Editing Decentralized underwriting (section)

=== 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.