Table 3 Notation used in the Paper
Parameter | Description |
|---|---|
t | Time |
r | Time to the next auction dispute arising from some breach of transaction protocol |
m,v | Mean and variance of the claims distribution \( \tilde{C}\left(m,v,k\right) \) |
k | Cap on claims to control moral hazard in \( \tilde{C}\left(m,v,k\right) \) |
z | Operating costs for insurance system |
d | The annuity for the net premium fund is computed at d the same opportunity cost of capital rate as I use for the NPV calculations. |
λ | Lagrange multiplier for optimization |
q | Cost adjustment depreciation rate (moral hazard declines the longer the insure has paid into the system through subscriptions) |
\( \tilde{C}\left(m,v,k\right) \) | Claims: mean, variance, caps on claims payouts |
P(t, z,q) | Net premium: premiums, costs (depreciating at q rate), moral hazard adjustment; this is ‘service systems’ charge is equivalent to an insurance premium; it is periodic, non-stochastic, stable over time. For the eBay parameter simulation either, one per auction listing or one per time period to cover all auctions within that time period \( P*\tilde{S}(t)\ or\ P*t \) For the charts in this section, the premium is set to 10 % of the average sales value of $85.95 or $8.60 premium for each sale; this is reasonable given the anecdotal evidence, but would require further investigation before finalizing pricing on an actual product |
\( \tilde{S}\left(t\Big|\nu \right) \) | Random variable of count of sales at time t where λ is the rate of sales. One listing every 6.36 days; t in days; v = 0.1572 * 57.53 % = 0.09046 sales per day \( P\left[ there\ are\ k\ sales\ in\ period\ of\ length\ t\right]=P\left[\tilde{S}\left(t\Big|\nu \right)=\mathrm{k}\right]=\frac{e^{-\nu t}{\left(\nu t\right)}^k}{k!} \) |
\( \tilde{C}\left(x\Big|\alpha, \beta \right) \) | Random variable of value of individual claim. Lomax or Normal (Gaussian). Average sale is $85.95 in 2013; and this is assumed to be the average claim (claim for full sales amount which is typical for auction insurance) |
\( \tilde{D}\left(k\Big|t,r\right) \) | Time between claims computed as every ϖ th sale; we assume every 221st sale; this is reasonable given the anecdotal evidence, but would require further investigation before finalizing pricing on an actual product |
\( \tilde{A}\left(t\Big|P,r,\tilde{S},\tilde{V},\tilde{C},K\right) \) | Annuity fund \( NP{V}_t\left(\tilde{S}(t)\times P - \tilde{D}(t)\times \tilde{C}\right) \) |
ϖ | Rate of auction failure as a proportion of auction listings |
γ | Quantile (parameter for the VaR and tVaR metrics) |
\( Va{R}_{\gamma}\left(\tilde{X}\right) \) | Value at risk for γ quantile; this is the quantile of the random variable \( \tilde{X}. \) VaR is not a coherent risk measure, because it violates the subadditivity property – i.e., it violates VaR(X + Y) ≤ VaR(X) + VaR(Y) |
\( tVa{R}_{\gamma}\left(\tilde{X}\right) \) | Tail value at risk for γ quantile; tVaR is a coherent risk measure |