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

Effective Rate of Discount: d = i/(1+i) and How to Convert

Updated 11 October 2026 · Fact-checked

The effective rate of discount d is the interest paid at the start of a year, as a proportion of the amount due at the end of that year. It equals i ÷ (1 + i), or 1 − v. To solve questions, convert between i, v and d, then discount or accumulate.

Understand Effective Rate of Discount

Interest can be paid at the end of a period or at the start. The effective rate of interest i is paid at the end. You invest 1 now and receive 1 + i at the end of the year. The interest is measured against the amount invested at the start.

The effective rate of discount d works the other way. It measures the interest earned over a year as a proportion of the amount at the end of the year. Suppose you will receive 1 at the end of the year. You pay 1 − d now. The amount d is the discount, taken off at the start.

The two views describe the same transaction. Paying v now gives 1 after one year, where v = 1 ÷ (1 + i). So 1 − d = v. This gives d = 1 − v = i ÷ (1 + i). Because d divides by the larger amount, d is always smaller than i when i > 0.

You can also see d as the interest per unit of the final amount, and i as the interest per unit of the starting amount. This is why i = d ÷ (1 − d) and why i − d = i × d. Interest of i paid at the end is worth the same as interest of d paid at the start.

The rate d is used for discounting over one period at the start. Over n years, the present value of 1 due at time n is (1 − d)^n = v^n. This is the compound discount idea, with a constant d each year.

Key rules to remember

Discount factor
v = 1 ÷ (1 + i)
Present value of 1 due in one year.
Rate of discount from i
d = i ÷ (1 + i)
Valid for effective annual rates with compound interest.
Rate of discount from v
d = 1 − v
Equivalent to 1 − d = v.
Interest from discount
i = d ÷ (1 − d)
Needs d < 1.
Link between i and d
i − d = i × d
Useful check on your answers.
Discount and interest factors
(1 − d)^−1 = 1 + i
So v = 1 − d and 1 + i = 1 ÷ (1 − d).
Compound discount over n years
PV = S × (1 − d)^n = S × v^n
S is the amount due at time n.
Discount as d × v^−1
d = i × v
Interest at the end, discounted back one year.

How to solve Effective Rate of Discount questions

Use this method for any question that gives or asks for d, i or v.

  1. 1Read whether the rate given is an interest rate i or a discount rate d, and whether it is annual effective.
  2. 2Write down the rate as a decimal, not a percentage.
  3. 3Convert to the form you need: d = i ÷ (1 + i), v = 1 − d, or i = d ÷ (1 − d).
  4. 4Decide the time: is the payment at the start or end of the period, and how many periods?
  5. 5Discount or accumulate: PV = S × (1 − d)^n, or accumulate with (1 − d)^−n.
  6. 6Check with i − d = i × d or by confirming d < i.
  7. 7Round only at the end and state the answer with units, such as ₹.

Quickest way: Convert to v first

When to use it: Use when a question mixes i and d, or asks for a present value over several years.

  1. From i, compute v = 1 ÷ (1 + i), then d = 1 − v.
  2. From d, compute v = 1 − d, then i = 1 ÷ v − 1.
  3. Use v^n for every present value and 1 ÷ v^n for accumulations.
  4. Quick check: d should be slightly less than i for small rates.

Common mistakes in Effective Rate of Discount

  • Using d = i ÷ (1 − i)

    The pattern looks like i = d ÷ (1 − d), so students swap the symbols badly.

    Fix: Remember d = i ÷ (1 + i) and i = d ÷ (1 − d). The sign matches the rate being divided into.

  • Treating d as a rate paid at year end

    Students think of all rates as end-of-period interest.

    Fix: d is paid at the start and measured against the final amount. Think of paying 1 − d now to receive 1 later.

  • Using 1 + d as an accumulation factor

    It mimics 1 + i.

    Fix: Accumulation over one year at discount rate d is 1 ÷ (1 − d). The factor 1 + d is wrong.

  • Forgetting to convert a percentage

    Rates are quoted as 8% and entered as 8.

    Fix: Always divide by 100 before using any formula.

  • Assuming d = i for nominal rates

    Students ignore the compounding frequency.

    Fix: Convert any nominal rate to an effective annual rate before finding d.

Worked examples

Example 1

The effective annual rate of interest is 8%. Find the effective annual rate of discount and the present value of ₹1,00,000 due in 1 year at this discount rate.

Show the solution
  1. i = 0.08.
  2. d = i ÷ (1 + i) = 0.08 ÷ 1.08 = 0.074074.
  3. v = 1 − d = 0.925926, which matches 1 ÷ 1.08.
  4. PV = 1,00,000 × 0.925926 = ₹92,592.59.

Answer: d ≈ 7.407%, PV ≈ ₹92,592.59

Example 2

A sum of ₹50,000 is due in 3 years. The effective annual rate of discount is 6%. Find the present value and the equivalent effective annual rate of interest.

Show the solution
  1. d = 0.06, so v = 1 − d = 0.94.
  2. PV = 50,000 × 0.94^3.
  3. 0.94^2 = 0.8836, and 0.8836 × 0.94 = 0.830584.
  4. PV = 50,000 × 0.830584 = ₹41,529.20.
  5. i = d ÷ (1 − d) = 0.06 ÷ 0.94 = 0.063830.
  6. Check: i − d = 0.003830 and i × d = 0.063830 × 0.06 = 0.003830.

Answer: PV ≈ ₹41,529.20 and i ≈ 6.383%

Exam tips

  • Show the conversion formula in your working, since written questions award method marks.
  • In MCQs, check which rate is given before applying any formula; the options often include the wrong conversion.
  • Use i − d = i × d as a fast check on your answer.
  • Keep at least five decimal places in v and d until the final answer.
  • State assumptions, such as annual effective rates and compound interest, where relevant.

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

Effective Rate of Discount in other exams

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

Effective Rate of Discount: frequently asked questions

What is the difference between the interest rate and the discount rate?

The interest rate i is interest paid at the end of the year as a proportion of the amount at the start. The discount rate d is interest paid at the start as a proportion of the amount at the end. For the same transaction, d is smaller than i.

How do I convert an interest rate to a discount rate?

Use d = i ÷ (1 + i). For example, i = 10% gives d = 0.10 ÷ 1.10 = 9.09%. You can also find v = 1 ÷ (1 + i) and then d = 1 − v.

How are i, d and v related?

v = 1 ÷ (1 + i) and d = 1 − v. Also i = d ÷ (1 − d), and i − d = i × d. Each one lets you find the others.

Is d always less than i?

Yes, when i is positive, because d = i ÷ (1 + i) divides i by a number greater than 1. The gap is small at low rates and grows as i increases.