PRICING ISSUES IN under $5.8 billion”1. This was more than twice the previous worst year, $2.3 billion in 1994, AVIATION even if adjusted for inflation, etc. (Ironically, AND absent the 9/11 events, 2001 was one of the best safety years ever.) To cover these claims Swiss Re estimated world wide premiums for 2001 of $1.9 billion.2 Losses would have to be paid out Airlines are in the business of transporting of capital or higher future premiums. passengers or freight from origin to destination The insurers responded quickly to as efficiently as possible. They do this with correct the cost of insurance that was mixed financial success. However, they do it immediately seen to be drastically under with remarkable physical success. The accident priced, given the exposures assumed. rate for airline travel is lower than for any other Swiss Re suggests that the premiums for mode of transportation, and it continues to 2002 would be in the range of $4-5 billion. decline. Nevertheless, when accidents do In addition, the “price” of insurance was happen they can cause considerable financial, as raised indirectly by changing coverage well as emotional, distress. Airlines choose to terms. Terrorism was excluded as a cause avoid the financial distress by purchasing of loss and third party coverage was insurance against loss-through-accident. limited to $50 million. Terrorism coverage Aviation insurers accommodate the desire of has since been reinstated and/or assumed airlines to get rid of loss-due-to-accident by by various government programs. The assuming all such losses. The remarkable thing question nevertheless remains, what is an is that the insurers have provided this cover on a appropriate price for aviation insurance ground-up basis for each and every loss, i.e., on coverage? What capital is necessary and an unlimited basis. The question such large and what return should insurance capital unlimited cover provokes is, how should it be providers expect? priced? To illustrate the magnitude of the problem, reflect back on the tragic events of 9/11. On that The Terms of Aviation Insurance day United Airlines and American Airlines each lost two aircraft to terrorism. The insurers Airlines buy insurance to cover both covering those two accounts faced the prospect hull loss and liability exposure on a twelve of claims for a) loss of hulls, b) liability for month basis. Until recently, typical passengers and crew, and last but by no means amounts might be a $250 million limit for least, c) liability for on-the-ground third party hull loss and a $1.5 billion limit for fatalities. Worse, at one time during the liability. Since 9/11, demand for higher harrowing hours of that day, as many as eight liability limits has increased and the other planes were thought to be under the introduction of larger Airbuses in the near control of terrorists, with the prospect of future will cause demand for higher hull financial losses several multiples of what was limits. These limits apply for each already known. One month later American aircraft, each and every loss. The policies Airlines insurers faced another (unrelated) loss over Queens. Airclaims, the aviation loss 1 adjuster, gives “[a] provisional estimate of The Airclaims Aviation Pocket Handbook. Airclaims, 2002. incurred aviation losses in 2001 [as] just 2 Flight to Quality – Financial security in the aviation insurance market, Swiss Re, September 2002.

1 are said to be “all risks” policies although all scenario where an accident occurs at a policies contain some exclusions. Similarly, the major airport destroying several fully policies can be viewed as a ground-up coverage loaded planes on an active taxiway. although small may apply in The 9/11 loss therefore represents a practice. The significant point is that stated marker of what is possible.4 Hopefully, it limits apply to each hull or each event, but in is a marker that is very unlikely to occur. total the coverage applies to the whole fleet and “How likely?” is a question of risk to multiple occurrences within the fleet. In estimation rather than exposure short, the insurer’s exposure is unlimited. The measurement. insurer’s premium is not, neither is their capital. Table [ 1] below summarizes the If a single accident totaled the limits, history of accidents 1980 to 2001.5 insurers would, under the above numbers, have Actuaries and statisticians can examine to pay claims of $1.75 billion. In other words, this data and its components in depth and insurers in 2001 collected premiums ($1.9 use it to develop a risk model billion) sufficient to cover 1.1 full limit losses characterizing both the amount of per year. Even at the elevated levels of 2002 exposure and likelihood of loss from premiums, only 3 full limit losses are covered aviation accidents. Using history to project annually. World wide aviation accidents occur into the future is a difficult science. at a rate of approximately two per month, even Perhaps it is more of a scientific art. For though few are full limit losses. The potential example, events that happened twenty exposure is huge compared to premiums years ago will have to be translated into collected. current dollars, but they should, ideally, also be adjusted to the current Aviation Exposure technological, economic and judicial environment. A passenger liability As of 2001 there were approximately settlement from 1980, even if put in 15,000 western built jets and 8,000 turboprops current dollars, would be much less than in annual scheduled airline operation. Those one that would be required today. planes move 1.6 billion passengers in 21.5 “Settlement creep” factors should be used million departures each year.3 Swiss Re to adjust the data. estimates the total exposure from hulls alone is $550 billion. And, as observed, liability limits are several times as great. Clearly, premium collected is tiny compared with total “sums insured” exposure. However, viewing the exposure on this basis, while dramatic, may not be helpful. Airplanes are, after all, not likely to all fall out of the sky in concert. To gauge the true exposure, underwriters use a combination of envisioning worst case scenarios of exposures together with historical experience. For example, even though the events of 9/11 are unlikely to be repeated, it is 4 still possible for multiple hulls to be lost and In this scenario, the excess third party liability beyond $50 million would inure to the airlines, or huge loss of life to take place in a disaster airports, under new terms and conditions. 5 Data provided by Global Aerospace and describes 3 2001 IUAU Statistics all losses greater than $10 million.

2 Historical Losses 1980-2001, Fully Developed. Year Basis, Including WTC. Characteristics of the 20,000 Scenario

Number of Annual Event Fixed Data Set* Developed Factor 5% Losses Losses Expected Annual Losses $2,079 per Year Total 351 $35,128 (Indexed to 1999) Standard Deviation of Losses $799 per Year 1980 2 $347 1981 2 $160 Maximum Aggregate Losses $6,877 1982 8 $1,253 1983 11 $1,370 1984 3 $183 Maximum Individual Event Loss $2,500 1985 13 $3,148 1986 5 $233 Expected Frequency of Losses 25.7 per Year 1987 10 $1,465 1988 9 $1,805 Expected Severity of Losses $81 1989 18 $2,096 1990 12 $1,967 1991 19 $1,227 1992 16 $1,262 * Assumes an inflation adjustment of 5% p.a. 1993 28 $1,614 1994 31 $2,930 1995 22 $1,281 1996 20 $2,254 1997 32 $1,411 Willis Data Base - Event Severity 1998 29 $1,817 1999 29 $1,667 2000 21 $1,117 1001 2001 11 $4,521 ity il b a b

Average (1980-2001) 16.0 $1,597 o r Standard Deviation 9.7 $1,022 P

501 10 Year Average (1992-2001) 23.9 $1,987 Standard Deviation 7.0 $1,042 Average Severity $81

1 Table 1 $1 $500 $999 $1,498 $1,997 $2,496 Size of Loss (000's) Aggregate Potential Losses, Willis 20,000 Year Data Set

1250 100.00%

1000 80.00%

y 750 60.00% nc

Typical risk modeling approaches will ue q e r involve examining the annual frequency of F 500 40.00% events and a separate study of the severity of accidents. Once separate frequency and 250 20.00% severity models are developed a simulation can take place to assess the risk 0 .00% 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 5 00 50 00 50 50 00 50 00 insurers face. Exact estimation of the risk 1 1 2 2 30 3 4 4 5 55 60 65 model is not the subject of this paper. Size of Loss Pricing is. Nevertheless it is impossible to adequately discuss pricing without some form of risk or uncertainty model. In what follows we have used a fixed set of 20,000 annual scenarios that have been generated

3 from a proprietary risk model6 to make price comparisons. A fixed set of scenarios is not as robust as a full Monte Carlo simulation (which will actually produce multiple 20,000 year or greater sets of scenarios) but is equally valid from a comparative point of view. More important are the characteristics of the data. It is detailed in Figure [1 ] below. Figures [1B ] and [1C ] illustrate the severity assumptions and the annual aggregate losses resulting from the frequency and severity assumptions contained in the table. As an illustration of the nature of the risk, the scenario set shows that there is one in eight chance that aggregate annual losses will exceed $3 billion.

6 Provided by Global-Aerospace, the scenarios were generated by Willis Re.

4 Pricing Using this, Rodney Kreps7 has theorized that an appropriate pricing formula is If an insurer were to insure the whole aviation industry, what price should it charge? Price = Expected Loss + Load Traditional approaches might be to charge expected losses plus some premium to cover Where overhead, brokerage and some profit margins. Load = [Fraction]*[Std Dev] Given the aggregate numbers above, the industry premium would be $2,079 million More formally, plus overhead etc. Setting the overhead, etc., to zero the premium would be exactly $2,079 P = EL + γ * σ million. This is not exactly the same as pricing at “burning cost”, but it is a forward looking EL is Expected Loss, and γ is the fraction version of it. (Remember, the suitably of σ, the standard deviation. [Astute adjusted ten year historic burning cost is readers will also note the similarity to the $1,987 million.) capital market’s Sharpe Ratio, where A consequence of pricing at expected γ=(P-EL)/σ.] losses, if the expectation estimate is an accurate one, is that underwriting profits will be made approximately half the time and Given the fixed data set, if γ = 100% underwriting losses the other half. This is not then the industry should pay a premium of, the reason investors enter into the risky world of insurance! It would be acceptable behavior, P = $2.079 + (1.0) * ($799) perhaps, if the profits when they occur were much greater in magnitude than the losses. = $2,878 million. Sadly, this is not the case. In insurance it is usually the reverse, with limited upside profit Similarly if γ = 150% the premium should and unlimited (or at least vastly greater) be $3.278. [γ is often referred to as the downside losses. mark up.] There is no agreed upon theoretical We can also look at the situation method for pricing insurance risk. Several in reverse. If the industry in 2003 was approaches have been designed but none can charged total premium of $3,400 million claim ascendancy over another. The feature that is most agreed upon, however, is that the then the implied load was γ = 165%. price should contain elements of “load” related to the riskiness of the subject insurance. (This would be in addition to overhead, etc., that we have herein assumed to be zero). 7 See Kreps, R. E., 1999, “Investment- One of the most recognized measures of Equivalent Reinsurance Pricing”, in O. E. Van risk is the standard deviation of outcomes. Slyke (ed), Actuarial Considerations Regarding Risk and Return in Property- Pricing, pp. 77–104, (Alexandria, VA: Casualty Actuarial Society). (This paper was awarded the CAS Dorweiler Prize ).

5 Several comments are in order. colleague, John Major,8 has also First, the conclusions about load and investigated another form. mark-up of the risk are relative to the fixed scenario set. Persons with a different Premium = γ (EL)^α view of the riskiness of the aviation world will have different views of expected loss And and mark-up percentages γ. And, since no 9 single independent or standardized risk Lane has investigated a power function model exists, comparisons of price levels that relies on both frequency and severity. can be problematic. However, the reverse Premium = observation, the implied γ can still be used γ *(Frequency)^α*(Severity)^β to some advantage for relative pricing. If price comparisons are made between one Both of these formulas require multiple insurer to another (or between insurer and observations to establish parameter fits, reinsurer, between one program and and may not be appropriate in this aviation another) with the same data set, then context. However, all have proved useful deductions about implied γ can gauge comparison tools in the context of Cat relative conservatism or cheapness. bonds. Parameter estimates from fits to the Examples of this will be provided below. pricing of these other insurance risks can A second observation is that also be a useful cross check to see how although a fixed data set is used here for aviation prices stack up. illustrative purposes it is neither recommended as accurate nor endorsed as appropriate. For example, the data set has a standard deviation of $799 million. The historical data in Table [ ] shows historical deviations of more than $1 billion. Now, to be sure, the history contains the 9-11 disaster but it cannot be ignored. Counting it in the sample as one in twenty two of the observations probably exaggerates its likelihood of recurrence, but the fixed set implies the probability of a year like 2001 as about one in one thousand, which is probably too low. Notwithstanding, the data set remains extremely valuable 8 Kreps R E.and Major J. M. “Reinsurance especially for comparison purposes. Pricing” in Lane, M, ed., 2002, Alternative Risk Strategies, (London: Risk Books), Chapter 10. A third observation concerns the pricing models. The Kreps model is the simplest model for credible risk pricing. It is not the only one. Kreps with a 9 Lane, M. N., 2000, “Pricing Risk Transfer Transactions”, ASTIN Bulletin, 30, (2), 2000, pp. 259–93 [This paper was awarded the 2000 CAS Hachemeister Prize].

6 Reinsurance

$1,500 $500 XS $1000 Showing The Inurance of Loss to each Layer $500 XS $500 (in a Hypothetical Sample Year with 26 Events) Reinsurance of aviation risks is $460 XS $40 Ground Up to $40 Million done on an aggregate excess of loss basis (AXoL) rather than a more $1,000 Loss familiar excess of loss (XoL) basis. vent E h c a This is necessary because insurers E y of t i require coverage for the aggregation of ever S $500 multiple events each year. Reinsurers do not, however, provide unlimited coverage for any number of events. $40 $0 Typically, the reinsurers will provide a 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 26 27 28 29 Number of Events single aggregate limit of coverage with $2,500 one or two reinstatements. Showing The Accumulative of Losses in 460 XS 40 Layer (in a Hypothetical Sample Year with 26 Events) AXoL works as follows. On a $2,000 per event basis, reinsurers agree to reimburse all losses above an agreed ses s o $1,500 e L v i attachment point up to a specified limit. t l u m This limit can be exhausted by the 40 Cu $1,000

60 XS aggregation of several events that each 4 penetrate the layer, but do not $500 individually exhaust it, or it could be penetrated by a single large event that $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 26 27 28 29 went through the limit itself. The Events $2,500 aggregation of event losses to the layer Showing The Accumulative of Losses in 460 XS 40 Layer (in a Hypothetical Sample Year with 26 Events) is depicted in Figs [ 3A and 3B ]. $2,000 SIDEWAYS If the agreed limit is exhausted EXPOSURE es es there are typically one or two ss ss o o $1,500 ve L ve L i i t t reinstatements. Once reinstated, the l l u u m m Net 2nd REINSTATEMENT Recovery second or third limits act as the first. 40 Cu 40 Cu

S S $1,000 60 X 60 X Reinstatement 4 4 Reinstatement premiums have to be Net 1st REINSTATEMENT Recovery paid, and this is done on a “pay as paid” $500 Reinstatement Premium basis. In other words, as claims are paid Net Recovery INITIAL LIMIT a fraction of the reinstatement premium $0 1 2 3 4 5 6 7 8 9 1011121314151617181920212223242526272829 Events is withheld from the claim. Naturally,

$2,500 as the limit is exhausted the full Showing The Accumulative of Losses in 460 XS 40 Layer SIDEWAYS (in a Hypothetical Sample Year with 26 Events) reinstatement premium for the next EXPOSURE

$2,000 limit will have been paid. But equally,

Net 2nd REINSTATEMENT if the first limit is not exhausted some sses sses Recovery o o $1,500 fraction of an unutilized reinstatement ve L ve L i i t t Reinstatement l l u u m m st limit will have been paid for u u 1 REINSTATEMENT C C Net Recovery 40 40 S S $1,000 unnecessarily. Reinstatement 460 X 460 X Net Recovery INITIAL LIMIT Certain reinsurances will also be $500 structured to include profit

DEDUCTIBLE commissions payable back to the $0 insured for non use. Alternatively, no- 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 26 27 28 29 Events

7 claims bonuses may be part of the say 50% higher then one is driven to the transaction instead of profit second conclusion – the insurer is commissions. underestimating the risk. All the other paraphernalia of Equally, if the reinsurance reinsurances may be available including proposals all cluster around the same additional premiums, deductibles, risk or mark-up, say 100% + or – 25% of clash covers. For example, the lower standard deviation, one could perhaps two panels of Fig [ 3] shows two deduce that competitive forces are conventional reinsurances, one with no keeping prices tight, but that within the , the second with a deductible range certain proposals will be cheaper equal to the first limit. than others – i.e. those with mark-ups at The important point is that 75% instead of 125%. reinsurances also contain price Load relative to risk (the mark- information that the insurer can consider up) is not the only measure to in appraising the array of proposals that discriminate between alternatives. he will have to consider. To say that Others measures might include load to insurers need to consider the cost of limit or the best case/worst case ratio. reinsurance in pricing insurance is The mark-up is, however, the most somewhat banal. What is being asserted important since it is a measure that here is slightly more important. If a affects profit directly and can be reinsurance proposal is deconstructed measured on both sides of the balance into its expected value and load, and the sheet. load is deconstructed as a mark up and risk (standard deviation) the insurer can Insurance vs Reinsurance see how the reinsurer views and values the risk. This is a form of reverse Aviation insurance is written on engineering that can inform the pricing an unlimited basis, reinsurance is written of the underlying insurance. with strict limits of liability. This Assume that a specific proposal mismatch is not the only difference. is evaluated on the same 20,000 scenario Reinsurance covers the event loss set that is used for the reinsurance. After and therefore inures to product all, this is what the insurer thinks manufacturers as well as airlines. The captures the risk he is taking. It must cause of loss of an aviation accident can necessarily characterize the risk he is fall on the airline, because of pilot error getting rid of to the reinsurer. for example, or it can fall on a Now if the price of insurance is manufacturer, because an engine fell off. 100% of the standard deviation assumed The reinsurance is paid to reimburse the and the price of reinsurance is 150% of settling party. Obviously, the limit standard deviation ceded, then one of applies to total payments – airline or two conclusions is possible. Either the manufacturer. This is not a problem for insurance is priced too low relative to analysis as long as the data set is the reinsurance, or the reinsurer sees a constructed to cover all accidents on a much riskier world than is captured by consistent basis. Also, it is worth noting the 20,000 scenario set. Further, if every that in contrast to airline insurance, reinsurer presented a proposal that was manufacturer’s insurance does usually contain a specific aggregate limit of

8 liability, typically around $1 billion per Put another way, if both insurers year. and reinsurers marked-up their expected Reinsurance is written on a losses by 100% of standard deviation, layered occurrence basis, insurance is and if it were possible to reinsure all written ground-up. This can cause a layers completely, the cost of problem for the deconstruction (of reinsurance would exceed the premiums expected loss and risk load relative to volatility) exercise suggested GENERALIIZED RAAEggggrrrIegIegNateateSUR PPootentententttiiiaaAlll LLLooBssesseLss 20E200000 L++ LLAaayyeeYrrr ERS OF EXPOEXPOSUSURERE 1000 100.00% 10001000 100.00%100.00%100.00% above. 80.00% 80.00%80.00% 750 Retained 750750

Retained y Mean $1.1 y

Retained y 60.00% Mean $1.1

y 60.00% Mean $1.1

y 60.00% c y 60.00% Mean $1.1 c 60.00% c 60.00% c c c n n n n n The issue is illustrated in Fig n ue

ue 500 ue 500

ue 500 ue 500 ue q 500 q 500 q q q e q e e

Layer e e Layer e Std Dev $16.4 Fr

Fr Std Dev $16.4 Fr 40.00% Fr 40.00% Std Dev $16.4 Fr [4 ] below. In looking at the ground Fr 40.00%40.00% 500 XS 2000 250 500 XS 2000 250250 1:1000* $250 20.00% 1:1000 $250 20.00%20.00%20.00% AAggrggregategatee PPoottteentntialialial LosseLossess 10100000 XXSS 10100000 LayLayeeerrr up risk it is relatively easy to 0 .00%.00% 100000 10.00%.00%.00%0% 10010000 1010100%0%0% 0 00 00 00 0000 0000 0000 0000 0000 0000 0000 50 0000 0000 0000 500500 0000 500500 0000 500500 0000 0000 5050 10 15 20 2 500500 30 3 500500 40 4 500500 50 5500 60 6500 1010 1515 2020 22 3030 33 4040 44 5050 55005500 6060 65006500 Siiizzee oooff LosLosLosss SSiiizzee ooofff LosLosLosss 80% unscramble expected loss and load 808080%% 750 Risk ExExcess 757500 MMeeanan $4$400...66 y y y 60% y 60% y 60% y 6060% nc nc nc nc nc (given the fixed data set). It also easy nc 505000 500 que 500 que que que que que e e

& Clash Layer e Std Dev $161.6 e

& Clash Layer r Std Dev $161.6 e r & Clash Layer e Std Dev $161.6 r

& Clash Layer r Std Dev $161.6 r r F F

F 40% F 40% F F 4040%

1000 XS 1000 250 1:100 $878.2 to measure load as a fraction of 1000 XS 1000 252500 1:100 $878..2 202020%% AAggggrrreeggatatee PoPottteenntttiiiaalll LLLoosssesess 505000 XSXS 550000 LLLayeayerrr

00 0%0%0% standard deviation. 175175000 1010100%0%0%0% 1750 100% 0 00 00 00 00 0000 0000 0000 0000 50 0 0000 0000 00 0 0000 5050 100 1500 2000 2500 30 3500 4000 45 5 00 5500 60006000 65006500 11 15001500 20002000 25002500 3030 35003500 40004000 4545 55 55005500 600060006000 650065006500 150150150000 SSSiiiizizzeee of ofofofof Lo Lo LoLoLossssss 808080%% Now to compare with 1250 Middlle 12512500 Mean $$116..3

y Mean $116.3 y y

y 60% y 60% y 10010000 6060%

nc 1000 nc 1000 nc 1000 nc nc nc que que que que que que e e reinsurance, the fixed data set must Layer e e Std Dev $207.5 Layer e r e

Layer r Std Dev $207.5 Layer r 750 Std Dev $207.5 r 750

r 750 Std Dev $207.5 r 750 F 750 F

F 40%

F 40%

F 40% F 4040% 500 XS 500 505050000 11::101000 $8$88899...99 be broken into layers, in particular to 202020%% 250 252500 AAggggrrreeggaatttee PoPottteenntttiiiaalll LLLoossss 446600 XSXSXS 404040 LLayerayer

00 0%0% 175175175000 10100%100%0%0% 0 0 0 0 0 the layers used by the reinsurers. A 000 00 0 00 00 00 00 00 00 00 00 0 00 00 00 00 0 0 0 00 00 00 00 00 0 0000 0000 0000 50 00 5 0000 0 0000 5 0000 00 5 0000 0 0000 50 00 5 000000 0 000000 5 000000 505050 10 1 55 2 00 2 55 3 0000 3 55 4 00 4 5050 50 5 555 6 000 6 555 101010 111 222 222 333 333 444 444 505050 555 6666 6666 150150150000 SSiiizzee ofofof LoLoLossss Siize ofof LoLoss 808080%% generalized set of complimentary Intermediate 125125125000 Mean $1,089.5

Intermediate y Mean $1,089.5 y Mean $1,089.5 y

y 60% Mean $1,089.5

y 60% y 60% 10010000 60%

nc 1000

nc 1000

nc 1000 nc nc nc que que que que que que e e e e e e

Layer r r Layer r 750 Std Dev $499.7 Layer r 750 r 750 Std Dev $499.7 Layer r 750 Std Dev $499.7 F 750 Std Dev $499.7 F F 40% F 40% F 40% layers is shown in Fig [4], together F 4040% 460 XS 40 505050000 11::101000 $2$2,,,447744...44 202020%% with the exposure statistics of each 252525000 AAggrggreeggateate PPPootenttenttentiiiaalll LLoosssseess """RReetttaaiiiinedned LayLayeerrr""" 4040 XXXXSS 10101010

000 0%0%0% 450450450000 1010100%0%0% 0 0 0 0 0 000 000 0000 00 00 00 000 00 00 000 0 000 00 00 00 0 000 000000 000000 000000 000000 000000 000 00000000 00000000 00000000 505050 555 000 555 000000 555 000 505050 555 000 555 layer. 40040000 50 101010 1115 2220 2225 33300 3335 4440 44450 505050 5555 6660 6665 40040000 10 1 2 2 3 3 4 4 50 5 6 6 SSiiiizzzeee ofofofof LoLoLoLossssss Size of Loss 808080%% 350350350000 80% Mean $832 3000 Mean $8$832 Retained 30030000

Since the whole data set is Retained y y y

y 60% y 60% y 60%

c 60%

c 60% c c c c n n n 2500

n 2500 n n 25025000 ue ue ue ue ue ue Std Dev $170.8

q Std Dev $170.8 q

q Std Dev $170.8 q q q e e e

e 2000 Layer e 2000 Layer e 20020000 Fr Fr Fr 40% Fr 40% Fr 40% broken into layers, the sum of the Fr 4040% 40 XSXS 00 150150150000 11::101000 $1$1,,,225555...55 10010000 1000 202020%% expected values of all layers is equal 505050000

000 0%0%0% 0 0 000 00 00 00 0 000 00 000000 0 0000 000000 000 0 000000 555 000 000 to the expected value of the industry 111 150015001500 200020002000 250025002500 30303030 3500350035003500 4000400040004000 45454545 5555 5500550055005500 6000600060006000 6500650065006500 SSSiiiizizzeee ofof ofofof LoLo LoLoLossssss Aggregate Potential Losses, Willis 20,000 Year Data Set data set treated as a single layer. WHOLE INDUASggrTegatRe YPo,t,e ntDDiaAl LoTAsses, WSESEillis 20T,00 CH0CH Year ARAData Set CTERISTIICS Readers may wish to check that the 12125050 10100.0.0000%% 10100000 8080..0000%% y Ground-up, y Mean $2,079

Ground-up, y Mean $2,079 layered sum is equal to the Ground-up, y 757500 6060..0000%% Mean $2,079 nc nc nc nc que que que Unlimited Sideways que Std Dev $799 e Unlimited Sideways e Std Dev $799 e e Fr Fr 500 40.00% Fr aforementioned $2,079. Whole Industry Layer Fr 500 40.00% 1:100 $4,$4,544 The same cannot be said of 252500 2020..0000%%

00 .0.00%0%

00 00 00 00 00 00 00 00 00 00 00 00 the volatilities. Standard deviations 0000 500500 00 5050 5050 5050 0000 5050 0000 0000 11 11 200200 22 300300 33 44 44 55 550550 66 650650 are not linearly additive. The sum of SiSizeze ooff LLoossss the layer volatilities will be greater than the volatility of the whole layer. Necessarily therefore, the mark-up of from insurance. Fully reinsured aviation volatilities of all layers will be less than insurance would be losing proposition. the mark-up for the whole layer, if one Obviously the insurer would have over- tries to equate loads. Remember this is reinsured. important because we are trying to Intuitively, over-reinsurance can equate pricing approaches and extract be seen another way. The level of consistency of mark-up. If reinsurance insurance can be viewed as covering, mark-ups are higher it again underscores say, losses in 99 years in one hundred. If the inadequacy of insurance pricing. each layer is reinsured at the same (1:100) level then the insurer is

9 effectively now reinsured to say one in a To illustrate, million. (The exact number will depend on the number of layers, correlations Expected Profit = etc). The ultimate insured benefits, but is not Premium – Expected Losses paying for the benefit. The investors in the aviation insurer who did this would And Premium = EL + γ * σ be the guaranteed losers. The aviation insurer would be subsidizing the insured. Therefore Given that reinsurance is not available on an unlimited basis the above Expected Profit = γ * σ. circumstance is at best theoretical. However, it is hopefully instructive. Any Calculating equity on the other time the deconstructed price of hand requires knowing exactly how reinsurance is higher than the much it is, or in a world of allocated deconstructed price of insurance, given capital, how much is allocated. Suppose the same data set, it is a warning flag the capital rule is to provide enough that some form of subsidy or cost is capital to cover losses in 99 years out of embedded in the operation. 100. And suppose that that γ = 100%; The other consequence of the then for the industry as a whole, on an coverage discrepancy between insurance un-reinsured basis, the return on equity and reinsurance is that however much is capital is as follows: actually reinsured, it is impossible to be fully covered. There is always going to RoE =Expected Profit/ be some residual sideways exposure Allocated Equity Capital. retained by the insurer. It is important that the insurer recognize that and allow i.e. $799/$4,544 for it in its capital calculations. = 17.6% p.a.

Return on Equity The chances of making a profit can also be calculated. With this data set Reinsurance is a powerful source it will be of the order of 85%. of capital for the aviation insurer. But as noted it can be an expensive one. The Now lets illustrate the cost of cost of reinsurance must be evaluated capital from the insured’s point of view. along with other forms of capital to Coverage from a provider with allocated maximize return on equity. Return on capital to cover 99 cases out of 100 is equity is, of course, a ratio – expected like investing in a BB+ bond - it is sub profit divide by equity. Its calculation investment grade. To feel secure that depends on a numerator and a insurance will be recoverable the insured denominator. The numerator can be well could buy credit default insurance at handled by Mr Kreps’ very useful some extra cost. Or he could require the pricing formula. The denominator may insurer have more capital allocated to depend on capital allocation rules. aviation. [Assume that this is visible directly, or indirectly via ratings.] The

10 consequence for the insurer is that the return on the larger capital allocation = 16.0% would fall – unless he received a higher This is higher than the 14.4% - so debt premium. would be cheaper capital.

Consider if capital was allocated Now consider reinsurance, which for the at the 1: 1000 level. This is about the sake of illustration we will assume is level of an AA- rated bond. Then the available on a no-limits basis at the 1000 RoE calculation would be, XS 1000 layer. Fig [4] shows that the standard deviation of this layer is $799/$5.542.9 = 14.4% $161.6, and if it is priced at 100% mark- up as the insurance is, then the return [$5,542.9 is from the 1:1000 level in the calculations are, data set] Expected Return on Equity To maintain a 17.6% RoE = ($799-$161.6)/$4,963 Profits would have to increase to = 12.8% $975.5 [With the purchase of the 1000 XS 1000 An extra $175 million profits is full reinsurance layer, 1:1000 capital required for the devotion of level has increased from $4,544 to approximately an extra $1 billion of $4,963, rather than to $5,544 without the capital. It seems exorbitant. Indeed it is. layer. Notice that the true cost of But it is the natural consequence of a) reinsurance is the load. The monetary the insured requiring a high rating, b) the outlay includes the expected loss, which insurer trying to maintain the RoE, and is then also picked up as a benefit.] c) the only source of capital being equity. Given this example the Reinsurance and debt capital may alternative capital structure answers may be cheaper. Then again they may not. now be arrayed as follows Consider the alternatives of buying reinsurance or borrowing to increase Return on Equity, when capital is capital to the 1:1000 level. Extra debt at provided to the 1:1000 level, is the BB+ level may cost LIBOR plus 700 basis points. Therefore a $1 billion loan All Equity 14.4% will cost $70 million10. The return on Equity and Reinsurance 12.8% equity now reads Equity and Debt 16.0%

Expected Return Clearly reinsurance is a more expensive alternative, given a mark-up = ($799 -$70)/ $4,544 of 100%. This underscores again that the insurance mark-up needs to be higher

10 LIBOR – the risk free investment return on than the reinsurance mark-up. capital is left out of these comparisons. It would be the same for all.

11 RETURN PROFILES FOR DIFFERENT REINSURANCE STRATEGIES This is a highly idealized and 300

200 somewhat impractical analysis. Not least ZERO REINSURANCE STRATEGY ALTERNATIVE of the reasons being that reinsurance is 100 REINSURANCE not available as described. Debt may not ▼ STRATEGY 0 IT 50.00% 55.00% 60.00% 65.00% 70.00% 75.00% 80.00% 85.00% 90.00% 95.00% 100.00% be available either. Also, as previously OF PR -100 ED T indicated the data set may not be C THE EFFEECT OF BUYING INCREASING AMOUNTS TRADITIONAL

XPE OF REINSURANCE IS TWO-FOLD; REINSURANCE E -200 conservative enough thereby overstating STRATEGY ▲ FIRST, TO REDUCE EXPECTED PROFIT ▼ the level of returns. And at other points -300 SECOND , TO ELIMINATE, OR AT LEAST in the cycle conclusions may reverse. -400 MITIGATE, EXTREME EVENTS ▲ Once again however the example is -500 hopefully instructive and can show CUMULATIVE PROBABILITY relative pricing behavior. shown. The underwriter can now The example suggests that it may examine the profile most appropriate to be advantageous to take reinsurance and the capital providers in addition to there debt into the insurer’s capital structure desire for high expected returns on on occasion. But, it is equally easy to see equity. Just like the capital structure that the results are quite sensitive to the question itself, trade-offs are necessary. relative cheapness or dearness of the Optimal choices are beautiful things but reinsurance and debt. Then again, they cannot be achieved without reinsurance may be at times the only honestly pricing , deconstructing and source of extra capital. When this is the evaluating all the possibilities. case, better change the price of insurance to keep the capital suppliers happy.

Conclusions and some Reinsurance Strategy Trade-Offs

The application of a simple pricing rule that links the price of insurance to risk has led to some powerful insights. Not only can it be applied to both sides of the balance sheet, it can be used to deconstruct prices offered by others, thereby making complicated offerings more transparent. The same methodology can be used at the operational level to discriminate between competing reinsurance proposals. While detailed consideration is beyond the scope of the present paper we close with a diagram, Fig [5], showing how comparisons can be made that take into account preferences not captured by simple expectations. Two such strategies are

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