# Time-Weighted Average Balance

Tracking a Time-Weighted Average Balance allows us to determine a users balance at any time in the past, or their average balance held between two times. Using their average balance and the average total supply, we can determine their share of the liquidity for that period of time.

This is critically important for PoolTogether, so that users can deposit and withdraw freely into a prize pool while having their liquidity contribution measured perfectly.

For example:

• if a user held 100 tokens between Dec 20 and Dec 27, then their average balance was 100.
• If instead they bought another 100 tokens halfway through, then their average between Dec 20 and Dec 27 would be 150.

The TWAB is wholly inspired by Uniswap's Time-Weighted Average Price, which is described here.

# Computing the TWAB

Each TWAB record is tuple of an amount a timestamp. The amount stores the cumulative time-weighted balance, and the timestamp is the time at which the twab was recorded. Immediately before a users balance changes, we record a new TWAB.

The formula is:

``new TWAB amount = last TWAB amount + current balance * (current time - last TWAB timestamp)new TWAB timestamp = current time``

## Example​

Let's assume a users transfer history looks like so:

timeactionamount
20send100
30send20

Let's go through it step-by-step.

The first twab would be:

``amount = 0 + 0 * (current time - 0) = 0timestamp = current time = 0``

The second twab would be:

``amount = 0 + 100 * (10 - 0) = 1000timestamp = current time = 10``

The third twab would be:

``amount = 1000 + 150 * (20 - 10) = 2500timestamp = current time = 20``

The fourth twab would be:

``amount = 2500 + 50 * (30 - 20) = 3000timestamp = current time = 30``

And so on. Our twab table would look like this:

timetwab amount
00
101000
202500
303000

What was the users average balance between t = 0 and t = 20?

We can calculate the average by calculating the difference in the cumulative amount and dividing by the elapsed time:

``average balance = (last TWAB amount - first TWAB amount) / (last TWAB timestamp - first TWAB timestamp)``

So we'd have

``average balance = (2500 - 0) / (20 - 0) = 125``

Does this intuitively feel correct? The user held 100 for 10 seconds, then held 150 for ten seconds. Yup. That's right.

# Measuring Liquidity Contribution

We can use a TWAB for the user's balance and a TWAB for the total supply to determine a user's share of the liquidity supplied in the past.

The formula is quite simple:

``users fraction of liquidity = average balance between (t1, t2) / average total supply between (t1, t2)``