Actuarial Mathematics for Modelling · Interest rates over different time periods

Present Value and Discounting Over Different Time Periods

Q: What is the difference between a rate of interest and a rate of discount?

Both measure the same growth over one period. The rate of interest i divides the interest by the amount at the start. The rate of discount d divides the same interest by the amount at the end. So d = i/(1+i), and d is smaller than i for i > 0.

Updated 11 October 2026 · Fact-checked

Present value is the amount today that grows to a given future payment. With effective annual interest rate i, PV = S × v^n, where v = 1/(1+i) = 1 − d. With a nominal discount rate d(m), PV = S × (1 − d(m)/m)^(mn). For an equation of value, put all cashflows at one time point and equate.

Understand Present Value and Discounting Over Time Periods

Present value answers one question: what is a future payment worth now? You find it by discounting. The discount factor for one year is v = 1/(1+i), where i is the effective annual rate of interest.

There are two ways to describe the same one-year growth. The rate of interest i is interest paid at the end of the year, as a proportion of the amount at the start. The rate of discount d is the same interest, but as a proportion of the amount at the end of the year. So d = i/(1+i). Both rates describe the same money; they only differ in the base they use. Because the base for d is larger, d is always smaller than i for i > 0.

A nominal rate of discount d(m) is quoted per year but applied m times a year. Each 1/m of a year, the amount is multiplied by (1 − d(m)/m). Over one year the discount factor is v = (1 − d(m)/m)^m. Compare this with the nominal interest rate i(m), where 1+i = (1 + i(m)/m)^m. Both give the same effective annual rate when they describe the same transaction.

For m > 1 the ordering is d < d(m) < δ < i(m) < i for... check carefully: with the same effective rate, d(m) is larger than d and below δ, and i(m) lies between δ and i. The full order is d < d(m) < δ < i(m) < i. As m grows, both d(m) and i(m) tend to the force of interest δ.

The equation of value says: at a chosen time point, the present value of all money in equals the present value of all money out (at the given rate). You choose the time point freely. The answer does not change if the rate is the same throughout, so pick the point that makes the working easiest.

Key rules to remember

Discount factor: v = 1 ÷ (1 + i); One-year discount factor at effective annual rate i. PV of S due in n years = S × v^n.
Effective rate of discount: d = i ÷ (1 + i) = 1 − v; d is interest at the end of the year as a proportion of the amount at the end of the year.
Interest from discount: i = d ÷ (1 − d); Use to convert d to i. Also 1 + i = 1 ÷ (1 − d).
Difference identity: i − d = i × d; Quick check that your i and d are consistent.
Nominal discount rate: (1 − d(m) ÷ m)^m = 1 − d = v; d(m) is payable m times a year in advance. Each 1/m year the factor is 1 − d(m)/m.
Link between nominal rates: (1 + i(m) ÷ m)^m = 1 + i = (1 − d(p) ÷ p)^(−p); Use this to move between i(m) and d(p) for any m and p, through the effective annual rate.
Nominal link, same m: 1 ÷ d(m) − 1 ÷ i(m) = 1 ÷ m; Valid when i(m) and d(m) have the same m and the same effective rate.
Force of interest: δ = ln(1 + i); v = e^(−δ); Both d(m) and i(m) tend to δ as m → ∞.
PV with nominal discount rate: PV = S × (1 − d(m) ÷ m)^(mn); S is paid at time n years. mn must be the number of discount periods.
Equation of value: Σ (inflows × v^t) = Σ (outflows × v^t); Taken at one time point. Cashflows after the chosen point are discounted; those before it are accumulated.

How to solve Present Value and Discounting Over Time Periods questions

Use this method for any question on rates of discount, nominal discount rates, present values or equation of value.

1Read the rate carefully. Note whether it is i, d, i(m) or d(m), and what m is. Note the time unit of each payment.
2Convert the given rate to the effective annual rate i (or v) using the links: v = 1 − d, or v = (1 − d(m)/m)^m, or 1 + i = (1 + i(m)/m)^m.
3Draw a timeline. Mark each payment, its amount and its time in years.
4Choose a valuation date. For a single unknown, pick the time of the unknown payment or time 0, whichever makes the working short.
5Write the equation of value: PV (or value at the chosen date) of inflows = that of outflows. Use v^t for payments after the date; use (1+i)^(−t) for payments before it.
6Solve for the unknown. If the unknown is a rate, rearrange for v first, then convert to i, d or d(m) as asked.
7Check the answer for sense: d < i, a later payment is worth less now, and the PV is smaller than the payment. State units and rounding.

Quickest way: Work through v

When to use it: Use when the question mixes i, d, i(m) and d(m), or when you need an answer fast in the multiple-choice section.

Turn the given rate into v in one line: v = 1/(1+i), v = 1 − d, or v = (1 − d(m)/m)^m.
Keep v in your calculator memory and compute every PV as S × v^t.
For an equation of value, put the unknown at time 0 or at its own date to avoid extra algebra.
Convert v to the requested rate only at the end: i = 1/v − 1, d = 1 − v, d(m) = m(1 − v^(1/m)).
Test the options: the answer must satisfy d < i and a PV less than the payment.

Common mistakes in Present Value and Discounting Over Time Periods

Using d(m) as if it were an effective rate over the year, for example PV = S × (1 − d(m))^n.
The word 'discount rate' makes students skip the division by m and the power of m.
Fix: Always divide by m and raise to the power mn: PV = S × (1 − d(m)/m)^(mn).
Writing i = 1 − v or d = 1/v − 1.
Students mix up which rate uses which base.
Fix: Remember d = 1 − v and i = 1/v − 1. Check with i − d = id.
Taking d = i when the rate is small.
The two look close, so the difference seems unimportant.
Fix: They are never equal for i > 0. Always convert using d = i/(1+i). Check that d < i.
Using the formula 1/d(m) − 1/i(m) = 1/m when the two rates have different values of m.
The formula is memorised without its condition.
Fix: It only holds for the same m and the same effective rate. Otherwise go through the effective annual rate 1 + i.
Counting the wrong number of periods, such as using n instead of mn, or months as years.
Terms are given in months or quarters but the rate is annual.
Fix: Write the term in years first, then multiply by m for the exponent. Check the units on the timeline.
Discounting a payment that lies before the chosen valuation date.
Students apply v^t automatically without looking at the sign of the time difference.
Fix: A payment before the valuation date must be accumulated: multiply by (1+i)^(time difference).

Worked examples

Example 1

A payment of ₹5,00,000 is due in 3 years. Find its present value at a nominal rate of discount of 8% per annum convertible quarterly. Also find the equivalent effective annual rate of discount.

Show the solution

The rate is d(4) = 0.08 and m = 4. So each quarter the factor is 1 − 0.08/4 = 0.98.
The term is 3 years, so mn = 4 × 3 = 12 quarters.
PV = 5,00,000 × 0.98^12.
0.98^4 = 0.92236816. Then 0.98^12 = 0.92236816^3 ≈ 0.784717.
PV ≈ 5,00,000 × 0.784717 ≈ ₹3,92,358.
The effective annual discount factor is v = 0.98^4 = 0.92236816, so d = 1 − v ≈ 0.07763, or about 7.76%.

Answer: PV ≈ ₹3,92,358 (the effective annual rate of discount is about 7.76%).

Example 2

₹1,00,000 is due in 2 years and ₹2,00,000 is due in 5 years. These are to be replaced by a single payment at time 4 years. The effective annual rate of discount is 6%. Find the single payment.

Show the solution

d = 0.06, so v = 1 − d = 0.94 and 1 + i = 1/0.94.
Choose time 4 as the valuation date. Equate the value of the original payments at time 4 to the single payment X.
The payment at time 2 is 2 years before time 4, so it is accumulated: 1,00,000 × (1/0.94)^2 = 1,00,000 ÷ 0.8836 ≈ 1,13,173.
The payment at time 5 is 1 year after time 4, so it is discounted: 2,00,000 × 0.94 = 1,88,000.
X = 1,13,173 + 1,88,000 = 3,01,173 (to the nearest rupee).
Check by valuing at time 0: 1,00,000 × 0.8836 + 2,00,000 × 0.94^5 ≈ 88,360 + 1,46,781 = 2,35,141. Then 3,01,173 × 0.94^4 ≈ 3,01,173 × 0.780749 ≈ 2,35,141. This agrees.

Answer: The single payment at time 4 is about ₹3,01,173.

Exam tips

In multiple-choice questions, convert everything to v first. Then you only need one power and one conversion.
Check that your answer has d < i and, if m > 1, d < d(m) < i(m) < i for the same effective rate. This quickly removes wrong options.
In written answers, state the rate you are using, the valuation date and the equation of value before calculating. Marks go to the method.
Watch the wording: 'convertible quarterly' or 'payable monthly' gives you m. Do not ignore it.
Where a computer-based question asks for solving a rate, set up the equation of value in the same way and use a solver or a short loop, then report the result to the requested accuracy.

Practice questions from Interest rates over different time periods

Present Value and Discounting Over Time Periods in other exams

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

CMA FinalTime Value of Money, Risk and Discount Rate Basics

Present Value and Discounting Over Time Periods: frequently asked questions

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

Both measure the same growth over one period. The rate of interest i divides the interest by the amount at the start. The rate of discount d divides the same interest by the amount at the end. So d = i/(1+i), and d is smaller than i for i > 0.

How do I find present value using a nominal discount rate d(m)?

Divide d(m) by m and subtract from 1 to get the factor for one 1/m-year period. Raise this to the power mn, where n is the term in years. Then multiply by the payment: PV = S × (1 − d(m)/m)^(mn).

How do I convert between d and i?

Use d = i/(1+i) to go from i to d, and i = d/(1−d) to go from d to i. You can also use d = 1 − v with v = 1/(1+i). The check i − d = id confirms your values.

Does the choice of time point matter in an equation of value?

If the same rate applies throughout, any time point gives the same answer. Pick the one that makes the working simplest, usually time 0 or the date of the unknown payment. If the rate changes over time, you must discount and accumulate carefully through each period.