21.2 Quantifying Stability and Its Value
Measuring the value of reinsurance is more than calculating the expected cashflow from the reinsurance program
Significant judgement is required to evaluate the cost-benefit tradeoff
Metrics to measure and value stability:
Ratio and difference of \(\text{Reinsurance Premium}\) and \(\mathrm{E}[\text{Recovery}]\)
Expected loss under different programs
\(\text{Premium} - \mathrm{E}[\text{Loss}]\) under different programs
\(\sigma\), percentiles, whole distribution
Space needle view,
Distribution of other metrics
- cost benefit diagram, pre tax net income, combined ratio
Efficient frontier charts
21.2.1 Reinsurance Premium and Expected Recoveries
Remark.
On a PV basis?
\(\mathrm{E}[Recovery]\) is net of reinstatement premiums, typically based on simulation results
- Should gauge how significant the net cost of reinsurance is by comparing to the firm’s expected earnings for the year
21.2.2 Amount of Protection
Definition 21.3 (Metrics 3) Compare \(\mathrm{E}[\text{Net Loss}]\) \(\forall\) programs
\[\mathrm{E}[\text{Net Loss}] = \mathrm{E}[\text{Gross Loss}] - \mathrm{E}[\text{Recovery}]\]21.2.3 Premium Less Expected Loss
Going forward we are considering Net \(\text{Premium}\) - Net \(\mathrm{E}[\text{Loss}]\)
Only consider premium and losses
Does not include expenses or investment income
Definition 21.4 (Metrics 4) \[\text{Premium} - \mathrm{E}[\text{Loss}]\]
- Compare \(\forall\) programs
Definition 21.5 (Metrics 5) \[\sigma_{\text{Premium} - \mathrm{E}[\text{Loss}]}\]
- Caveat: Can be lowered by removing favorable outcomes
Definition 21.7 (Metrics 7) Worst simulated outcome \(\text{Premium} - \mathrm{E}[\text{Loss}]\)
- Might be too extreme to look at 1-in-25K (99.996%)
Remark.
- Here we are sticking to high percentile means bad, similar to how we used to look at loss, but can flip it for looking at earnings
21.2.4 Distribution Based
Density of \(\text{Premium} - \mathrm{E}[\text{Loss}]\)
Compare shapes for different programs
See if it’s giving up the upside and like how does the tail look
CDF of \(\text{Premium} - \mathrm{E}[\text{Loss}]\)
Pick a percentile and read it across the graph to find the value at that percentile for each curve
Look at the difference between different programs at each percentile
Note on difference of distribution:
Event that generate a given percentile is different across programs
- Reinsurance changes the order of the outcomes
We are interested in the difference of the distributions
- NOT interested in the distribution of differences (as it doesn’t make sense? due to the above bullet point)
Goal here is the choose a reinsurance program and its associated distribution \(\Rightarrow\) More useful to look at the distn themselves
Interested in the cost-benefit trade off, cost is the net cost of reinsurance and benefit is the protection against adverse deviation
Space Needle View on \(\text{Premium} - \mathrm{E}[\text{Loss}]\)
Shows different percentiles
Colored section is proportional to the probability that the result is in that range
Easy to compare each quantile across programs
Shows the shape of the distribution
Cost Benefit Diagram
Cost of reinsurance (for each program) vs loss @ each percentile
X-axis being the cost of reinsurance
Y-axis is the loss amount (@ given percentile)
We should expect the loss to decrease as we increase the cost of reinsurance
Pre-Tax Net Income
Include expenses and investment income
Look at value at each iterations of the sim (percentile)
Get a perspective on overall profitability e.g. probability of negative earnings
We can look at the probability of negative earnings
CDF on Combined Ratio
Caveat: Can give distorted results when the net premium is reduced due to significant ceded premium
Expense ratio will be higher due to lower net premium (denominator)
Can make results look worst for program that have big cession
21.2.5 Efficient Frontier Charts
Construct the efficient frontier with a given risk metric (x-axis)and return metric (y-axis):
Based on different reinsurance structure options
For different percentile of the return metrics
Remark.
Plot increasing risk to the right so it looks more familiar
- Look for programs that are up and left along the efficient frontier with the lowest risk metric for a given return metric
Return metrics:
- e.g. Earnings
Risk metrics:
Probability of making plan
Probability of:
Surplus < 2 \(\times RBC\)
Surplus < BCAR score supporting a target rating
Expected loss in a 10 year period, \(TVaR_{90\%}\), exceeds a threshold level or surplus
(e.g. \(TVaR_{90\%}\) should not exceed 20% of surplus)
x% drop in quarterly EPS