Changing Settlement Rate Model

Changing Settlement Rate Model (CSR) is Meyers' Bayesian model for paid loss triangles that lets the payment pattern speed up or slow down from one accident year to the next. It adds a single parameter $\gamma$ to the cross classified (CRC) model, and scales every development parameter by $(1-\gamma)^{w-1}$ .

 $\begin{aligned} \mu_{w,d} = {} & \log(\text{Premium}_w) + \mathit{logelr} \\ & + \alpha_w + \beta_d \cdot (1-\gamma)^{w-1} \end{aligned}$

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

C_{w,d} is the cumulative paid 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 $\beta_{10} = 0$
- $\gamma \sim \text{normal}(0, 0.05)$
- $\sigma_d^2 = \sum_{i=d}^{10} a_i$ with $a_i \sim \text{uniform}(0, 1)$ , so $\sigma_1^2 > \dots > \sigma_{10}^2$
Reading $\gamma$ . Cumulative paid losses grow with $d$ and $\beta_{10} = 0$ , so the paid $\beta_d$ are usually negative for $d < 10$ . A positive $\gamma$ moves the "log-development factors" $\beta_d(1-\gamma)^{w-1}$ toward zero as $w$ increases, so later years pay a larger share early. That is a speedup in the claim settlement rate. A negative $\gamma$ is a slowdown, and $\gamma = 0$ gives back the CRC model.
What it fixes. On paid data the Mack, bootstrap ODP and CRC models all tend to overestimate ultimate losses. A settlement speedup that the development factors ignore would do exactly that: factors from slower years, applied to the faster recent years, overproject them. The CSR model was proposed to correct this.
Simulating the outcome. Because $\beta_{10} = 0$ , $\gamma$ drops out at development year 10. The Estimate and the outcome percentile are calculated exactly as for the CRC model.
Results for the illustrative insurer. The posterior mean of $\gamma$ is $0.0446$ (standard deviation $0.0282$ ), a speedup. The estimated total ultimate falls to $37{,}597$ from the CRC's $40{,}121$ , and the actual $40{,}000$ lands at the $86.26$ th percentile. $\widehat{elpd}_{loo}$ favors CSR over CRC.
Validation on 200 triangles:
- A speedup in claim settlement is fairly common across all four lines.
- CSR beats CRC in over half the triangles on both $\widehat{elpd}_{loo}$ and $\widehat{elpd}_{test}$ ( $119$ of $200$ on the test data). One would expect CRC to win for insurers that are not changing their settlement rate.
- Every p-p plot lies within the Kolmogorov-Smirnov band: $D = 3.1$ against $9.6$ for all lines combined. Among the models before the integrated one, it performs best on paid data.
- Its standardized residual box plots are slightly better than the CRC model's.
Where it is used next. The CSR model is the paid half of the integrated paid and incurred model, which shares $\mathit{logelr}$ and $\alpha_w$ with a CAY model on incurred data. With the paid side of that model, it drives Meyers' payout patterns and cost of capital risk margins. Meyers also suggests letting $\gamma$ change linearly with $w$ as one refinement.
First edition (2015). The CSR model is also in the monograph's first edition, in a different parameterisation:
- $\mu_{w,d} = \alpha_w + \beta_d(1-\gamma)^{w-1}$ , with $\alpha_w \sim \text{normal}(\log(\text{Premium}_w) + \mathit{logelr}, \sqrt{10})$ and $\mathit{logelr} \sim \text{uniform}(-1, 0.5)$ .
- $\beta_d \sim \text{uniform}(-5, 5)$ , $\gamma \sim \text{normal}(0, 0.025)$ , and $\sigma_d = \sum_{i=d}^{10} a_i$ .
- There it corrected the paid bias for three of the four lines. Personal auto remained biased ( $D = 25.1$ against $19.2$ ), though significantly less than under the other models, and the combined plot passed ( $D = 4.2$ ).

How Much Faster Does the Latest Year Pay? For Meyers' illustrative paid triangle the CSR posterior means include $\beta_1 = -1.3794$ and $\gamma = 0.0446$ .

Compute the log-development factor at development year $1$ for accident years $1$ , $5$ and $10$ . Convert each to the ratio of the median loss at year $1$ to that at year $10$ , and compare with the triangle: accident year $1$ paid $952$ in its first year of an eventual $3{,}912$ , and accident year $10$ paid $1{,}413$ of an eventual $4{,}139$ .
Answer
 $\begin{align*} w = 1: \quad \beta_1 (1-\gamma)^0 &= -1.3794 \\ w = 5: \quad \beta_1 (0.9554)^4 &= -1.3794 \times 0.8332 \\ &= -1.1493 \\ w = 10: \quad \beta_1 (0.9554)^9 &= -1.3794 \times 0.6632 \\ &= -0.9149 \end{align*}$ 
$\beta_{10} = 0$ , so $e^{\beta_1(1-\gamma)^{w-1}}$ is the ratio of median year-1 paid to median year-10 paid:

AY Log-dev factor Year 1 as share of year 10

$1$ $-1.3794$ $25.2\%$

$5$ $-1.1493$ $31.7\%$

$10$ $-0.9149$ $40.1\%$

The data agree in direction: $952/3{,}912 = 24.3\%$ for the first year against $1{,}413/4{,}139 = 34.1\%$ for the latest. A CRC model applies one first-year factor, estimated from all ten years, to every year. It reads a year that pays faster than that average as a large year and overstates its ultimate, which is the paid bias the CSR model corrects.

AY	Log-dev factor	Year 1 as share of year 10
$1$	$-1.3794$	$25.2\%$
$5$	$-1.1493$	$31.7\%$
$10$	$-0.9149$	$40.1\%$

Choosing Between CSR and CRC For the illustrative paid triangle, Meyers reports these leave-one-out statistics:

Model $\widehat{elpd}_{loo}$ $p_{loo}$

CSR $49.76$ $15.09$

CRC $47.80$ $14.97$

Compute each model's LOOIC and say which model the upper triangle favors.
Answer
 $\begin{align*} \text{LOOIC}_{CSR} &= -2 \times 49.76 \\ &= -99.52 \\ \text{LOOIC}_{CRC} &= -2 \times 47.80 \\ &= -95.60 \end{align*}$ 
A higher $\widehat{elpd}_{loo}$ , or equivalently a lower LOOIC (the AIC-like deviance scale), is preferred, so the triangle favors CSR. Meyers prints $-99.53$ from unrounded values.

The gain costs little: $p_{loo}$ , the effective number of parameters, rises only from $14.97$ to $15.09$ for the extra $\gamma$ . This statistic uses only the upper triangle, so it can be computed at the reserve date. The retrospective check, $\widehat{elpd}_{test}$ on the lower triangle, favored CSR in $119$ of the $200$ triangles.