Securitization of life insurance policies

Transcription

1 Insurance Markets and Companies: nalyses and ctuarial Computations Volume Issue Snorre Lindset (Norway) ndreas L. Ulvaer (Norway) ertel nestad (Norway) Securitization o lie insurance policies bstract In this paper we develop price and analyze a securitization structure o lie insurance policies. y transerring term insurance policies to a special purpose vehicle all risk is transerred rom insurers to the capital market. With mortality rates as the only underlying source o uncertainty the structure is easy to analyze. We calibrate our model to the Swiss Re/Vita III-deal and ind that insurers may transer mortality risk to the capital market at a reasonable cost. Keywords: lie insurance securitization mortality risk. Introduction In this paper we construct analyze and price a possible securitization structure o lie insurance policies hereater abbreviated SSLI. We calibrate it to market data using the Vita III-deal rom 7. Securitization is a relatively new and important innovation in the history o modern inance. he technique has developed rapidly and the scope has expanded rom mortgage loans to odd cash low streams such as uture royalties rom rock music. dmittedly the inancial turmoils and the recession that have ollowed in the wake o the subprime crisis have put securitization in a dim light. he idea o securitization is to isolate speciic uture cash low streams and make them tradable. In general all uture cash lows have potential or securitization. he rights or obligation to uture cash lows may be physically transerred to a single purpose vehicle (SV) as an o-balance-sheet transaction or simply ust held as collateral against the SV in an on-balance-sheet transaction. Next the SV issues a number o tranches o securities with dierent seniority to the cash lows. he most unior tranche is oten called the equity class and typically receives what is let ater the other tranches have received their parts. his tranche is thus the most risky but also the one with the highest expected return. Normally one or several o the acknowledged rating agencies are hired to rate the dierent tranches beore they are sold to investors through inancial institutions. So ar the lie insurance industry has been outside the scope o mass securitization. here have been some deals in recent years but most o these have been either closed book value in orce or XXX deals (see Cowley and Cummins 5; or Garnsworthy 6 or an overview). he irst securitization directly linked to mortality rates was the Swiss Re/Vita deal in 3 ollowed by Vita II in 5 and Vita III in 7. With growing demand or lie insurance and new and stricter regulatory reserve requirements the Snorre Lindset ndreas L. Ulvaer ertel nestad. industry needs more ree capital. Lie insurers will need 8- billion o new reserves the next 7-8 years according to research by KMG s actuarial service practice in 5 (see Lie Insurance International 5). here are regulations on what types o capital classes may be utilized as reserves and the most common have been equity or subordinate debt which both are costly. ccording to Lie Insurance International (6) alternatives as reinsurance and letters o credit are limited in volume costly and are only available or a ew years at a time. s a consequence the presence o bankruptcy cost and regulation can ustiy securitization as a new way to cope with the capital problems both as a new source o inancing and as a risk management tool that can lower the need or holding reserve capital. However the main advantage o securitization over traditional solutions such as reinsurance and letter o credit is the possibility to develop long-term solutions. Most lie insurance companies hold large blocks o policies with expected proits to emerge in the uture. Securitization may allow insurers to unlock part o these proits today and at the same time shit the risk associated with the policies to the capital market. he released capital may be used to write new policies which again may be securitized. Securitization can thereore make it possible to undertake new business at a higher pace and utilize capital more eiciently. Insurers may ocus on selling new policies while the capital market takes care o investment management and risk bearing. he policy servicing unction could either still be handled by the originator or outsourced to a specialized third party. Depending on how a securitization is structured mortality risk or longevity risk may be the main drivers. hese risk drivers oten have low correlation with other risk drivers investors are exposed to. Securities based on lie insurance products thereore add little systematic risk to a well diversiied portolio and investors should demand a low rate o return. he unsystematic risk is diversiiable. Investors may thereore easier be able to bear this mortality risk and longevity risk than e.g. reinsurers and may thereore oer more attractive conditions. 37

2 Insurance Markets and Companies: nalyses and ctuarial Computations Volume Issue Milevsky romislow and Young (6) show that when mortality rates are stochastic not all mortality risk is diversiiable. For a more thorough discussion regarding securitization see Cowley and Cummins (5). Cairns lake and Dowd (6) develop a twoactor model or the development over time o mortality rates that can be used when pricing assets exposed to mortality and longevity risk. he irst actor aects mortality-rate dynamics at all ages in the same way. he second actor aects the tilt o the mortality curve. Cairns lake Dowd Coughlan Epstein Ong and alevich (9) analyze eight stochastic mortality-rate models using data rom England and Wales (EW) and the United States (US). hey ind that both actors in the model by Cairns et al. (6) are important and also that cohort eects are important. Cox Lin and Wang (6) model mortality rates as a ump-diusion process. hey urther take into account that changes in mortality rates across countries are correlated. his work is extended by Chen and Cox (9) to take into account correlation between mortality-rate changes over time. Lin and Cox (8) develop a model or analyzing securitization o catastrophe mortality risks. In this paper we ocus on how term insurance policies can be securitized. he paper is organized as ollows: In section we describe our securitization structure in section we discuss mortality rate modelling in section 3 we extract the market price o risk rom the Vita III-deal and in section 4 we analyze the securitization structure we propose in this paper. Finally we conclude in the last section.. he securitization structure In this section we present a way insurers can securitize term insurance policies and also how to ind a price on these mortality-linked securities. We want to transer all the mortality risk to the capital market by setting up a so-called risk transer securitization. his is in principle similar to a closed-book securitization as the insurer does not have to worry about the risk o holding these policies anymore. Furthermore the securitization allows lie insurers to realize uture proits today. Fig.. Illustration o the securitization structure reerred to as SSLI In practice an insurer typically only transers part o the mortality risk to reduce the problem with asymmetric inormation. See e.g. is and lake (8) or a discussion o asymmetric inormation and securitization when there is longevity risk. 38

3 Insurance Markets and Companies: nalyses and ctuarial Computations Volume Issue ll cash lows t SSLI are at time t. Investors are thereore protected against all counterparty risk. he collateral is invested in risk-ree assets. Mortality and beneits ( t ) paid upon the underlying portolio o policies are the only risk drivers inluencing V. o protect policyholders s (the saety margin) is set such that there is very low probability that the collateral is less than the beneits. he terminal value o the SV V is divided among the classes ater the typical waterall principle where has seniority over and C gets what is let. here are no coupon payments rom the SV. eneits are paid via the insurer or directly to the policyholders. We present this structure by setting up an example (see also Figure ). n insurer takes a number m equal single premium n -year term insurance policies with ace value v p sold at the same date t to people in the same region o a certain gender and age x. he policies are transerred to a special purpose vehicle (SV). In addition to the air premiums mz xn : the insurer transers a saety margin o size sm Z xn : to the SV. he capital o the SV is invested at the risk-ree rate r and has the value V at maturity = t + n ater paying beneits t = qx tmt vp each year rom t to where qxt = q( x t) is the probability that a person aged x at t = t who was alive at t dies between t and t and mt is the number o policyholders still alive at t. Hence V = V ( + r ) t =... () t t t and t = + t + t= V V ( r ) ( r ) () where V = ( s+ ) m Z xn : i.e. all single premiums and the saety margin. he SV is split into three classes that have dierent claims on V. he two most senior classes and are set up as zero-coupon bonds that mature at with ace values v and v. he third class C can be considered an equity class that receives We denote the present value o expected uture cash lows by Z x:n. what is let ater and have received their ace values. he claim unctions on V or the three classes are: Φ ( V) = min( v V) Φ ( V) = min( v V Φ( V)) and (3) Φ ( V ) = V Φ ( V ) Φ ( V ). C Class has the payo structure o a short put with strike price v in combination with a risk-ree investment paying v. Class has the payo structure o a long bull spread in combination with a risk-ree asset while class C has the payo structure o a long call. he value o the SV at time could in principle become negative i beneits are very large due to extreme changes in mortality rates. he SV structure is meant to protect the insurer rom mortality risk but also make sure investors cannot lose more than invested. o avoid losses or policyholders one can either use a guarantor or increase the saety margin s. For SSLI we choose the latter and set s big enough to make sure r( V > ). t t = t investors are oered to buy these claims on the SV and the proceeds go to the insurer. he value o the claims are : Π ( ) t = t E [ Φ ( V)] or = C (4) where E is the expected value under the riskadusted probability measure. he discount actor t ( ) is the price at time t or a risk-ree zero-coupon bond that pays at. he cost or the insurer to transer the risk related to the policies at t is thus ( s+ ) m Z. xn : Π = C t Under the real world probability measure we get δ ( t) t Π = te ( ) E [ Φ ( V)] or = C (5) where δ can be interpreted as an average risk premium per annum (see e.g. Cairns et al. 6). y setting equations (4) and (5) equal we get e ( t) E [ ( V)] = or = C. E [ Φ ( V )] δ Φ (6) We make the same assumption as Cairns et al. (6 page 7 ssumption 3) that mortality rates and interest rates are stochastically independent. Note in particular that the ormulation in equation (4) implies deterministic interest rates since a zero-coupon bond is used or discounting and the -measure and the orward measure thereore coincide. 39

4 Insurance Markets and Companies: nalyses and ctuarial Computations Volume Issue he risk premium δ is oten reerred to as the price o the investment product because it relects how much more (or less) an investor expects to get compared to investing the money with no risk and earning the risk-ree rate. he yield to maturity (YM) the two bond classes by solving or e v = or =. Π y ( t) t y can be ound or y in (7). Mortality rate modelling We use inormation contained in the prices rom the Vita III-deal to price SSLI. o this end we need a mortality rate model which can be applied both to the Vita III-deal and to SSLI. ccording to lbertini (6) term insurance policies the underlying o SSLI are mainly bought by middle-aged males. s a consequence the Vita III index has most weight on males between ages o 3 and 6. Mortality risk is the only underlying source o risk in SSLI. We need a model or mortality risk that uses age-speciic mortality rates and not ust the population as a whole. he two-actor model presented by Cairns et al. (6) does that. his model includes two actors one that aects all ages (a shit in the mortality curve) and one that is linked proportionally to age (a tilt in the mortality curve). From Figure we see that over the EW years there have been downward shits in mortality rates and mortality rates or the younger decreased earlier than mortality rates or the older. We also ind there to be correlation between the mortality rates or EW and US or these eects. he two-actor model includes data rom one region only. Our model should include at least two regions to it Vita III. lso the two-actor model is constructed to price longevity bonds and thereore only considers the age group o 6-89 years old. For the purpose o this paper the model should include the age group o 3-6 years old. Cairns et al. (9) expand the two actor model by Cairns et al. (6) in three dierent ways by adding: a) a constant cohort eect term; b) a quadratic age eect term; c) an age-dependent cohort eect term. When comparing our data to the data used by Cairns et al. (9) it is clear that a quadratic term is not as appropriate or ages o 3-6 as it is or ages o Figure shows logit( q ) or our age group or three dierent years with linear trend lines and corresponding R or EW and US. R or the quad- ratic trend line is slightly higher than R or the linear trend line but they are both very good. Even i the dierence between the two R s would prove to be signiicant it means very little or the results when adding a stochastic term to predict the uture. We thereore do not include a quadratic term in our mortality rate model. US R = 998 R = 9968 R = ge R = 995 R = 9965 R = ge Fig.. Logit(q) or males aged 3-6 years in EW and US with linear trend line and corresponding R or three dierent years When analyzing EW and US data rom 933 to 3 or cohort eects we ind some o the same cohort eects as Cairns et al. (9) ind in EW and US data rom 964 to. For US data we ind no cohort eects beore 959 but or EW data the same eects can be traced through all our data 3. It is also diicult to ind any cohort eects or the generations born ater 948 and beore 96. Cairns et al. (9) show that including cohort eects yields a bet- With q being the mortality rate the mortality curve is deined as how logit(q) varies with age x. R or the quadratic trend line is not shown here. 3 It is unlikely that cohort eects suddenly appeared in the US in 959 while it has been present in EW through all our data. However we choose to not research this urther as validating data is not the ocus o this paper. 4

5 Insurance Markets and Companies: nalyses and ctuarial Computations Volume Issue ter model or data rom 964 to but we still choose to exclude cohort eects in our model because we want to keep it as parsimonious as possible. he challenge is to ind a mortality rate model that works with both Vita III and SSLI. he underlying o Vita III is a mortality rate index that is weighted or country age and gender to reduce basis risk related to the portolio o policies in the Vita III SV. However in SSLI we want to be able to look at only one country one age group and one gender. We ind the original two-actor model presented in Cairns et al. (6) to be suitable or our purpose mainly because it allows us to use any age we like as input. lthough the model suits us we need to expand it in a ew ways. o model the dynamics o the mortality rate index used by Vita III we need to include more than one region and also model the correlation between regions. y including two regions the principle is shown and i the two regions are the US and EW most o the index is covered. Cox et al. (6) also use these two regions when working with Vita I. We now have a our-actor model and can ind (the market price o risk) or two dierent countries in Vita III. When we know s or dierent countries we are able to price SSLI by using the corresponding to the country used as input in SSLI. We ignore gender and use data or males both when we ind and when estimating the price o SSLI. Cox et al. (6) use total population or both US and EW in their model and Cairns et al. (6) use males only so these simpliications are quite common... Speciication o the two-region-two-actor mortality rate model. he measure or mortality applied here is the mortality rate q (t x). q (t x) = the probability that an individual aged x at t dies between t and t or t =. Cairns et al. (6) show that logit( q ) is close to linear in age or males older than 6 years. lots in Figure indicate that this holds or males between 3 and 6 years old in both EW and US in and 3. he plots also show that the level and the tilt develop over time. We thereore use a stochastic mortality rate model with two stochastic parameters. he model is o the orm: logit( qtx ( )) = () t + ()( t x x) (8) or rearranged: () t + ()( t x x) e qtx ( ) = (9) () t + ()( t x x) + e where x is the mean age over the range o ages being used in this analysis ( x = 45.5 ). For each year t and are estimated using least squares. he plots in Figure 3 show how the parameters develop over time. he level parameter have a downward drit or both EW and the US. he tilting actor develops more randomly and the sign o the drit is not very clear. For both actors we ind correlation between EW and US. o orecast uture mortality rates we need to model how () t = [ () t () t () t ()] t () EW US EW US develops. Here we assume that () t is a random walk with drit Fig. 3. Developing and over time or EW and US () t = ( t ) + µ + CZ() t () where µ is a constant 4 vector C is a constant 4 4 upper triangular matrix and Z() t is a ourdimensional standard normal random variable. his model speciication allows or covariance among all 4

6 Insurance Markets and Companies: nalyses and ctuarial Computations Volume Issue our actors in our two-region-two-actor model. I we do not allow or covariance we are likely to underestimate the risk when applying the mortality rate model on Vita III. Data o irst-dierences is used to estimate the drit µ and the covariance matrix V. V = CC have ininitely many solutions. Similarly to Cairns et al. (6) we restrict C to be upper triangular and can thus easily derive a unique solution by applying Cholesky decomposition... Selection o mortality rate data. s we choose to work with a two-region model and since the obective now is to extract risk premiums rom the Vita III-deal data rom EW and US is used to calibrate the model. he Vita III payout structure is based on an index where death rates rom EW and the US account or 8% o the total. Cox et al. (6) use the same simpliication in their analysis o Vita I. ˆ µ EW.684E ˆ µ US.348E ˆ µ = = and ˆ µ EW.5E 4 ˆ µ US.E 4 4.4E E 4 3.4E E 6 ˆ 4.53E E E 5.378E 5 V =. 3.4E E E 5.8E E 6.378E 5.8E 6.896E 6 s experience a decrease in the mortality level parameter. he positive sign o ˆµ EW and ˆµ US indicates that mortality rates decrease less or the older than or the young. From the V ˆ matrix it is clear that the level actor () t and the tilt actor () t have higher 4 EW ˆµ and US ˆµ are negative both EW and US volatility or EW than US. he correlation matrix in able reveals high negative correlation between and or both EW and US indicating that when mortality rates improve they improve more or the young than or the old. Furthermore we observe positive correlations between EW and US as expected. able. Correlation matrix on irst dierences o (t) EW US EW US EW US EW..9 US. 3. Extracting the market price o mortality risk rom Vita III he stochastic mortality rate model developed in section can be used to evaluate SSLI under the real-world probability measure. However to Lie tables or EW and or US are published as interim lie tables or three years at a time. t Human Mortality Database (8) one can ind mortality rates in lie tables or one year at a time or EW and US. EW lie tables by year o death are available or 84-3 while the same tables or the US only are available or lso or EW the volatility o the mortality rates was larger beore the mid 95s than the volatility or the next 5 years. o have data rom the same time period or both regions we use data rom In the calibration o the model data is limited to contain mortality rates or males aged o 3 to Calibration o the two-region-two-actor model and considerations. When mortality rates or males between 3 and 6 years old or EW and US between 933 and 3 are used to calibrate the model we get the ollowing results: () (3) estimate the value o SSLI a -probability measure needs to be developed. More precisely the - dynamics o the two-region-two-actor model needs to be derived so that expected payos o SSLI under the -measure can be discounted at the riskree rate o return to ind the value o SSLI. he time t value o SSLI is given by Π tv = Π t = t ( ) E [ V] (4) = C with the -dynamics o the underlying stochastic t () process given as [ ] t () = t ( ) + µ + C Z() t + (5) where = EW US EW US. Rearranging we have t () = t ( ) + % µ + CZ() t (6) where % µ = µ + C. (7) Note that equation (6) is equal to C added to equation ().

7 Insurance Markets and Companies: nalyses and ctuarial Computations Volume Issue he dierence between the - and -probability measures depends on the market price o risk. s is not speciied within our model it must be extracted rom the market price o assets with the same underlying source o uncertainty i.e. t (). 3.. Method applied to ind. o estimate the market price o risk a payo model in accordance with Vita III is built based on the stochastic mortality rate model in section. Vita III is evaluated yearly and i trigger levels are reached beore maturity Swiss Re receives payouts at the end o that year. We simpliy and say that all payouts happen at maturity but still evaluate every year separately. hen we have the ollowing payo structure and loss or Vita III series : L = max[( q + q ) / q ] (8) t t min[ %] t= t q ( E ) where t is 7 ( or ) the exhaustion point under the -measure at t and q = ( q + q )/. 4 5 is the maturity o series is the attachment point E is q t is the mortality rate index Simulations with dierent values o are conducted until the discounted expected risk-adusted payo is in accordance with the actual price the bonds were sold at. When this is true we have an estimate o. Or in mathematical terms and the corresponding -measure are ound when equation (9) holds = t t t = t + ( ) E [ L ] v + ( t t)( r + δ ) v (9) where t is the time o the irst interest payment. For Vita III the value o bond series at t = t is the same as the ace value v ( Π = v ). hus t investors are only compensated or possible losses by being paid a spread δ over LIOR/EURIOR (see ppendix ). L is the percentage loss on the bond series (see equation (8)). y using constant interest rates equation (9) may be rewritten as o simpliy we do not distinguish between LIOR/EURIOR and the risk-ree rate o return r. Issues regarding interest rates are not the main ocus o this paper and as a simpliication we set LIOR/EURIOR and r at all maturities to 4.5% annually compounded or all urther calculations. t EL = E L = δ ( + r). () t= t EL is the simulated risk-adusted payo. We vary until equation () holds. Or by deining the annual loss rate under as d = t= t E L ( + r ) we try to achieve d t () = δ () when running simulations or dierent values o. here are three securities rom Vita III we can use to estimate Vita III series and 7 (see ppendix ). Series 3 is identical to series except or the currency while series 4 5 and 6 include a guarantor and thereore have a much lower δ ssumptions made to ind. he two-regiontwo-actor model allows or two regions hence a total o our s: EW US EW and US. From equations (6) and () we can see that represents the market price o risk o a shit in the mortality curve and the market price o risk o a tilt in the mortality curve. I only one o the bond series o Vita III is used there are ininitely many possible combinations o EW US EW and US that ulill equation (). Four variables cannot be uniquely speciied with only one equation or by applying only one price rom the Vita III series. s a compromise we set EW = US =. It is then in principle possible to solve or three unique sets o EW and US by combining the three series o Vita III. Unortunately none o the combinations could ulill equation () or both series with one unique set o EW and US. his is addressed urther in section 3.3. hereore we also assume that EW = US = (3) or all urther analysis. he elements in the variance-covariance matrix in equation (3) are small in magnitude. hroughout the paper we run 5 simulations which are suicient to obtain estimates with low standard errors and also lead to acceptable computation times. 3 I we knew how much Swiss Re paid or the guarantee we could have used these series as well but this inormation has not been available to us. 43

8 Insurance Markets and Companies: nalyses and ctuarial Computations Volume Issue nother important restriction used in this paper is that is time independent. Without this restriction there is too little data to estimate. his is a common assumption in inancial economics (or urther discussion see Cairns et al. (6 ootnote )). s explained in section we have chosen to only have two regions in the mortality rate model. Vita III is linked to indices in ive countries: the US EW Germany Japan and Canada. In section. the US and EW are chosen as the two regions because they account or 8% (6.5% and 7.5% respectively) o the Vita III index. he volatility o EW mortality rates is larger than the volatility o the US mortality rates. One o the reasons or this is larger luctuations in mortality rates around World War II speciically in 939 and 945. Germany and Japan were also hit hard during and ater World War II. For EW the volatility is even greater when looking at data as ar back as 9 due to World War I and the Spanish lu both beore 933. We like SSLI to work as an insurance against catastrophes. ecause o these acts we keep the weight o US at 6.5% and set the weight o EW (with the highest volatility) to 37.5%. o ind we need to calculate an age-weighted mortality rate index corresponding to Vita III. In addition to weighting or gender and country the Vita III mortality rate index is weighted or age. weighted average o mortality rates or 5-year age groups or EW and -year age groups or the US is ound. We use the same weights but exclude ages below 3 and above 6 years old as our model is not calibrated or these ages 3. o get each age group assigned to one speciic mortality rate we split the US age groups to match the EW age groups and then ind the middle age value. he mortality rates or these middle values are weighted to ind the index mortality rate. he target equation equation () shows that the EL we seek and the corresponding depend on r. However is not very sensitive to r. For instance or Vita III series EL varies rom 4.53% to 4.96% as r varies rom % to 8%. When removing 939 and 945 rom the dataset the variance o irst dierences o and or decreases by 8% and 85% or EW and only by % and 3% or US. When including 9-93 in the dataset the variance o irst dierences o and increases by 77% and 4% respectively. 3 For EW 3-6 year-old males account or 86.5% o the total male population in the portolio. For the US 5-64 year-old males account or 95.5% o the total male population and we assume 3-6 year old males to account or 86.33% Estimates o and discussion. In theory should be equal or all securities on the same underlying (and the same time to maturity) which means series and 7 should yield the same. able shows d annual loss rate under as deined in equation () ater simulations with dierent values or. he correct is ound when d = δ ( δ equals the number o basis points series was issued at). able. he tables below show d as basis points (bps) when running simulations or dierent values o or three dierent series o Vita III. We search or the that yields d = δ. Series 4 years class δ = bps d [bps] Series 5 years class δ = bps d [bps] Series 7 5 years class δ 7 = 8 bps d 7 [bps] s able shows the three values or do not match exactly. lso in the previous subsection we ind that there are no unique solutions to a set o EW and US. here are at least two possible reasons:. he stochastic mortality rate model is misspeciied.. Some o the assumptions we make or pricing do not hold. here are a number o stochastic mortality rate models and they are all inluenced by the data used to calibrate them. Including umps could be reasonable when pricing C bonds like Vita III. However we have chosen not to include umps in the analysis. he other reason may be violations o assumptions o the pricing theory applied. One o the assumptions is that there exists a liquid and rictionless market where assets on the same underlying are traded. Obviously this assumption is violated. Dierent estimates rom the dierent series o Vita should in theory give arbitrage possibilities. In the present market though with ew securities and low 44

9 Insurance Markets and Companies: nalyses and ctuarial Computations Volume Issue liquidity the theoretical arbitrage possibilities may be diicult to exploit. s a result dierent investors with dierent appetite or risk may buy the dierent series and no one is able or willing to exploit the theoretical arbitrage possibility. Other assumptions like e.g. constant over time may also not hold and thus explain why the estimates dier between series and which have dierent maturities. he rating and spread paid on series 7 (the most secure series) may not relect the underlying mortality risk but rather the credit quality o the collateral and the swap counterparty. hus the results rom series 7 (see able ) are likely to be overestimated. Estimates o rom series and can also be aected by these non-mortality risks although not by as much as or series 7. We thereore place less emphasis on the corresponding to series nalysis o the securitization structure We now have an estimate o the market price o mortality risk and can hence analyze and price SSLI. When we reer to the price o SSLI we mean the risk premium (expected rate o return) investors demand on the three classes. he prices o classes and are presented as basis points (bps) above r ( δ and δ ) and or C the price is presented as the total required rate o return ( r + δ C ). For classes and one can also ind the yield to maturity YM. his is the rate o return investors earn i there is no loss and investors receive ace values at maturity and is thereore greater than the risk premium. o analyze and ind the prices o the dierent classes o SSLI we run simulations. s the only stochastic underlying is mortality rates we run simulations with dierent development o mortality rates or ages rom 3 to 6. For each simulation we use to derive the risk-adusted mortality rates under. In that way we use the same random numbers or all calculations under both and. Given a set o speciied input parameters we calculate the terminal value o the SV V under both and or each simulation. Furthermore or each simulation we use V to calculate the payos on classes and C according to the claim unctions given in equation (3). y averaging over the simulated payos we obtain estimates or the expected values E Φ ( V) and E Φ ( ) V or all classes. 4.. ase case parameters. We irst present results rom a simulation with base case parameters. hen we analyze how prices vary when some o these parameters are changed. Our base case parameters have values as shown in able 3. able 3. he parameters used when simulating the base case. arameter Region Gender greed beneit Number o contracts Market price o risk o tilt actor Value US Males v p $ m Market price o risk o shit actor. Saety margin s 55% Face value class Face value class Years in term policy n ge at t x Risk-ree interest rate v 55% v 5% years 35 years old r 4.5% he irst ive parameters in able 3 are considered constant throughout the paper while the last seven are changed when doing sensitivity analysis. he US is chosen as base case. Mortality rates are dierent or males and emales. nalyse o dierent mortality rates are done by varying age not gender. ccording to Standard & oor s (7) 65% o Vita III policyholders are males and we thereore use mortality rates or males in the base case. greed beneit is proportional to the value o the SV at ( vp ~ V ) and does thus not aect the results. he number o contracts in the securitization only aects the basis risk and is thereore only changed when analyzing this risk. For all other cases basis risk is ignored and hence the number o contracts does not aect the prices o the classes. he market price o risk o the tilt actor is set to as explained in section 3.. In section 3.3 able we get three dierent values or. Series is more similar to SSLI than series and 7 and we have thereore set =.4 c.. able. For Vita III class has lower trigger levels than class and is thereore more similar to SSLI in that less extreme events aect the payo. lso the estimate rom series 7 might be biased because o strong inluence rom other risk actors than mortality (see discussion in section 3.3). ecause SSLI has a maturity o ten years in the base case the ive-year bonds o Vita III are more similar than the ouryear bonds. Hence series is used in the base 45

10 Insurance Markets and Companies: nalyses and ctuarial Computations Volume Issue case. lthough we ind to the nearest thousandth in section 3.3 we round to the nearest hundredth here and use.. he saety margin s is set to 55%. With s = 55% r ( V > ). he total cost or the insurer to transer risk to investors is not aected by s (see section 4.3) as opposed to the δ s. 55% yields sensible prices o the three classes. y sensible prices we mean that prices and rating roughly correspond to other previously traded assets. he ace values o classes and are set to 55% and 5% o E [ V ] respectively. lthough the total cost or insurers is independent o these numbers they have great inluence on the δ s. ter trying many dierent combinations o v and v we ound these values to yield sensible prices and ratings or all three classes. -year term insurance policies are chosen or the base case. In ten years mortality rates might change a lot and insurers are hence exposed to a signiicant risk. lso -year term contracts are typical or the US lie insurance market. 35 years old is chosen as age or the base case. eople with amilies are more likely to buy lie insurance products than other people and 35 year old males are good representatives or this group. he risk-ree interest rate r is as beore set to 4.5% annually compounded or the base case. 4.. ase case results. Results rom simulating the base case are reported in able 4. able 4. Results rom 5 simulations with base case parameters he price o class δ Expected loss class Moody s rating class YM class he price o class δ Expected loss class Moody s rating class 8 bps EL.5% aa y 4.58% bps EL.4% a3 YM class y 6.7% he price o class C δ C + Weighted average prices Insurer s minimum markup r.4% wa prices 3.6% im min 4.34% Number o SV deaults StdDev( E [ V ]).69% he prices are ound as explained above using equation (6). nother interesting measure is the expected loss or classes and E [ Φ ( V)] EL = or =. (4) v Investors are concerned with expected loss and it is possible to make some assumptions about what kinds o investors are willing to buy the classes and at what price level. ccording to Moody s Investors Service () the expected loss rate has become the primary measure o credit quality particularly or structured inance securities. Numerical simulations are oten conducted to estimate expected loss rates. Dierent levels o expected loss rates correspond to dierent rating levels that give an impression o relative credit quality o the dierent classes. he cumulative expected loss rates estimated in this paper are mapped with Moody's Idealized Loss Rate able which is shown in ppendix. YM is ound using equation (7). he weighted average o prices wa prices is the total risk premium investors demand or the SV as a spread above r and is ound by solving or wa prices in e [ V ] ( wa prices + r )( t ) E =. (5) Π t V Insurer s minimum markup im min is the cost or the insurer to transer the risk to the capital market as percent o received air premiums. Hence im sm Z Π x: n t V t V min = = s (6) mz xn : mz xn : and is one o the measures we ind the most interesting. Number o SV deaults is or how many o the simulations we observe V < and StdDev( E [ V ] ) is the standard deviation o the expected values o V under Sensitivity analysis. o analyze how the results depend on the dierent parameters simulations are perormed when varying one parameter at a time. Some parameters aect the underlying mortality rates and hence the cost or the insurer to transer the risk to the investors. Other parameters do not aect this value but changes the prices investors are willing to pay or the dierent claims. In all tables presented in Π ll interest rates and prices are presented as annually compounded. Equation (6) gives the prices as continuously compounded so annually compounded rates have been computed rom these beore presented here. ecause YM does not take into account the possibilities o losses it is greater than or equal to the price added the risk-ree rate y δ + r. 46

11 Insurance Markets and Companies: nalyses and ctuarial Computations Volume Issue this subsection the slanted characters are used or the base case results he market price o risk o a shit in the mortality curve. he market price o risk o a shit in the mortality curve aects the mortality rates under the -measure and hence all valuations o SSLI. From able 5 we see that the higher the market price o risk is the higher compensation do investors require to take on mortality risk. Even i is as high as.5 the insurer only needs a markup o.9% to transer the risk to the capital market. able 5. Sensitivity analysis o the market price o risk o a shit in the mortality curve wa prices im min.7.83% 8.94%.9.46%.6%. 3.6% 4.34% % 7.7%.5 4.8%.9% Saety margin s. When varying the saety margin s im min is constant. his makes sense because the underlying risk and the premiums do not vary with im min and hence should im min neither do so. able 6 shows the results when varying s. he only actual requirement is that s is big enough to obtain r ( V < ). his is true or all trials with s except when s = %. It is thus no problem to use s = 3% as the saety margin. In theory (assuming a liquid and eicient market) the attractiveness o the securities is not aected by these values. However we want to create securities that are rated as investment grade and thus StdDev(E [V ]) cannot be too high. We ind that when StdDev(E [V ]) % and hence wa prices 3% we achieve the ratings we seek on the structure. I one wants to create classes with aa ratings or both classes and and at the same time one wishes to avoid too high prices or class C one needs to set s even higher than what is done in the base case. able 6. Sensitivity analysis o the saety margin s s Deaults wa StdDev( prices E [ V ]) % % 9.63% 3% 6.94% 9.69% 4% 4.68% 4.75% 5% 3.54%.76% 55% 3.6%.69% 6%.85% 9.78% 7%.39% 8.4% 8%.6% 7.35% 9%.8% 6.5% Note: Deaults is the number o times V < out o the 5 simulations Face value class and v and v. he v are ace values o classes and v and expressed as percentages o E [ V ]. Class C receives what is let and thereore has no ace value. How the investors split the SV at maturity does not aect the value o the SV at maturity V. Hence wa prices and im min are constant when the ace values are varied. For the capital market to be interested in buying the dierent classes the classes should be as transparent as possible i.e. the payo proiles should resemble other securities in the market. Figure 4 shows the prices and Moody's ratings o the three classes or dierent values o v and v. he rightmost bar shows the base case with v = 55% and v = 5%. he prices and corresponding ratings or the classes are attractive and also compare somewhat to similar securities traded or example the ueensgate securities presented in Lane (6). Fig. 4. he prices and Moody's ratings o the three classes or dierent values o v and v expressed as percentages o E [V ]. he rightmost bar shows the base case Years in term n. erm insurance policies are usually sold with maturity o 5 5 or 3 year(s). When modelling the two-regiontwo-actor model rom section. we look at mortality rates ten years ahead and hence are not able to analyze maturities longer than ten years. We do however look at results or -year and 5- year policies. able 7 shows the results when varying years in term. Results show that when reducing n im min decreases. his makes sense because one is now exposed to the underlying mortality risk or only one or ive years and hence the cost o transerring risk is less. he StdDev( E [ V ] ) also decreases with decreasing maturity because s is kept constant. he value or s is set to it n = and could be lower when n = 5 to get the same StdDev( E [ V ] ) as or the base case. 47

12 Insurance Markets and Companies: nalyses and ctuarial Computations Volume Issue able 7. Sensitivity analysis o years in term n n im min StdDev( [ ] 4.34%.69% % 9.%.4% 6.7% E V ) ge x. ge is important because it directly inluences the only underlying mortality rates. Mortality rates increase as age increases. he same absolute increase in mortality rates thereore results in a smaller relative increase or high ages compared to lower ages. able 8 shows results when varying age x. It is clear that as x increases wa prices im min and StdDev( E [ V ] ) decrease. hey do not vary much though and apparently SSLI works or all ages presented here. he small dierences also make it unproblematic to combine several age groups without changing the properties o SSLI much. Note that the variation o StdDev( E [ V ] ) indicates that a lower saety margin s may be applied when securitizing policies or higher ages. able 8. Sensitivity analysis o age x x wa prices im min StdDev( [ ] 3 3.8% 4.5%.6% % 4.34%.69% 4 3.3% 4.7% 8.77% % 3.99% 7.69% 5 3.3% 3.79% 7.7% E V ) Risk-ree interest rate r. he risk-ree interest rate aects what returns the SV gets on invested capital. ecause the ocus o this thesis is mortality risk and not interest rate risk we assume interest rates to be constant. We still run simulations to analyze scenarios with higher or lower interest rates. Results are shown in able 9. able 9. Sensitivity analysis o the risk-ree interest rate r r EL EL δ δ δ C + r im min.%.6%.477% % 4.86% 4.5%.5%.4% 8.4% 4.34% 7.%.3%.66% % 3.85% he consequence to V o unexpected deaths is greater i it happens in year one relative to year ten. his is due to lost return on the collateral when beneits are paid out early. Interest rates strongly inluence how important the development in mortality rates in the irst years is relative to the last years. Higher interest rates make the irst years more important while lower interest rates make the last years more important. he mortality rate model applied is random walk and hence big deviations rom expected mortality rates are most likely in the last years. s a consequence lower interest rates make V relatively more dependent on the last years volatile mortality rates and hence make V more volatile. Increased volatility o V under caused by lower interest rates results in higher expected loss on classes and. he adustment between and mortality rates increases every year. With relatively more weight on the last years due to lower interest rates the payos under are reduced compared to under. Hence when the dierence between E [ V ] and E [ V ] increases the risk premium δ increases c. equation (6). Most importantly the sensitivity analysis shows that im min increases with lower interest rates but is not very sensitive asis risk. o analyze basis risk is not straightorward. asis risk is deined as the remaining risk associated with hedging a position in SSLI with mortality index linked securities. Or put dierently the risk o deviation between realized mortality in SSLI and oicially published population mortality rates. here are two reasons or this risk. One is that the policyholders do not represent the population and hence are subect to dierent mortality rates. We do not analyze this type o basis risk and continue to assume it can be ignored. he second source o basis risk is simply statistical sample risk. When the number o policyholders in the sample portolio is increased the realized sample mortality rate is likely to vary less rom the population mortality rate. Hence this risk is diversiiable and investors should not demand a risk premium or this risk. However basis risk results in increased volatility o V. ecause o the non-linear payo structure o the dierent classes the increased volatility aects the expected payo or all classes under both and. So even i investors do not demand a risk premium basis risk aects YM on and. o analyze this risk we use the binomial distribution to simulate sample mortality under. It is not straightorward to simulate sample mortality under the risk adusted -measure. hus we are not able to value the claims directly. We are however able to look at StdDev(E [V ]) EL and EL and rom that we can draw some conclusions o how the values and YM o the dierent classes are aected. Hereater when we reer to basis risk we mean only this second statistical type o basis risk. Dahl (4) reers to this type o risk as unsystematic mortality risk. 48

13 he maturity o the policies n is important when considering basis risk. s n increases basis risk becomes less important because the variation tends to even out over the years. When including basis risk in the model some results change 4 % Insurance Markets and Companies: nalyses and ctuarial Computations Volume Issue while others are unchanged. E [V ] is not aected by basis risk but StdDev(E [V ]) increases as shown in Figure 5. lthough basis risk is small or m = when n = years it is still present. years 5 years 4 % % % StdDev(E [V] % 8 % 6 % 4 % StdDev(E [V] % 8 % 6 % 4 % % % % % 5 5 Without basis risk With basis risk Without basis risk With basis risk Fig. 5. hese charts show that when including basis risk the standard deviation o the expected value o the SV at maturity StdDev(E [V ]) decreases as number o contracts m increases and maturity o policies n increases. asis risk is thus more important or 5-year bonds than or -year bonds When StdDev(E [V ]) increases expected loss under the -measure o the two bond classes and EL and EL also increase. hus ratings might worsen and the YM demanded by investors y and y increase. For class C it is opposite and the initial value o class C Π increases due to its call option characteristics t C. Figure 6 shows the mortality risk and the basis risk or class and or n = years with dierent number o policies included in the SV. It is clear that basis.5 % risk aects the expected loss o the bond classes. One should include at least policies in the SV to reduce the basis risk. are o course even better. I this is diicult or one insurer to achieve alone several insurers can cooperate. However the expected loss and corresponding rating do not change much. For years only class with 5 policies changes rating one class. he others are unchanged. Note that one should ocus on expected loss and that whether the rating changes or does not depend on the choice o base case parameters. years class years class.3 %. %.5 % EL.5 %. % EL. %.5 %. %.5 %.5 %. %. % 5 5 Mortality risk asis risk Mortality risk asis risk Fig. 6. hese charts show how basis risk aects the expected loss and corresponding ratings or classes and or n = when including a dierent number o policies m in the SV s is known rom option pricing theory call options increase in value with increasing volatility. hus the value o class C Π StdDev(E [V ]) which is the volatility o the underlying o class C increases. t C increases when 49

14 Insurance Markets and Companies: nalyses and ctuarial Computations Volume Issue Conclusion his paper contributes to the literature on mortality rate modelling and on securitization. In the paper we have proposed a possible securitization structure or term insurance policies. he securitization structure has also been calibrated to market data using the Vita III-deal and mortality data or England and Wales and the United States. We ind that the structure makes it possible or lie insurance companies to reduce their exposure to mortality risk i.e. changes in mortality rates at a reasonable cost. o make the dierent classes Reerences o the structure transparent or investors we have constructed classes and so as to have ratings aa and a3. High ratings combined with decent expected rates o returns that have low correlation with other asset classes are likely to make securitization structures o lie insurance policies attractive investments. he only uncertainty included in our model is changes in mortality rates. his simpliication makes the structure easy to analyze and hopeully will oster a stronger interest among lie insurers and investors towards securitization o lie insurance policies.. lbertini L. (6). pplications o Structured Finance echniques to Insurance Companies alance Sheet URL: < is E. and lake D. (8). Scrutinizing and ranching Longevity Exposures Discussion aper I-84 ensions Institute. 3. Cairns.J. lake D. and Dowd K. (6). wo-factor Model or Stochastic Mortality with arameter Uncertainty: heory and Calibration Journal o Risk and Insurance 73 (4) Cairns.J. lake D. Dowd K. Coughlan G. D. Epstein D. Ong. and alevich I. (9). quantitative comparison o stochastic mortality models using data rom England & Wales and the United States North merican ctuarial Journal 3 () Chen H. and Cox S.H. (9). Modeling Mortality with Jumps: pplications to Mortality Securitization Journal o Risk and Insurance 76 (3) Cowley. and Cummins D.J. (5). Securitization o lie insurance assets and liabilities Journal o Risk and Insurance 7 () Cox S.H. Lin Y. and Wang S. (6). Multivariate exponential tilting and pricing implications or mortality securitization Journal o Risk and Insurance 73 (4) Dahl M. (4). Stochastic mortality in lie insurance: market reserves and mortality-linked insurance contracts Insurance: Mathematics and Economics 35 () Garnsworthy C. (6). Innovative inancing: lie insurance securitization ricewaterhousecoopers URL: pwcpublications.ns/docid/d6e84e f55e4fc.. Human Mortality Database (8). University o Caliornia erkeley (US) and Max lanck Institute or Demographic Research (Germany) vailable at or Lane M. (6). ricing Lie Securitizations and their lace in Optimal ILS ortolios URL: bowles/lane.pd.. Lie Insurance International (5). Capital markets: Lie insurers look to securitisation or capital Lie Insurance International Lie Insurance International (6). Industry-trends: Securitisation gains Ground Lie Insurance International Lin Y. and Cox S.H. (8). Securitization o catastrophe mortality Risks Insurance: Mathematics and Economics 4 () Milevsky M.. romislow S.D. and Young V.R. (6). Killing the law o large numbers: Mortality risk premiums and the sharpe ratio Journal o Risk and Insurance 73 (4) Moody's Investors Service (). Rating Methodology: he meaning o Moody s non-lie insurance ratings Moody s Global Credit Research. New York. 7. Standard & oor s (7). New Issue: Vita Capital III Ltd. RatingsDirect. ppendix. Vita Capital III overview able. Overview o the nine dierent bonds issued on January st 7 called Vita III. Series Rating - rincipal $9 $5 3 $ $ $ Class ttachment % % % 5% 5% 5% 5% % 5% Exhaustion 5% 5% 5% 45% 45% 45% 45% 5% 45% Maturity Currency L L E L L L E E E Margin 8 Guaranteed X X X X X Source: Standard & oor s (7). 5

15 ppendix. Moody's idealized loss rate table Insurance Markets and Companies: nalyses and ctuarial Computations Volume Issue Rating -Year -Year 3-Year 4-Year 5-Year 6-Year 7-Year 8-Year 9-Year -Year aa % % 4% % 6% % 9% 36% 45% 55% a 3% 7% 55% 6% 7% 3% 97% 369% 45% 55% a 7% 44% 43% 59% 374% 49% 6% 743% 9% % a3 7% 5% 35% 556% 78% 7% 49% 496% 799% % 3% 4% 644% 4% 436% 85% 33% 64% 35% 385% 6% 385% % 898% 569% 37% 395% 456% 54% 66% 3 4% 85% 98% 97% 45% 55% 65% 75% 836% 99% aa 495% 54% 38% 4565% 65% 7535% 985% 835% 485% 43% aa 935% 585% 4565% 66% 869% 835% 355% 5675% 78% 98% aa3 3% 5775% 945% 39% 6775% 35% 385% 7335% 3635% 3355% a 4785% % 75% 3% 94% 34375% 3883% 43395% 47795% 57% a 858% 985% 849% 374% 4655% 53735% 5885% 643% 69575% 745% a3 5455% 335% 4385% 53845% 653% 7495% 84% 8645% 995% 973% 574% 469% 6369% 7675% 8866% 98395% 55% 65% 68% % 3938% 6485% 8555% 9975% 395% 4575% 355% 3835% 44% 496% 3 639% 9355% 5665% 3% 48775% 66% 75% 799% 8579% 995% Caa 95599% 7788% 575% 78634% 9976% 437% 76% 43% 595% 635% Caa 43% 7875% 45% 434% 685% 86% 33875% 375% 33965% 3575% Caa3 8446% 33548% % 36433% 3847% 3966% 4887% 4669% 4396% 44385% Note: Cumulative idealized loss rates or Moody's rating classes. Sourse: Moody s Investor Service pril 7 Special Comment on Deault & Loss Rates o Structured Finance Securities: New York. Horizon 5