When you do anything with data, you should think about the intuition of each thing you do, and what it represents **“in the real world”.**

Let’s take the example of log returns, which some people tell me they find confusing.

Consider an asset whose price goes from $100 to $200

Assume there are no other cash flows like dividends associated with this thing.

What are the returns for being long?

The intuitively obvious answer is 100%.

It went up $100, the same as its price at the start of the period.

But let’s be careful we understand what that represents.

That represents the return:

- Over the whole period, and
- Calculated on the initial value at the start of the period.

To be tediously clear:

- The asset was worth $100 at the start.
- It increased by $100.
- So the total return over the period on the initial value is 100%.

We might call this the periodic arithmetic return. Or simple return.

N**ow let’s consider log returns.**

We calculate log returns for this period as log(200/100) = 69%

That’s lower.

What does it represent?

It represents the return that we’d apply continuously throughout the period (rather than to the price at the start) to get to the end price.

I told you the answer, but if it’s your first time thinking about this seriously, that probably won’t have made any sense.

**So let’s go through it slowly.**

It’s obvious that 69% isn’t the return over the period on the initial value because that would only get us to $169, not $200.

N**ow let’s split our period into two.**

We’ll apply a return of 69% / 2 to the first period to get $135.

Then apply a return of 69% / 2 to $135 to end up at $181:

Which still isn’t $200, but it is closer.

N**ow, let’s split our period into four.**

And apply a return of 69% / 4 to each of the four periods.

Now we end up at $190:

Which still isn’t $200, but is closer still.

**Now, split the period into eight equal periods.**

And apply a return of 69% / 8 to each of the eight periods.

Now we end up at $194, which is closer still:

N**ow we split it into thirty-two equal periods**.

We apply a return of 69% / 32 to each of the 32 periods.

Now we end up at $199, which is really close.

I won’t draw this, but if we divide it into 64 periods, we end up at $199.30.

And if we do this with 200 periods, we end up at $199.76

And if we do it **over an infinite number of tiny periods we get to exactly $200.00.**

**This is what the log return represents.**

Log returns are useful because:

- You can add them up over time.
- They’re easy to compare over different time periods of different lengths because they scale linearly.
- You can easily convert them into simple returns and back (see image at the top).
- They’re symmetrical on the upside and downside (if you make 10% and lose 10%, you end up at the same place.)

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