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Actuarial Mathematics for Modelling · Time value of money: compound interest and discounting

Simple and Compound Interest for IAI CM1

Updated 11 October 2026 · Fact-checked

Interest is the reward for lending money. Under simple interest, only the original capital earns interest: A(t) = C(1 + it). Under compound interest, interest also earns interest: A(t) = C(1 + i)^t. To solve a question, find the rate and time, choose the right formula, then accumulate or discount.

Understand Simple and Compound Interest

Interest is the payment a borrower makes to a lender for the use of money. Money today is worth more than the same amount later, because you can invest it and earn interest. This is the time value of money, and the whole of CM1 interest theory rests on it.

The accumulation of a sum is its value at a later date. If you invest capital C at time 0, its accumulated value at time t is A(t). The accumulation factor is A(t) ÷ C, the amount one rupee grows to. It is always at least 1 when interest is non-negative.

With simple interest at annual rate i, interest is calculated only on the original capital. Each year adds the same amount, C × i. Growth is a straight line: A(t) = C(1 + it). Interest earned is never reinvested.

With compound interest at effective annual rate i, interest is added to the capital at the end of each year and then earns interest itself. Each year the balance is multiplied by (1 + i). So A(t) = C(1 + i)^t. Growth is exponential.

For whole years after the first, compound interest beats simple interest at the same rate. Over exactly one year they are equal. For part of a year, simple interest gives a larger value than compound interest at the same rate i, because (1 + i)^t < 1 + it for 0 < t < 1 and i > 0. The gap widens with time and with the rate. In CM1, compound interest is the default unless the question says otherwise.

Key rules to remember

Simple interest accumulation
A(t) = C(1 + it)
i is the annual rate, t is time in years. Interest earned = C × i × t.
Compound interest accumulation
A(t) = C(1 + i)^t
i is the effective annual rate. Works for fractional t when interest is compounded at rate i per year.
Accumulation factor
A(t) ÷ C = (1 + i)^t
Value at time t of one rupee invested at time 0.
Present value (compound)
C = A(t) × (1 + i)^(-t) = A(t) ÷ (1 + i)^t
Discounting is the reverse of accumulation. Define v = 1 ÷ (1 + i), so C = A(t) × v^t.
Present value (simple)
C = A(t) ÷ (1 + it)
Use only when the question states simple interest.
Interest earned in the nth year (compound)
A(n) − A(n − 1) = C × i × (1 + i)^(n − 1)
For simple interest the interest in every year is Ci.
Rate from two values (compound)
i = (A(t) ÷ C)^(1/t) − 1
Use to find an unknown effective annual rate.

How to solve Simple and Compound Interest questions

Use this method for any question on simple or compound interest, accumulation or discounting.

  1. 1Read the interest basis. Is it simple or compound? Is the rate effective annual, or nominal or something else? If nothing is said, assume compound effective annual.
  2. 2Draw a time line. Mark each payment and the date at which you want the value.
  3. 3Convert time into years. Express months or days as fractions of a year, stating your assumption.
  4. 4Pick the formula: A = C(1 + it) for simple, A = C(1 + i)^t for compound. Use v^t for discounting.
  5. 5Substitute carefully. Convert the percentage to a decimal, so 8% becomes 0.08. Keep the full calculator value until the end.
  6. 6If the unknown is i, t or C, rearrange first, then calculate. For t use logs: t = ln(A ÷ C) ÷ ln(1 + i).
  7. 7Check the answer is sensible: accumulated values exceed capital, present values are below the future amount, and units are right.
  8. 8Write the formula, the substitution and the result, rounded as the question asks.

Quickest way: Multiply by (1 + i) each year

When to use it: Use for compound questions with whole years and a small number of periods, and for sense-checking.

  1. Write the growth factor as 1 + i, for example 1.08 for 8%.
  2. Raise it to the power of the number of years on your calculator.
  3. Multiply by the capital to accumulate, or divide to discount.
  4. To compare simple and compound, remember that at t = 1 they are equal, and compound is larger for t > 1.
  5. For a rough check, the rule of thumb is that interest at 8% roughly doubles money in about 9 years under compound growth. Use it only to check, never as your answer.

Common mistakes in Simple and Compound Interest

  • Using the simple interest formula when the question is compound.

    Simple interest is learned first in school, so students use it by habit.

    Fix: Look for the word 'compound' or 'effective'. In CM1 assume compound unless the question says simple.

  • Writing (1 + i)t instead of (1 + i)^t.

    Mixing up the two formulas, or typing the calculator input wrongly.

    Fix: Remember that the exponent signals compounding. Check by testing t = 2: the answer should be C(1 + i)², not C(1 + 2i).

  • Entering the rate as 8 instead of 0.08.

    Rushing and forgetting to convert percentages.

    Fix: Write i = 0.08 on the first line of your working before substituting.

  • Rounding the accumulation factor too early.

    Students round (1.07)^5 to 1.40 and then multiply by a large capital.

    Fix: Keep all digits in the calculator memory and round only the final answer.

  • Applying compound interest to a fractional year and then assuming simple interest gives less.

    Students assume compound is always larger.

    Fix: Remember that for 0 < t < 1 and i > 0, simple interest gives a larger value than compound at the same rate. For t > 1, compound is larger.

  • Treating the nth-year interest as the total interest.

    Confusing interest in one year with interest earned over the whole period.

    Fix: Total interest is A(n) − C. Interest in year n is A(n) − A(n − 1).

Worked examples

Example 1

₹2,00,000 is invested for 4 years. Calculate the accumulated amount (a) at 6% per annum simple interest and (b) at 6% per annum effective compound interest. Find the difference.

Show the solution
  1. Simple: A = 2,00,000 × (1 + 0.06 × 4) = 2,00,000 × 1.24 = ₹2,48,000.
  2. Compound: A = 2,00,000 × (1.06)^4.
  3. (1.06)^2 = 1.1236, so (1.06)^4 = 1.1236 × 1.1236 = 1.26247696.
  4. A = 2,00,000 × 1.26247696 = ₹2,52,495.39 (to the nearest paisa).
  5. Difference = 2,52,495.39 − 2,48,000 = ₹4,495.39.

Answer: Simple: ₹2,48,000. Compound: ₹2,52,495.39. Compound interest gives ₹4,495.39 more.

Example 2

A sum invested at compound interest grows from ₹50,000 to ₹68,000 in 5 years. Find the effective annual rate of interest. Then find the present value of ₹1,00,000 due in 3 years at 8% per annum effective.

Show the solution
  1. Accumulation factor over 5 years = 68,000 ÷ 50,000 = 1.36.
  2. So (1 + i)^5 = 1.36, giving 1 + i = 1.36^(1/5).
  3. ln(1.36) = 0.307485, divided by 5 = 0.061497.
  4. 1 + i = e^0.061497 = 1.06342, so i ≈ 6.34% per annum.
  5. For the second part, PV = 1,00,000 × (1.08)^(-3).
  6. (1.08)^3 = 1.259712.
  7. PV = 1,00,000 ÷ 1.259712 = ₹79,383.22.

Answer: The effective annual rate is about 6.34%. The present value is about ₹79,383.

Exam tips

  • Always state your assumption on the interest basis and the time unit in one line. Examiners award marks for clear method even if arithmetic slips.
  • In MCQs, test the answer options with t = 1 and t = 2 to catch simple versus compound traps quickly.
  • Show the formula in standard notation, such as A(t) = C(1 + i)^t, before substituting numbers.
  • For written questions, compare simple and compound by reasoning about (1 + i)^t versus 1 + it, not only by numbers.
  • In Paper B, use R or Excel to compute accumulation factors, and write down the formula used in the cell or code.

Practice questions from Time value of money: compound interest and discounting

Simple and Compound Interest in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Simple and Compound Interest: frequently asked questions

What is the main difference between simple and compound interest?

Simple interest is earned only on the original capital, so growth is linear. Compound interest is earned on capital plus past interest, so growth is exponential. Over one year they give the same amount.

What is the accumulation factor?

It is the value at time t of one rupee invested at time 0. Under compound interest at effective annual rate i it equals (1 + i)^t. Under simple interest it equals 1 + it.

Does CM1 assume compound interest by default?

Yes, in most CM1 work you use compound interest unless the question clearly states simple interest. Always read the wording and state your assumption.

When is simple interest larger than compound interest?

For a period shorter than one year at a positive rate, the simple interest accumulation is larger than the compound one. Over exactly one year they are equal, and beyond one year compound is larger.