Mean coin age

Definition

Recall that a coin-age model allows us at each given point in time to split all coins into segments where each segment \(x\) has an assigned amount \(v_x\) and creation timestamp \(ot_x\). The set of all segments that exist at time \(t\) will be denoted by \(S_t\).

Then, the formula for computing the mean coin age \(MCA(t)\) is:

\[ MCA(t) := \frac{\sum_{x\in S_t} (t-ot_x)v_x}{\sum_{x\in S_t}v_x} \]

Let us call the quantity in the numerator, the total coin age and let us denote it by \(TCA(t)\). The quantity in the denominator is the total supply existing at time \(t\). We will call it \(TS(t)\)

Total supply

We can already compute the coin circulation. For each period \(p\) we can compute the amount of coins that have been active in the last \(p\) days. Let us denote this amount by \(Circ_p(t)\).

Lemma

\[ TS(t) = Circ_p(t) \]

for \(p\) sufficiently large. More precisely the equality holds if \(p\) is larger than the total life of the coin.

Total age

Let’s focus on computing the total age of each coin. We have

\[ TCA(t) = t\left(\sum_{x\in S_t} v_x\right) - \sum_{x\in S_t} ot_x = TS(t)t - TCT(t) \]

We call the second summand, the total creation timestamp. If we divide it by the token supply we will get the mean creation timestamp \(MCT(t)\). Hence we have the following formula:

\[ MCA(t) = t - MCT(t) \]

In other words the mean coin age is equal to the current timestamp minus the mean creation timestamp.

Total creation timestamp

According to the theory that we have already developed, we can efficiently compute the total creation timestamp. It is the metric \(M_f(t)\) associated to the function \(f(x,t) = ot_x v_x\).

Lemma

Let \(E\) denote the event stream associated to the coin age model \(S\). Then \[ \Delta TCT(t) = \sum_{e\in E \\ t_e = t} \sigma_e ot_x v_e \]

The SQL statement for computing the real-time total creation timestamp delta is

SELECT
  asset_id,
  dt,
  sum(sigma*odt*amount) as value
FROM {events}
GROUP BY asset_id, dt

In practice we don’t use the real-time delta functions. Instead we use daily or five-minute deltas. Those functions must be constructed in such a way that the value of the daily TCT is the same as the value of the real-time TCT at the start of each day. For the five-minute analogue a similar condition must hold. We have:

Lemma

The daily TCT delta is computed using the following SQL:

SELECT
  asset_id,
  toStartOfDay(dt + 86400) AS daily_dt,
  sum(sigma*odt*amount) AS value
FROM {events}
GROUP BY asset_id, daily_dt

The five-minute TCT delta is computed using the following SQL:

SELECT
  asset_id,
  toStartOfFiveMinutes(dt + 300) AS five_minute_dt,
  sum(sigma*odt*amount) AS value
FROM {events}
GROUP BY asset_id, daily_dt

Mean age computation

From the facts above we can easily derive the mean coin age. We first compute the daily or five-minute \(\Delta TCT\). Then we use a cumulative sum to compute the daily of five-minute total creation timestamp \(TCT\). From that we can compute the mean creation timestamp as a composite metric with the SQL formula

total_creation_timestamp/circulation_20y

Finally we can compute the mean coin age as a composite metric with the SQL formula:

dt - mean_creation_timestamp

Relation to age consumed

There is a relation between the total creation timestamp delta and age consumed. The age consumed (or coin-days destroyed) is computed by the formula:

\[ AC(t) = \sum_{e\in E \\ t_e = t} -\sigma_e (t-ot_e)v_e \]

So you have

\[ AC(t) = \Delta TCT(t) - t\sum_{e\in E \\ t_e=t} \sigma_e v_e \]

The latter summand is the timestamp multiplied by the total supply delta at the time \(t\). If we choose a sufficiently large period \(p\) it is also equal to \(\Delta Circ_p(t)\). So if we already compute the age consumed we can compute the delta total creation timestamp as a composite metric:

age_consumed + dt * circulation_delta_20y

However our age consumed metric is measured in days. If we want the delta TCT to be measured in seconds the formula is

age_consumed*86400 + dt*circulation_delta_20y

If we want it to be measured in days, then we have

age_consumed + dt/86400 * circulation_delta_20y

In conclusion once we have age_consumed and circulation we can compute the mean coin age using only cumulative sums and composite metrics.