In the evolving world of cryptoeconomics, one tool that has always been of interest from the early curation market primitives is the bonding curve. They have been modified and used in AMM’s (automatic market makers) and CFMM’s (constant function market makers) with popular tools like Uniswap, and in fact implemented in real world use cases such as speculating on collectibles and novelty items. Though since their initial popularity around 2017, it had been difficult to find more practical use cases of bonding curves which had certain tendencies to incentivize ponzi-scheme-like behavior and over speculation. Somewhere along the lines, it seemed interesting to harness some of the dynamic nature of these instruments while applying a rate to limit the desire to overspeculate. This paper proposes the idea of a modified version of the bonding curve that issues tokens that receive value from fixed income cash flows. I wish to call these instruments Distribution Curves.
When bonding curves are interconnected with a fixed income generating asset, an equilibrium exists within the system dynamic in which the price increase from additional buyers into the bonding curve is slowed down due to a decrease in yield of the fixed income component of the curve to the point in which you encounter an equilibrium between price and yield.
Equilibriums form based on economic incentives around the speculative price of a token issued from a bonding curve, and the yield that the token receives from sources such as fixed income, staking rewards or project revenue. The equilibrium ranges can create micro economies within the interactions of bonding curves and yielding instruments.
What is a bonding curve? (Feel free to skip to the next section if this is a review)
Bonding curves are a cryptoeconomic primitive originally introduced by Simon De La Rouvier during his exploration of curation markets and the use of game theoretical mechanisms to incentivize certain behaviors within a token economic ecosystem as well as through research done at Bancor in creating their Smart Token issuance system. These cryptoeconomic primitives were designed towards achieving certain dynamics such as continuous token issuance models that avoided the inconsistencies and fallacies of fundraising tools such as ICO’s, while still allowing for the issuance of tokens in a usable and economic activity driven fashion.
Dynamics of these curves can be found here:
The basic premise relies on a few key concepts:
- Tokens are issued on a continuous basis. There is no fixed initial supply limit. A user can purchase tokens, through a smart contract designed to issue tokens. You put in some asset, you get a token; you put in more crypto and you get more tokens.
- The price to buy an additional token in the system increases and is determined by an algorithmic curve that is a function of the supply (We will refer to these as Constant Pricing Functions) eg. as supply increases, price per token increases.
- An example formula would be price = supply * 2; exponential and logarithmic formulas can also be used for the Constant Pricing Functions to reflect the difference in velocity of the price/supply relationship.
- When new buyers emerge, the supply responds to the demand (based on the function) and creates the exact amount of token in need.
- The price to purchase a new token from the bonding curve is also algorithmically determined by total supply multiplied by 2 based on the price = supply * 2 example.
- Users are able to sell their tokens back into the system at the current price of the bonding curve algorithm.
- If a user purchased the token when the supply was low, and then sells the token once the supply reaches a higher state, then they make a profit from the price appreciation that complemented the supply increase.
The key feature that makes this model useful is that the bonding curves can be overlaid into various types of applications. One concept revolves around appending a bonding curve into a non-fungible token where additional tokens can be issued from the non-fungible token, which will become fungible again. This can be categorically determined as a refungible token.
As a product, bonding curves were a very interesting tool for speculating and curating artwork or placing emphasis on certain NFT’s. A few initial use cases:
- Dispensing tokens in a continuous process
Unfortunately, the plain vanilla bonding curves proposed in 2017 proved difficult to practically implement because they were designed primarily to incentivize speculation in areas like curation markets.
Bonding Curves Going Forward
It is now 2020 and throughout the years, curation market tools including bonding curves and token curated registries have found difficulty in achieving mass adoption.
The method proposed here is to combine the concept of a bonding curve with a constant stream of payments. Historically, bonds and debt instruments can be constructed with various permutations of complexity, though at the end of the day, these instruments all boiled down into relatively simple cash flows. In this paper, I propose the dynamic of combining the use of bonding curves with one of the oldest financial instruments in the world: the fixed income product.
The concept revolves around the idea of digitizing a fixed income cash flow. We will not go into significant detail, though the general idea of implementing an amortizing fixed income instrument (loan, lease, bond, debt etc…) would be:
- Create a digital representation of the instrument
- Convert the monthly principal and interest cash payments into a digital trust
- Convert payments into stablecoins that will be distributed to token holders
- Add bonding curve
In this mechanism, consider a scenario such as this:
- Loan A was just issued
- Loan A pays payments of principal and interest equaling $215 USD every month
- Loan A is expected to make these payments for 5 years
Enter bonding curve:
 The Constant Pricing Function from a bonding curve is used to issue tokens that receive that $215 USD per month cash flow pro rata between all the tokens issued
 For simplicity, let’s use the price = supply * 2
- If supply is 1, price is 2; if supply is 2, price is 4; if supply is 3, price is 6 etc…
- As supply of tokens increases, the price increases linearly
 The way this contract works is that the token holders will receive that $215 USD fixed income payment distributed pro rata between each other in proportion to tokens held.
- In this mechanism, as the supply of tokens increases, each token will receive a smaller piece of the $215 USD fixed income.
- Eg. if the supply of tokens was 1 and the price of tokens was $2 USD, having $215 USD of fixed income allocated directly to that 1 token results in a tremendous yield to an investor.
- As supply of tokens increases, the amount distributed per token decreases
In this ecosystem, token purchasers will have two incentive schemes for wanting to purchase tokens early.
1) They wish to buy tokens early because as subsequent buyers enter into the bonding curve (increasing the supply of tokens), the price per token will concurrently increase, resulting in a return for early buyers.
2) Early on, the yield per token will be substantially higher because the amount of fixed income generated (based off of the principal and interest payments from the digitized loan) will be evenly distributed among the existing supply of tokens. Smaller token supply means more pro rata distribution of the initial fixed income return.
One problem of the originally proposed bonding curve is that there was never really an incentive to sell tokens except for the fact that holders were tired of speculating, and they wanted to lock in some of their gains. Or perhaps they were no longer interested in the artwork or NFT they were curating and wished to sell the token back. Such incentive is likely to drive price volatility and is not helpful to create a balanced equilibrium of buyers and sellers into a bonding curve.
In this modified version, as the supply of tokens increases, the yield per token will decrease as the same amount of fixed income will be distributed pro rata among a larger supply of tokens. This yield degradation is enough of an incentive to cause participants to stop speculating and start judging the economic incentive of holding the token. Early investors who were attracted by high yields will no longer be attracted by the lower yields at high token supplies. Therefore there will be enough incentive to sell back into the bonding curve as yields become unattractive.
In bonding curves, users are incentivized to purchase tokens early as well as to convince others to buy tokens later which will subsequently result in price increases in the tokens (as a ponzi scheme would be designed to do). Though in this fixed income distribution model, as the price and supply of the token increase, the yield that gets allocated to token holders decreases because of the pro rata distribution of the fixed income.
The theoretical phenomenon that should occur would be an equilibrium range of price and yield per bonding curve instrument. As the supply of the token increases, price increases (based on the constant pricing function of the bonding curve) though yield decreases. As yields become unattractive, earlier buyers of the fixed income tokens will sell their tokens into the bonding curve at a profit, thus decreasing the supply and price of the fixed income token, and increasing the yield. As higher yields result from selling, the instruments will be attractive investments again, and buy pressure will increase the price and supply, while concurrently decreasing the yield. These counterbalancing forces create a cycle/feedback loop where there should be a natural equilibrium that forms.
Future Use Cases
We have always discussed the idea around staking derivatives in relationship to Interchain Finance (plus DeFi) in the Cosmos ecosystem, and one thing to explore is whether the fixed income component, could be replaced with any constant stream of payment such as the staking yield that you would expect from a proof of stake network or revenue from a business. The basic premise is that one could take a chunk of atoms, and delegate them. The staking rewards can be siphoned to a pool. Claim of those rewards is distributed based on the token issued via the distribution curve. This would create a new dynamic between bonding curves and any form of continuous payment.
Future work will also take into consideration concepts like default & credit risk among fixed income payments and slashing risks around proof of stake networks. Generally the idea is to have predictive models and risk parameters on the reliability of the fixed income components that Distribution Curve token buyers would expect to receive.
There is also interest in further exploring work such as what has been published via the Gauntlet team to identify these risk components:
Overall, we think that much innovation can come from Distribution Curves and how they are applied to the modern world of new cryptoeconomic primitives. This mechanism could create new relationships for any source of consistent income.
Wanted to thank folks for their friendly acknowledgements and feedback: Mingfang Duan, Ryan Yi, Dmitriy Berenzon, Sunny Aggarwal, Dogemos Lee, Jim Yang, Ashwin Ramachandran, Billy Rennekamp, and Chjango Unchained