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

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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 non-

participating 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

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

0 20 40 60 80 100 120 140 160 180 200

Month

Reven

ue i

tem

s;

pre

miu

m,

cla

ims

Reins Net P

Reins Claims

Grp Life P

Grp Life C

Layering effect ~ New Business

claims higher than premiums, 100%

initial commission

Group Life premium,

expected claims (3 years)

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%)

60/75 60/80 60/85

adjustment required to non-

par reinsurance rates

+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

Undiscounted reinsurance claims have the following distribution (with mean =

$7.2M)

Frequency distribution for undiscounted reins. claims, 5-years NB

0.5 2.5 4.5 6.5 8.5 10.5 12.5 14.5

$M claims

mean = $7.2M

skewness = 0.33

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;

Average profit share distributed, 100/80 formula, 230

simulations

-

200,000

400,000

600,000

800,000

1,000,000

1,200,000

1,400,000

1,600,000

0 5 10 15 20 25

Year

Dis

trib

uti

on

MEAN

95TH CENTILE

From the above graph we note the following conclusions;

The 95th

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

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

0

20

40

60

80

100

120

140

160

$ amount of PC distribution

nu

mb

er

of

ob

serv

ati

on

s

Mean $0.28M

Skew = 1.82

146/230 (63%)

observations are zero

95th centile =

$1.4M

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 revenues vs average profit share

distribution

-

500,000

1,000,000

1,500,000

2,000,000

2,500,000

3,000,000

3,500,000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Year

Re

ins

ura

nc

e p

rem

ium

s,

cla

ims

,

pro

fit

sh

are

dis

trib

ute

d

Mean profit commission

distributed x 9

Reins P

E(Claim)

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 qx 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;

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

qx,t = F(t). qx and t is years from the starting point and qx 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 qx,t

for all t is equal to qx.

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

0

0.2

0.4

0.6

0.8

1

1.2

1.4

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

mu

ltip

le t

o p

res

en

t Q

x

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%)

60/75 60/80 60/85

adjustment required to non-

par reinsurance rates

+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)

0.5 2.5 4.5 6.5 8.5 10.5 12.5 14.5

$M claims

mean = $7.4M

skewness = 0.40

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.

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;

Average profit share distributed, 100/80 formula, 250

simulations, with systemic risk

-

200,000

400,000

600,000

800,000

1,000,000

1,200,000

1,400,000

1,600,000

0 5 10 15 20 25

Year

Dis

trib

uti

on

MEAN

95th CENTILE

Note a very similar pattern to the average profit share distribution from Graph 4.1.3.

Again, the 95th

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.

Profit share iso-cost curve

50

55

60

65

70

75

80

85

90

0 20 40 60 80 100

Outside term (X)

Ins

ide

te

rm (

Y)

2% profit share cost

frontier

1.5% profit share

cost frontier

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.

Average profit distribution, Stochastic only simulations vs

stochastic+systemic (+10% to premium, 100/80 terms)

0 5 10 15 20 25

Year

pro

fit

dis

trib

uti

on

SYSTEMIC

NON-SYSTEMIC

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.