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FRM Exam Part I · Measuring Return, Volatility, and Correlation

Simple Return vs Log Return: Formulas and Conversion

Updated 11 October 2026 · Fact-checked

A simple return is (P₁ − P₀ + income) ÷ P₀. A log (continuously compounded) return is ln(P₁ ÷ P₀). Convert with R = e^r − 1 and r = ln(1 + R). Log returns add across periods; simple returns compound by multiplication. Use the conversion formulas to solve most exam questions.

Understand Simple and Log Returns

A return measures how much an investment gained or lost relative to what you put in. The FRM exam uses two versions, and you must know which one a question is asking for.

The simple return (also called discrete or arithmetic return) is the percentage change in price, plus any income, over one period. If a share goes from 100 to 110 and pays no income, the simple return is 10%. It is easy to read and it is what you see on a statement.

The log return (continuously compounded return) is the natural log of the price ratio: r = ln(P₁ ÷ P₀). For the same move from 100 to 110, r = ln(1.10) = 0.09531, or 9.531%. It is the rate that, compounded continuously, takes you from P₀ to P₁. The two are linked by 1 + R = e^r.

The key reason risk managers like log returns is time additivity. Over several periods, price ratios multiply, and the log of a product is the sum of the logs. So the multi-period log return equals the sum of the single-period log returns. Simple returns do not add; you must compound them: (1 + R₁)(1 + R₂) − 1.

Log returns also cannot take a price below zero when you convert back, because e^r is always positive. A log return can be any real number, so it fits the normal distribution assumption better than a simple return, which is bounded below by −100%. For small returns, r ≈ R. For large moves, the gap is big, and simple return is always greater than log return (R > r) for any non-zero move.

Key formulas to remember

Simple return (no income)
R = (P₁ − P₀) ÷ P₀ = P₁ ÷ P₀ − 1
With income D paid during the period: R = (P₁ + D − P₀) ÷ P₀.
Log (continuously compounded) return
r = ln(P₁ ÷ P₀) = ln(P₁) − ln(P₀)
Uses the natural log. With income, use ln((P₁ + D) ÷ P₀) if the question treats income as reinvested at the end.
Log to simple
R = e^r − 1
Use the e^x key on your calculator.
Simple to log
r = ln(1 + R)
Enter 1 + R as a decimal, e.g. 1.10 for 10%.
Multi-period log return
r(0,T) = r₁ + r₂ + … + r_T
Valid because logs turn products into sums.
Multi-period simple return
1 + R(0,T) = (1 + R₁)(1 + R₂)…(1 + R_T)
Do not add simple returns to get a multi-period return.
Relationship between the two
R > r for any non-zero return; r ≈ R − R² ÷ 2 for small R
The gap grows with the size of the move.

How to solve Simple and Log Returns questions

Use this method for any question on simple or log returns.

  1. 1Identify which return the question wants: simple (discrete) or log (continuously compounded). Also note the horizon.
  2. 2List the inputs: prices P₀ and P₁, any income paid, or given period returns.
  3. 3If prices are given, compute the price ratio (P₁ + income) ÷ P₀.
  4. 4For a log return, take ln of the ratio. For a simple return, subtract 1.
  5. 5For multiple periods, add log returns, or multiply (1 + R) terms for simple returns.
  6. 6To convert, use R = e^r − 1 or r = ln(1 + R). Keep returns in decimals while calculating.
  7. 7Check reasonableness: R should be larger than r, and a negative return must give a ratio below 1.
  8. 8Round only at the end and match the form (decimal or percent) the options use.

Quickest way: Ratio, ln, and the additivity shortcut

When to use it: Use when the question gives prices or several period returns and asks for a total or an annualised figure.

  1. Turn everything into a price ratio first: 1 + R.
  2. If you need a total over many periods and returns are log returns, just add them.
  3. If returns are simple, take ln of each (1 + R), add, then convert back with e^x − 1 only if asked for a simple return.
  4. To annualise a log return, multiply by periods per year. Do not use this for simple returns.
  5. Eliminate options where a simple return is lower than the log return for the same positive move.

Common mistakes in Simple and Log Returns

  • Adding simple returns over several periods to get the total return.

    Additivity is true for log returns, and students carry the habit over.

    Fix: Compound simple returns: multiply (1 + R) terms and subtract 1. Add only log returns.

  • Taking ln of the simple return, ln(R), instead of ln(1 + R).

    Students forget that the log applies to the price ratio, not the percentage change.

    Fix: Always add 1 first. For 10%, compute ln(1.10), not ln(0.10).

  • Using R = e^r instead of R = e^r − 1.

    The −1 is dropped when converting back.

    Fix: Remember e^r is the price ratio. Subtract 1 to get the return.

  • Using log base 10 or a wrong calculator key.

    The log key and the ln key sit side by side.

    Fix: Use ln, the natural log. Test: ln(e) should give 1.

  • Believing simple and log returns are nearly equal for large moves.

    They are close for small returns, and students generalise.

    Fix: For a 50% gain, R = 50% but r = ln(1.5) = 40.55%. Always convert for large moves.

  • Ignoring income such as dividends or coupons when computing a return.

    Only the two prices are looked at.

    Fix: Add income to the end-of-period value when the question gives it.

Worked examples

Example 1

A stock is priced at USD 80 at the start of the year and USD 92 at the end. It pays no dividend. Calculate the simple return and the continuously compounded return.

Show the solution
  1. Price ratio = 92 ÷ 80 = 1.15.
  2. Simple return R = 1.15 − 1 = 0.15 = 15%.
  3. Log return r = ln(1.15).
  4. ln(1.15) = 0.13976.
  5. So r ≈ 13.98%. Check: R > r, as expected.

Answer: Simple return 15.00%; continuously compounded return about 13.98%.

Example 2

A portfolio has monthly log returns of +4.0%, −6.0% and +5.0% over three months. Find the three-month log return and the three-month simple return.

Show the solution
  1. Log returns add: r = 0.04 − 0.06 + 0.05 = 0.03.
  2. Three-month log return = 3.00%.
  3. Convert to simple: R = e^0.03 − 1.
  4. e^0.03 = 1.030455.
  5. R = 0.030455, about 3.05%.

Answer: Three-month log return 3.00%; three-month simple return about 3.05%.

Exam tips

  • Read the question for the words continuously compounded, log, or ln. They signal the log return.
  • When several period returns are given as log returns, add them first. This saves time.
  • Expect questions that ask you to convert between R and r and then choose between close options. Compute to four decimals.
  • Remember that the log return is symmetric: a +ln(1.2) move followed by −ln(1.2) returns you to the starting price. Simple returns of +20% then −20% do not.
  • Know the directional rule R > r for non-zero moves. It can eliminate options without a calculation.

Practice questions from Measuring Return, Volatility, and Correlation

Simple and Log Returns: frequently asked questions

What is the difference between simple and log returns?

A simple return is the percentage change in value, P₁ ÷ P₀ − 1. A log return is the natural log of the price ratio, ln(P₁ ÷ P₀). The log return is the continuously compounded rate that links the two prices, and log returns add over time.

Why are log returns additive?

Multi-period growth is a product of price ratios. The log of a product equals the sum of the logs, so the total log return is the sum of the period log returns. Simple returns must be compounded instead.

How do I convert a log return to a simple return?

Use R = e^r − 1. For r = 5%, R = e^0.05 − 1 = 5.127%. To go the other way, use r = ln(1 + R).

Are simple and log returns ever the same?

Only when the return is zero. For small returns they are very close, but the simple return is always larger than the log return for any non-zero move.