23.3 Projection Risk
Different ways to project trends
23.3.1 Simple Trend
Forecast one trend using historical average and into the future
Caveat: Additional uncertainty due to estimate of ultimate losses is uncertain
23.3.2 Severity Trend and Inflation
Claim severity \(\neq\) general inflation
\[\begin{equation} [\text{Claim Severity Trend}] \approx [\text{General Inflation}] + [\text{Superimposed Inflation}] \tag{23.2} \end{equation}\]2 approaches:
Model the severity trend independent of general inflation
Adjust the data for general inflation and model the residual superimposed inflation
For ERM where general inflation is modeled, severity trend should be dependent on the general inflation
23.3.3 Trend as a Time Series
More realistic, allow for trends to change over time
AR(1) with mean reverting form:
\[\begin{array}{ccccc} r_{t+1} &= &m + \alpha_1(r_t - m) &+ &\epsilon_{t+1} \\ &= &(m - \alpha_1 m) + \alpha_1 r_t &+ &\epsilon_{t+1} \\ &= &\alpha_0 + \alpha_1 r_t &+ &\epsilon_{t+1} \\ \end{array}\]
\(m\) is the long term mean estimated from data
\(\alpha_1\) is the correlation from one period to the next
\(\epsilon_t \sim N(0,\sigma)\)
Simple trend understate the projection risk especially for long tail LoBs
- Simple trend had a 10 year forecast 99th percentile prediction error of +45%