RepRisk Index for Countries

Mathematical Decription



1. Calculation of Average Severity of Incidents
2. Normalization of Average Severity among countries
3. Calculation of Incident Index
4. Normalization of Inversed PRR
5. Calculation of Scaled PRR by using WGI
6. Combining Incident Index and scaled PRR

The Country RRI for a country $n$ at time $t$, $\text{RRI}_{n,t}$, is calculated in 6 steps:

Step 1: Calculation of Average Severity of Incidents

$$ \varnothing\text{ Severity}_{n,t} = \frac{\sum\limits_{i \in \text{I}_{n,t}} w_{sev}(i) w_{t}(i)} {\sum\limits_{i \in \text{I}_{n,t}} w_{t}(i)} $$$$\text{where}$$$$ w_{sev}(i) = \left\{\begin{matrix} 8 & \text{if Severity of incident $i$ is 1} \\ 16 & \text{if Severity of incident $i$ is 2} \\ 32 & \text{if Severity of incident $i$ is 3} \end{matrix}\right. $$


$$ w_{t}(i) = \left\{\begin{matrix} 1 & \text{T}_{i,t} \leq 14 \\ (502-3\text{T}_{i,t})/460 & 14 \lt \text{T}_{i} \leq 60 \\ (5110-7\text{T}_{i,t})/6700 & 60 \lt \text{T}_{i} \leq 730 \\ 0 & \text{else} \end{matrix}\right. $$


  • $i$ : An incident
  • $\text{I}_{n,t}$: The set of incidents for a country $n$ obtained by:
    • identifying the most recent two-year window containing a minimum of ten incidents;
    • selecting all incidents that occurred within that window.
  • $\text{T}_{i,t}$: The time difference (in days) between the publication date of the risk incident $i$ and the RRI calculation date $t$.

Step 2: Normalization of Average Severity among countries

$$ \text{Norm}(\varnothing\text{ Severity}_{n,t}) = 37.5 + 12.5 \times \frac{\varnothing\text{ Severity}_{n,t} - \text{Median}(\{\varnothing\text{ Severity}_{n,t}\}_{n \in \text{N}^{\text{PRR}}_t})}{\text{MAD}(\{\varnothing\text{ Severity}_{n,t}\}_{n \in \text{N}^{\text{PRR}}_t})} $$
  • $\text{Median}(x)$ : The median of a series of values $x$.
  • $\text{MAD}(x)$ : The median of the absolute deviation of a series of values $x$.
  • $\text{N}^{\text{PRR}}_t$ : A set of countries whose value of $\varnothing\text{ Severity}_{n,t}$ are calculatable in Step1.

Step 3: Calculation of Incident Index

$$ \text{Incident Index}_{n,t} = \left\{\begin{matrix} \text{Norm}(\varnothing\text{ Severity}_{n,t}) & \text{if } \text{I}_{n, t} \ge 10 \\ \text{Decay}(\text{Incident Index}_{n,t-1}) & \text{if } \text{I}_{n,t} \lt 10 \end{matrix}\right. $$
  • $\text{Decay}(x) = \textbf{max} (0, x - 100/730)$, which makes the decay from 100 to 0 would take 730 days.

Step 4: Normalization of Inversed PRR

$$ \text{Standardized PRR}_{n,t} = \frac{ \text{Inversed PRR}_{t,n} - \mu_t^{\text{InvPRR}}}{\sigma_t^{\text{InvPRR}}} \\ $$
  • $\text{Inversed PRR}_{n,t} = 100 - \text{PRR}_{n,t}$
  • $\mu_t^{\text{InvPRR}} = \text{Mean}(\{\text{Inversed PRR}_{n,t}\}_{n \in \text{N}^{\text{PRR}}_t})$, the mean of Inversed PRR among countries at time $t$.
  • $\sigma_t^{\text{InvPRR}} = \text{Std}(\{\text{Inversed PRR}_{n,t}\}_{n \in \text{N}^{\text{PRR}}_t})$, the standard deviation of Inversed PRR among countries at time $t$.

Step 5: Calculation of Scaled PRR by using WGI

$$ \text{Scaled PRR}_{n,t} = \big( \text{Standardized PRR}_{t,n} \times \sigma^{\text{Scaled WGI}}_t + \mu^{\text{Scaled WGI}}_t \big) \Big|_{[0,100]} $$$$\text{where}$$$$ \text{Scaled WGI}_{n,t} = \left\{\begin{matrix} 100 & \text{if } \varnothing\text{WGI}_{n,t} \le -1.5 \\ 0 & \text{if } \varnothing\text{WGI}_{n,t} \ge 1.5 \\ 50 - \frac{100}{3}\varnothing\text{WGI}_{n,t} & \text{otherwise} \end{matrix}\right. $$
  • $\varnothing\text{WGI}_{n,t}$ : The average of WGI's six indicators for country $n$ at time $t$.
  • $\mu_t^{\text{Scaled WGI}} = \text{Mean}(\{\text{Scaled WGI}_{n,t}\}_{n \in \text{N}^{\text{WGI}}_t})$, the mean of Scaled WGI among countries at time $t$.
  • $\sigma_t^{\text{Scaled WGI}} = \text{Std}(\{\text{Scaled WGI}_{n,t}\}_{n \in \text{N}^{\text{WGI}}_t})$, the standard deviation of Scaled WGI among countries at time $t$.
  • $\text{N}^{\text{WGI}}_t$ : A set of countries whose value of $\varnothing\text{WGI}_{n,t}$ are calculatable.
  • Note: For a country whose $\text{Standardized PRR}_{n,t}$ is not available, $\text{Scaled WGI}_{n,t}$ is used instead.

Step 6: Combining Incident Index and scaled PRR

$$ \text{RRI}_{n,t} = \frac{\text{Incident Index}_{n,t} + \text{Scaled PRR}_{n,t}}{2} $$


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