1000(1 + 0.044/12)³⁶ ≈ $1 140.83. Here the higher annual rate wins, by about 33 cents.
  3. (c) 4000(1.05)⁵ ≈ $5 105.13.
  4. (d) 4000(1 + 0.049/12)⁶⁰ ≈ $5 107.94. Here monthly compounding wins, by about $2.81.
  5. Lesson: you cannot compare offers by the stated rate alone. Calculate the amounts (or the effective annual rates) with the same principal and time.

(a) beats (b), but (d) beats (c). Always calculate before you choose.

Worked example

Example 3: Debt grows the same way

A credit card charges 19.99% compounded monthly. Suppose $800 is owed and no payments are made for 1 year (a bad idea; real cards also require minimum payments and may charge fees). How much is owed?

  1. i = 0.1999 ÷ 12 and n = 12.
  2. A = 800(1 + 0.1999/12)¹² ≈ $975.42.
  3. Interest: about $175.42, more than 19.99% of $800 (which is $159.92), because the interest compounds.

About $975.42 is owed.

Worked example

Example 4: Two stages

$3 000 is invested at 3% compounded quarterly for 2 years. The total is then moved to an account paying 4.5% compounded monthly for 3 more years. Find the final amount.

  1. Stage 1: i = 0.0075, n = 8. A₁ = 3000(1.0075)⁸ ≈ $3 184.80. Keep the unrounded value in your calculator.
  2. Stage 2: i = 0.00375, n = 36. A₂ = A₁(1.00375)³⁶ ≈ $3 644.20.

About $3 644.20.

Calculator habits

Type the growth factor in brackets: 2500 × (1 + 0.036 ÷ 12) ^ 48. On many calculators the power key is ^, xʸ or yˣ. Do not round i before you raise it to a power: 0.049 ÷ 12 = 0.0040833…, and rounding it to 0.0041 changes the answer.

Just for fun

Put $1 in an imaginary account paying 100% a year. Compounded annually it becomes $2.00. Monthly: about $2.61. Daily: about $2.71. Every second of the year: still only about $2.72. No matter how often you compound, you can never pass 2.71828…, a famous number called e that the mathematician Jacob Bernoulli ran into while studying compound interest. You will meet e again in later math courses.

Hands-on activity

How often is often enough?

You need: a scientific calculator (or a phone calculator turned sideways) and paper.

  1. Choose a rate between 3% and 8% and a principal of $10 000. Use a 10-year term.
  2. Calculate the amount when interest is compounded annually, semi-annually, quarterly, monthly, weekly and daily. Record them in a table.
  3. Find how much extra each step adds compared with the row above.
  4. With a grown-up, look up the rates on two real savings accounts or GICs from Canadian banks or credit unions. Write each one as "__% compounded __" and calculate what $10 000 would grow to in 10 years.

What to look for: at 5%, the amounts are about $16 288.95 (annually), $16 386.16 (semi-annually), $16 436.19 (quarterly), $16 470.09 (monthly), $16 483.25 (weekly) and $16 486.65 (daily). The extra amounts shrink quickly. Real advertised rates change often; that is why we calculate instead of guessing.

Your turn

1

Find i and n for 6% compounded monthly for 5 years.

2

Find i and n for 5.2% compounded quarterly for 3 years.

3

Find the amount when $2 500 is invested at 3.6% compounded monthly for 4 years.

4

How much interest was earned in the last question?

5

Find the amount when $10 000 is invested at 4.4% compounded semi-annually for 7 years.

6

Find the amount when $1 200 is invested at 2.92% compounded daily for 2 years (use 365 days a year).

7

Which gives more on $1 000 after 3 years: 4.5% compounded annually or 4.4% compounded monthly?

More practice: warm-up

1

How many times a year is interest compounded quarterly?

2

How many times a year is interest compounded semi-annually?

3

What is i for 12% compounded monthly?

4

What is n for 8 years compounded quarterly?

5

In A = P(1 + i)ⁿ, which letter is the principal and which is the future value?

More practice: on your own

1

$800 is owed on a credit card at 19.99% compounded monthly, with no payments for 1 year. Find the amount owed and the interest.

2

$3 000 earns 3% compounded quarterly for 2 years, then 4.5% compounded monthly for 3 more years. Find the final amount.

3

Compare $4 000 for 5 years at 5% compounded annually and at 4.9% compounded monthly. Which is better, and by how much?

4

Write A as a function of t (years) for $750 at 6% compounded monthly, then find A(2.5).

5

Explain why compounding more often gives more interest, and why the extra amount gets smaller.

Challenge

1

$1 is invested at 100% per year for 1 year. Find the amount if interest is compounded annually, quarterly, monthly, weekly and daily. What seems to happen?

2

Does doubling the rate double the interest? Test with $1 000 for 10 years at 3% and at 6%, compounded annually.

3

You deposit $1 000 today and another $1 000 in 2 years. The account pays 4% compounded semi-annually. What is the balance 5 years from today?

Check your understanding

Quiz

1.

For 8% compounded quarterly for 5 years, what are i and n?

2.

Find the amount when $1 000 is invested at 6% compounded monthly for 2 years.

3.

All else equal, which compounding frequency gives the largest amount?

4.

The interest earned on an investment is:

5.

Which calculator entry is correct for $500 at 3.6% compounded monthly for 3 years?

6.

A(t) = 2000(1.0025)¹²ᵗ describes an investment. What is the stated annual rate and compounding?

For parents and teachers

Big idea: Expectation C3.3 asks students to use a scientific calculator and A = P(1 + i)ⁿ to solve for the future value, the present value or the rate per compounding period. This lesson focuses on the future value; Lessons 4 and 5 solve for P and i.

How to help: Have your teen say i and n out loud before touching the calculator: "4.8% compounded quarterly for 6 years means i = 0.012 and n = 24." Most errors happen at this step.

Common difficulties:

  • Forgetting to divide the annual rate by the number of periods, or forgetting to multiply the years by it.
  • Missing brackets on the calculator.
  • Rounding i early (0.049 ÷ 12 should stay unrounded).
  • Believing a higher stated rate is always better. Example 2 shows it depends on the compounding.