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Towards the Risk-Free Curve: Logarithmic vs. Arithmetic Returns

Enrique Millán

30/10/2019

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As Nassim Taleb states, ideas come and go, stories stay.

So today Maximiliano and myself are going to build for you a story which hopefully will carve in your mind the importance of doing things right; or put differently, of using logarithmic returns instead of arithmetic returns when you should. To do so, we will use once again a common process carried out in Finance: return annualization. We will build on top of it some concepts and finally end up showing the importance of using logarithmic returns.

As a personal disclaimer, this is a purely mathematical post, expect no direct application for the moment, but rather understanding of the internal nature of the concepts we deal with every day.

The Annualization Process

The return annualization process answers the following question: for a given time period and a total return achieved in it, for a given partition of this period, which is the constant return for each segment of this partition such that I end up with the same return?

For example, if my asset (understood as a sequence of returns) has achieved a total return of 152% in 10 years, what is the return it should have had every year in the past 10 years to end up with the same 152% return?

The solution to this question is a simple geometrical mean, where the exponential factor will be the number of segments in which we divide our period. For our example, it would be:

\(
r_{1Y} = (1+1.52)^{\frac{1}{9}} – 1 = 10.82\%
\)

Indeed, if you compound that figure for 10 years you end up with the exact 152%.

10 years returns tables

(Watch out for the typical fence post mistake, the annualization factor is 1/9 and not 1/10!).

Generalization

The previous operation is the usual one when analysts compare funds, portfolios or indices with several years of track record. However, if you stop and think about it, you can take it even further and breakdown the return to the daily scale, which will give you the daily return an asset would have each trading day to arrive to the same point 10 years later.

We will call this return \(a_T\), the annualized return to a daily basis, and as you might have guessed, it can be computed by setting the annualization factor to \(\frac{1}{T-1},\) where \(T\) is the total number of days.

The (Mathematical) Risk-Free Curve

And when you think about, you have just defined a mathematically perfect risk-free curve that gets you to the same point as the original market-like price series. Why is it risk-free? Because the daily returns of this series have zero variance for the whole period of evolution.

Original vs. risk-free curves

Of course, this isn’t the risk-free curve you will encounter out there. In the industry, most will refer to the risk-free asset or curve as the one drawn by sovereign bonds, or the interest rate of the central bank of the region. But those are not purely risk-free curves, as their value fluctuates with time.

Summing up, it is risk-free in mathematical terms, but it is also an ex-post curve, which means you won’t ever achieve such path, as we only found it once the market realization had taken place.

The Sequence of Annualization Curves

Great, so now we have a price series and its risk-free equivalent. Both of them have the same total return, but one has had volatility completely sucked out from it.

And so a funny question comes through: what is going on in the gap between these two curves?

Whatever lies in there, it must satisfy two conditions:

  1. Preserve the total return.
  2. Have less volatility than the original series but more than the risk-free.

Let’s have a look at how we can compute that collection of annualization curves.

Returns Interpolation

To generate those sequences of curves, we will directly attack the daily returns of the original curve. In fact, we will modify each return via a simple linear interpolation between the original value and the annualized one.

Arithmetic Returns

And you know what, there is nothing as breaking things to understand how they work. So let’s go ahead and naively start with the arithmetic returns:

$$
\hat{r}_t(\varepsilon) = (1 – \varepsilon) \cdot r_t + \varepsilon \cdot a_T
$$

The cumulative product of this new sequence of returns \(1 + \hat{r}_t\left(\varepsilon\right)\) will lead to new price series which will hopefully lie in the space between the original and the risk-free curves.

Space of curves induced by the interpolated arithmetic returns

This looks like the kind of thing we are after, but something went wrong … they do not end up in the same point!

Logarithmic Returns

Before going into what happened there, let’s give it another try, this time with the logarithmic returns:
$$
\ln\left(1+\hat{r}_t(\varepsilon)\right) = (1-\varepsilon) \cdot \ln \left(1 + r_t\right) + \varepsilon \cdot \ln \left(1 + a_T\right).
$$

See below the comparison between the previous interpolation and this new one.

Comparison of the spaces for arithmetic and logarithmic returns

Now the transformation seems to be working properly! The curves decrease in volatility as the parameter \(\varepsilon\) increases, but they also end up in the same spot as the original price series.

What Is Going On?

So now that we have solved our problem, let’s analyze what went wrong with the first attempt. Let’s go back into our own steps, to see if we can spot where is the subtle mistake:

  1. We took a price series.
  2. We computed its total return.
  3. We computed the geometrical mean of that total return with a factor equal to the number of days minus one. We named this the annualized return on a daily basis.
  4. We linearly interpolated the arithmetic/logarithmic returns for each day between the original return and its annualized counterpart.
  5. We compounded these new returns to generate new price series.

The issue is between steps three and four. To see why, we expand and rearrange the two previous formulas making use of logarithmic properties. For the arithmetic returns we get
$$(1+\hat{r}_t) = (1 + r_t) + \varepsilon\cdot(a_T – r_t),$$

and for the logarithmic returns

$$(1+\hat{r}_t) = \left(1 + r_t\right)^{1-\varepsilon} \cdot (1+a_T)^{\varepsilon}.$$

The first equation is the new absolute return using the annualized return as a linear correction factor. Instead, the second equation is the new absolute return as a geometrical average between the absolute returns of the original series and the annualized one.

If you take logarithms in the second expression and expand the terms, the \(\varepsilon\) term cancels out and you end up finding that for any of its values you will always preserve the total return.

Instead, the first equation is using the annualized return, which is nothing but a geometrical mean, in the middle of arithmetic returns! That’s why the total return was not preserved, we are not adding the same kind of returns.

When you think about it, in the realm of the log-returns, the world is linear, as you can compound returns by simply adding them up. That’s the reason our linear interpolation was not working properly on the first attempt, conceptually we where using tools from one world into another one where it doesn’t belong.

Volatility

Interestingly, and as a last remark, when the volatility of each new sequence of returns for each interpolation strategy is computed, we get a surprising result: both strategies generate series with the same volatility for each value of \(\varepsilon\); and what is most, volatility decreases linearly. Did you expect this?

Volatilities comparison

Final remarks

I hope you have enjoyed today’s walk through the internal mechanisms of the annualization process and the clear distinction between the use of arithmetic and logarithmic returns.

Many argue that daily returns are so small that it doesn’t really make any sense between using the arithmetic or the logarithmic versions, but we have clearly seen that it can change your analysis.

The correct use of the logarithm has allowed us to induce a well-behaved space of curves between any given price series and its mathematical risk-free counterpart.

Original vs. risk free with the right space of curves in the middle

See you next time!