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tags: tokenomics
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# Token distribution and protocol's price dynamics

## Introduction

In many ways, protocol's tokens serve the same purposes as firm's shares in traditional finance. Both entitle their owners to firm/protocol's income and both give them a say in governance through voting rights.

A fundamental difference is the way the token supply is distributed in crypto environment. While traditional firms rely on initial and seasoned public offering to attract capital, protocols gradually disseminate their ownership rights to stake-holders and distribute a large part of the supply to users to promote adoption and lever network effects. Moreover, instead of distributing dividends to shareholders once a year, most protocols let investors stake their tokens in a staking vault in which revenues continuously accrues. To stress how fundamental is the difference, let us imagine the company Amazon distributing part of its shares to its customers over several years to bootstrap its market share. How does this difference affect the way protocols are valued by the market? What are the implications for the tokenomics?

Whereas fundraising by share emission has been extensively studied in traditional finance, we are not aware of similar studies for blockchain-based protocols, like defi applications or crypto-games. The aim of this section is to provide a deep economic understanding of how token emission affects investors' profitability and price dynamics. To do so we apply standard financial tools like the discounted cash flow model. We conclude by making some recommendations about the best way to manage token emission.

## Profitability

Consider a protocol which gradually distributes its tokens to the market. Circulating supply at dates $t=0,1,...$ is denoted $s_t$. Initial circulating supply is $s_0$, then gradually converges to its maximum $\bar{s}$ ($\lim_{t \rightarrow \infty} s_t = \bar{s}$). In this simplified economy, the protocol distributes each period two kinds of income to stakehoders through the staking module:
- fees $y_t$ collected from protocol's use
- freshly minted supply valued at current price : $p_t(s_t-s_{t-1})$

With $r$ the discount rate, the discounted cash flow model prices the protocol as the present value of future incomes discounted at rate $r$:
$$
p_t = \sum_{z=1}^{\infty} \dfrac{1}{(1+r)^z} \Big( \dfrac{{s_{t+z}-s_{t+z-1}}}{s_{t+z-1}} p_{t+z} + \dfrac{y_{t+z}}{s_{t+z-1}} \Big)
$$

$({s_{t+z}-s_{t+z-1}})/s_{t+z-1}$ is the time-decaying token emission rate. Stake-holders receives protocol's revenue akin to a dividend $y_{t+z}/s_{t+z-1}$, but contrary to traditional firms, they also benefit from the continuous distribution of tokens.

The price earning ratio (PER) is a convenient measure of performance. It is derived by dividing the price $p_t$ of the token by protocol's earnings per token $y_t/s_t$. A high PER ratio means that investors are expecting high growth rates in the future which reflects in the price although not in current earnings yet.

The PER denoted $\pi_t = p_t/(y_t/s_t)$ is
$$
\pi_t = \sum_{z=1}^{\infty} \dfrac{1}{(1+r)^z} \dfrac{y_{t+z}}{y_t} \Big( \dfrac{s_{t+z}-s_{t+z-1}}{s_{t+z-1}} \dfrac{s_t}{s_{t+z}} \pi_{t+z} + \dfrac{s_t}{s_{t+z-1}} \Big) \quad \quad \quad (1)
$$

It has two components: an earning component coming from protocol's organic profitability and a token emission component.

Long-term, all tokens are distributed and circulating supply reaches its maximum. With emission $s_{t+z}-s_{t+z-1}$ converging to zero and both $s_t$ and $s_{t+z-1}$ converging to $\bar{s}$, the asymptotic PER is the discounted sum of income growth rates:
$$
\pi_t = \sum_{z=1}^{\infty} \dfrac{1}{(1+r)^z} \dfrac{y_{t+z}}{y_t}
$$

Assuming income $y_t$ grows at constant rate $g$, pricing is given by the classical Gordon growth formula:
$$
\pi_t = \sum_{z=1}^{\infty} \bigg( \dfrac{1+g}{1+r} \bigg)^z = \dfrac{1+g}{r-g}
$$

The higher the protocol's growth rate $g$, the higher its PER. In short and medium terms, token emission adds a second source of income, preventing closed-form solutions, but the PER, return rate and price can still be numerically simulated.

## Simulations

### Token supply

The supply is assumed to be gradually released over time:
$$
s_t = \alpha s_{t-1} + (1-\alpha) \bar{s} \quad \; \; t=1,2,...
$$

The parameter $\alpha \in (0,1)$ governs the rate at which the circulating supply converges to its maximum $\bar{s}$. Based on a weekly fequency, the emission parameters are initial supply $s_0=0.1$, $\alpha=0.97$ and $\bar{s}=1$. They are chosen so that the supply is approximately fully distributed after four years:


### Price earning ratio

Long-term valuation is given by Eq. (1):
$$
\pi_t = \sum_{z=1}^{\infty} \bigg( \dfrac{1+g}{1+r} \bigg)^z \Big( \dfrac{s_{t+z}-s_{t+z-1}}{s_{t+z-1}} \dfrac{s_t}{s_{t+z}} \pi_{t+z} + \dfrac{s_t}{s_{t+z-1}} \Big)
$$

For simulation purposes, $\pi_{t+z}$ is approximated by $(1+g)/(r-g)$ for $t$ far enough in the future (numerically greater than 5000). Those terminal values are then used to compute the sequence of $\pi_t$ backward. Selecting $r=0.00183$ (equivalent to a 10% APR) and $g=0.00165$ (giving an annualized 9% growth rate), we get:


The PER is higher the first year reflecting the additional return to stakeholders coming from the supply distribution.
<!--The excess of PER is very small compared to its long-term value though. -->

### Return rate

Return rate is computed as:
$$
\rho_w = \dfrac{y_t + p_t({s_t-s_{t-1}})}{p_{t-1} s_{t-1}}
$$

then annualized: $\rho_a = (1+\rho_w)^{52}-1$. Annualized return rate is decreasing:


While the PER returns to its long-term value after the first year, return rate is still positively impacted by the token emission the second year. Yet, the impact remains small compared to a long-term return rate of 10%.

### Price

The price of the token can be retrieved from the PER formula: $p_t = \pi_t y_t/s_t$:


Contrary to the PER and the return rate which show little sensitivity to the initial token distribution program, the price strongly reacts the first year starting from 5453 and gradually decreasing to 650.

The simulated model is now used to study and compare several designs of token launch.

### The small float problem

The overshooting dynamics of the price is getting worse when the initial circulating supply (or float) is small. The float refers to tokens that exist on-chain, and which are not encumbered by any programmatic or legal contracts. A small float means that newly emitted tokens are given away to a small circulating set, entailing high return rate and eventually high prices.

### Fair launch

Many protocols launch their token with a small or non-existent float. This is especially true for "fair launch" protocols which don't rely on external funding and give away most of their tokens to the community. A low initial float produces undesirable effects on price dynamics. To see it, we now assume that the initial supply is 1% at launch instead of 10% ($s_0=0.01$ instead of 0.1). The impact on return is similar to the one with $s_0=0.1$:


As for the price, it sky-rockets at launch to 51,000, then rapidly decreases, to reach 750 at the end of the first year:


A simple calculation shows that the initial price level is in direct relation to the initial supply $s_0$. For an initial supply of 1%, the price is:
$$
p_0 = \dfrac{\pi_0 y_0}{s_0} = \dfrac{546.3 \times 1}{0.01} = 54,630
$$

An initial supply ten times greater ($s_0$=0.1) does not affect the numerator much (the PER shows little sensitivity to this parameter and incomes $y_t$ are exogenous by assumption) but reduces the initial price by $10\times$. If the initial supply were fully distributed at launch ($s_0=1$), the price would be divided 10x again. However, in all scenarios, the price returns to the same long-term level once the supply is fully distributed.

Several fair launches conformed to this pattern. The price of \$CRV, the [Curve Finance](https://curve.fi) token, sky-rocketed shortly after its launch on August 14th 2020 and the start of the liquidity mining program:


The price started at $54 the first day (not visible on the Coingecko graphic), then rapidly declined to around $0.5. Investors attracted by high APR were led to buy the token to profit from the abnormal yield.

A second example is given by the launch of \$LQTY, the token of [Liquity Finance](https://www.liquity.org):


Liquity Finance provides \$LQTY holders with miniting fees. Starting with zero float, the resulting high APR was arbitraged away by a skyrocketting price, which went from $10 to $100 in one day then back to around $20 a few days later. A $100 price briefly gave a fully dilluted market cap of $10b, which was more than four times the market cap of Maker DAO, its direct competitor.

A last and extreme example is given by the lending protocol [Geist Finance](https://geist.finance/markets), a fork of Aave living on the Fantom blockchain. Geist Finance launched on October 6th, 2021 with an agressive liquidity mining program. The price of \$GEIST started at $35, that is more than ten times the fully diluted market cap of Aave. It then rapidly decreased to around $0.3, a 100x reduction rate:

<!--Interstingly, another protocol, Blizz finance, replicated the launch model of Gesit Finance a few weeks later on the Avalanche blockchain and suffered the same price overshooting problem but to a much lesser extent -->


To be fair, this price pattern is not necessarily a sign of irrational exuberance. As our fundamental analysis shows, the return rate associated with buying the token at current price is consistent with a stable return rate. Abnormal return is rapidly arbitraged away by buyers resulting in higher price. Yet, the path to profitability is narrow and risky for investors, as they have to continuously balance a high return from liquidity mining rewards and a negative return from holding a depreciating token. It is undoubtful that many investors incurred large losses during those price episodes.

## Managing the emission rate

We have shown how a token launch with low circulating supply may produce price overshooting and excess volatility. This is an undesirable event as many early buyers who believe in protocol's value proposition are led to sell at a loss.

We propose two ways to mitigate the strong effect on the launching price: a sizeable initial supply and a lower emission rate at launch.

### Initial supply

Previous analysis recommends to launch a token with a large initial float. This can be done through various means: airdrop, public sale via an initial dex offering (IDO) or exchange offering (IEO, involving a centralized exchange), unlocked tokens distributed to the team and investors.

Another way of stabizing the initial price is to start the distribution of the token for incentives purposes without allowing transferability between wallets. This way, the token has no market value in the early period, until a decision of the DAO allows transferability. This is for example the solution adopted by Euler Finance.

The protocol could also make the token freely tradable but delay the distribution of its revenues to stake holders until the circulating supply is large enough.

To see the effects of the supply at launch, we assume an initial supply of 50%. The PER is similar to the case with an initial supply of 10%:


The excess return rate is approximately halved with an excess return of around 0.15% instead of 0.30%:


The main effect is on initial price which starts at 1090.


The price still overshoots as eliminating overshooting altogether would mean distributing 100% of the supply at launch. The amplitude is however considerably reduced compared to an initial price of 5453 with $s_0=0.1$ and 51,000 with $s_0=0.01$.

Even so, this strategy faces limits. Protocols cannot free up a large portion of tokens at once as they have to keep most of the supply aside for liquidity mining incentives, treasury and vested emission.

### Soft launch

A complementary approach is to optimize the emission rate $(s_{t+z}-s_{t+z-1})/s_{t+z-1}$ which enters the PER formula (1). A large emission rate means that recipients are rewarded with a large share of tokens compared to the circulating supply. This translates into high APR, which articifially pumps the price.

We define the ratio float to maximum supply (FMS) as:
$$
\dfrac{s_t}{\bar{s}} = \dfrac{\text{circulating supply}}{\text{maximum supply}}
$$

For an initial FMS $s_0$ of 10%, the emission rate is higher than 0.25 before steadily decreasing to more sustainable levels:


The initial emission rate is as high as 3 for a FMS of 0.01. At the other extreme, it is only 0.03 for an initial FMS of 0.5 as new emissions are diluted by a large supply from the start.

A decreasing emission rate comes form the concave shape of the distribution schedule $s_t$. All protocols adopt a concave distribution schedule, as illustrated by the UNI tokenomics:


In this example, the emission rate is the highest the first year, then decreases over the four year distribution period.

### Linear emission schedule

To smooth the initial emission rate, a linear distribution schedule might be more appropriate:
$$
s_t = s_0 + \beta t \quad \; \; t=1,2,...
$$

with the supply reaching $\bar{s}=1$ in 200 weeks or approximately four years. The emission rate is still decreasing but starts from a lower level. For an initial FMS equal to 10% it starts at 0.045 (instead of 0.27 in the concave distribution schedule):


As for the price, it starts at 5000 before slowly decreasing to less than 1000:


This is much less than 50,000 in the concave distribution scheme but still very high.

### S-shaped emission schedule

The initial emission rate could be lowered further by adopting a S-shaped distribution schedule. Interestingly, this option has been recently adopted by the lending protocol [Euler Finance](https://www.euler.finance/) for the incentives in \$EUL given to liquidity providers (accounting for 25% of the total supply):


<!-- The amount of EUL distributed in each epoch has been determined in advance according to a non-linear schedule (see below). The schedule tries to match increasing protocol user numbers after launch with a concomitant increase in the EUL distribution, which should help to decentralise supply of the token rather than allocating a disproportionate amount to early users. -->

A S-shaped schedule can be simulated by the following function:
$$
s(t) = s_0+\dfrac{1-s_0}{1+\bigg( \dfrac{1-(t/T)}{t/T} \bigg)^{\gamma}}
$$

with $T$ the number of weeks until the entire supply is distributed (as $s(T)=1$). With $s_0=0.1$ and all supply distributed after 200 weeks ($t$=200), the simulated distribution schedule has a S-shaped form:


The emission rate is now hump-shaped, first increasing then decreasing:


The return rate is also hump-shaped with a maximum delayed by more than two years:


The price is still significantly affected with a starting price of 5450 (equivalent to the initial price in the concave case), but the decrease rate is much slower compared to the concave case:


The price takes more than three years before returning to its long-term value compared to one year in the concave case.

## Conclusion

This note has studied in details the impact on various measures of profitability (PER, return rate and price) of progressively distributing the token supply. The distribution adds a second source of return to token-holders beyond protocol's revenues. The upward pressure on return rate is arbitraged away by buyers which propels the price up. With the emission rate declining over time, the price returns to its long-term path which ultimatley depends on protocol's capacity to generate revenues.

The resulting price over-shooting pattern has the potential to create huge losses among early supporters and is clearly an undesirable consequence of many tokenomics. The problem is particularly acute when the token is launched with a low initial supply. This is why we recommend to launch a token with a large enough initial distribution. Adopting a linear or S-shaped distribution schedule mitigates the most adverse consequences by smoothing the price landing over several years.

Another problem faced by protocols during the first phase of token emission, not reviewed here, is that the price may over-react to information. When the circulating supply is small, the AMM pool in which the token is exchanged has low liquidity. Any sized order volume entails significant price deviations. This is why many protocols experience a highly volatile price during the early period, until enough tokens have been emitted.

Some important dimensions of the token economics have been left aside. It is possible that an aggressive initial token distribution may bootstrap the protocol's adoption, which are not studied here due to the assumption of exogenous growth rate of protocol's revenues. Also, all investors have the same information and ability to make optimal investment choice which is obviously a simplification.

<!--

The same problem arises for gaming protocols. For instance, the game Star Atlas launched its share token POLIS on August 25th 2021 with less than 5% of its total supply initially in circulation. The same price dynamics can be observed at launch: