Quantifying the relation between reserves and ruin

Scratch page for noodling out the math... Needs to be fixed and deleted.

??The following analysis is flawed. An iDAO defaulting is independent of how many or few claims, since exposure is limited to the particular underwriters. The iDAO will only default if the previously established value of REP decreases while the group is exposed to risk, in which case the DAO would need to mint more BONDs to cover the excess risk. We need to compare the change in REP value from decreasing premia. So we make an assumption of a certain equilibrium amount of premia (equivalent to picking a value for REP tokens, though the different contracts will have different quantities of REP with different values at different contract initiation points. So we simplify, by assuming all contracts had a short enough negotiation time to assume the REP value in all contracts is the same. That all risk C was covered at that time, but now the premia have dropped. Now what is the relationship between K and the probability of iDAO ruin? Then we can show how exponential P' assumptions can be correlated with changing K to keep the probability fixed. ??

The probability of default

$$Pr_{C}(default)=Pr(T_{K,f}<\infty )=\lim _{t\to \infty }Pr(T_{K},f<t)$$

is the probability that the stopping time

$$T_{K,f}:=inf\left\{t:K(t)+\int _{t}^{\infty }e^{-r(s-t)}P'(s)ds>\int _{t-1}^{t}C'(s)ds\right\}$$

is finite. Here 1 is the unit of time which represents 1 period of the BOND market. The term$\int _{t-1}^{t}C'(s)ds$ represents the amount of claims made since the previous BOND market period. The term $\int _{t}^{\infty }e^{-r(s-t)}P'(s)ds$ is the value of the BOND market at time $t$.  $K(t)$ is the size of the reserve treasury at time $t$. We want to capture the idea that at this time the claims might exceed the value of the market, which would lead to a collapse in the value of REP and the ruin of the iDAO if the capital reserve  is insufficient to cover the difference.

Evidently, the probability of ruin depends on the size of the reserve, the rate of premia the iDAO earns, the discount rate, and the claims. Given the maximum possible claims in the period $C_{max}=\int _{t-1}^{t}C'_{max}ds$ we assumed when the contracts were written that the estimate of the REP value was based on an equilibrium premium rate $P_{0}'$ which satisfied

$${\frac {q_{0}}{R_{0}}}K_{0}+\int _{t}^{\infty }{\frac {q_{0}}{R_{0}+(s-t)mP'_{0}}}e^{-r(s-t)}P'_{0}ds=C_{max}$$

The fraction ${\frac {q_{0}}{R_{0}}}$ represents the fraction of total REP tokens that were encumbered in contracts to cover the risk. ${\textstyle V_{0}:=K_{0}+\int _{0}^{\infty }e^{-rs}P'_{0}ds}$ is the estimate of the value of the iDAO at time ${\textstyle t}$ under the (flawed) assumption of constant premia. The term

$$\int _{t}^{\infty }{\frac {q_{0}}{R_{0}+(s-t)mP'_{0}}}e^{-r(s-t)}P'_{0}ds$$

represents the estimated present value of $q_{0}$ REP tokens since new tokens will be minted beyond the initial $R_{0}$ at premia enter the DAO. Assuming the rate of premia is constant $P'(t)=P'_{0}$ then we need $K$ to satisfy $K(t)$

In order to guarantee 0% chance of ruin, we need to make sure $K(t)$ is larger than the difference between the