Simple and Compound Interest Accumulation: Effective and Nominal Rates

Understand Simple and Compound Interest Accumulation

Key rules to remember

Simple interest accumulation

A(t) = 1 + it, so accumulated value = P(1 + it)

Interest is on the original principal only. i is the annual rate and t is in years.

Compound interest accumulation

A(t) = (1 + i)^t, so accumulated value = P(1 + i)^t

i is the effective annual rate. t can be fractional.

Effective rate from accumulation

i = A(1) − A(0) = A(1) − 1

More generally, the effective rate in period n is i_n = [A(n) − A(n−1)] ÷ A(n−1).

Nominal to effective

1 + i = (1 + i(m)/m)^m

i(m) is the nominal annual rate convertible m times a year. Each period earns i(m)/m.

Effective to nominal

i(m) = m[(1 + i)^(1/m) − 1]

Use this to find the nominal rate that matches a given effective rate.

Equivalent rates for different periods

(1 + i(m)/m)^m = (1 + i(p)/p)^p

Both sides equal 1 + i. Use it to convert between two nominal rates.

Rate for a fractional period

Effective rate over 1/m year = (1 + i)^(1/m) − 1

This is the compound-equivalent rate, not i/m.

How to solve Simple and Compound Interest Accumulation questions

1. 1Read the wording of the rate. Decide whether it is simple, effective or nominal, and note the compounding frequency m.
2. 2Choose one time unit for the whole question, usually one year.
3. 3If the rate is nominal, convert it to an effective annual rate using 1 + i = (1 + i(m)/m)^m. Keep the full decimal value in your calculator.
4. 4Choose the accumulation function: 1 + it for simple interest, (1 + i)^t for compound interest.
5. 5Multiply by the principal and substitute t in the same time unit as the rate.
6. 6If you must find an unknown rate or time, set up the equation first, then rearrange. For time, use logarithms: t = ln(A/P) ÷ ln(1 + i).
7. 7Check that the answer is sensible. Compound should exceed simple for t > 1. State the answer with units and sensible rounding.

Quickest way: Convert once, then use powers

Common mistakes in Simple and Compound Interest Accumulation

• Using the nominal rate as if it were the effective annual rate.

  The quoted rate looks like an annual rate, so it is applied directly with t in years.

  Fix: Always check for the subscript or the words 'convertible'. Convert to effective with 1 + i = (1 + i(m)/m)^m first, or use the per-period factor with m × t periods.

• Finding the monthly effective rate as i ÷ 12 from an effective annual rate.

  Dividing by 12 works for a nominal rate, so students apply it to the wrong kind of rate.

  Fix: For an effective annual rate i, the monthly effective rate is (1 + i)^(1/12) − 1. Divide by 12 only when you have a nominal rate i(12).

• Using compound interest when the question says simple interest, or the reverse.

  Compound is the default in most CM1 questions, so students stop reading carefully.

  Fix: Underline the word simple or compound. Under simple interest, A(t) = 1 + it and the effective rate falls over time for the same i.

• Assuming compound interest always exceeds simple interest.

  The rule is true for most examples with t above 1, so it is remembered as always true.

  Fix: Remember the crossover at t = 1. For 0 < t < 1 and the same i, simple interest gives the larger accumulation.

• Rounding the effective rate early.

  Students write i = 0.0618 and then use it in a power with a large exponent.

  Fix: Store the unrounded value in the calculator memory. Round only the final answer.

• Mismatching the time unit, for example a half-yearly rate with t in years.

  The rate period and the time period are not written side by side.

  Fix: Write the rate period under the rate and the time period under t. If they differ, change t to the number of periods or convert the rate.

Worked examples