# FAQ

## What is 420

420 is a DAO. Our mission is to collectively build the right decentralized digital future for everyone.

## Why is it number 420

420 has become a magic Internet number. It symbolizes community cohesion and empowerment.

## How do I become a DAO member

One becomes a DAO member by owning an asset-backed token called 420. It is an ERC-20 token on a EVM-compatible blockchain.

## The First Decentralized Quantitative Easing Program

420 token emission schedule and monetary policy is guaranteed by a smartcontract protocol.
The protocol resembles to a Quantitative Easing (QE) program. Initially the protocol will stimulate economic growth by a generous bond purchase program in the $\gamma$-phase. Then it will gradually taper by reducing duration and number of tokens released in the successive phases. This will introduce more stability and harden the Store of Value property of the tokens.

## What are the properties of 420 token

420 tokens have built-in properties such as Store Of Value, Membership Value and Governance value.

## How do I secure my membership in the DAO

One can join 420 DAO by participating in the token's daily auctions. Every day during the first 420 days, there will be 100,000 tokens available in an auction. The participants commit their capital in a pool which will be matched against the 100,000 allocated for that day.

You can participate in the 420 distribution.

## How do I play to maximize my number of tokens

The most simple way to maximize the number of 420, which also means maximize the share in the asset-backed Treasury, is to acquire 420 early and consistenly maintain staking.

The distribution mechanism is designed to benefit long-term focused members. Several game-theoretical components are introduced to safeguard the health of the DAO token holders.

Early DAO members and stakers benefit from a more generous token emission. The staking rewards are paid only to the stakers. Moreover, Harmonic reward is there to channel rewards from the people who leave staking to the committed stakers. The reward being paid to the stakers further separates the valuable members from the quick flippers.

If you already have some 420, you can stake them.

If you don't have any yet, you can participate in the auctions.

## What is $\gamma$-phase

The first token release phase, which lasts for 420 days, is called $\gamma$-phase. It aims to stimulate and expand the membership base of the DAO.

As the staking rewards form a Harmonic series, the non-compounded staking reward of an early member for the whole period is the sum of the series over 420 days, which is approximately equals $log(420) + \gamma$ where $\gamma$ is the well-known Euler constant.

Please refer to mathematics section for more details.

## What is Double-halving

The double-halving or tapering is a mechanism which halves both the number of issued tokens and the duration of each phase. This aims to programmatically reduce the initial stimulus and harden the Store of Value property of the tokens.

The first phase will last 420 days and there are maximum 420,000 tokens emitted each day.
The second phase will last 210 days and there are maximum 210,000 tokens emitted each day.
The third phase will last 105 days and there are maximum 105,000 tokens emitted each day.
The fourth phase will last 52 days and there are maximum 52,000 tokens emitted each day (note 52 is number of cards in a deck).

After the fourth phase, new tokens will be minted at the same rate as in the fourth phase until the supply cap of 420,000,000 tokens is reached. The cap will be reached in about 12 years.

## What is Floor price

The floor price is the minimum price one can acquire new tokens from the protocol's daily auctions.

On a given day , if the committed capital is lower than expected, the protocol will issue less than the planned number (e.g. daily 100,000 tokens in the first phase), and each of the tokens will be auctioned at the floor price. If there is sufficient demand, then the planned number of tokens will be issued, each will be auctioned at higher price than the floor price.

This mechanism is there to make sure that under weaker demand regimes, there will be less issuance of tokens until demand recovers.

## What is a DAO-owned and Asset-backed Treasury

The capital committed in the daily auctions form the Treasury. It is a pool of assets owned by the DAO, so ultimately the governance token holders decide how to manage the Treasury. The DAO using its Treasury can buy back and burn the governance tokens.

## What is an Insurance fund

The concept of insurance fund originated from the derivatives exchanges such as Bitmex, Binance, Deribit etc. The exchange needs a separate pool of capital called the insurance fund to maintain operational integrity during market stresses: the fund is used as the counterparty of the forced liquidations when no one else is ready to take the positions in the order book; and also to pay winning traders when the margins collected from the losing ones are not sufficient.

Insurance funds are central to crypto trading as there is no other (regulatory or not) intermediary body who is responsible for counterparty risks and settlement insurance. Interestingly the insurance funds in the currently dominating derivatives exchanges tend to increase in value as the exchanges collect margin from liquidated traders in a very conservative manner.

In the design of 420 DAO, we propose to maintain a native insurance fund as a part of the Treasury. It will be used to handle counterparty, credit or smart contracts crisis, and helping other protocols that need capital injection, etc. In the future, it will also take value from imprudent players in the Dapps ecosystem, as standardly implemented in the current centralized exchanges.

Since the insurance fund is a part of the Treasury, the governance token holders naturally own the fund.

## What is the allocation of funds and tokens in different pools

If the auction concludes successfully, 80% of the capital collected will become part of the DAO-owned Treasury. The remaining 20% will be distributed to an Operations Fund to sustainably cover operational cost. The Operations fund will be used to further develop the DAO, reach symbiotic relationship with friendly protocols, and provide community grants etc.

During the first 420 days, there are 220,000 tokens issued every day to reward the stakers only. As mentioned before, thanks to the double-halving mechanism, the number of tokens successively halves during the next phases.

The DAO will match 1:1 of the auctioned tokens, i.e. 100,000 tokens per day during the first 420 days, to allocate to different pools, namely: 60,000 tokens to the Reserve Pool, 30,000 tokens to Development and Marketing pool, and 10,000 tokens to Early supporters.

The picture below illustrates the allocation methodology:

## What is staking token s420

Traditional staking mechanism locks the user's asset in a pool and incentivizes some benefits when users withdraw their asset from the pool. It is inconvenient because usually users want to have exposure to the staking interest, while not being locked a large amount of capital. We implement s420 ERC-20 conformed token to allow users to stake to our pool and still can do something with their stake (buy/sell/collateralize their stakes, fully or partially).

When user put 420 token to our staking pool, we mint and return them s420 as a receipt. When user withdraw from staking pool, they have to return the equivalent amount of s420 to the staking pool, that will get burned and then the equivalent 420 tokens amount will be returned to them. The value relationship between 420 and s420 is 1:1. However, the token s420 is elastic, meaning it allows users to have their stake dynamically grows as the interest of their stake.

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The DAO matches the auction with 100,000 tokens to be allocated in different pools: 60,000 to the Reserve Pool, 30,000 tokens to Development and Marketing pool, 10,000 to Early Supporters. The daily staking rewards are 220,000 tokens to be distributed to stakers only. Overall there is a maximum of 420,000 tokens released every day. The capital collected from the auction will form the Treasury: 30% goes to the Insurance Fund and 50% goes to the Asset Fund, the remaining 20% goes to the Operations fund. After the first 420 days, one proceeds to the double-having phases. The second phase will last for 210 days, with the token emission halves to 210,000 tokens per day. The relative allocation of tokens and assets stays the same as in the first phase. Similarly, the third phase will last for 105 days with a maximum of 105,000 tokens released every day. The fourth phase will last for 52 days with 52,000 tokens released every day. After the fourth phase, the DAO emits tokens at constant amount (i.e. 52,000 tokens per day), until the cap of 420,000,000 supply limit is reached. The ongoing stakers get additional from the unstakers called Harmonic reward. An unstaker pays a fee depending on when she starts staking. The earlier and the less frequent she stakes, the less fee there is for her. The fee decreases linearly in time and reaches at 0% at the end of the 4th phase. The governance token holders have the voting right proportionally to her number of tokens, including voting on decisions on how to spend the Treasury. ### Mathematical formulation To formalize and make the calculations concise we will use the below notations: $T_i$: the duration of the phase $i$ in number of days (with convention $T_0$ = 0), i.e. $T_1 = 420, T_2 = 210, T_3 = 105, T_4 = 52$. $Q_t$: the circulating supply at the end of the day $t$. $Q^a_t$: the circulating supply after the auction of the day $t$, before the staking rewards is distributed. $n$: total number of phases, i.e. $n = 4$. $a_t$: the actual amount of tokens auctioned on day $t$. E.g. $a_1 = 100000$. $r_t$: the amount of staking rewards on day $i$. E.g. $r_1 = 220000$. $b_t = 2a_t + r_t$ is the total number of tokens released on day $t$. $A_t$: the amount of assets in the Treasury, including the Insurance fund. $Z_t$: the amount of tokens in the Reserve pool on days $t$. $f_t$: the fee if the staking time starts at $t$. The charged fee will be distributed among stakers and is called Harmonic reward. $s_t$: the non-compounded staking reward return on day $t$. $S(t, T)$: the non-compounded cumulative staking reward from day $t$ to day $T$. $P(t, T)$: the compounded cumulative staking reward from day $t$ to day $T$. $v_t$: the intrinsic value of a token on day $t$. $o_t$: the auction price of a token on day $t$. $m_t$: the market price of a token on day $t$$.
$l_t$: the membership value of a token on day $t$.
$e_t$: the fidelity value of a token on day $t$.
$\pi_t$: inflation rate on day $t$.
$X$: percentage allocation to the Treasury from the auction fund collected, i.e. $X = 80\%$.

#### Token emission schedule

The total duration of the 4 phases in days:

D_n = \sum_1^n T_i = 787

which is $\frac{787}{365} \approx 2.1$ years.

The circulating supply is given by:

Q_t = \sum_1^t 2a_i + r_i = \sum_1^t b_i

At the end of the $n$-th phases, the circulating supply is:

Q_{D_n} = \sum_1^n (2 a_{T_i} + r_{T_i}) \times T_i = \sum_1^n b_{T_i} \times T_i = 234229000

The table below shows the maximum number of tokens over time. Notice that more than half of the total supply is minted during the first 2 years.

Time Circulating supply
420 D = 1.1 Y 176,400,000
630 D = 1.7 Y 220,500,000
735 D = 2 Y 231,525,000
787 D = 2.1 Y 234,229,000
1825 D = 5 Y 288,205,000
3650 D = 10 Y 383,105,000
4360 D = 12 Y 420,000,000

#### Staking reward

The non-compounded staking return on a given day $t \leq 420$ is given by:

s_t = \frac{r_t}{Q_t} = \frac{r_t}{b_t t} = \frac{r_t}{b_t}\frac{1}{t}

The non-compounded staking return from inception to day $t$ is obtained by summing up $s_t$:

S(0, t) = \sum_1^t s_i = \sum_1^t \frac{r_i}{b_i} \frac{1}{i}

Under this form $S(0,t)$ is approximately a Harmonic series. At the end of the $\gamma$-phase, i.e. on day 420th, one has that:

S(0, 420) = \sum_1^{420} \frac{r_t}{b_t} \frac{1}{t} = \frac{r_{420}}{b_{420}} \sum_1^{420} \frac{1}{t} \approx 0.52 (\mathrm{log}(420) + \gamma) \approx 346\%

which is equivalent to $346\% / 420 \times 365 = 300\%$ annualized.
The last equality is a well-known result which links a Harmonic series to the Euler constant.
Note that this rate is non-compounded, meaning that the staker always start her day by having the same amount of coins. If she does nothing but remains staked and let her coins do the compounding work on their own, she will get much more rewards. Indeed, the compounded staking return from inception to day $t$ writes:

P(0,t) = \Pi_1^t (1 + s_i) - 1 = \Pi_1^t (1 + \frac{r_i}{b_i}\frac{1}{i}) - 1

For the first phase of 420 days, as we assume $r_i/b_i = 220000/420000 \approx 0.52 \ \forall i = 1..420$, $P(0, t)$ can be further simplified as follows:

P(0, 420) = \Pi_1^{420} (1 + s_i) - 1 \approx \Pi_1^{420} (1 + \frac{0.52}{i}) - 1

As a side note: one can prove that $S(0, t) \leq 1 + P(0, t) \leq \mathrm{e}^{S(0, t)}$. Consequently, as $S(0, t)$ is divergent, $P(0, t)$ also is.

#### Inflation rate

The inflation rate is straightforward to compute.

\pi_t = \frac{b_t}{Q_t} = \frac{b_t}{\sum_1^t b_i}

For example, at day 420, it reads:

\pi_{420} = \frac{b_{420}}{\sum_1^{420} b_i} = \frac{420000}{420000 \times 420} \approx 0.23\%

which is equivalent to $0.23\% \times 365 = 86\%$ annualized.

\pi_{787} = \frac{b_{787}}{\sum_1^{787} b_i} = \frac{52000}{420000 \times 420 + 210000 \times 210 + 105000 \times 105 + 52000 \times 52} \approx 0.022\%

which is equivalent to $0.022\% \times 365 \approx 8\%$ annualized.

The long-term inflation rate is even lower and terminates at zero when the supply reaches its limit of 420,000,000 coins. On day 4360th, which is the last day with positive emission, the inflation rate is:

\pi_{4360} = \frac{b_{4360}}{\sum_1^{4360} b_i} = \frac{52000}{Q_{787} + 52000 \times 3573} \approx 0.012\%

which is equivalent to $0.012\% \times 365 \approx 4.5\%$ annualized.

#### Harmonic reward

If a user unstakes at $T$ an amount $K$, the protocol charges a percentage fee $f_T$ of the total capital and this amount will be distributed to the remaining stakers (called the Harmonic reward for the committed stakers).
Recall that the total payout before fee when unstake at $T$ is:

G(t, T) = P(0, T) \frac{K}{P(0, t)}

After fee, the user will get back a total payout:

H(t, T) = G(t, T) \times (1 - f_T)

For $f_T$, we will implement the following form: $f_T$ will decrease linearly from $T = 0$ to $T = 787$ (the end of phase 4) from $42\%$ to $0\%$:

f_T = (1 - \frac{\mathrm{min}(T, 787)}{787}) \times 0.42

#### Intrinsic value

The intrinsic value of each token is given by the quotient between the current Treasury value and the circulating supply, which is always strictly positive:

v_t = \frac{A_t}{Q_t} > 0

#### Floor price and cheapness

The arbitrage price is the auction price at which new members immediately breaks even after the auction, i.e. when the auction price is equals to the new intrinsic value.
The floor price is an auction price at which the intrinsic value stays the same before versus after auction $v_t = v^a_{t+1}$. In this case, new members don't contribute positive value to the DAO as a whole.
At day $t+1$, if the current auction price is $o_{t+1}$, right after the auction, the Treasury will see a new value:

A^a_{t+1} = A_t + X o_{t+1} a_{t+1} = v_t Q_t + X o_{t+1} a_{t+1}

The number of token immediately after the auction is (i.e. when staking rewards of the day $t+1$ is not yet distributed):

Q^a_{t+1} = Q_{t} + 2a_{t + 1}

The intrinsic value right after the auction:

v^a_{t+1} = \frac{A^a_{t+1}}{Q^a_{t+1}} = \frac{v_t Q_t + X o_{t+1} a_{t+1}}{Q_t + 2a_{t+1}}

The arbitrage price is $\tilde{p}$ such that $\tilde{p} = v^a_{t+1}$.
The floor price is $p_{m}$ such that $p_{m} = 2 \frac{v_{t+1}}{X}$.

#### Membership value

Is the value that a member has via her direct token ownership and her share in the Reserve pool:

l_t = v_t + \frac{Z_t}{Q_t - Z_t} v_t

The membership value is a monetary value but is not immediately fully realizable; only the intrinsic value part of it is realizable. One loses the fidelity value when one realizes the intrinsic value.

420 tokenomics design has a desired yet surprising property: the membership value of the current members will increase not only when someone joins the DAO but also when someone decides to leave the organisation.

#### Fidelity value

Is the difference between membership value and intrinsic value, which is always positive:

e_t = l_t - v_t = \frac{Z_t}{Q_t - Z_t} v_t > 0

When someone realizes the intrinsic value, her tokens are burned, hence making the ratio $\frac{Z_t}{Q_t - Z_t}$ increased. It means that when someone decides to burn their tokens, it results in an increased membership and fidelity value for the remaining members. This incentivizes long-term mindset and prevent a reflexive bank-run scenario.