Cross Classified Model

Cross Classified Model (CRC) is the basic Bayesian model of Meyers (2019). It gives each accident year and each development year its own parameter and treats each cumulative loss $C_{w,d}$ as lognormal, with the parameters fitted by MCMC. The other models in the monograph are built by adding a feature to it.

$\mu_{w,d} = \log(\text{Premium}_w) + \mathit{logelr} + \alpha_w + \beta_d$

$C_{w,d} \sim \text{lognormal}(\mu_{w,d},\ \sigma_d)$

$\sigma_d^2 = \sum_{i=d}^{10} a_i$

$w = 1, \dots, 10$ is the accident year and $d = 1, \dots, 10$ the development year. $\mu_{w,d}$ and $\sigma_d$ are the log mean and log standard deviation.
Priors, exactly as the monograph states them:
- $\mathit{logelr} \sim \text{normal}(-0.4, \sqrt{10})$
- $\alpha_w \sim \text{normal}(0, \sqrt{10})$ for $w = 2, \dots, 10$ , with $\alpha_1 = 0$
- $\beta_d \sim \text{normal}(0, \sqrt{10})$ for $d = 1, \dots, 9$ , with $\beta_{10} = 0$
- $a_i \sim \text{uniform}(0, 1)$ for $i = 1, \dots, 10$
- That is 29 parameters; $\mu_{w,d}$ and $\sigma_d$ are "transformed parameters".
Reading the parameters. $\beta_{10} = 0$ and $\alpha_1 = 0$ , so $e^{\mathit{logelr}}$ is approximately the final loss ratio of the first accident year. The $\alpha_w$ show how much the loss ratio varies by year. Because $\beta_{10} = 0$ , $e^{\beta_d}$ is the ratio of the median loss at development year $d$ to the median at year 10; Meyers calls the $\beta_d$ log-development factors.
Why the variance falls with $d$ . Summing the $a_i$ from $d$ upward forces $\sigma_1^2 > \dots > \sigma_{10}^2$ . A cumulative loss mixes claims with different settlement dates, and the proportion settled grows with $d$ .
Predictive distribution. For each of 10,000 posterior parameter vectors, set $\mu_{w,10} = \log(\text{Premium}_w) + \mathit{logelr} + \alpha_w$ , simulate $C_{w,10}$ for $w = 2, \dots, 10$ , and add the known $C_{1,10}$ . The "Estimate" is the mean of the simulated totals, and the outcome's percentile is the share of them at or below it.
Diagnostics on the upper triangle. Standardized residuals $r_{w,d} = (\log C_{w,d} - \mu_{w,d})/\sigma_d$ , from 100 random posterior draws, go into box plots by accident year and by development year. The interquartile range should contain 0, and should lie close to that of a standard normal.
Validation on 200 Schedule P triangles (p-p plots, KS critical value 9.6 for all lines combined):
- On paid losses the plot is worse than Mack's or the ODP bootstrap's ( $D = 25.5$ ), and shares their reputation for predicting losses too high. All four lines fail.
- On incurred losses it is better than Mack ( $D = 11.9$ against $15.4$ ), but it still understates the variability of the outcomes. Only Other Liability fails its own line test.
- Against the stochastic Cape Cod model, which drops the $\alpha_w$ and so holds the loss ratio constant, the CRC model has the higher $\widehat{elpd}_{loo}$ on all 200 paid and incurred triangles.
What it leads to. Two fixes follow from these results: the CSR model for the paid bias and the CAY model for the light incurred tail. Each reduces to the CRC model when its extra parameter is 0.
First edition. The 2015 first edition's leveled chain ladder (LCL) model has the same $\alpha_w + \beta_d$ structure on cumulative lognormal losses. It differs in its priors, and it sums the $a_i$ to give $\sigma_d$ rather than $\sigma_d^2$ .
"Cross-classified" also describes the over-dispersed Poisson model behind the bootstrap, which likewise has a row and a column parameter; see Stochastic Reserving.

Why the Estimate Beats the Plug-In For accident year $10$ of Meyers' illustrative paid triangle, premium is $4{,}962$ . The CRC posterior means are $\mathit{logelr} = -0.3965$ , $\alpha_{10} = 0.3435$ (posterior standard deviation $0.3316$ ) and $\sigma_{10} = 0.0202$ .

(a) Compute the lognormal mean of $C_{10,10}$ at the posterior means.

(b) Meyers' simulated Estimate for the year is $4{,}976$ . Explain the difference, and approximate it by treating $\alpha_{10}$ as normal and the other parameters as fixed.
Answer (a)
 $\begin{align*} \mu_{10,10} &= \ln 4{,}962 - 0.3965 + 0.3435 + 0 \\ &= 8.50956 - 0.0530 \\ &= 8.45656 \\ E[C_{10,10}] &= e^{8.45656 + 0.0202^2/2} \\ &= 4{,}707 \end{align*}$ 
(b) The Estimate averages $e^{\mu + \sigma^2/2}$ over the posterior, and the exponential is convex, so uncertainty in $\alpha_{10}$ raises the mean. That is parameter risk entering the expected value. With $\alpha_{10}$ normal, $E[e^{\alpha_{10}}] = e^{0.3435 + 0.3316^2/2}$ :
 $\begin{align*} e^{0.3316^2/2} &= e^{0.0550} \\ &= 1.0565 \\ 4{,}707 \times 1.0565 &= 4{,}973 \end{align*}$ 
That is within a few units of $4{,}976$ . The newest year has almost no data, so its $\alpha$ is uncertain and the gap is largest there. For older years, with tight posteriors, the plug-in and the Estimate agree.

One Standardized Residual Accident year $7$ of the illustrative paid triangle has premium $4{,}992$ and a cumulative paid loss of $2{,}980$ at development year $2$ . The CRC posterior means are $\mathit{logelr} = -0.3965$ , $\alpha_7 = 0.4354$ , $\beta_2 = -0.5751$ and $\sigma_2 = 0.2073$ .

Compute the standardized residual for this cell at the posterior means, and say how it would be used.
Answer
 $\begin{align*} \mu_{7,2} &= \ln 4{,}992 - 0.3965 + 0.4354 - 0.5751 \\ &= 8.51559 - 0.5362 \\ &= 7.97939 \\ r_{7,2} &= \frac{\ln 2{,}980 - 7.97939}{0.2073} \\ &= \frac{7.99968 - 7.97939}{0.2073} \\ &= 0.098 \end{align*}$ 
The cell's paid loss is about a tenth of a standard deviation above the model's log mean ( $e^{7.97939} = 2{,}920$ ). Meyers repeats this for all $55$ upper-triangle cells under each of $100$ posterior draws, giving $5{,}500$ residuals. He then draws a box plot for each accident year and each development year. A year whose box sits wholly above or below 0 flags an effect the model has missed. That is how the stochastic Cape Cod model's fixed loss ratio shows itself.