Correlated Accident Year Model

Correlated Accident Year Model (CAY) is Meyers' Bayesian model for incurred loss triangles that correlates successive accident years. It adds to the cross classified (CRC) model a term proportional to the previous year's log residual in the same development year, and so thickens the tails of the predicted distribution.

$\mu_{1,d} = \log(\text{Premium}_1) + \mathit{logelr} + \beta_d$

 $\begin{aligned} \mu_{w,d} = {} & \log(\text{Premium}_w) + \mathit{logelr} + \alpha_w + \beta_d \\ & + \rho \cdot \big(\log(C_{w-1,d}) - \mu_{w-1,d}\big), \quad w > 1 \end{aligned}$

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

C_{w,d} is the cumulative incurred loss of accident year w at development year d (w, d = 1, \dots, 10). The priors in Meyers (2019) are:
- $\mathit{logelr} \sim \text{normal}(-0.4, \sqrt{10})$
- $\alpha_w \sim \text{normal}(0, \sqrt{10})$ for $w = 2, \dots, 10$ , with $\alpha_1 = 0$
- $\beta_d \sim \text{normal}(0, \sqrt{10})$ for $d = 1, \dots, 9$ , with $\beta_{10} = 0$
- $\rho = 2\rho_{pos} - 1$ with $\rho_{pos} \sim \text{beta}(2, 2)$ , so $\rho$ can take any value in $(-1, 1)$ and is centred on $0$
- $\sigma_d^2 = \sum_{i=d}^{10} a_i$ with $a_i \sim \text{uniform}(0, 1)$ , so $\sigma_1^2 > \dots > \sigma_{10}^2$
The correlation it creates. A proposition credited to John Major gives the correlation between the log losses of accident years $k$ apart. Its proof also gives $\mathrm{Var}[\log C_{w,d}] = \sigma_d^2(1+\rho^2)$ .

$\operatorname{Corr}\big[\log C_{w,d},\ \log C_{w-k,d}\big] = \begin{cases} \dfrac{\rho}{1+\rho^2} & k = 1 \\[6pt] 0 & k > 1 \end{cases}$

What it fixes. The Mack and CRC models both predict a tail that is too light on incurred losses, and a positive correlation between accident years thickens it. The model does not favor a sign for $\rho$ , and $\rho = 0$ gives back the CRC model.
Simulating the outcome. The years must be simulated in order, because each year's log mean uses the previous year's loss. Accident year 1's development-year-10 loss is known. For $w \geq 3$ , the simulated $\tilde C_{w-1,10}$ feeds $\mu_{w,10}$ , so a high draw in one year pulls the next year up.
Results for the illustrative insurer. The posterior mean of $\rho$ is $0.1709$ , with a wide posterior (standard deviation $0.2071$ ). The standard error of the total rises to $1{,}859$ from the CRC's $1{,}642$ . $\widehat{elpd}_{loo}$ still favors CRC for this triangle ( $70.97$ against $68.65$ ).
Validation on 200 triangles:
- Positive mean $\rho$ s are an overwhelming majority, and CAY tends to raise the standard error.
- The p-p plots improve noticeably for Commercial Auto, Personal Auto and Workers' Compensation, and are almost unchanged for Other Liability. The combined plot is still just outside the band ( $D = 10.8$ against $9.6$ ), and Meyers notes the model still appears a bit light in the tails.
- $\widehat{elpd}_{loo}$ favors CAY over CRC in only $26$ of $200$ triangles, but $\widehat{elpd}_{test}$ on the lower triangles favors it in $121$ . The wide posterior of $\rho$ makes the two hard to tell apart from the upper triangle alone, so the choice of CAY rests mainly on its reputation.
Where it is used next. CAY is the incurred half of Meyers' integrated paid and incurred model, which shares $\mathit{logelr}$ and $\alpha_w$ with a CSR model of the paid triangle. One suggested refinement lets $\rho$ decay exponentially toward zero as $d$ increases.
First edition. The 2015 edition's correlated chain ladder (CCL) model has the same $\rho$ term, with a $\text{uniform}(-1, 1)$ prior on $\rho$ .

A Good Year Pulls the Next One Up Two accident years each have premium $10{,}000$ . The posterior draw has $\mathit{logelr} = -0.40$ , $\alpha_2 = 0.05$ , $\beta_{10} = 0$ , $\sigma_{10} = 0.02$ and $\rho = 0.30$ . Accident year $1$ is fully developed at $C_{1,10} = 7{,}300$ .

Compute $\mu_{2,10}$ and the expected $C_{2,10}$ , and compare with the CRC model ( $\rho = 0$ ) at the same parameters.
Answer
 $\begin{align*} \mu_{1,10} &= \ln 10{,}000 - 0.40 \\ &= 8.81034 \\ \ln 7{,}300 - \mu_{1,10} &= 8.89563 - 8.81034 \\ &= 0.08529 \\ \mu_{2,10} &= 8.81034 + 0.05 + 0.30 \times 0.08529 \\ &= 8.88593 \\ E[C_{2,10}] &= e^{8.88593 + 0.02^2/2} \\ &= 7{,}231 \end{align*}$ 
With $\rho = 0$ the log mean is $8.86034$ and the expected loss is $e^{8.86034 + 0.0002} = 7{,}048$ .

Accident year $1$ came in $0.085$ above its log mean, and $30\%$ of that carries into year $2$ : $e^{0.30 \times 0.08529} = 1.026$ , a $2.6\%$ increase. Over thousands of draws the years move up and down together, which widens the distribution of the total.

How Correlated Are Adjacent Years? (a) Meyers' illustrative incurred triangle has a posterior mean of $\rho = 0.1709$ . What correlation does that imply between the log losses of adjacent accident years, and between years two apart?

(b) What is the largest adjacent-year correlation any CAY model can produce?
Answer (a)
 $\begin{align*} \operatorname{Corr} &= \frac{0.1709}{1 + 0.1709^2} \\ &= \frac{0.1709}{1.0292} \\ &= 0.166 \end{align*}$ 
Years two apart are uncorrelated. In the proof, $\log C_{w,d}$ carries only its own shock and $\rho$ times the previous year's shock, so years $w$ and $w-2$ share no shock.

(b) $\rho/(1+\rho^2)$ increases on $(-1, 1)$ and approaches $1/2$ as $\rho \to 1$ . So the adjacent-year correlation of log losses is always less than $0.5$ in size, even though $\rho$ itself can approach $1$ . For $\rho$ near zero the two are almost equal, which is the usual case.