A token bonding curve model has interesting properties, including:
* **Limitless supply**. There is no limit to the number of tokens that can be minted.
* **Deterministic price calculation**. The buy and sell prices of tokens increase and decrease with the number of tokens minted.
* **Guaranteed and immediate liquidity**. The bonding curve contract is the counterparty of the transaction and always holds enough ETH in reserve to buy tokens back. So tokens can be bought or sold instantaneously at any time, the bonding curve acting as an automated market maker.
* **Continuous price**. The price of token n being inferior to the token n+1 and superior to the token n-1, calculating the number of tokens minted for a given amount of ETH (or the number of ETH sent back for a given amount tokens) requires some integral calculus.
It is important to note that in a bonding curve model, the x-axis represents the **number of tokens issued**. To give a simple example, let's say `B(x)=x` and `S(x)=0`. The cost `C` to buy the first 10 tokens is given by the surface between the buy curve and the sell curve that can be expressed as the following integral:
So, in our example: `C=10*10/2=50`.
The bonding curve contract of a _Decentralized Autonomous Trust_ issues _FAIR Securities_ (_FAIRs_). These _FAIRs_ represent a claim on the _DAT_'s cash reserve. It is important to note that, unlike a stock, a _FAIR_ does not represent a claim on the organization's ownership, it only carries a financial right to the cash reserve managed by the _DAT_. And the _DAT_'s cash reserve is a function of the organization's revenues. So, by buying _FAIRs_, an investor gets a financial exposure on the organization's future revenues.
The function _B_ defines the price at which _FAIRs_ can be bought from the _DAT_. _B_ is a linear function and has a positive slope **_b_** such that `B(x)=b*x` where 
_Value flow when an investment occurs_
_Impact on the Bonding Curve Contract of the DAT when an investment occurs_
The investors buying _FAIRs_ are doing so to invest money in the underlying organization. Investors don't want their money to be held in reserve by the _DAT_, they want their money to be put to good use by the organization. Consequently, the value of _s_ must be an order of magnitude lower than _b_, which means that **I** should ideally be low. **I** could also be `0` if the organization's characteristics (revenues, growth...) can justify it.
_Example_: Let's say that an investor sends 10 ETH to the _DAT_, if I=10% then the _DAT_ will transfer 9 ETH to the organization's wallet and will keep 1 ETH in its cash reserve.
The rules described above do not apply if the investor is the beneficiary organization, that is, if the organization is technically _investing in itself_. In that case, **I** is always equal to `100%`. It means that whenever an organization is investing in its _DAT_, 100% of the amount invested by the organization to buy _FAIRs_ goes to the buy-back reserve. For more information, see the 
_Value flow when the beneficiary organization purchases FAIRs_
_Impact on the Bonding Curve when the beneficiary organization purchases FAIRs_
The difference between an investment by an external investor and a _FAIRs_ purchase by the beneficiary organization is their respective _contribution ratio_ to the _DAT_'s reserve:
1. **investment by external investor**: an amount M contributes `I*M` to the _DAT_'s reserve while minting the value equivalent of M, thus a contribution ratio of `(I*M)/M=I` and by construction I<<100%
2. **FAIRs purchase by beneficiary organization**: an amount M contributes `M` to the _DAT_'s reserve while minting the value equivalent of `M`, thus a contribution ratio of `M/M=100%`
After each transaction, _s_ can be recalculated from the amount in reserve Rt:
_Representation of the impact of burning FAIRs on the Bonding Curve_
_Value flow when a FAIR sale occurs_
_Impact on the Bonding Curve Contract of the DAT when an investor sells its tokens_
_Value flow when the CO relies on the DAT to receive its payments_
Optionally, the customer can specify a parameter of the `pay()` function to sent the newly minted _FAIRs_ to an address of his choice (most likely the address of his wallet) in which case the value flow would look like this:
_Value flow when the customer specifies his wallet address to pay()_
_Example_: Suppose D=5%, if the _Continuous Organization_ receives a payment of 100 ETH, 5 ETH will be funneled to the "buyback" cash reserve, increasing the collective value of _FAIRs_.
_Note_: For some _Continuous Organizations_ (_COs_ with no underlying legal entity, for example), it can make sense to receive their customers' payments (i.e. the _CO_'s revenues) directly through the _DAT_. It is important to note that it is not mandatory for the organization's revenues to funnel through the _DAT_ as the organization can also decide to _only_ reward _FAIRs_ holders through _FAIRs_ purchase.
For organizations that already have a running business, they will very likely prefer to _first_ receive a payment from their customer in fiat (as they usually do, without changing their selling process) and will _then_ purchase _FAIRs_ to transfer a fraction of their perceived revenues to the _DAT_ to increase the _FAIRs_ value, as illustrated here:
This way, the _DAT_ is made completely invisible for the customer (no change in UX) and the organization does **not** have to modify any of its highly optimized selling processes.
_Impact of pre-minted tokens, everything else being equal_
So, as an organization, you might have good reasons to pre-mint some _FAIRs_ but beware because pre-minting too much will make your _FAIRs_ become less attractive for investors. A good rule here is to only pre-mint the _FAIRs_ needed before generating revenues. Once revenues starts rolling in, the organization will accrue _FAIRs_ naturally, through the funneling of its revenues to the _DAT_.
Investment (buy)
▪ The DAT mints new FAIRs and send them to the buying investor.
▪ The sum invested is in part distributed to the beneficiary organization and in part saved in the DAT cash reserve according to a pre-defined immutable function I (for investment).
FAIR purchase (buy)
▪ the DAT uses the funds to mint new FAIRs and sends them back to the beneficiary organization.
▪ The funds used to mint the FAIRs are entirely funneled in the DAT cash reserve.
FAIR burn (burn)
▪ The DAT destroys the received FAIRs
▪ The lowest value of the burnt FAIRs is being reaffected equally to all FAIRs holders via the `sell()` function.
Investment (sell)
▪ The DAT burns the received FAIRs and sends ETH back to the selling investor according to a function S (for sell). S has a slope s that increases discretely over time, every time the DAT receives a payment.
▪ The ETH sent back to the investor is taken from the DAT cash reserve and does not affect the organization's treasury.
Revenues (pay)
▪ The DAT transfers the revenues to the organization but retains a fraction D (for distribution) of the revenues that are funneled to the cash reserve, issuing new FAIRs.
▪ The organization (or optionally the customer) receives the newly minted FAIRs.
_Initialization phase of a Continuous Securities Offering_
Until the _MFG_ is reached, all funds are escrowed and investors can decide to withdraw their investment at any time (by calling the `sell()` function) and will receive 100% of their investment back. Once the _MFG_ is reached, the bonding curve starts, a fraction **I** of the MFG is funneled to the cash reserve and the complement (`MFG*(1-I)`) is being transfered to the beneficiary organization.
Also, before the _MFG_ is reached, the beneficiary organization can unilateraly decide to cancel the _CSO_ in which case investors can then withdraw 100% of the funds they individually invested.
It's important to note that once the _MFG_ is reached, then the organization cannot cancel the _CSO_ anymore and it will now continue to be live for a minimum period of time (defined in the smart-contract by the organization). Equally, after _MFG_ is reached, investors cannot withdraw their funds anymore as the bonding curve started. They can now only call the `sell()` function will operates as described in the previous section.
The _MFG_ protects both investors and the organization. It protects the investors because if there is low appeal from investors, the _MFG_ won't be reached and investors can withdraw their money. Plus, the fact that all early investors get averaged priced _FAIRs_ means that no early investors will get unreasonably low price _FAIRs_. But the _MFG_ also protects the organization as the organization can use it to gauge the market appetance for its _CSO_ and can decide to cancel it if investors interest is below its expectations.
The organization should not set the _MFG_ too high though, otherwise it would have the effect of transforming the _CSO_ into a simple crowdfunding campaign and defeat the purpose of the _CSO_. In other words, the _CSO_ must reflect the **minimum** amount the organization expects to validate its _CSO_. It should definitely not be set to the entire value an organization expects to raise.
_Visualization of the exit fee required to close a Continuous Securities Offering_
By doing this, it means that the last investor will make a white operation (bought at `buy` price and sold at the same price moments after) while early investors will hopefully turn a profit (it is obviously not guaranteed as the last issuance price is not necessarily the highest price).
Said otherwise, an investor will always be better off buying or selling their _FAIRs_ in the secondary market, as the price will likely be better than the price proposed by the _DAT_.
Interestingly, the recent rise of automated market mechanisms for secondary markets (like [Uniswap](https://uniswap.exchange) or [Kyber network](https://kyber.network)) means that one could completely blend the primary market (the _DAT_) and the secondary market together from a UX perspective: a user would enter the value of _FAIRs_ he wants to buy and his trade could be automatically optimized between the primary and secondary market. This is a feature [Fairmint](https://fairmint.co) provides, which is important to reduce price volatility and maximize investors' returns.
The zones in **blue** correspond to upward trends of the _FAIR_ price, which translates into the _Continuous Organization_ raising funds. Alternatively, the white zones are downward trends which translates into the _DAT_ (**not** the organization) buying back the _FAIRs_ that are being sent to it using its buyback reserve.
Let's calculate the number of _FAIRs_ `x` received when `d` is invested when `a` _FAIRs_ have already been minted. We have the following:
So the number of tokens minted for an investment `d` is:
Let's calculate the number value `M` received when `x` _FAIRs_ are sold when `a` _FAIRs_ have already been minted. We have the following:
Which gives:
As `s` is easy to calculate:
We ultimately get:
We have:
Now, we do not want this value `R'` locked forever. We want to redistribute it to current _FAIRs_ holder so let's "spread" `R'` accross all current _FAIRs_ holders:
We have:
Which means we can now express the complete `sell()` function, including the burn factor:
## Notes
[1]: the definition of Returns to Scale on [Wikipedia](https://en.wikipedia.org/wiki/Returns_to_scale)
[2]: FAIR: Frictionless Agreement for Investments and Returns
[3]: FAIR Securities undeniably pass the [Howey test](https://en.wikipedia.org/wiki/SEC_v._W._J._Howey_Co.)
[4]: To illustrate: an Uber driver is at the same time a user of Uber and a worker for Uber. Same goes for the renter of a flat on Airbnb. A Facebook user is also a Facebook (non-paid) worker.
[5]: Nicolas Colin is a regular contributor to the [Financial Times](https://www.ft.com/stream/3fd492f4-09b7-3f07-a744-b322b5bd015c) and [Forbes](https://www.forbes.com/sites/nicolascolin/#7ae88301c3b1)
[6]: [The Family](https://thefamily.co) is a european organization educating, protecting, and financing ambitious Entrepreneurs
[7]: [Hedge, A Greater Safety Net for the Entrepreneurial Age](https://www.amazon.com/Hedge-Greater-Safety-Net-Entrepreneurial/dp/1718917082/)
[8]: [L'age de la multitude - Entreprendre et gouverner après la révolution numérique](https://www.amazon.com/gp/product/B00XDVC5X8/ref=dbs_a_def_rwt_bibl_vppi_i2) (not translated in english)
[9]: [Airbnb asks SEC to let it give hosts equity](https://www.axios.com/airbnb-asks-sec-to-let-it-give-hosts-equity-a7d99495-0782-4bce-92bb-4c692ef1b621.html)
[10]: In the 'best case scenario' liquidity event which is the IPO, "[the average founder ownership at IPO was 17% and the average VC ownership at IPO was 56%](https://avc.com/2016/11/founder-dilution/)"
[11]: [The End of Employees](https://www.wsj.com/articles/the-end-of-employees-1486050443?mod=djmc_pkt_ff&tier_1=21128300&tier_2=dcm&tier_3=21128300&tier_4=0&tier_5=4508749)
[12]: See Multicoin Capital "[Venture Capital Economics with Public Market Liquidity](https://austinstartups.com/venture-capital-economics-with-public-market-liquidity-c2dd5cd29ab6)"
[13]: Quote from [Taxation of the Digital Economy](https://www.hldataprotection.com/files/2013/06/Taxation_Digital_Economy.pdf) - Pierre Collin & Nicolas Colin - 2013
[14]: See [Tragedy of The Commons definition on wikipedia](https://en.wikipedia.org/wiki/Tragedy_of_the_commons)
[15]: Some projects have on-chain governance or used a DAICOs to raise funds which gave token holders some governance rights, but there are more the exception than the rule
[16]: ...and also for the so-called sophisticated investors!
[17]: [ICOs raised a record $8.3B in Q2 2018 but most of them were abject failures](https://www.ccn.com/icos-raise-8-3-billion-last-quarter-and-most-of-them-were-abject-failures)
[18]: [Public - ICO Returns 2014-2018](https://docs.google.com/spreadsheets/d/1ioxvJgJiZui9ZD1nud2oLpClYCFcnzvgvMucI56uy-g/edit#gid=1772156349)
[19]: [Meltem Demirors explains the ShitCoin Waterfall on Laura Shin's podcast](https://soundcloud.com/unchainedpodcast/meltem-demirors-and-jill#t=51:03)
[20]: [Multibillion dollar ICO market down to a few hundred million](https://coincentral.com/multibillion-dollar-ico-market-down/)
[21]: [Token Bonding Curves Explained](https://medium.com/@justingoro/token-bonding-curves-explained-7a9332198e0e)
[22]: [Sponsored Burning for TCRs](https://medium.com/@avsa/sponsored-burning-for-tcr-c0ab08eef9d4)
[23]: [equilibrium bonding market](https://blog.oceanprotocol.com/introducing-the-equilibrium-bonding-market-e7db528e0eff), [dynamic bonding curve](https://tokeneconomy.co/token-bonding-curves-in-practice-3eb904720cb8), [fomo3d](https://medium.com/@hayeah/code-analysis-of-fomo3d-pricing-and-dividends-6fb267bbf3a7) [etc](https://medium.com/thoughtchains/on-single-bonding-curves-for-continuous-token-models-a167f5ffef89)...
[24]: [Simon De La Rouvrière](https://twitter.com/simondlr)
[25]: [Tokens 2.0 - Curved Token Bonding in Curation Markets](https://medium.com/@simondlr/tokens-2-0-curved-token-bonding-in-curation-markets-1764a2e0bee5)
[26]: Excerpt adapted from the article [Token Bonding Curves Explained](https://medium.com/@justingoro/token-bonding-curves-explained-7a9332198e0e)
[27]: See the definition of [Irrevocable Trust](https://www.investopedia.com/terms/i/irrevocabletrust.asp) on investopedia
[28]: [How to make bonding curves for continuous tokens](https://blog.relevant.community/how-to-make-bonding-curves-for-continuous-token-models-3784653f8b17)
[29]: [The main benefits of the DutchX mechanism](https://blog.gnosis.pm/the-main-benefits-of-the-dutchx-mechanism-6fc2ef6ee8b4)