Note—This is ported over from an old post written before launch. Please note that since this was originally written, the value of rebase_reaction_lag has changed from 30 to 10. Basically any time you see the number 30 replace it with 10, the same general principles apply

Those of you who have been following us already know that the Ampleforth protocol propagates price exchange-rate information into user wallets by proportionally adjusting wallet balances through a publicly callable rebase() method up to once every 24 hours.

We've talked a bit about how this will cause the AMPL token to move differently from today's digital assets, but I'd also like to discuss the dynamics how it affects market cap and user-balances from another perspective.

How Might This Affect Market Cap and Balances?

In addition to applying countercyclical pressure—if we assume price is held constant—the Ampleforth protocol has the effect of:

  • Compounding user balance changes upon Expansion
  • Attenuating user balance changes upon Contraction

As a quick reminder, the Ampleforth protocol computes supply changes is by taking as input, the 24 hr volume weighted average price (VWAP) of Amples. And then calculating a supply change output by:

  • Inferring the supply change necessary to offset the price difference
  • Smoothing out this difference, as though it will occur over 30 days

By example:

If the reported exchange_rate = $3 and the price_target = $1 the protocol would infer this price difference could be offset by a ($3-$1)/$1 = 200% supply increase. Smoothing this as though it would distribute uniformly over 30 days, means on the next rebase the protocol will increase user balances by 200%/30 = 6.66%.

Further Explanation

To help explain, let's look at the equation for how supply changes are calculated.

Now let's imagine what happens if the volume weighted average price, P_avg, were to hold steady outside the price target threshold for n rebase periods. To see this, we can separate out the rate of growth r like so:

In this case, during dynamic periods when supply is expanding or contracting, the market cap M, will change geometrically as follows:

This means that under expansion, when 1 < r < inf. the curve will look something this, asymptoting vertically:

And under contraction, when 0 < r < 1 the curve will look something like this, asymptoting horizontally:

Note that the two curves above, are not actually symmetric (the contraction curve does not asymptote at -inf. Since r > 0 always, this means the second derivative is always positive.

In other words, upon expansion it becomes increasingly difficult to resist sell-pressure since the absolute quantity of user balance changes, increases with the number of rebases over this period.

And upon contraction, it becomes increasingly easy to find support since the absolute quantity of supply change, decreases over rebases in this period. This means in absolute dollars it costs less and less to restore price to its threshold over time.

Please remember the notion of price holding perfectly constant in expansion or contraction is for illustration purposes only.