Actuarial Notes

Integrated Paid and Incurred Model

Integrated Paid and Incurred Model (IPI) is the model Meyers builds in "Combining the CAY and CSR Models". It fits the paid and the incurred triangle of one line together. A CSR model describes the paid losses and a CAY model the incurred losses, and the two share the loss-ratio level \mathit{logelr} and the accident-year parameters \alpha_w.

\begin{aligned}
{}_P\mu_{w,d} = {} & \log(\text{Premium}_w) + \mathit{logelr} \\
& + \alpha_w + {}_P\beta_d \cdot (1-\gamma)^{w-1}
\end{aligned}
\begin{aligned}
{}_I\mu_{w,d} = {} & \log(\text{Premium}_w) + \mathit{logelr} + \alpha_w + {}_I\beta_d \\
& + \rho \cdot \big(\log({}_I C_{w-1,d}) - {}_I\mu_{w-1,d}\big)
\end{aligned}
\begin{aligned}
{}_P C_{w,d} &\sim \text{lognormal}({}_P\mu_{w,d},\ {}_P\sigma_d) \\
{}_I C_{w,d} &\sim \text{lognormal}({}_I\mu_{w,d},\ {}_I\sigma_d)
\end{aligned}
  • The left subscripts P and I mark the paid-only and incurred-only parameters. The incurred equation holds for w > 1; for w = 1, {}_I\mu_{1,d} = \log(\text{Premium}_1) + \mathit{logelr} + {}_I\beta_d. The 15-step specification in Meyers (2019), Section 9, has these priors:
    • Shared: \mathit{logelr} \sim \text{normal}(-0.4, \sqrt{10}), and \alpha_w \sim \text{normal}(0, \sqrt{10}) for w = 2, \dots, 10 with \alpha_1 = 0.
    • Paid (CSR steps): {}_P\beta_d \sim \text{normal}(0, \sqrt{10}) for d = 1, \dots, 10; \gamma \sim \text{normal}(0, 0.05); {}_P\sigma_d^2 = \sum_{i=d}^{10} {}_P a_i with {}_P a_i \sim \text{uniform}(0, 1).
    • Incurred (CAY steps): {}_I\beta_d \sim \text{normal}(0, \sqrt{10}) for d = 1, \dots, 9 with {}_I\beta_{10} = 0; \rho = 2\rho_{pos} - 1 with \rho_{pos} \sim \text{beta}(2, 2); {}_I\sigma_d^2 = \sum_{i=d}^{10} {}_I a_i with {}_I a_i \sim \text{uniform}(0, 1).
  • Why combine. The CSR model performs best on paid data and the CAY model on incurred data. In both, \mathit{logelr} and \alpha_w have the same interpretation and possibly the same values. The rest of the parameters should differ, since each model fits a different but related set of losses. Fitting both triangles at once gives more data behind the shared parameters. The idea is credited to Ned Tyrrell: estimates from paid and incurred data together should have lower variability.
  • The one change to the CSR half. {}_P\beta_{10} is not fixed at 0. That reduces the distortion when the paid triangle's loss ratio level differs significantly from the incurred one's, which often happens in Workers' Compensation. So the paid and incurred expected losses at development year 10 can differ.
  • Results for the illustrative insurer.
    • The shared parameters are similar to the separate CSR and CAY fits, but with noticeably smaller standard deviations. For \mathit{logelr} they are 0.0109, against 0.0246 (CSR) and 0.0150 (CAY); for \alpha_{10}, 0.1947 against 0.3467 and 0.2984.
    • The standard error of the total falls to 1{,}253 (paid, from the CSR's 2{,}401) and 1{,}225 (incurred, from the CAY's 1{,}859).
    • \widehat{elpd}_{loo} is 63.54 (paid) and 78.36 (incurred), above every other model for this triangle. It is computed from 55 \times 10{,}000 log-likelihood matrices for the paid and incurred data.
  • Validation on 200 triangles:
    • The standard error falls for almost all of them.
    • \widehat{elpd} favors IPI over CSR in 146 (\widehat{elpd}_{loo}) and 156 (\widehat{elpd}_{test}) paid triangles, and over CAY in 107 and 111 incurred ones. The advantage is stronger on the paid side.
    • Its residual box plots are slightly worse than the CSR's and CAY's, but still reasonable: zero lies within the interquartile range for every accident year.
    • Seven of its eight p-p plots fall within the Kolmogorov-Smirnov bounds, including both combined plots (D = 8.7 paid, 9.4 incurred, against 9.6). The exception is Workers' Compensation incurred, which is light-tailed (D = 19.3 against 19.2).
    • It works best for the auto lines, where paid and incurred are almost equal by development year 10. It works less well for Workers' Compensation, whose long-term annuity benefits leave a large part of losses unpaid after 10 years.
  • Use in risk margins. Meyers computes payout patterns and cost of capital risk margins with the CSR model and with the IPI's paid parameters. Under the IPI, expected incurred at development year 10 will differ from expected paid, so the difference is treated as one more payment:

P_{w,11} = e^{{}_I\mu_{w,10} + {}_I\sigma_{10}^2/2} - e^{{}_P\mu_{w,10} + {}_P\sigma_{10}^2/2}

  • The ultimate uses the incurred parameters. For the illustrative insurer the IPI needs an initial capital of 3{,}087 against the CSR's 7{,}139, and its mean risk margin is 233 against 597.
  • In practice. Meyers' suggested workflow ends here. Once a paid and an incurred model are satisfactory and in close agreement on the ultimate, use them as parts of an IPI model to obtain a more accurate estimate of the liability.

Paid and Incurred Log Means From One Draw One IPI posterior draw for accident year 10 (premium 5{,}000) has \mathit{logelr} = -0.40, \alpha_{10} = 0.10, {}_P\beta_{10} = -0.05, \gamma = 0.04, {}_P\sigma_{10} = 0.03, {}_I\sigma_{10} = 0.02 and \rho = 0.2. Accident year 9's incurred loss at development year 10 is 0.03 above its log mean.

Compute the expected paid and incurred losses at development year 10, and the extra payment P_{10,11} used in the risk margin.

Answer The shared part is \ln 5{,}000 - 0.40 + 0.10 = 8.21719. The paid development factor is scaled by (0.96)^9 = 0.69253:

\begin{align*}
{}_P\mu_{10,10} &= 8.21719 - 0.05 \times 0.69253 \\
&= 8.18257 \\
E[{}_P C_{10,10}] &= e^{8.18257 + 0.03^2/2} \\
&= 3{,}580 \\
{}_I\mu_{10,10} &= 8.21719 + 0 + 0.2 \times 0.03 \\
&= 8.22319 \\
E[{}_I C_{10,10}] &= e^{8.22319 + 0.02^2/2} \\
&= 3{,}727 \\
P_{10,11} &= 3{,}727 - 3{,}580 \\
&= 147
\end{align*}

Both triangles use the same \mathit{logelr} and \alpha_{10}. The gap comes only from {}_P\beta_{10} \neq 0 and the CAY's correlation term. Fixing {}_P\beta_{10} = 0, as in the stand-alone CSR, would tie the paid level at year 10 to the shared loss ratio. Meyers drops that constraint to reduce the distortion when the paid and incurred loss ratio levels differ.

Where the Lower Standard Error Comes From Meyers' Table 9.1 gives the posterior standard deviation of \alpha_{10} as 0.3467 (CSR), 0.2984 (CAY) and 0.1947 (IPI).

Treating \alpha_{10} as the only source of uncertainty, approximate the coefficient of variation of accident year 10's ultimate under each model. Compare with Meyers' reported CVs of 0.3736 (CSR paid), 0.3282 (CAY incurred) and about 0.20 (IPI, paid and incurred).

Answer A lognormal with log standard deviation s has \mathrm{CV} = \sqrt{e^{s^2} - 1}:

\begin{align*}
\text{CSR}: \sqrt{e^{0.3467^2} - 1} &= 0.357 \\
\text{CAY}: \sqrt{e^{0.2984^2} - 1} &= 0.305 \\
\text{IPI}: \sqrt{e^{0.1947^2} - 1} &= 0.197
\end{align*}

Each is close to the reported CV, so the newest year's uncertainty is almost all parameter risk in its level \alpha_{10}. The IPI estimates \alpha_{10} from the paid and the incurred triangle together, so its posterior is narrower and the CV nearly halves. The more accurate IPI also needs far less capital: 3{,}087 / 7{,}139 = 43\% of the CSR's for this insurer.