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CFA Level I Exam · Statistical Distributions for Financial Asset Prices and Returns

Lognormal Distribution and Continuously Compounded Returns Explained

Updated 7 October 2026 · Fact-checked

A variable is lognormal if its natural log is normally distributed. Asset prices are modelled as lognormal because continuously compounded returns, ln(S1 ÷ S0), are modelled as normal. To solve questions, convert the price ratio to a log return, apply normal rules, then convert back with eˣ.

Understand Lognormal Distribution and Continuously Compounded Returns

Start with the problem. A normal distribution runs from minus infinity to plus infinity. A stock price cannot fall below zero, so a normal distribution is a poor model for the price itself. Prices are also skewed: a price can rise by any amount but can fall by at most 100%.

The fix is to model the return, not the price. The continuously compounded return over a period is r = ln(S1 ÷ S0), where ln is the natural logarithm. If you assume this return is normally distributed, then the price S1 = S0 × e^r can never be negative, because eˣ is always positive. A variable whose natural log is normal is called lognormal.

The shape of a lognormal distribution follows from this. It is bounded below at zero, it is positively skewed, and it has a long right tail. The normal distribution is symmetric and unbounded. That is the key contrast for the exam: normal for returns, lognormal for prices.

Continuously compounded returns have a useful property: they add over time. If the return is 2% in month one and 3% in month two (both continuously compounded), the two-month return is 5%. Simple holding period returns do not add; they multiply. Because of this, if single-period log returns are independent and identically distributed, the multi-period log return has mean equal to the single-period mean times T and variance equal to the single-period variance times T.

The link to simple returns is: 1 + HPR = e^r, so r = ln(1 + HPR). For the same period, for any non-zero return, the continuously compounded return is less than the simple return. Also, the continuously compounded return is the rate that, compounded continuously, turns S0 into S1.

Key formulas to remember

Continuously compounded return
r(0,T) = ln(S_T ÷ S_0) = ln(1 + HPR)
Uses the natural log. Works for price relatives and for holding period returns.
Price from a continuously compounded return
S_T = S_0 × e^r
The reverse of the log return. Always gives a positive price.
Simple return from a log return
HPR = e^r − 1
Use this to move back to a holding period return.
Additivity over periods
r(0,T) = r(0,1) + r(1,2) + ... + r(T−1,T)
Log returns add. Simple returns compound by multiplying (1 + HPR) terms.
Scaling mean and variance
Mean over T = T × μ; Variance over T = T × σ²; Std dev over T = σ × √T
Holds when periodic log returns are independent and identically distributed.
Definition of lognormal
X is lognormal if ln(X) is normal
Lognormal values are positive, positively skewed and right-tailed.
Annualizing from a holding period
Annual cc return = (1 ÷ T years) × ln(S_T ÷ S_0)
Divide the total log return by the number of years.

How to solve Lognormal Distribution and Continuously Compounded Returns questions

Use this order for any question on lognormal prices or continuously compounded returns.

  1. 1Identify what is given: prices, a simple return, or a log return, and the time period.
  2. 2Decide which quantity is normal. The log return is normal; the price is lognormal.
  3. 3Convert to a log return if needed: r = ln(S1 ÷ S0) or r = ln(1 + HPR).
  4. 4If the period changes, scale: add log returns, multiply the mean by T, and multiply the standard deviation by √T.
  5. 5Do any normal-distribution work (z-score, probability) on the log return, not on the price.
  6. 6Convert back if the question asks for a price or simple return: S = S0 × e^r or HPR = e^r − 1.
  7. 7Check the answer: price positive, log return below the simple return (for any non-zero return), units annual or periodic as asked.

Quickest way: Calculator shortcut with the ln and eˣ keys

When to use it: Use it when the stem gives two prices or a return and asks for a log return or a future price. It takes under 90 seconds.

  1. TI BA II Plus: enter S1 ÷ S0, then press the LN key. For the reverse, enter r, then press 2ND then eˣ (the key above LN).
  2. HP 12C: press S1, ENTER, S0, ÷, then g and the %T key (LN is the blue g function printed on the %T key). For the reverse, enter r, then press g and the 1/x key (eˣ is the blue g function printed on the 1/x key). Check the key labels on your calculator before exam day.
  3. If you start with a simple return, use 1 + HPR as the ratio before pressing LN.
  4. Eliminate options: the log return is always smaller than the simple return, since ln(1 + x) < x for x ≠ 0.
  5. Options run smallest to largest, so check the sign and rough size first and cross out two choices.

Common mistakes in Lognormal Distribution and Continuously Compounded Returns

  • Using the simple return in place of the log return

    Both measure growth, and the numbers look close for small moves.

    Fix: If the stem says continuously compounded or log, compute ln(S1 ÷ S0). Do not use (S1 − S0) ÷ S0.

  • Saying asset returns are lognormal and prices are normal

    The two words get swapped under time pressure.

    Fix: Remember: log of price ratio is normal, so the price is lognormal. Normal is for returns.

  • Forgetting to scale volatility with √T

    Students scale the standard deviation by T, as they do with the mean.

    Fix: Mean scales with T. Variance scales with T. Standard deviation scales with √T.

  • Using ln when the question gives a percentage and not a ratio

    A 5% return gets entered as 5 or 0.05 directly.

    Fix: Enter 1.05, the price relative, before pressing LN. Then ln(1.05) = 0.04879, or 4.879%.

  • Adding simple returns across periods

    Additivity belongs to log returns only, and students apply it to everything.

    Fix: Add only continuously compounded returns. For simple returns multiply the (1 + HPR) factors.

  • Stating that a lognormal distribution is symmetric or can take negative values

    It is confused with the normal distribution.

    Fix: A lognormal variable is bounded at zero, positively skewed and has a long right tail.

Worked examples

Example 1

A share is priced at €40.00 at the start of the year and €46.00 at the end, with no dividends. What is the continuously compounded annual return? Options: A) 13.98%, B) 15.00%, C) 16.19%.

Show the solution
  1. Price relative = 46 ÷ 40 = 1.15.
  2. Log return = ln(1.15).
  3. ln(1.15) = 0.139762, which is 13.98%.
  4. The simple return is 15.00%, which is option B and the trap.
  5. The log return must be below the simple return, which also rules out C.

Answer: A) 13.98%

Example 2

A stock trades at $50. Its continuously compounded return over the next year is 8%. What is the price at the end of the year, and the implied simple holding period return? Options for the price: A) $54.00, B) $54.16, C) $58.00.

Show the solution
  1. S1 = S0 × e^r = 50 × e^0.08.
  2. e^0.08 = 1.083287.
  3. S1 = 50 × 1.083287 = $54.16.
  4. HPR = e^0.08 − 1 = 8.33%.
  5. $54.00 is the simple-return trap (50 × 1.08) and $58.00 adds 8 to 50 as a mistake.

Answer: B) $54.16, with an implied simple holding period return of about 8.33%.

Exam tips

  • Questions often ask only for the property of the distribution: bounded at zero, positively skewed, right tail. Learn those three words.
  • Check whether the stem says continuously compounded. If so, use ln and eˣ, not simple division.
  • For multi-period questions, add log returns first. Do not compound simple returns unless the stem gives them.
  • Eliminate options using sign and size. The log return is always below the simple return for the same period.
  • With three options and no penalty for wrong answers, always answer. Narrow to two using the ln(1 + x) < x check, then choose.

Practice questions from Statistical Distributions for Financial Asset Prices and Returns

Lognormal Distribution and Continuously Compounded Returns: frequently asked questions

What is the difference between normal and lognormal distribution?

A normal variable is symmetric and can take any real value. A lognormal variable is the exponent of a normal variable, so it is always positive, positively skewed and has a long right tail. If ln(X) is normal, X is lognormal.

Why are asset prices modelled as lognormal?

Prices cannot go below zero, and a normal model would allow that. If continuously compounded returns are normal, the price S0 × e^r is always positive and right-skewed, which fits the pattern seen in real prices.

How do I calculate a continuously compounded return?

Divide the ending price by the starting price and take the natural log: r = ln(S1 ÷ S0). If you have a holding period return instead, use r = ln(1 + HPR). Include dividends in S1 if the question gives them.

Why do continuously compounded returns add across periods?

Because ln(a × b) = ln(a) + ln(b). Price growth over two periods is the product of two ratios, so the log of the product is the sum of the two log returns. This makes multi-period analysis simple.