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Bond Price Maths

Bond control mechanism

Financial maths control the decentralized finance
In order to determine the amount of interest to pay to each bonder when they redeem their bond shares, the protocol uses a control mechanism. Mechanistically, the ELO bond mechanism is similar to a linear actuator that aims to achieve a target that is set over a duration of time by voting. To describe the bond control mechanism, we assume that all quantities are indexed by bond durations, j ∈ 1, . . . , nd, and we drop dependence on j for readability. That is, the calculations below are for a single bond duration.
The interest rate paid out to each bonder varies as a function of the debt ratio for that bond duration. The debt ratio, dr(n) ∈ R, is defined as the ratio of the total amount of ELO that is needed to pay existing debt holders, D(n) ∈ R, to the total supply of ELO at block height n, S(n) ∈ R. If the inflation rate is high and going to staking, then the debt ratio will shrink if no new bonders join the market. On the other hand, if staking inflation is low or zero, then the debt ratio will generally increase as new bonders join.
Instead of directly computing an interest coupon (e.g. amount of ELO to pay out a bond- holder upon loan termination), ELO DAO uses a pricing discount model to imply the coupon value. If the implied ELO price from the LP share at the time of bond j’s share creation is pi(t), then the discounted price pd(t) used to compute the ELO coupon payment is:
(1)
where c(n) ∈ R is a dynamic variable known as the control variable that is adjusted by ELO governance. Note that here, we have let n = ⌊t⌋, by letting the variables depending discrete clock be piecewise constant while the continuous variables run.
We note that by definition pd(t) ≤ pi(t) for all t and that the units of pi and pd are ELO/USD .
If the value of the USD collateral initially deposited by the bond is X0 USD, then the bond coupon payout C(t) is computed as,
C(t) = pd(t) × X0
where C is in units of ELO. Since pd(t) ≤ pi(t), this implies a net bond yield γb(t) of
(2)
as 1/(pi(t)) represents the per unit of USD value of ELO without a discount. If a bond is of duration d ≥ 0, then at time t ∈ [0, d], a bondholder’s shares will have value X(t) equal to
(3)
Once a bond is created, the total outstanding debt is updated as D(n) = D(n − 1) + X0(1 +γb(t)) and the debt ratio is updated to
(4)
Upon expiry at time τb, the debt in the bond is decremented by:
For other custom ELO bonds that allow the user to withdraw their LP collateral in the future, the debt is updated as
Finally, the control variable c(t) is updated over a duration τ according to the formula:
(5)
when t ≤ τ, where c′, the reference control to track, is set using a multisig voting mechanism in which many users vote on what the appropriate discount rate should be. Then, there is a nonlinear update on the control value that tries to adjust c(n) until it has converged to c′ over τ duration of time, at which point a new c′ is voted on by users. This approach is akin to solving a sequence of optimization problems, each of duration τ , where the optimization is done in a decentralized fashion by voting (philosophically, this is similar to model predictive control).
Equations (3), (4), and (5) represent the state and control variables for an adjustment mechanism that controls the discount price. As we can compare, a simple linear feedback controller given by the solution to a tractable Hamilton-Jacobi-Bellman equation can find optimal solutions to a related control problem, when the goal is set to be price tracking and volatility minimization. We propose this problem as a proxy for what the ELO system is approximating.
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