CAS Exam 7 Spring 2019, Question 1

Study guide: Exam 7

An insurance company historically never attempted to recover salvage and subrogation ("S&S") on claims. On January 1, 2018, the insurer enters into a one-year agreement with an S&S recovery vendor, requiring the vendor to pursue all S&S opportunities for accident year 2018.

The insurance company's IT department generated the following loss development triangles as of December 31, 2018 (assume no development after 36 months):

Incremental Paid Loss Gross of S&S ($000s):

Accident Year	12 Months	24 Months	36 Months
2016	16,500	6,000	2,500
2017	17,000	5,000
2018	14,000

Incremental Paid Loss Net of S&S ($000s):

Accident Year	12 Months	24 Months	36 Months
2016	16,500	6,000	2,500
2017	17,000	5,000
2018	11,000

Prior to 2018, the Actuarial department estimated unpaid losses using the following Bayesian model:

$C_{ij}$ represents the incremental losses for accident year $i$ as of development year $j$ which follow an over-dispersed Poisson ("ODP") distribution with mean $x_i y_j$ and variance $\varphi x_i y_j$ .
$x_i$ represents the expected ultimate losses for accident year $i$ .
$y_j$ represents the proportion of ultimate losses that emerge in development year $j$ .
The prior distribution for $x_i$ is gamma with mean $\frac{\alpha_i}{\beta_i}$ and variance $\frac{\alpha_i}{\beta_i^2}$ .
$\varphi$ represents the dispersion parameter for the ODP distribution, which is 93.
$\lambda_j$ represents the incremental chain ladder loss development factor for development year $j$ .
$D_{ij}$ represents the cumulative losses for accident year $i$ as of development year $j$ .
$M_i$ represents the value for ultimate losses for accident year $i$ that is obtained using expert knowledge about the losses.
The mean of $C_{ij}$ for this Bayesian model is:

$Z_{ij}\,(\lambda_j - 1)\,D_{i,j-1} + (1 - Z_{ij})\,(\lambda_j - 1)\,M_i\,\frac{1}{\lambda_j \lambda_{j+1} \cdots \lambda_n}, \quad \text{where } Z_{ij} = \frac{\sum_{k=1}^{j-1} y_k}{\beta_i \varphi + \sum_{k=1}^{j-1} y_k}$

The Actuarial department believes that the S&S recovery vendor agreement has resulted in a slowdown of the gross loss payment pattern for calendar year 2018 and will continue in calendar years 2019 and 2020.

To estimate the total unpaid losses gross of S&S for accident year 2018, the Actuarial department is considering the following prior gamma distributions for the Bayesian model above:

Parameter	Option 1	Option 2	Option 3
$\alpha$	50	200	1,250
$\beta$	0.002	0.008	0.100

To estimate the total S&S recoveries for accident year 2018, the Actuarial department analyzed S&S data from competitors of comparable size to find:

The expected S&S recoveries per accident year were $5M.
The standard deviation of the S&S recoveries was $1M.
The expected percent of the S&S recoveries received through 12 months was 75%.
The standard deviation of the percent of S&S recoveries received through 12 months was 10%.

Part a (1 point)

Select the prior gamma distribution most appropriate to address the concern that the vendor agreement has slowed the gross loss payment pattern. Justify the selection.

Part b (2 points)

Calculate the incremental losses gross of S&S for accident year 2018 expected to emerge between 12 and 36 months using the Bayesian model with the most appropriate prior distribution from part a. above.

Part c (0.5 points)

Describe model risk with respect to the Bayesian model for incremental losses gross of S&S.

Part d (1.5 points)

Calculate the total S&S recoveries for accident year 2018 using Bayesian credibility.

Part e (0.5 points)

Describe estimation risk with respect to the Bayesian model for S&S recoveries.

Part f (0.5 points)

Calculate the total unpaid losses net of S&S for accident year 2018.

Part g (1 point)

Describe two types of operational risk introduced by the vendor agreement and recommend a unique key risk indicator ("KRI") to monitor each risk.

Answer this question and see the worked solution