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Explain continuous compounding and the lognormal distribution

CFA® Level I · Quantitative Methods · Statistical Distributions for Financial Asset Prices and Returns · QM.6.c

The idea

A variable is lognormally distributed when its natural logarithm is normally distributed. The distribution is bounded below at zero and skewed right, which makes it a natural model for asset prices: a price cannot go negative, but it can in principle rise without limit. Continuously compounded returns are the logs of price relatives, so if those log returns are normal the prices themselves are lognormal. This is the assumption underneath the Black-Scholes model, and it is also why compounding log returns is simply a matter of adding them.

Formulas

Continuously compounded return

continuously compounded return = ln(1 + HPR)

HPR
the holding period return over the period

Continuously compounded returns add across periods where holding period returns must be multiplied, which is why they are the ones used in models. They are always below the holding period return, and further below it the larger it is.

A practice question

From the question bank, with the reason each wrong option is wrong.

What is true of a lognormally distributed variable?

  1. It can take any value, positive or negativeWhy not: The log of a negative number is undefined, so a lognormal variable cannot go below zero. That bound is the reason it suits prices.
  2. It is bounded below at zero and skewed right — correct
  3. It is symmetric about its meanWhy not: It is skewed right: bounded at zero on one side and unbounded on the other, which cannot be symmetric.

Since the log of a negative number is undefined, a lognormal variable cannot fall below zero, and its upper tail stretches out. That shape is why it is used for prices.

A harder one

Why are continuously compounded returns convenient for multi-period analysis?

  1. They can be added across periods rather than compounded — correct
  2. They are always larger than simple returnsWhy not: They are smaller for positive returns — ln(1.10) is 0.0953, not 0.10. Size is not the attraction; additivity is.
  3. They remove the need to know the holding periodWhy not: The period still has to be known, and log returns are still per-period. What they remove is the need to multiply rather than add.

Log returns are additive: the two-period log return is the sum of the two single-period log returns. Simple returns must be chained multiplicatively instead.

The hardest tier

If continuously compounded returns are normally distributed, what follows about prices?

  1. Prices are also normally distributedWhy not: A normal variable can go negative, and prices cannot. Exponentiating a normal gives a lognormal, not another normal.
  2. Prices are uniformly distributedWhy not: A uniform distribution gives every value in its range equal probability, which no exponential transformation of a normal produces.
  3. Prices are lognormally distributed — correct

Prices are the exponential of the log returns, and exponentiating a normal variable produces a lognormal one. This pairing is the distributional foundation of the Black-Scholes framework.

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