Editing Decentralized underwriting (section)

== Tokenomics ==

=== REP token valuation ===
In this section, we give formulas for valuating REP tokens in an iDAO. Since the value of a REP token depends on the performance of the DAO in the future, in particular the amount of fees it attracts, such valuation is necessarily a probabilistic estimate. But such formulas can help traders valuate the REP-to-BOND market, making it more efficient, and it will help iDAO members more accurately estimate the amount of REP required to cover claims, making the insurance contracts and the entire iDAO more secure. 

==== Components ====
# iDAO <math>=\{Underwriters\}=\{U_i\}=</math> ledger of all REP tokens
# <math>C(t)=</math> Capital reserves
# <math>\{active\  contracts\}=\{+ \ premia\} \cup \{-\  risk\ of\ potential\ claims\}</math>  

From [[Reputation tokenomics|REP tokenomics]], we have the following definitions:
#<math>R(t)=</math> the total number of REP tokens in the iDAO at time <math>t</math>.
# The rate of total premia <math>f'={df \over dt}</math> that the iDAO earns. Therefore <math>f(t)</math> denotes the '''total fees''' earned from the beginning of the iDAO until time <math>t</math>.
#<math>f_0^1 (t)</math> is the cumulative [[Reputation#REP salary mechanism|reputational salary]] collected for one REP token.
#<math>m</math> is the minting ratio.
#<math>r</math> is the base discounting rate.
#<math>L</math> is the lifetime after which a REP token expires.

==== Fundamental formulas ====
The basic results of [[Reputation tokenomics|REP tokenomics]] give the [[present value]] of a single REP token with finite lifetime <math>L</math>. 

'''Theorem 3.'''  (''Finite Life Tokens'')

''The total number of active REP tokens at any time <math>t</math> is <math display="block">R(t)=\int_{t-L}^t m*f' (s)ds.</math>''The premia REP salary fees it earns is given by <math display="block">f_0^1 (t)=\int_0^{min\{t,L\}}\frac{f' (s)}{R(s)}ds.</math>''Assuming an iDAO has exponentially growing fees <math>f'(t)=f_0' e^{ct}</math>'' ''and risk is always covered accurately by the REP market, then after the iDAO has been running <math>L</math> units of time, the number of active REP tokens will grow at a proportional exponential rate. The income stream of a single token is then <math display="block">f_0^1 (t)=\frac{c}{m(1-e^{-cL} )}  min\{t,L\}</math>with present value <math display="block">PVf_0^1=\frac{1}{m} \biggl(\frac{1-e^{-rL}}{rL}\biggr) \biggl(\frac{cL}{1-e^{-cL}}\biggr) </math>''

Actual fees are stochastic, not exponential. I.e., <math>f(t)</math> is a random variable. So, accurately valuating ''<math display="inline">PVf_0^1 </math>'' of REP is a difficult problem for the market. The best valuation depends on information about the set of all active contracts, the health of the marketplace (actuarial statistics), and on the history of the DAO and the talent of its underwriters in securing good customers (avoiding [[wikipedia:Adverse_selection#Insurance|adverse selection]]). The larger and more decentralized the market and the iDAO membership becomes, however, the more predictable <math>f(t)</math> becomes, and the more accurately the formulas will reflect reality.

=== Risk pricing ===
Basic theory of premium pricing gives the fees (i.e., the premia, i.e., the price of insurance) as<math display="block">f\propto\Pi_X=E(X)(1+\theta)</math>using the simplest pricing principle<ref name=":1" /> where <math display="inline">\Pi_X</math> is the ''price'' of the contract which covers ''risk'' <math display="inline">X</math>, which is the random variable given by a claim. The term <math display="inline">\theta</math> denotes the ''premium loading factor'', which contains the extra [[wikipedia:Transaction_cost|transaction costs]] that an underwriter demands in order to issue a contract. 

This pricing principle guarantees that an underwriter who sells enough independent insurance policies, each with risk <math display="inline">X</math>, will break even according to the [[wikipedia:Law_of_large_numbers|law of large numbers]].

=== 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>.