Log Returns Vs Normal Returns

Log Returns Vs Normal Returns

So, why there is so much hype about Log Returns, Normal Returns in the investment community? What do these returns mean and why we need to worry about them? Where do they get used and why do we need to convert from one form to another or vice-versa??

We as finance savvied are used to announce few difficult terms in the community such as heteroskedasticity, homoskedasticty and so on to earn some respect in the community (show off mostly or show how cool are we).

The return concept mentioned in the Blog does have some strong meaning in finance community and it makes life much easier while making investment decisions or doing financial modeling.

So, what are these return types and why do we worry about them?
Let's start with an example:

You invested $100 today and earned 12% return in year 1, 15% in year 2, 23% in year 3, 10% in year 4 and 18% in year 5. Now, if someone asks you what was your total accumulated return in these 5 years? The traditional way to calculate this (trust me, you don't need to be a Math Savvy for this) is as follows:
(1.12) * (1.15) * (1.23) * (1.10) * (1.18)  - 1 = 1.05 * 100 (Converting Decimal to Percentages) = 105% or an average of 21% per year.

Now, imagine there exists another yearly return series in which numbers are added and not multiplied in order to get the cumulative effect. We call that return series as continuous compounding series where returns are added and not multiplied. Key point is, these numbers are not same but they bring out the same final outcome. In Mathematics, it's called continuous compounding which is different than Discrete compounding.

Here is how you calculate one from another.
In the example above, 12%, 15%, 23% etc... are all Discrete returns and we can convert them to continuous compounded returns as follows:

Here is a little bit math behind it so excuse me for that:

You simply add 1 to the original return in decimal format and take Natural Log for it to convert to continuous compounded return.
So, 12% = Ln(1.12), 15% = Ln(1.15), and 23% = Ln(1.23)

Upon calculation,
12% = 11.33%
15% = 13.97%
23% = 20.70%

Now, one can simply add them to find multi-period continuous return. If you then want to know the SIMPLE return, you need to take the exponent of the final number.

Let me show you how this works mathematically and how multiplications are converted into additions:

For t periods Simple Returns:
1 + R (0, t) = (1+R1) * (1+R2) * (1+R3) * (1+R4)*.........*(1+Rt)  ----------(1)

Now, taking Natual Logs on both sides:
Ln(1+R(0,t) ) = Ln[(1+R1) * (1+R2) * (1+R3) * (1+R4)*.........*(1+Rt)  ]  ------------(2)

Logs have the property of converting multiplication into addition and it is this property which makes this concept so simple and yet highly innovative

Ln(1+R(0,t)) = Ln(1+R1) + Ln(1+R2) + Ln(1+R3) +...............................+ (1+Rt) ------------(3)

Let's Say using Logarithmic series, you found out what was the final sum on the right side, then all you need to do is:

Ln(1+R(0,t)) = x ----- where x is the final solved value from the express on right side of equation (3)

Taking Exponential on Both sides,
1+R(0,t) = Exp(x)

and R(0,t) = Exp(x) - 1

Primary use of such application is in Financial Modeling where a lot of numbers need to be manipulated and computer programs work much faster for additions than multiplications. Moreover, computer variables have tendency to produce arithmetic overflow errors when precisions go too far and as you know, precisions only go over in multiplications and divisions BUT NOT  in additions and subtractions so this techniques helps a lot.

Now, how to use Natural Log and Exponents is another question for another time but I hope you liked the concept.