REINSURANCE PROFIT SHARE

REINSURANCE PROFIT SHARE Prepared by Damian Thornley Presented to the Institute of Actuaries of Australia Biennial Convention 23-26 September 2007 Christchurch, New Zealand This paper has been prepared for the Institute of Actuaries of Australia s (Institute) Biennial Convention 2007. The Institute Council wishes it to be understood that opinions put forward herein are not necessarily those of the Institute and the Council is not responsible for those opinions. The Institute will ensure that all reproductions of the paper acknowledge the Author/s as the author/s, and include the above copyright statement: The Institute of Actuaries of Australia Level 7 Challis House 4 Martin Place Sydney NSW Australia 2000 Telephone: +61 2 9233 3466 Facsimile: +61 2 9233 3446 Email: actuaries@actuaries.asn.au Website: www.actuaries.asn.au

ABSTRACT This paper examines the projected cost of competing reinsurance profit share formulae in relation to traditional reinsurance business of individually underwritten yearly renewable, stepped premium term business of the type that is most common in the Australian Life Insurance market. The paper considers the stochastic risks that impact on a mortality reinsurance portfolio and derives estimated premium loadings for various profit share formulae. We find that the cost of typical profit share formulae that tend to be offered in the Australian market is around +5% reinsurance premium charge to the equivalent nonparticipating terms. The paper also measures the impact of unplanned systemic risks and finds that the profit share estimated cost is not largely affected by the introduction of systemic risk, however this may be partly due to the simplicity of the model and also due to the measurement basis being return on capital. Keywords: profit share, profit commission, profit share formulae, life reinsurance 1 INTRODUCTION This paper considers the value of reinsurance profit share to try and determine what allowance the Insurer and Reinsurer should make when they are calculating valuation or embedded value financial projections. The paper considers the value of the reinsurance profit share in the context of both the stochastic insurance risk that affects the risks within the portfolio, and also the systemic risk that impacts on the portfolio as a whole such as mortality improvement, links between TPD incidence rates and economic conditions. The methodology used is a model with stochastic projections of profit tests for a block of mortality reinsurance business. The paper presents the results of profit share modelling just with the stochastic risk measured and then hypothesizes a way to include systemic risk over and above the stochastic risk and the paper presents results with both the stochastic & systemic risks measured. The paper then discusses the relationship between the two and explains the results.

2 BACKGROUND 2.1 Definition of reinsurance profit share By reinsurance profit share, we refer to a reinsurance agreement covering a defined block of risks or covering the continuous issuance of defined type of new business. The reinsurance agreement contains a clause whereby annual payments are made from the Reinsurer to the Insurer depending on the claims experience of the business. The profit share is determined by a formula which at it s simplest looks something like this; Profit share = X% (Y% P C LCF) Where, X%, Y% as negotiated between Insurer / Reinsurer P = earned premium or revenue premium for the year. Note specifically that initial commission / renewal commission are deducted prior to calculation of P C = claims incurred or claims paid LCF = losses carried forward from the previous calculation (in excess of Y% of P) Sometimes the reinsurance profit share has X%, Y% varying according to the level of P, so that the terms improve as the size of business under the reinsurance agreement increases. Note: the term PC is used as an abbreviation throughout the paper for profit commission and has the same meaning as profit share. 2.2 Differences between Group Life profit commission (PC) and reinsurance PC Others papers have suggested methods for valuing group life PC (refer Bibliogrpahy). Group life is a stable sized group of lives, no financing built into the premium and with the PC formula operating across a limited timeframe of 1-3 years. The Reinsurance PC has a fundamentally different behaviour for many reasons; 1. reinsurance premiums typically have financing built in. The Insurer usually pays zero reinsurance premium in the first policy year of each risk. The reinsurer increases up the subsequent years reinsurance premiums to recoup the initial financing that they provided. 2. risk quantity varies across time. When a reinsurance agreement commences the volume builds up across time and then the volume runs down as the new business ceases and the business runs off to its natural expiry (lapse, claim or policy reaches maturation). 3. reinsurance PC has losses carried forward. This considerably reduces the value of the profit commission to the Insurer by requiring that the sum of the past years combined Claims & Premium result is profitable to the reinsurer

Revenue items; premium, claims before any PC payment is made. LCF is usually defined as the amount of claims (if any) which is in excess of Y% x P. Contrasting the structuring of the profit share between these two different business lines, the differences arise out of the timeframe and ownership characteristics of each business type. Group business tends to be remarketed every three years so losses cannot be carried forward for any period longer than three years. On the other hand individually underwritten business is reinsured in a binding manner where the policy is reinsured to the natural expiry of the business. These characteristics show that reinsurance PC is fundamentally more complicated than Group Life PC. This is illustrated conceptually as follows; Reinsurance vs Group Life premiums, claims Group Life premium, expected claims (3 years) Reins Net P Reins Claims Grp Life P Grp Life C Layering effect ~ New Business claims higher than premiums, 100% initial commission 0 20 40 60 80 100 120 140 160 180 200 Month 3 CONSTRUCTING THE MODEL This paper uses a stochastic profit test model to investigate the behaviour of reinsurance profit share. For simplicity the analysis is restricted to an assumed block of mortality reinsurance. The profit test model was built up in the following way;

1. initially a deterministic model is used which contains model points grouped by age, sex, policy size (ranging up to $2M). The deterministic model uses economic, lapse, mortality, expense, capital assumptions to measure return on capital, the aim being to determine the required reinsurance premium rate for a non-participating arrangement. 2. change the deterministic model from above into a stochastic model by allowing the number of claims to vary in each month of the projection. Add in the tracking of the profit share, and then determine the premium loading additional to (1) that is required to give the same return on capital when profit commission is included. 3. add in systemic variation to the mortality basis across the future duration of the projection term and again determine the premium loading additional to (1) that is required to give the same return on capital. All the assumptions for assumed reinsurance agreement and the business covered under the agreement are set out in Appendix A as regards; new business volume, number of policies and distribution of sums insured pricing of the reinsurance business, mortality assumptions. 4 RESULTS 4.1 Results of the stochastic model For the sample portfolio and assumptions that I have used, the table below shows the costs of the some different profit share terms (these terms still allow 15% return on capital for the Reinsurer). These are only approximate estimates because they depend on the volume of new business assumed and they are based on a limited number of simulations. Results are rounded to nearest ½% Profit share terms (X% / Y%) adjustment required to nonpar reinsurance rates 60/75 60/80 60/85 +1.5% +2.0% +3.0% This shows that for example, 60% (85%P C LCF) needs a +3% loading to the reinsurance rates to put the reinsurer in the same expected position as a non-par quote, with respect to attaining the same expected NPV of transfers. 4.1.2 Examining the claims distribution

Distribution Undiscounted reinsurance claims have the following distribution (with mean = $7.2M) Frequency distribution for undiscounted reins. claims, 5-years NB mean = $7.2M skewness = 0.33 0.5 2.5 4.5 6.5 8.5 10.5 12.5 14.5 $M claims This is the distribution of undiscounted claims for the total lifespan of the portfolio arising from 5-years written new business. As expected the claims distribution is slightly skewed due to amounts volatility arising from the sum insured distribution. 4.1.3 Examining behaviour of the profit commission distributed The graph showing the profit commission distributed is shown below; 1,600,000 Average profit share distributed, 100/80 formula, 230 simulations MEAN 95TH CENTILE 1,400,000 1,200,000 1,000,000 800,000 600,000 400,000 200,000-0 5 10 15 20 25 Year From the above graph we note the following conclusions; The 95 th centile is the simulation where total distributions exceed 95% of other simulations. Total distributions is $11.3M compared to the average which totals $4.9M across the range

number of observations Average distribution peaks at Year 8 and then decreases steadily in line with runoff of the portfolio. This is explained by the fact that if you go back to the Graph in section 2.2 you can see that year 6 is the year where expected net revenue (premiums minus claims) to the reinsurer is the highest. Note that in 20% of simulations a profit share distribution is never paid across the total projection period. For all other cases profit share distribution is paid at some point even if only once. This raises a practical point that even where profit share applies, once the distribution is made it can t be clawed back. Also, it is interesting to examine the skewness of the profit commission distributions within any particular Year, across all the simulations. If we select out Year 10 from the above graph, and examine the frequency distribution of PC distributions for that single Year across the 230 simulations, this is shown below. Probability distribution frequency of {Year 10 PC distribution}, 230 simulations, 100/80 formulae 160 140 120 100 80 60 Mean $0.28M Skew = 1.82 146/230 (63%) observations are zero 95th centile = $1.4M 40 20 0 $ amount of PC distribution Note the skewness of the distribution in any one Year. The mean for Year 10 consists of approximately 2/3rds zero observations balanced by a couple of outliers (in each Year) that are 4-5 times larger than the mean. It would be interesting to bring together Graph 2.2 and Graph 4.1.3 above to see how they look combined. This is shown below.

Reinsurance premiums, claims, profit share distributed Reinsurance revenues vs average profit share distribution 3,500,000 3,000,000 2,500,000 Mean profit commission distributed x 9 Reins P E(Claim) 2,000,000 1,500,000 1,000,000 500,000-1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Year Reinsurance premiums is net of renewal / initial commission so starts from zero at Year 1. Note that the profit share distribution is magnified x 9 so that it graphed better against the reinsurance revenues. Unsurprisingly the shape of the average profit share distribution follows closely the shape of premiums, expected claims. 4.2 Results of the complete (stochastic and systemic) risk model Section 4.1 only considers the insurance risk, randomness attributable to number of claims and the size of the claim. In the real world and over long periods of time systemic risk dominates, especially pricing risk (that assumptions prove wrong). This possibility will be even more so if Trauma, TPD and/or Disability Income are included in the portfolio. It s difficult to allow for systemic risk in our model since by it s nature the forces that will shape life insurance risks [over and above any predicted trends allowed for by the Actuary in their pricing best estimate] are unknown. Extra volatility should increase the value of the profit share since there is greater option value to the profit share. For this reason, I do not claim to estimate the size of the systemic risks just to add in some arbitrary estimate of volatility in the q x rates going forward and to examine how much this impacts the profit share costs derived above. Therefore, I re-ran the stochastic projections with a random walk added in so that for any single run of the projection the future mortality varies from the mortality best estimate. The model adopted below is very simple and might fail to capture the true nature of systemic risk in the following ways;

multiple to present Qx 1. model I have used assumes that the starting best estimate mortality assumption is correct and then drifts from there. Doesn t allow for the possibility that systemic risk could emerge quite quickly and be quite deep. A practical example of this would be mis-pricing of Disability Income risks in the mid to late 1990 s in the Australian market. 2. The random walk varies up or down with equal probability. One could argue that once systemic risk emerges it is likely to be a force that will continue on the same path. 3. Doesn t allow for one off shocks such as pandemic or catastrophe risks. Therefore, in considering the results from the addition of the systemic model, I would encourage the reader to be aware of the limitations in applying any model for systemic risk. 4.2.1 Systemic mortality model q x,t = F(t). q x and t is years from the starting point and q x is the starting decrement assumption; where F(t+1) = F(t)+/-0.03 with equal probability, starting from F(0) = 1 By selecting this method of allowing for systemic risk I note that the average of q x,t for all t is equal to q x. The graph below shows the funnel of doubt ; the 95% confidence interval for the mortality estimates in future years, expressed as a multiple of the best estimate assumptions in each future year. 95% confidence interval of future mortality changes around BE 1.4 1.2 1 0.8 0.6 0.4 0.2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 time

4.2.2 Results of the stochastic & systemic model For the sample portfolio and assumptions that I have used, the table below shows the costs of the some different profit share terms (these terms still allow 15% return on capital for the Reinsurer). Results are rounded to the nearest 0.5% Profit share terms (X% / Y%) adjustment required to nonpar reinsurance rates 60/75 60/80 60/85 +1.5% +2.0% +3.5% As before I would caution that these results are approximate and they depend on the volume of new business and the figures are estimated from a simulation approach which has an upper bound of accuracy due constraints on the number of projections. 4.2.3 Claims distribution for the stochastic and systemic risk model The following graph shows the sum of the undiscounted reinsurance claims over the 5-year new business period that was modelled. Frequency distribution for undiscounted reins. claims, 5-years NB (systemic risk included) mean = $7.4M skewness = 0.40 0.5 2.5 4.5 6.5 8.5 10.5 12.5 14.5 $M claims The distribution displays higher skewness now that the systemic risk component is included. The coefficient of skewness has increased up from 0.33 (see graph in section 2.4.1) up to 0.40. I also note that the Mean of the undiscounted claims distribution is higher after the inclusion of the systemic risk. This is surprising since we noted earlier that the Qx,t were unbiased variables of Qx for all t, but could just be due to the limited number of simulations.

Distribution 4.2.4 Profit commission distributed The aim is to examine how the systemic risk distributions compare to the same analysis from section 4.1.3. For 250 simulations of the 100/80 profit share calculation, the distributions are as follows; 1,600,000 Average profit share distributed, 100/80 formula, 250 simulations, with systemic risk MEAN 95th CENTILE 1,400,000 1,200,000 1,000,000 800,000 600,000 400,000 200,000-0 5 10 15 20 25 Year Note a very similar pattern to the average profit share distribution from Graph 4.1.3. Again, the 95 th centile is the specific simulation that exceeds 95% of the others (using undiscounted total). Average total undiscounted profit distribution is $4.8M

5 DISCUSSION AND EXPLANATION OF RESULTS 5.1 Effect on profit share terms following the systemic model Switching from the purely stochastic model the estimated costs incorporating systemic risks increase slightly, for example, the 60/85 costs +3% from the stochastic only model, and +3.5% from the model including systemic risk. The additional cost due to the systemic variance is perhaps less than one would anticipate. The reasons for this are likely three-fold; 1. measuring costs using a target Return on Capital (ROC) means that cashflows in the early years of the projection are more important than those used in the later years of the projection. In the early years the claims cost profile will be almost completely dominated by the stochastic risks, in the later years the systemic risk component will increase in importance. Therefore the measurement basis puts more weight on the earlier part of the projection where only stochastic risk is dominant. 2. the losses carried forward mechanism doesn t allow the higher variance arising from the systemic risks to suddenly impact on the profit distributions in the later years. 3. simplicity of the systemic model fails to capture the true magnitude and pattern of systemic risk, ie possible early emergence of pricing error and / or catastrophe/pandemic shocks. It s worth noting however that the considerations discussed in (1) above are largely attributable to the estimated size of the new business falling under the profit share arrangement. If the size of the block is much larger than what has been used in the projections here, then stochastic risk would relatively diminish and systemic risk would increase. In this sense it s also worth noting that a profit share of any type divides the risk pool, providing less scope for the reinsurer to pool this block of risk with others of the same type to ameliorate the stochastic risks. This has obvious implications for the Reinsurer. A related point to the above seems to be that there is less difference between the stochastic and the systemic costs when the profit share terms inside the brackets is smaller. For example, at 60/75, the cost is estimated to be +1.5% under both models and the difference is not larger than the 0.5% limit of accuracy. At the 60/85 level the cost is either +3% or +3.5%. This seems to be explained by the fact that adding in systemic risk will improve the profit share distributed only if the risk improves by such a degree so as to cause the option to strike at the lower % of Premium level. 5.2 Different combinations of profit share terms Transforming the results from the table 4.2.2 we can derive a graph showing a profit share iso-cost curve. Note that this cost graphed below is dependant on the size of the portfolio and all the other assumptions used, however the intention is to show how equivalent outcomes trade-off in deriving the X / Y profit share terms.

Inside term (Y) Profit share iso-cost curve 90 85 1.5% profit share cost frontier 80 2% profit share cost frontier 75 70 65 60 55 50 0 20 40 60 80 100 Outside term (X) For example, to move from 60 to 80 outside the brackets, have to only trade-off from 80 to 77 inside the brackets. This reflects an intuitive result that the generosity of the profit share terms are more dependant on the number inside the brackets of the profit share formulae. 5.3 Behaviour of the profit share distribution To compare the average profit share distribution between the initial stochastic model and the final stochastic and systemic model, juxtapose the charts from 4.1.3 and 4.2.4 and look at the differences if any in the behaviour of the distributions.

profit distribution Average profit distribution, Stochastic only simulations vs stochastic+systemic (+10% to premium, 100/80 terms) SYSTEMIC NON-SYSTEMIC 0 5 10 15 20 25 Year Visually you can see that there is little discernible difference between the shape of the profit share distribution. 5.3 Real world comparison Typical terms in the reinsurance market would be 60-70% outside the brackets with perhaps 85% inside the brackets. If we refer back to the pricing derivation for this block of business that was shown in Appendix A, the reinsurance premium rates charged were 127% x Mortality best estimate. After allowing for the initial financing, the expected undiscounted claims / premium ratio is 84%. Difference between the reinsurance premium and the expected loss ratio consists of expenses and return on capital. Tying back to the pricing model it is interesting to note that the term inside the brackets tends to be set at a level close to the theoretical point consistent with the level that would determine experience profits or losses for this portfolio. From looking at our cost estimates in Table 4.2.2, 60/85 would cost about a +3.5% loading to the non-participating Reinsurance premium. 6 CONCLUSION A simulation model can be used effectively to estimate the value of proposed profit share terms, although the model needs to be carefully calibrated to the particular portfolio under consideration. We found that the cost of typical profit shares would be around a +4-5% premium loading to the equivalent non-participating reinsurance terms. The addition of systemic risk did not greatly change the cost estimates because systemic risk takes hold over a longer period of time and for the reasons discussed above in section 5.1.

Further interesting work that could be developed from the principles discussed in the paper would be to vary the assumed new business quantity and/or sum insured distribution of the business to see how the change in the stochastic risk impacts the cost estimates and to model systemic risk differently. Acknowledgements Although final responsibility rests with the Author for the quality of the work and the opinions expressed herein, I would like to thank Rod Berry for peer reviewing this paper. Disclaimer The real world is infinitely more complex and interesting than a simulation model. The costs indicated in the paper are specific to the assumptions made particularly as to the risk type, distribution of anticipated risks, new business volume and the selection of an appropriate systemic risk model. Bibliography Other earlier papers on the topic of Group Life profit share Bozenna Hinton (nee Steele), 1992, Self experience profit share formulae, Quarterly Journal of the Institute of Actuaries of Australia, pg39 Damian Thornley, 2001, Pricing for profit shares in Group Life Risk Business, Australian Actuarial Journal, 2001 volume 7 issue 2, pp 411-421

APPENDIX A Block of business assumptions Mortality best estimate = 50% ALT2003-2005 Distribution of age / sex / sums insured as set out in the attached table Number of new policies per annum = 5,000 per annum Lapses best estimate = 15%pa Reinsurance pricing assumptions reinsurance arrangement is surplus $300,000 (reinsured proportion of each policy = ((SI-300,000)/SI) with minimum zero, where SI = original sum insured. block of business is profit tested to achieve 15% return on capital investment income 6% 100% reinsurance commission is payable in Year 1 capital margins allocated to the business = 75% x Net reinsurance premium (ignoring initial commission) less PV(profit share loading margins) mortality basis = same as BE described above lapse basis = 15%pa reinsurers expenses = 5% x Net reinsurance premium (after commissions, excl. the 100% initial commission.) initial select effect for mortality is assumed to be 50% Year 1, then ultimate from Year 2+ Reinsurance rates that satisfy the return on capital are equivalent to 127% x Best estimate mortality basis.