Discrete returns versus log returns of assets

Question

There have been similar posts here already but nevertheless I find the question worth posting: why do some people claim that log returns of assets are more suitable for statistics than discrete returns.

E.g. in the ESMA CESR guidliens about SSRI log returns are used. I personally think that discrete returns are as good for means of risk management as continuous returns. Furthermore in portfolio context I can calculate the portfolio return by weighting the discrete returns of the assets which does not work with log returns. The time-aggregation of log returns is easier that's true. But people rather think in discrete returns. If my NAV drops from $100$ to $92$ then I have lost $8\%$ and that's it.

Is there any study on this - any good reference? Anything that I can tell my regulator why I stick to discrete returns.

Answer 1

Arithmetic returns allow for easier cross-sectional aggregation and log returns allow for easier time-aggregation.

The reason people use log returns (for equities) is that they are approximately invariant and hence easier to work with in estimating distributions. Meucci does better justice in describing invariance here. The basic idea (again, for equities) is that the distribution of security prices is log-normal, so the arithmetic returns will also be. However, making a log transformation results in approximately normal returns, which are easier to work with. Also, if you do assume them to be normally distributed, then there are convenient results for the convolution of multivariate normal series. This is what allows for easier time-aggregation.

However, you shouldn't take log returns and use them to obtain the arithmetic portfolio return. This is because while you can link them through time, the math doesn't work out, particularly at long horizons, cross-sectionally. Hence, after estimating the distribution of the log returns, proper procedure is to convert the them to arithmetic returns for the purposes of portfolio optimization and risk management.

Answer 2

To fill in the details of what "John" just explained above:

Say that you have stock portfolio for several years: $t_0, t_1, \ldots, t_m$. Say that you have $n$ stocks, so that stock $i$ has a vector of prices $X_i$. The length of each price vector is $m$ because there are $m$ years.

Then, for the first year $t_1$: Calculate the $n$ different arithmetic returns for each stock. This means that stock $i$ will have an arithmetic return during the first year as $r_i = (X_i(1) - X_i(0)) / X_i(0)$. Add all these stock returns, which gives you the total sum of all the arithmetic returns for the first year $t_1$ like this

$r(1) = r_1 + r_2 + ... + r_n$.

This means, for the first year, you have added returns of the first stock, plus returns of the second stock, ..., plus returns of the last stock $n$. This gives you the total return for all stocks, for the first year.

For year $t_2$: Do the same. This will give you a total sum of all the arithmetic stock returns during year 2, call it $r(2)$.

$\vdots$

Up to year $t_m$: which gives you total arithmetic return $r(m)$, summing all stock returns during year $t_m$.

Now, to deduce the total portfolio returns for the entire time period, you want to sum the returns for every year:

$r(1) + r(2) + ... + r(m)$ (*)

BUT! You cannot sum the aritmetic returns for all years. You must add the logarithmic returns instead, to calculate the total portfolio returns. Because when you add returns for different years, and you want to pass through time, you must use the logarithmic returns as they are time invariant.

So you must transform $r(1)$ to logarithmic returns like this:

$r_{log}(1) = \ln (1 + r(1))$

So to calculate the total returns for the entire portfolio for all years, do expression (*) like this instead

$\ln (1 + r(1)) + \ln (1 + r(2)) + ... + \ln (1 + r(m))$

and let us call this sum for $R$. This is a sum of logarithmic returns. To transform $R$ back to the normal simple returns, you need to do like this:

$e^R - 1$

and this is your answer, i.e. the total returns for the entire portfolio, during years $t_0,\ldots,t_m$.

So you use arithmetic returns to calculate the stock returns within a given year (because the arithmetic returns preserve the weights of each stock). But when you want to add the returns traveling through time, one year to the next, you must add the logarithmic returns.