CSR - Posterior Distribution of Parameters
Feb 17, 2018

# Goal

Following on from a prior post, in this post we will investigate the simulated posterior distribution of a few key parameters from our CSR model. Our goal is to evaluate the reasonableness of the prior distributions that we used and understand our model results better by evaluating the posterior distributions.

# Plotting the posterior

We will take a look at the logelr, alpha, gamma, beta, and sigma parameters from our model for five companies across our four lines of business. The results for these companies are representative of the rest of the data set.

## Expected loss ratio

The posterior distribution of the expected loss ratio looks as expected - mostly normal with slightly fatter tails because we modeled on the log scale.

Since the alpha parameters account for accident year variations, this parameter represents something like an underwriting cycle adjusted volatility.

## Alpha - accident year effects

Alpha provides the underwriting cycle effect. We can see the general downward trend for most of the lines as we move from the early accident years (late 1980s), which had poor experience, to the later years that were better.

The volatility also increases as we move towards more recent accident years as we have fewer and fewer data points (just one in the final accident year).

We can also clearly see how the volatility for PA and WC is much lower than OL and CA during this time period.

## Gamma - speedup of claims

The gamma parameter indicates whether the claim payments sped-up or slowed-down through the triangle. The effect is relatively small, so the posterior distributions are all in the range of [-0.15, 0.10].

For several lines it is clear that there is a non-zero effect, but several are inconclusive. It may be worth investigating a more regularizing prior on this variable than the current normal distribution.

## Beta - development pattern

We see pretty clean estimated development patterns in the graph below. A few of the company / line combinations have noticeably more volatility in their patterns, but none appear unreasonable.

## Sigma

The sigma parameters control the volatility of each cell in our loss triangle, and they vary by development year. The posterior distributions look okay. The model structure imposes decreasing volatility as development period increases.

# Conclusion

None of the parameter posterior distributions raise any huge red flags. Maybe investigate a regularizing prior on gamma and alpha.