Editing Decentralized underwriting (section)

=== Reserves vs. risk of ruin ===
[[Quantifying the relation between reserves and ruin|Calculations]]

We can use the basic tokenomics formulas to make precise the relation between the size of reserve holdings and the risk of an iDAO defaulting on its customers' claims. Given an accurate estimate of the future income of the iDAO, an estimate of the risk of that future income changing, and the risk of the customers making claims, we can quantify the relationship between the size of the iDAO's reserve holdings and the risk of the iDAO's ruin. We employ standard financial concept of [[wikipedia:Value_at_risk#Mathematical_definition|value at risk]] (VaR). 

Let <math display="inline">X</math> be a profit and loss random variable for the period under consideration (usually an insurance cycle, such as 1 year). Loss means <math display="inline">X</math> is negative and profit means <math display="inline">X</math> is positive. The VaR at level <math display="inline">p</math> is the smallest number <math display="inline">x=VaR_p(X) </math> such that the probability that <math display="inline">X</math> does not exceed <math display="inline">x</math> is at least <math display="inline">1-p</math>. Mathematically, <math display="inline">VaR_p(X) </math> is the <math display="inline">p</math>-quantile of <math display="inline">-X</math>, i.e.,  <math display="block">VaR_p(X)=-inf\{x|F_X(x)>p\}=-F_X^{-1}(p).</math>Intuitively, this formula captures the idea that there is a greater than <math display="inline">1-p</math> chance that the iDAO will not lose more than <math display="inline">Var_p(X) </math> during the insurance cycle.  

Now the size of the reserve can be tied to the risk <math display="inline">p</math> that the iDAO is willing to bear using the bound<math display="block">VaR_p(X)\leq \Tau(t)+E[PV(f(\infty)-f(t))]\qquad (1)</math>where <math display="inline">\Tau (t)</math> is the value of the treasury at time <math display="inline">t</math>, and <math>E[PV(f(\infty)-f(t))]</math> represents the estimated value of the BOND market, since it is the present value of all future fees. 

Therefore, Condition <math display="inline">(1) </math> guides an iDAO in determining the size of the reserve <math display="inline">\Tau (t)</math> at any given time <math display="inline">t</math> to maintain a bound on the risk of ruin they are exposed to. Whenever their estimated <math display="inline">VaR_p(X) </math> exceeds this bound, then all premia fees from insurance contracts should be diverted from the REP salary to the treasury. When the bound is next satisfied, the fees are again paid to the REP salary along with any excess reserves. At all times, the workers who sold the contracts that gained the fees for the iDAO will still earn the newly minted REP tokens. And the treasury is still technically the property of the REP holders, because as soon as the risk decreases, the treasury is liquidated to the level that keeps the risk of ruin below <math display="inline">p</math> and shared with all REP holders.  

In the ideal case, we see that there is no need for a treasury. The ideal case assumes the fees keep pace with inflation and the iDAO is properly policed--i.e., underwriting of contracts are always sufficient to cover all claims. The treasury is unnecessary in this case, since the valuation of REP tokens can be made certain, so that <math display="inline">VaR_p(X)=0 </math> so <math display="inline"> \Tau (t)=0</math> satisfies Condition <math display="inline">(1) </math>.