FRM Exam Part I · Pricing Conventions, Discounting, and Arbitrage

Time Value of Money and Compounding for FRM Part I

Updated 11 October 2026 · Fact-checked

Time value of money says a unit of currency today is worth more than the same unit later because it can earn interest. You move values through time with FV = PV × (1 + R/m)^(mT) or FV = PV × e^(RT), and compare rates by converting them to one compounding frequency.

Understand Time Value of Money and Compounding

Money today can be invested and earn interest. So ₹1 or $1 received today is worth more than the same amount received later. This is the time value of money. Every bond, swap and forward price in the FRM rests on it.

The future value (FV) is what an amount grows to. The present value (PV) is what a future amount is worth today. Finding PV is called discounting. It is the reverse of compounding.

A quoted rate is meaningless without its compounding frequency. A rate of 8% per year compounded semiannually means 4% every six months. It is not the same as 8% compounded annually. The more often you compound, the more you earn, because interest earns interest sooner.

As the frequency grows (monthly, daily, every instant), the growth factor approaches e^(RT). This is continuous compounding. The FRM uses it constantly because it makes the maths clean: returns add over time and discount factors multiply.

To compare rates fairly, convert them to the effective annual rate (EAR), the rate with annual compounding that gives the same growth. Two rates are equivalent when they give the same growth factor over the same period.

Key formulas to remember

Future value, discrete compounding: FV = PV × (1 + R/m)^(m×T); R is the annual quoted rate, m is compounding periods per year, T is years.
Present value, discrete compounding: PV = FV ÷ (1 + R/m)^(m×T); Same growth factor, used in reverse.
Future value, continuous compounding: FV = PV × e^(R×T); R is the continuously compounded annual rate.
Present value, continuous compounding: PV = FV × e^(−R×T); The discount factor is e^(−RT).
Effective annual rate: EAR = (1 + R/m)^m − 1; For continuous compounding, EAR = e^R − 1.
Continuous rate from discrete rate: Rc = m × ln(1 + Rm/m); Rm is the rate compounded m times a year.
Discrete rate from continuous rate: Rm = m × (e^(Rc/m) − 1); Inverse of the line above.
Equivalent rates: (1 + R1/m1)^m1 = (1 + R2/m2)^m2; Equal annual growth factors mean equivalent rates.

How to solve Time Value of Money and Compounding questions

Use this routine for any time value or compounding question.

1Identify what you must find: PV, FV, a rate, a time, or an equivalent rate.
2Read the compounding frequency of the quoted rate: annual, semiannual, quarterly, monthly or continuous.
3Write the time T in years, and set the number of periods as m × T for discrete compounding.
4Choose the matching formula: (1 + R/m)^(mT) for discrete, e^(RT) for continuous.
5Plug in the numbers. Keep the rate as a decimal and divide by m only for discrete compounding.
6For conversions, equate the two growth factors over one year and solve for the unknown rate.
7Check the answer: a higher compounding frequency should give a lower quoted rate for the same growth, and a higher EAR than the quoted rate.

Quickest way: Convert with ln and e, then compare

When to use it: Use when the question asks you to convert between compounding frequencies or compare rates.

Convert everything to a one-year growth factor.
Semiannual to continuous: Rc = 2 × ln(1 + R/2).
Continuous to semiannual: R = 2 × (e^(Rc/2) − 1).
Use the calculator keys ln and e^x. Do not approximate.
Eliminate options where the converted rate moves the wrong way: the continuous rate is always a little below the discrete quoted rate.

Common mistakes in Time Value of Money and Compounding

Using the quoted annual rate directly in (1 + R)^T when compounding is semiannual or monthly.
Students forget that the quoted rate is a nominal rate split across m periods.
Fix: Always divide R by m and multiply T by m.
Dividing the rate by m in the continuous formula.
Habit from discrete formulas.
Fix: Continuous compounding uses e^(RT) with no division by m.
Treating EAR as equal to the quoted rate.
Ignoring the effect of interest on interest.
Fix: Compute EAR = (1 + R/m)^m − 1. It is higher than the quoted rate whenever m > 1.
Entering the rate as a percentage, such as 5, in the formula.
Rushing under time pressure.
Fix: Convert to a decimal (0.05) before using the formula.
Mixing months and years, such as T = 18 with an annual rate.
Time units in the question differ from the rate's units.
Fix: Convert T to years first: 18 months is 1.5 years.
Converting between rates by comparing nominal rates directly instead of growth factors.
Students assume 6% semiannual equals 6% continuous.
Fix: Equate growth factors over one year, then solve.

Worked examples

Example 1

A deposit pays 6% per year compounded semiannually. What is the equivalent continuously compounded annual rate? Choose the closest: A) 5.82% B) 5.91% C) 6.00% D) 6.09%

Show the solution

Growth factor over one year: (1 + 0.06/2)^2 = 1.03^2 = 1.0609.
Continuous rate: Rc = ln(1.0609).
Equivalently Rc = 2 × ln(1.03) = 2 × 0.029559 = 0.059118.
Rc ≈ 5.91%.

Answer: B) 5.91%

Example 2

A bank offers $50,000 invested for 3 years at 4% compounded continuously. What is the future value, and what EAR does this represent?

Show the solution

FV = 50,000 × e^(0.04 × 3) = 50,000 × e^0.12.
e^0.12 ≈ 1.127497.
FV = 50,000 × 1.127497 ≈ $56,375.
EAR = e^0.04 − 1 = 1.040811 − 1 = 0.040811, about 4.08%.

Answer: FV ≈ $56,375 and EAR ≈ 4.08%

Exam tips

Questions often hide the compounding frequency in the wording. Underline it before calculating.
Practise ln and e^x on the approved calculator until the key sequence is automatic.
Expect conversion questions between continuous, semiannual and annual rates. Equate growth factors.
Sense-check: for the same growth, continuous rate < annual rate with fewer compounding periods, and EAR > quoted rate.
Later topics such as bond pricing, forwards and swaps all reuse these formulas, so build speed here.

Practice questions from Pricing Conventions, Discounting, and Arbitrage

Time Value of Money and Compounding in other exams

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

Time Value of Money and Compounding: frequently asked questions

What is the difference between discrete and continuous compounding?

Discrete compounding adds interest at fixed intervals such as annually or semiannually. Continuous compounding is the limit as the intervals shrink to zero, giving the growth factor e^(RT). It is mostly used in derivative pricing and risk models.

How do I convert a semiannual rate to continuous compounding?

Use Rc = 2 × ln(1 + R/2), where R is the semiannual-compounded annual rate. For example, 6% semiannual gives 2 × ln(1.03), about 5.91%. The continuous rate is always slightly lower.

How do I calculate the effective annual rate?

Use EAR = (1 + R/m)^m − 1 for discrete compounding, or e^R − 1 for continuous compounding. EAR shows the annual-compounding rate that gives the same growth.

Is the continuous rate higher or lower than the discrete rate?

For the same growth, the continuous rate is lower than the quoted discrete rate. If both rates have the same number value, the continuous one gives more growth.