Normal vs Lognormal Distributions for Asset Prices

Updated for the 2026-2027 CFA® Level I curriculum.

Normal and lognormal distributions are closely connected in asset-price modeling. Continuously compounded returns may be modeled with a normal distribution, while the future prices produced from those returns follow a lognormal distribution.

For CFA Level I, focus on how the exponential relationship between returns and prices keeps modeled prices positive and creates a right-skewed price distribution.

Key Takeaways About Normal vs Lognormal Distributions

• A normal distribution is symmetric and can include negative values.

• A lognormal distribution is right-skewed and includes only positive values.

• Continuously compounded returns can be negative, zero, or positive.

• If the continuously compounded return is normal, the resulting future asset price is lognormal.

• The exponential function connects continuously compounded returns with asset prices.

• A lognormal price can approach zero, but it cannot become negative.

• CFA questions commonly test whether you can separate the distribution of returns from the distribution of prices.

What You Need to Know for CFA Level I

For CFA Level I, focus on:

• Comparing the shapes and possible values of normal and lognormal distributions.

• Explaining why the lognormal distribution is used to model asset prices.

• Connecting continuously compounded returns with future asset prices.

• Using the relationship between , and .

• Recognizing that normally distributed returns do not imply normally distributed prices.

• Explaining why a lognormal distribution has a longer right tail.

• Applying the relationship in a basic simulation or calculation.

Normal vs Lognormal Distribution: Main Difference

A lognormal variable comes from exponentiating a normally distributed variable. The exponential transformation changes the shape of the distribution and keeps every resulting value positive.

How Are Normal and Lognormal Distributions Related?

The continuously compounded return from time 0 to time is calculated as:

Rearranging the formula gives the future asset price:

When follows a normal distribution, follows a lognormal distribution. Multiplying this value by the positive starting price produces a positive, lognormally distributed future price.

The exponential function is positive for every real input:

Therefore, when :

This relationship allows the return model to include negative outcomes while keeping the resulting asset prices above zero.

Where:

• = continuously compounded return from time 0 to time

• = asset price at the beginning of the period

• = asset price at the end of the period

• = natural logarithm

• = base of the natural logarithm

• = end of the holding period

Why Are Asset Prices Modeled With a Lognormal Distribution?

A continuously compounded return may be negative, zero, or positive. For example, a continuously compounded return of -20% represents a decline in the asset’s value.

An asset price requires a different range of possible values. A normal price distribution would extend below zero and assign some probability to negative prices. The lognormal model keeps every simulated price positive.

The lognormal distribution is also right-skewed. A price can move closer to zero on the downside, while its potential increase has no fixed upper limit. This creates a longer right tail and places the mean above the median.

Worked Example: Converting Continuous Returns Into Prices

Assume an asset currently trades at .

Its future price is calculated as:

Consider three possible continuously compounded returns:

A positive and negative continuous return of the same size produces different dollar price changes.

The 20% continuously compounded return increases the price by $22.14:

The -20% continuously compounded return decreases the price by $18.13:

The returns are equally distant from zero, but the corresponding prices are not equally distant from $100. This asymmetry comes from the exponential relationship between returns and prices.

Arithmetic Returns vs Continuously Compounded Returns

A simple, or arithmetic, holding-period return is calculated as:

The continuously compounded return associated with that simple return is:

You can convert the continuously compounded return back into a simple return using:

A simple return has a lower limit of -100% because an investor cannot lose more than the asset’s full starting value under the standard price model.

A continuously compounded return has no finite lower bound. As becomes increasingly negative, moves closer to zero. The corresponding simple return approaches -100% without falling below it.

How Are Normal and Lognormal Distributions Used in Simulation?

A basic asset-price simulation follows four steps:

1. Draw a continuously compounded return from an assumed normal distribution.

2. Convert the return into a price multiplier using .

3. Multiply the starting price by the price multiplier.

4. Repeat the process to produce a range of possible future prices.

For one simulation outcome:

Repeating the process produces a distribution of future asset prices. When the modeled continuously compounded returns are normal, the simulated prices follow a lognormal pattern.

This approach gives each simulation path a positive future price. The quality of the results still depends on the model assumptions, including the chosen mean, volatility, and return distribution.

Common Exam Traps

Common mistakes include:

• Assigning a normal distribution to both continuously compounded returns and future prices.

• Forgetting that a lognormal variable takes only positive values.

• Describing a lognormal distribution as symmetric.

• Assuming a lognormal price can equal a negative value.

• Using instead of as the continuous-return price multiplier.

• Confusing a continuously compounded return with a simple holding-period return.

• Forgetting that the price is lognormal only when the underlying continuously compounded return is assumed to be normal.

• Treating the normal and lognormal models as perfect descriptions of observed market returns and prices.

Practice Question

An analyst assumes that an asset’s one-year continuously compounded return is normally distributed. Which statement about the implied future asset-price distribution is most accurate?

1. The future price is normally distributed and may take a negative value.

2. The future price is lognormally distributed, takes positive values, and is right-skewed.

3. The future price is lognormally distributed and symmetric around its mean.

FAQs About Normal and Lognormal Distributions