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

Present Value and Discount Factors: How to Discount a Future Payment

Updated 11 October 2026 · Fact-checked

Present value is the amount today that is equivalent to a payment due in the future. You find it by multiplying the payment by the discount factor v = 1/(1+i), raised to the number of years n: PV = C × vⁿ. Here i is the effective annual interest rate.

Understand Present Value and Discount Factors

Money today is worth more than the same money later, because money today can earn interest. This is the time value of money. To compare payments made at different times, you must move them all to the same date.

Suppose you invest ₹1 for one year at effective annual rate i. You get 1 + i at the end. So a payment of 1 due in one year is worth 1 ÷ (1 + i) today. This quantity is the discount factor, written v = 1/(1+i).

For n years, you divide by (1+i) n times. So the present value of a payment C due in n years is C × vⁿ = C ÷ (1+i)ⁿ. A higher interest rate gives a smaller v, so the present value falls. A longer time also lowers the present value.

The principle of equivalence says two sets of payments are equivalent at a given rate if their present values (at the same date) are equal. This is the idea behind the equation of value. You write present values of money in and money out on one side each, set them equal, and solve for the unknown.

In CM1, always state the interest rate basis. Unless told otherwise, i is an effective annual rate and time is measured in years.

Key rules to remember

Discount factor: v = 1 ÷ (1 + i); i is the effective rate of interest per time unit. Then 1 + i = 1/v.
Present value of a single payment: PV = C × vⁿ = C ÷ (1 + i)ⁿ; C is paid at time n, with constant effective rate i per period.
Accumulation (the reverse): A = P × (1 + i)ⁿ; Present value and accumulated value are inverse operations over the same period.
Relation to effective rate of discount: d = i ÷ (1 + i) = 1 − v; d is the effective rate of discount per period.
Equation of value: Σ (payments in) × v^t = Σ (payments out) × v^t; All terms are valued at the same date, using the same rate.

How to solve Present Value and Discount Factors questions

Use this method for any question on present value, discount factors or equivalence of payments.

1Draw a timeline. Mark each payment, its amount and its time in years.
2Identify the interest rate. Check whether it is effective annual, nominal or a force of interest. Convert to an effective rate per time unit if needed.
3Choose the valuation date. Often it is time 0, but it can be any date that makes the equation simpler.
4Compute v = 1/(1+i), or use the rate for the correct period.
5Multiply each payment by v raised to its time distance from the valuation date. Use negative powers for payments before the valuation date.
6Add the values. For an equation of value, put values of inflows equal to values of outflows.
7Solve for the unknown amount, time or rate.
8Check the answer: it should be smaller than the future payment if you are discounting, and the sign and units should make sense.

Quickest way: Divide by (1+i) once per year

When to use it: Use when there is one or a few payments and the rate is a simple effective annual rate.

Compute 1 + i and use the calculator power key: C ÷ (1+i)^n.
Do not round v early. Keep the full calculator value, or at least 6 decimal places.
For several payments, value each from the same date and store the running total.
To solve for the unknown payment, put it on one side: x = (PV of other side) ÷ vᵗ.
Sanity check: the present value must be less than the future sum when i > 0.

Common mistakes in Present Value and Discount Factors

Multiplying by (1+i)ⁿ instead of dividing when discounting.
Students mix up accumulating forward and discounting back.
Fix: Ask: am I moving to an earlier date? Then divide by (1+i)ⁿ, or multiply by vⁿ.
Using a nominal rate directly as if it were effective.
The question says 8% p.a. convertible quarterly and students use 0.08.
Fix: Convert first: i = (1 + i⁽⁴⁾/4)⁴ − 1, or use the rate per quarter with time in quarters.
Mixing the valuation dates in an equation of value.
Some terms are discounted to time 0 and others to another time.
Fix: Choose one valuation date and compute every term's distance from it.
Rounding v too early.
Students round v to 2 or 3 decimals and then raise it to a power.
Fix: Keep full calculator precision until the final answer.
Confusing v with d.
Both are used for discounting, and d = 1 − v looks similar.
Fix: Remember v = 1/(1+i), the factor for one period. d is the discount rate and equals i/(1+i).

Worked examples

Example 1

A payment of ₹5,00,000 is due in 4 years. The effective annual rate of interest is 6%. Find its present value.

Show the solution

i = 0.06, so v = 1 ÷ 1.06 = 0.943396.
PV = 5,00,000 × v⁴ = 5,00,000 ÷ 1.06⁴.
1.06⁴ = 1.262477.
PV = 5,00,000 ÷ 1.262477 = 3,96,047 (to the nearest rupee).

Answer: The present value is about ₹3,96,047.

Example 2

You will receive ₹1,00,000 in 2 years and ₹2,00,000 in 5 years. At an effective annual rate of 5%, find the present value today and the single payment at the end of year 3 that is equivalent to these two payments.

Show the solution

Present value at time 0: 1,00,000 × 1.05⁻² + 2,00,000 × 1.05⁻⁵.
1.05² = 1.1025, so 1,00,000 ÷ 1.1025 = 90,702.95.
1.05⁵ = 1.276282, so 2,00,000 ÷ 1.276282 = 1,56,705.23.
Total present value = 90,702.95 + 1,56,705.23 = 2,47,408.18.
Let x be the equivalent payment at time 3. Equation of value at time 0: x × 1.05⁻³ = 2,47,408.18.
Using unrounded values, x = 1,00,000 × 1.05 + 2,00,000 × 1.05⁻² = 1,05,000 + 1,81,405.90 = 2,86,405.90, which is about ₹2,86,406 (to the nearest rupee).

Answer: The present value is about ₹2,47,408, and the equivalent single payment at the end of year 3 is about ₹2,86,406.

Exam tips

Always write the timeline and the valuation date before computing. Written questions give marks for the set-up even if the arithmetic slips.
In multiple-choice questions, estimate first. If i > 0, a present value must be smaller than the future payment, which removes some options quickly.
State the formula in standard notation, such as PV = C vⁿ, and show the substituted values in written answers.
Check whether the rate is effective, nominal or a force of interest before starting. Many errors in this topic come from the rate, not the discounting.
Keep full calculator precision through the working and round only at the end.

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

Present Value and Discount Factors in other exams

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

Present Value and Discount Factors: frequently asked questions

What is the discount factor v in actuarial mathematics?

It is v = 1/(1+i), where i is the effective rate of interest per period. It is the present value of 1 due one period from now. For n periods the factor is vⁿ.

How do I calculate the present value of a future payment?

Multiply the payment by vⁿ, where n is the number of periods until it is paid. This is the same as dividing by (1+i)ⁿ. Make sure i and n use the same time unit.

What is the difference between v and d?

v is the discount factor, 1/(1+i). d is the effective rate of discount, d = i/(1+i), and d = 1 − v. You use v to find present values directly.

What does the principle of equivalence mean?

It means that two sets of cashflows are equivalent if their present values are equal at the same date and rate. You use it to form an equation of value and solve for an unknown payment, time or rate.