The lognormal case, and why returns use logs

Chapter 6 assumed returns might be normal. There are two structural reasons they cannot be, and one change fixes both.

Problem one: prices cannot go negative

A normal distribution assigns positive probability to every real number, including values below zero. Applied to a price, it says a ₹100 share might be worth −₹40, which is not a modelling inaccuracy but a statement about an impossible world. Limited liability means the floor is zero.

Problem two: simple returns do not add

A simple return is

$R_t = \frac{P_t - P_{t-1}}{P_{t-1}}$

Over two periods the returns compound rather than add:

$1 + R_{\text{total}} = (1 + R_1)(1 + R_2)$

So +20% then −20% is $1.2 \times 0.8 = 0.96$, a 4% loss, while the simple returns sum to zero. This is the geometric-mean problem from chapter 3, and it makes every piece of additive statistics — means, variances, the central limit theorem — the wrong tool for simple returns.

The fix: take logarithms

Define the log return:

$r_t = \ln\!\left(\frac{P_t}{P_{t-1}}\right) = \ln(1 + R_t)$

Both problems disappear at once.

Returns now add. Because $\ln(ab) = \ln a + \ln b$:

$r_{\text{total}} = r_1 + r_2 + \cdots + r_n$

Check the example: $\ln(1.2) + \ln(0.8) = 0.1823 - 0.2231 = -0.0408 = \ln(0.96)$. Exactly.

Prices stay positive. If $r$ is normal, then $P_t = P_{t-1}e^{r}$, and $e^{r} > 0$ for every real $r$. A price can approach zero but never reach or cross it.

A variable whose logarithm is normally distributed is lognormal. So the standard model is: log returns are normal, prices are lognormal.

What the lognormal looks like

Not symmetric. It is bounded below by zero and unbounded above, so it is right-skewed — which matches reality in an important way: a stock can lose at most 100% and gain more than 100%. The distribution has that asymmetry built in rather than bolted on.

A consequence that surprises people: for a lognormal, the median and the mean differ.

$\text{median} = e^{\mu} \qquad \text{mean} = e^{\mu + \sigma^2/2}$

The mean exceeds the median, and the gap grows with volatility. The typical outcome is worse than the average outcome, because the average is pulled up by a thin tail of very large gains. For a long-horizon projection this is the difference between the return most people get and the return the brochure quotes — the same half-variance term that separated the geometric and arithmetic means in chapter 3, reappearing because it is the same phenomenon.

This is the arithmetic behind SEBI requiring compounded annualised returns for mutual fund periods beyond a year: a compounded figure is a statement about the median path, not the mean of a skewed distribution.

When the two measures differ enough to matter

For small returns they are nearly identical, because $\ln(1 + x) \approx x$ when $x$ is small:

Simple return	Log return	Difference
1%	0.995%	negligible
10%	9.53%	small
50%	40.5%	material
−50%	−69.3%	large
100%	69.3%	large

So for daily data the choice barely matters, and for a doubling or a halving it matters a great deal. Note that log returns are symmetric in a way simple returns are not: a log return of $+0.693$ and $-0.693$ are exactly a doubling and a halving, which is the right symmetry for a multiplicative process.

Which to use, and when

Log returns for anything that aggregates across time, any statistical modelling, and any volatility calculation. They add, which is what the maths needs.

Simple returns for anything that aggregates across assets at a point in time — a portfolio's return is the weighted average of its holdings' simple returns, because rupees add. Log returns do not aggregate across a portfolio.

That split is the whole rule: logs across time, simple across holdings.

The point

Simple returns compound rather than add, and a normal distribution permits negative prices. Taking logarithms fixes both: log returns add across time and the implied price distribution is lognormal, bounded below by zero and right-skewed so that a maximum loss of 100% sits naturally beside an unbounded gain. The lognormal's mean exceeds its median by a term in the variance, which is why the typical outcome is worse than the average one.