Last week we learnt about returns, how to compute and how to use them. However the story would not be complete if I did not talk about logarithmic returns.

Most of us came across the logarithm function during school days and, back then, you were probably wondering why you should even bother knowing about this mathematical expression. Time has come to appreciate its value in finance :)

Key takeaways:

  • Logarithm of the price is handy and natural in finance. It guarantees that prices never go below zero.
  • The baseline model in financial theory assumes log-returns are normally distributed so that prices follow the log-normal distribution.
  • Transposing the mathematics to a Gaussian distribution allows for simpler and more solvable equations.
  • The other main benefits of log returns derive from the properties of the logarithm function.

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Why logarithmic returns?

Last week, we defined returns as:
\[ r_t = \frac{p_t - p_{t-1}}{p_{t-1}} \tag{1} \]

Where \(p_t\) and \(r_t\) denote price and return on day \(t\). Then, using the logarithm function, we can derive the logarithmic returns:
\[ \log(1+r_t) = \log(\frac{p_t}{p_{t-1}}) \tag{2} \]

Why use (2) instead of (1)? Well first, it is not always better to use (2). It is better in most cases and we will see why.

In the realm of asset pricing, it is often handy and natural to use the logarithm of the price: it guarantees that price never go below zero.

The other main benefits of log returns derive from the properties of the logarithm function.

Log-normality

The baseline model in financial theory assumes log-returns are normally distributed so that prices follow the log-normal distribution. (e.g. this is used in the so-called Black-Scholes-Merton option pricing model).

In mathematics, the distribution that allows to simplify most of the calculation is the Gaussian. It makes calculus easier and solvable.

It is simple to derive from (1):

\[ 1+r_t=\frac{p_t}{p_{t-1}}=\exp^{\log(\frac{p_t}{p_{t-1}})} \]

Indeed, if we assume returns to be log-normally distributed, then \( \log(1+r_t) \) is normally distributed.

Raw-log equality

When returns are very small (\(r\ll1\)) (almost always true for short holding durations, such as daily returns), log-returns are close (i.e. comparable) to the value of raw returns:
\[ \log(1+r)\simeq r,r\ll1 \]

Time-additivity

It is probably the major advantage of log-returns. To compute the compound returns of \(n\) trades with raw returns you would have to deal with this very unpleasant formula:
\[ (1+r_{0})(1+r_{1})\ldots(1+r_{t})=\prod_{t}(1+r_{t}) \]

Instead, we can use the logarithmic identity:
\[ \log(1+r_{t})=\log(\frac{p_{t}}{p_{t-1}})=\log(p_{t})-\log(p_{t-1}) \]

And, because we know that the sum of normally-distributed variables is normal (when all variables are uncorrelated), compounded returns are normally distributed. Then the magic happens, a simple formula for calulating compound returns:

\[ \sum_{i}\log(1+r_{t})=\log(1+r_{1})+\ldots+\log(1+r_{t})=\log(p_{t})-\log(p_{0}) \]

Therefore, the compound return over \(n\) periods is the log difference between final and initial periods.

Numerical stability

Adding small numbers is very safe numerically, while multiplying is not and is subject to arithmetic underflow.

Bijectivity

The logarithm is a bijective function (mapping the set of positive real numbers to the set of all numbers). Therefore log returns can easily be converted back to raw returns applying the exponential function:

\[r_t=e^{\log(1+r_t)}\]

The harsh reality

The real world is not that simple. Assuming the distribution of log-returns to be a Gaussian is an approximation.

The reality is different: it has a non-Gaussian and a heavy-tailed characters. (A heavy-tailed distribution has a heavier tail than the exponential distribution). The Gaussian is just an approximation that comes at a cost: it underestimate extreme events and therefore risk.

To illustrate this let's look at the logarithmic returns distribution of Google:

Fat-tail distribution of Google returns distribution
Empirical distribution of the log-returns of Google stock price, gray boxes. Normal/Gaussian fit of the distribution, black line.

The normal distribution (the bell curve we learned about in high school) stretches off to either direction, getting skinnier. The further away you go from the center the less likely the result will be (high returns are very unlikely events).

It means that if the empirical distribution was indeed a Gaussian, daily returns of ±10% would never happen. We see that the reality is different: extreme events are much more common.

In order to be closer to the real distribution you need to pull it up a little at each end, and stretch it at the top. In simple terms: events closer to the center and farther away from it are more likely to happen than in a normal distribution. The 2008 financial crisis is one of these events, and it was way more likely to happen than any Gaussian model ever predicted.


That's all for today.

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