# the intuition of log returns

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 cashflows 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
- calculated on the initial value at the start of the period

to be tediously clear...

- the asset is worth $100 at the start
- it increased $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.

**now 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 slow.**

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

**now 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 is closer.

**now, let's split our period in four.**

and apply a return calc 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 in eight equal periods.**

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

now we end up at $194, which is closer still...

**now we split it into thirty two equal periods**

we apply a return calc 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.**

they're useful cos:

- you can add them up over time
- they're easy to compare over different time periods of different lengths cos 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.)

beep...boop.