Actuarial Notes

Correlated Incremental Trend Model

Correlated Incremental Trend Model (CIT) is the Bayesian model for incremental paid losses in the 2015 first edition of Meyers' Stochastic Loss Reserving Using Bayesian MCMC Models. It adds a payment-year (calendar-year) trend \tau and correlation between accident years. It uses a mixed lognormal-normal distribution, which is skewed to the right yet allows negative incremental losses. With \rho = 0 it becomes the Leveled Incremental Trend (LIT) model.

\mu_{w,d} = \alpha_w + \beta_d + \tau \cdot (w + d - 1)

Z_{w,d} \sim \text{lognormal}(\mu_{w,d},\ \sigma_d), \quad \sigma_1 < \sigma_2 < \dots < \sigma_{10}

\tilde I_{1,d} \sim \text{normal}(Z_{1,d},\ \delta)

\begin{aligned}
\tilde I_{w,d} \sim \text{normal}\big( & Z_{w,d} + \rho\,(\tilde I_{w-1,d} - Z_{w-1,d})\, e^{\tau}, \\
& \delta\big), \quad w > 1
\end{aligned}
  • \tilde I_{w,d} is the incremental paid loss of accident year w in development year d, and w + d - 1 is its payment year. Z_{w,d} is a lognormal latent mean, and \delta is the standard deviation of the normal layer. \beta_{10} = 0 to prevent overdetermining the model.
  • Priors, as the first edition states them:
    • \alpha_w \sim \text{normal}\big(\log(\text{Premium}_w) + \mathit{logelr},\ \sqrt{10}\big), with \mathit{logelr} \sim \text{uniform}(-5, 1)
    • \beta_d \sim \text{uniform}(0, 10) for d = 1 to 4, and \beta_d \sim \text{uniform}(0, \beta_{d-1}) for d > 4, so \beta_d decreases after d = 4
    • \rho \sim \text{uniform}(-1, 1)
    • \tau \sim \text{normal}(0, 0.0316), a JAGS precision of 1{,}000
    • \sigma_1^2 \sim \text{uniform}(0, 0.5) and \sigma_d^2 \sim \text{uniform}(\sigma_{d-1}^2,\ \sigma_{d-1}^2 + 0.1)
    • \delta \sim \text{uniform}(0, \text{average premium})
    • Two priors are deliberately narrow. A wider prior on \tau often gave values below -0.1, offset by high \alpha or \beta values. Unrestricted growth in \sigma_d produced unreasonably high simulated losses.
  • Why it was built. On paid data the bootstrap ODP, Mack and CCL models all predicted results that were too high. A payment-year trend, as in Zehnwirth's models, was a plausible fix. A trend has two consequences:
    • The model must use incremental losses, because cumulative losses include settled claims, which do not change with time.
    • Incremental paid losses are skewed to the right and occasionally negative.
    • The skew normal distribution can never be more skewed than a truncated normal (coefficient of skewness 0.995), and in trial fits its shape parameter sat near that limit. Meyers therefore mixes a lognormal into a normal instead: X \sim \text{normal}(Z, \delta) with Z \sim \text{lognormal}(\mu, \sigma).
  • How it differs from the CCL.
    • \sigma_d rises with d, since smaller, less volatile claims settle earlier.
    • The correlation acts outside the log space, on the incremental losses themselves, so that they can be negative. For given parameters, \rho is the coefficient of correlation between \tilde I_{w-1,d} and \tilde I_{w,d}.
    • Because the trend enters in log space but the correlation does not, the prior year's deviation is multiplied by e^{\tau} to trend it.
  • Findings (paid data, 200 triangles):
    • The posterior means of \tau were predominantly negative, but a negative \tau may be offset by higher \alpha_w and \beta_d. The level, development and trend parameters stayed correlated, and the models found it hard to separate them.
    • Posterior means of \rho were not as overwhelmingly positive as on incurred data, and CIT and LIT standard deviations differed only a little.
    • The p-p plots still show estimates that are too high: combined D = 25.3 for CIT and 29.6 for LIT, against a critical value of 9.6. Only a handful of triangles, mostly Other Liability, had noticeably lower estimates than under the CCL.
    • The CIT and LIT models did not achieve the desired improvement. Meyers read this as support for the view that incurred data carry real information that paid data lack. A reviewer's point that claims now settle faster led to the changing settlement rate (CSR) model, which did correct the bias.
    • The MCMC chains converged noticeably more slowly than for the other models (burn-in raised to 50{,}000 iterations).
  • Not in the current reading. The second edition, Meyers (2019), assigned for Exam 7, drops the incremental paid models. Meyers tried to build a new one, but none of his attempts validated.
  • Source: Meyers, first edition, CAS Monograph No. 1 (2015), Sections 6 and 8 and Appendix C.

The Trend in the Log Mean A CIT posterior draw has \alpha_4 = 5.0, \beta_3 = 3.2, \tau = -0.02 and \sigma_3 = 0.3. Compute \mu_{4,3} and the expected incremental paid loss E[Z_{4,3}], and the effect of the trend.

Answer Accident year 4, development year 3 is payment year 4 + 3 - 1 = 6.

\begin{align*}
\mu_{4,3} &= 5.0 + 3.2 + (-0.02)(6) \\
&= 8.08 \\
E[Z_{4,3}] &= e^{8.08 + 0.3^2/2} \\
&= 3{,}378
\end{align*}

The trend multiplies this cell by e^{-0.12} = 0.887, an 11.3\% reduction over six payment years. The normal layer has mean Z, and the correlation term averages to zero, so 3{,}378 is also the unconditional expected incremental loss. A negative \tau lowers every future diagonal, which is how the model was meant to pull down paid estimates that ran high.

A Negative Incremental Payment For a late development year, a draw has latent means Z_{w,d} = 50 and Z_{w-1,d} = 60, and the previous accident year's simulated payment was \tilde I_{w-1,d} = 20. Also \rho = 0.4, \tau = -0.02 and \delta = 30.

Find the mean of \tilde I_{w,d} and the probability that it is negative. Repeat for the LIT model (\rho = 0).

Answer

\begin{align*}
E[\tilde I_{w,d}] &= 50 + 0.4\,(20 - 60)\, e^{-0.02} \\
&= 50 - 15.68 \\
&= 34.32 \\
P(\tilde I_{w,d} < 0) &= \Phi\!\left(\frac{-34.32}{30}\right) \\
&= \Phi(-1.144) \\
&= 0.126
\end{align*}

Under the LIT model the mean stays at 50, and P = \Phi(-50/30) = \Phi(-1.667) = 0.048.

The previous year paid 40 below its latent mean, and with \rho = 0.4 that shortfall carries forward. A lognormal alone could never produce the negative payments that salvage and subrogation cause late in development. The normal layer allows them while the lognormal keeps the right skew.