Stablecoins as Perpetual Convertible Bonds with AMMs Features
Stablecoins as constant product AMMs
AMMS (Automated Market Makers) like Uniswap enable permissionless exchange of cryptocurrencies via a single invariant function: x * y = k. New stablecoins with elastic monetary policies, like Chi Protocol, behave similarly to an AMM. More specifically, the stablecoin’s protocol defines a rule function to change the supply Q(t) as soon as the current price P(t) deviates from its target K (i.e. K = $1):
Respectively, if the stablecoin’s price is trading above, below or at K, the supply must be expanded, contracted or remain constant:
The similarity with the constant product of a typical AMM arises from rearranging Equation 1 and making K as the subject of the equation:
Thus, Equation 6 tells us that the responsive change in the stablecoin’s supply given by Δ must be of equivalent magnitude and direction of the price deviation from its target for the price to re-stabilise at K.
Stablecoins as perpetual convertible bonds
Convertible Bonds
In traditional finance, hybrid debt instruments that can be best compared to algorithmic stablecoins are convertible bonds. After the firm issues convertible bonds, the bondholder can decide to convert them into shares of the company or continue to hold them to collect coupons. At any time before maturity, the firm may also call (repurchase) the bonds back for the same conversion value in cash, forcing the holder to surrender.
We say that the firm’s value is equivalent to the stock’s value given by V(t). At any time, bondholders can convert, in which case they will receive a fraction θ of the company’s equity value θV(t). Similarly, the firm may call the bond for a fixed value K of its assets equivalent to the value if converted into shares. However, the firm does not call back the bond if its value is not sufficient for reissuing future debt (i.e. V(t) < K). Thus, for V(e)= inf (V(t): V(t) ≥ K) denoting the least value of the assets for which the firm calls back the bond, and V(o)= sup (V(t): V(t) < K) representing the highest value of the assets for which the holder converts in shares, then it must be the case that V(e) > V(o), since the firm does not repurchase bonds if the assets are insufficient to perform a call back. Furthermore, for V(o) ≤ x < V(e), the bond is converted into shares and for x ≥ V(e), the firm repurchases it back. We define γ as the call factor subject to γ < θ since it requires more asset’s value to call back the bond than convert into shares, and we visualise these relationships as follows:
Perpetual convertible bonds like stablecoins
This is a topic I have written about in the Chi Protocol Whitepaper (see [1]). In the past, there have been various attempts to recreate semi-perpetual convertible bonds like stablecoins. The most notable of those is certainly UST. However, very bad things can easily happen to the stablecoin if the stability mechanism simply relies on seigniorage shares. This is true for two primary types of reasons:
- Suppose the stablecoin’s demand moves inversely with respect to one of the markets for volatile currencies. In that case, this is counterproductive since it creates a disequilibrium where the governance token cannot support the volatility of the stablecoin.
- If the goal is solely for the governance token to compensate for de-pegs, then there will always be the risk of a death spiral since seigniorage shares can continue with no formal definitions of stopping times.
There is a way to get around this: having reserves backing the stablecoin and alternating between two stability mechanisms depending on the solvency of the protocol. Rather than having a single condition on the price of the stablecoin, the protocol must impose additional rules which activate or deactivate a stability mechanism in response to changes in market dynamics. First, suppose the protocol lacks the reserves to back the stablecoin’s supply, then its stability mechanism must provide economic incentives to translate a temporary de-peg into an opportunity to incentivise solvency. For example, if the stablecoin is above the target price, the AMM will accept asset deposits to mint more stablecoins, reducing the deficit by expanding reserves. Conversely, if the price is below target, the AMM will burn stablecoins and mint new shares, narrowing the gap between assets and debt with debt reduction. Second, and more favourably, suppose the protocol has excess reserves to back the stablecoin’s supply. Then, it can use these as revenue distribution mechanisms for its stakeholders. For instance, if the price faces a positive de-peg, shares can be burnt in exchange for newly minted stablecoins. Similarly, if the price faces a negative de-peg, stablecoins can be returned to the protocol in exchange for excess assets. But the underpriced stablecoins are not burnt; they are distributed to its stakers. Hence, similar to coupon payments, rewards are a distributed in the form of a stable asset to those who stake the stablecoin.
How does it work in practice?
Let’s now look at the details of the stability mechanism and notice how this best compares to a perpetual convertible bond with $1 face value (i.e. K = $1). Let V(t) denote the value of the protocol’s reserves and C(t) the value of the outstanding stablecoins’ supply. Let’s also assume the stablecoin’s price is trading above $1 so that Δ>0. Given this latest conditions holds, the protocol must mint Δ stablecoins at the total cost ΔK paid by the user. To identify if to use seigniorage shares (shares mechanism) or reserves convergence (assets mechanism), the protocol must compute the following inequalities:
Now, it’s worth noting what these conditions tell us. Assuming the value ΔK is added to the total reserves and these new reserves exceed the value of the stablecoins in circulation (i.e. Condition 1), then the protocol does not need assets anymore, and it can afford to burn shares to mint new stablecoins to stabilise the price at $1. Conversely, if the new reserves persist in being insufficient to back the stablecoins (i.e. Condition 2), then the protocol will accept ΔK worth of asset’s deposit by the user. In both cases, the governance token benefits the most from the mechanism since if Condition 1 is satisfied, shares become more scarce; otherwise, with Condition 2, the protocol’s assets will grow, increasing any of the yield they generate for the shares stakers.
Let’s consider the inverse of the above conditions. Suppose the stablecoin’s price trades below $1, so Δ < 0. In this case, the protocol must either repurchase back |Δ| stablecoins by paying |Δ|K worth of assets to the holder who returns them. Alternatively, holders can convert them into share tokens. To obtain this, we identify the conditions for which the stablecoins are called back by the protocol (reserves convergence) or the holders of stablecoins exercise their option to become shareholders (seigniorage shares).
Assume that |Δ|K is deducted from the protocol’s assets. If post-call, there is no sufficient value to collateralise the stablecoins fully (i.e. Condition 3), then the users can convert their stablecoins into the governance token. On the other hand, if, following redemptions, the protocol continues to have excess reserves (i.e. Condition 4), then the protocol can purchase back stablecoins by paying |Δ|K to the user.
It’s now worth looking at the benefits of having a dual stability mechanism and how this behaves in any of the market conditions. To do so, we must first have a visual idea of how the hierarchy of repayment in a typical firm looks like:
For a stablecoin protocol, like Chi Protocol, this structure is much simplified since it issues convertible debt in the form of stablecoins (USC) and has equity expressed with the governance token (CHI):
The idea applied is that when USC trades below the target price, its holders receive priority on the protocol’s excess reserves with respect to the governance token holders. More specifically, if Chi protocol has sufficient assets to satisfy its debt obligations (i.e. Condition 4), then it can repurchase stablecoins for a premium (i.e. user returns 1 USC and receives $1 worth of staked ETH). On the other hand, when USC trades above $1, the value of the excess reserves is distributed to the CHI token holders. This is because there are already sufficient assets to back the outstanding debt value (i.e. Condition 1), and so the protocol can afford to burn governance tokens and expand the stablecoin’s supply (i.e., user burns $1 worth of CHI in exchange for 1 newly minted USC).
In the same way, the hierarchy of repayment and debt financing can be applied when the protocol faces a reserve deficit. Given this circumstance holds and USC trades below the desired price (i.e. Condition 3), the protocol prioritises debt holders by allowing conversion of USC into newly issued CHI tokens (i.e, user burns 1 USC and receives $1 worth of newly minted CHI). This occurs as part of the stability mechanism and the protocol not having sufficient assets to allow for debt re-issuance, so redemptions with reserves would further boost the protocol’s insolvency. Additionally, by enabling USC to be converted into CHI, the protocol retains the ability to close its deficit gap in market downturns through debt burning while leaving the reserves constant. Correspondingly, if USC trades above the target price and there is still a deficit (i.e. Condition 2), then the protocol can eliminate this by letting users expand the stablecoin’s supply with deposits of new assets (i.e. user mints 1 USC and deposits $1 worth of staked ETH).
So what happens if the expected future activity drops to near zero following a fall in cryptocurrency prices? In the case of a fully algorithmic stablecoin (e.g., UST), the market cap of the volatile coin drops until it becomes quite small relative to the stablecoin. At that point, the system turns exceptionally delicate: a minor decrease in demand for the stablecoin could trigger the targeting mechanism to generate a substantial amount of governance tokens. This, in turn, could result in hyperinflation of the volatile coin, ultimately causing the stablecoin to depreciate as well. In Chi Protocol, the risk of hyperinflation on the governance token is completely eliminated. There can be periods in which the market cap of CHI is lower than the outstanding stablecoin debt, however, the occurrence of this event will not imply any fragility in the system. Obviously, this occurs because the value of the assets in the system will always reflect the outstanding stablecoin debt C(t) ≈ V(t) so that the deficit is brought to a minimum. Another way to interpret this is to always think of the stablecoin’s debt being always over-collateralised as it is backed by the sum of the reserves V(t) and governance tokens E(t):
and by design, we have that the deficit is minimised so that the limited inflation on CHI tokens can support it:
It should be pretty clear that as long as the value of the assets in the system is greater than zero, the governance token can support the volatility of the stablecoins even in rapid drops in ETH prices.
Conclusion
As we saw with Terra in May 2022, systems with seigniorage shares as a single mechanism are subject to collapses. During periods of decreased activity, this reduced the expectation of future fees, which served as the foundation for the volcoin’s value. As a result, the market capitalisation of LUNA declined below the one of UST, further weakening the system and triggering that very collapse.
As of today, the crypto space thought it would not have been possible to solve the stablecoin trilemma. It turns out that the application of the theory behind convertible bonds and AMMs makes Chi Protocol an elegant solution to this problem. Certainly, this is achieved by relying on seigniorage shares to close the deficit gap in the most pessimistic case but the risk of death spirals is eliminated by having assets to impose predictable stopping times on the inflation of the volatile token.
Instead of relying on the user’s user-based overcollatteralisation, Chi Protocol prioritises capital efficiency while not requiring the market capitalisation of the governance to be greater than the stablecoin. Whilst we certainly should hope for growth, we evaluated how safe the system is by looking at the steady state and even the pessimistic state of how it should fare under extreme conditions. Ultimately, this solves the most important problem crypto faces today and allows for a safe dual stability mechanism that can go through every market state without being subject to a collapse.
